research article matrix theory, ads/cft, and...

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 604637 34 pageshttpdxdoiorg1011552013604637

Research ArticleMatrix Theory AdSCFT and GaugeGravity Correspondence

Shan Hu1 and Dimitri Nanopoulos123

1 George P and Cynthia W Mitchell Institute for Fundamental Physics Texas AampM University College Station TX 77843 USA2Astroparticle Physics Group Houston Advanced Research Center (HARC) Mitchell Campus Woodlands TX 77381 USA3Division of Nature Sciences Academy of Athens 28 panepistimiou Avenue Athens 10679 Greece

Correspondence should be addressed to Shan Hu hushanmailustceducn

Received 29 April 2013 Accepted 10 August 2013

Academic Editor Kingman Cheung

Copyright copy 2013 S Hu and D Nanopoulos This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

With 119873 rarr infin being fixed 119877 rarr infin the free energy of the Matrix theory on a supergravity background 119865 is a functional of 119865119882 = 119882(119877 119865) We try to relate this functional with 119878eff (119877 119865) the effective action of 119865 where 119865 is translation invariant along 119909

minusThevertex function is then associated with the connected correlation function of the current densities From119882(119877 119865) one can constructan effective action Γ(119884) for the arbitrary matrix configuration 119884 Γ(119884) is 119877 and thus 119901+ independent If 119882(119877 119865) = 119878eff (119877 119865) Γ(119884)

will give the supergravity interactions among 119872 theory objects with no light-cone momentum exchange We then discuss theMatrix theory dual of the 11119889 background generated by branes with the definite 119901

+ as well as the gauge theory dual of the 10119889

background arising from the 119909minus reduction Finally for SYM

4

with the 4119889 background field 1198650

we give a possible way to induce theradial dependent 5119889 field 119865(120590)

1 Introduction

Up to present two kinds of nonperturbative formulation of119872string theory are developed The one is Matrix theoryThe typical examples are BFSS Matrix model [1] and theplane wave Matrix model (PWMM) [2] describing a sectorof M theory with the definite light-cone momentum on flatbackground and pp-wave background respectivelyM theoryon a generic weakly curved background is described byBFSS Matrix model with the corresponding vertex operatorperturbations added [3 4] PWMM could just be derivedin this way [5] For backgrounds that cannot be taken asthe perturbations of the flat spacetime the correspondingMatrix models are also known provided that certain amountof supersymmetries is preserved The other is the AdSCFTcorrespondence [6ndash9] for which the AdS

5

SYM4

correspon-dence is intensively-studied SYM

4

gives a nonperturbativedescription of string theory onAdS

5

times1198785 It is natural to expect

that SYM4

with the 4119889 vertex operator perturbations addedthen describes the string theory on AdS

5

times 1198785 with the corre-

sponding 5119889 field perturbations turned on Although Matrixtheory and AdSCFT are obtained in entirely different waysboth of them use a gauge theory to describeMstring theory

on a particular background If the M theory background is11987711minus119899

times119879119899 the dual gauge theory will become SYM

119899+1

whichis Matrix theory compactified on 119879

119899 [10ndash12]As the nonperturbative description ofMstring theory on

a particular background theHilbert space of the gauge theoryshould be isomorphic to the Hilbert space of the Mstringtheory on that background For AdS

5

SYM4

the one-to-onecorrespondence should exist between the state (spectrum) ofthe second-quantized string theory on AdS

5

times1198785 and the state

(spectrum) of SYM4

ForMatrix theory it is easier to establishthe correspondence between configurations The transitionamplitude between the matrix configurations should beequal to the transition amplitude between their M theorycounterparts When 119873 rarr infin Matrix theory is the discreteregularization of the supermembrane theory in light-conegauge [13] The matching is explicit It is natural that thetransition amplitude for membranes is defined in the sameway as the transition amplitude for strings since the formerwhen wrapping 119878

1 with the vanishing radius reduces to thelatter Matrix theory compactified on 119878

1 gives the Matrixstring theory [14 15] The off-diagonal degrees of freedomare KKmodes of membrane along 119878

1 [16] In strong coupling

2 Advances in High Energy Physics

limit (the radius of 1198781 approaches 0) matrices commute (KKmodes could be dropped) and Matrix string theory reducesto the second quantized type IIA string theory in light-cone gaugeThe configurations aremultistring configurationswith the transition amplitude given by the integration of allintermediate string joining and splitting processes [14 15]

Since the state the spectrum and the transition ampli-tude are all in one-to-one correspondence the partitionfunction of the gauge theory equals the partition function oftheMstring theory For SYM

4

we have [8 9]

119885SYM4

(120573 1206010

) = 119885AdS5times119878

5 (120573 1206010

) (1)

where 120573 = 1119879 is the radius of the time direction The zerotemperature partition function is only the functional of 120601

0

for which [8 9]

119885SYM4

(1206010

) = 119890119882[120601(120601

0)]

(2)

120601(1206010

) is the on-shell supergravity solution with the boundaryvalue 120601

0

119882 is the type IIB supergravity action on AdS5

times 1198785

If the gravity dual of SYM4

with the source 1206010

added is thetype IIB string theory on AdS

5

times 1198785 with the background

field 120601(1206010

) turned on in zero temperature limit the partitionfunction will only contain the contribution of the groundstate geometry so 119885SYM

4

(1206010

) = 119890119882[120601(120601

0)]

Except for (2) 119885SYM4

(1206010

) also has another expression onAdS

5

times 1198785 Let 119882

119904

be the free energy of the type IIB stringtheory on AdS

5

times 1198785 and 119885SYM

4

the partition function ofSYM

4

it is expected that

119885SYM4

= 119890119882

119904 (3)

where 119882 = sum119899

ℎ=0

119882119899

ℎ

When the background field 1206010

varies SYM4

undergoes a change of the coupling constantsCorrespondingly there is also a change of the couplingconstants for strings living in AdS

5

times 1198785 which is in fact a

modification of the 5119889 background For (3) to be valid a one-to-one correspondence 120601

0

harr 120601(1206010

) should exist

119885SYM4

(1206010

) = 119890119882

119904[120601(120601

0)]

(4)

On AdS5

side the string free energy 119882119904

[120601(1206010

)] cannot bedefined without a definite 5119889 background 120601(120601

0

) Also forthe state (spectrum) correspondence to be valid the definite5119889 background is necessary otherwise it is impossible todetermine the string spectrum If (2) and (4) both hold

119882[120601 (1206010

)] = 119882119904

[120601 (1206010

)] (5)

The free energy of the string theory on a given backgroundequals the effective action of the background fields

For Matrix theory similarly

119885Matrix (120573 119901+

119865) = 119885119872

(120573 119901+

119865) (6)

119865 is the 11119889 supergravity background and is the translationinvariant along the 119909

minus direction 119885119872

(120573 119901+

119865) is the par-tition function of the M theory sector with the light-conemomentum 119901

+ which is supposed to be the supermembrane

Equation (6) is trivially satisfied sinceMatrix theory is just theregularization of the supermembrane with the definite light-cone momentum When 120573 = infin

119885Matrix (119901+

119865) = 119890119882(119901

+119865)

(7)

119882(119901+

119865) is the 119901+-parameterized functional of 119865 with the

10119889 covarianceIn string theory one can also calculate the free energy of

the strings on a given background 119865

119882119904

(119865) =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 (8)

In [17 18] it was shown that for 119865 satisfying the free fieldequation 119882

119904

(119865) could be taken as the effective action ofthe renormalized background field 119865(119865) that is 119882

119904

(119865) =

119878eff[119865(119865)]It is tempting to establish a relation between119882(119901

+

119865) andthe effective action of the 11119889 supergravity However Matrixtheory no matter if it was taken as the DLCQ formulationof the M theory or as the discrete regularization of thesupermembrane theory in light-cone gauge only describesthe sub-Hilbert space of theM theory with the definite light-cone momentum 119901

+ without capturing all the information ofthe covariant theory For different 119901+ 119882(119901

+

119865) is differentNevertheless let119865 = 119865

0

119865minus

119865minusminus

where1198650

119865minus

119865minusminus

representfields with zero one and two 119909

minus indices respectively one canfind that 119882(119901

+

119865) = 119882(1198650

119865minus

119901+

119865minusminus

119901+2

) Let 119901+ = 119873119877

with 119877 being the radius of 119909minus 119873 rarr infin 119877 rarr infin and

119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus

) On the other hand for 11119889

supergravity fields 119865 that are translation invariant along 119909minus

suppose 119878eff(119877 119865) is the effective action of 119865 and there isalso 119878eff(119877 119865) = 119878eff(1198650 119877119865

minus

1198772

119865minusminus

) The 119877 dependence of119882(119877 119865) is consistent with 119878eff(119877 119865)

One may want to consider the complete partition func-tionwith all light-conemomentum taken into account whichis roughly 119890

119882(119865)

= int119889119901+

119890119882(119901

+119865) Each 119882(119901

+

119865) only differsby a rescaling of (119865

minus

119865minusminus

) so the summation does not givemore information It is enough to consider 119882(119901

+

119865) withthe definite 119901

+ In fact 119865 is the translation invariant along119909minus as a result a sector with the definite 119901

+ has the enoughdegrees of freedom to produce 119878eff(119877 119865) The complete Mtheory degrees of freedom including sectors with all light-cone momentum is necessary only when 119865 is the field withthe 11119889 spacetime dependence

For the arbitrary matrix configuration 119884 we may defineΓ(119884) via

119890Γ(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= 119890119882(119877119865)+119878

119877119865(119884)

(9)

with 119865(119909) solved from

minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865

=120575119878

119877119865

(119884)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865

(10)

It is easy to see that Γ(119884) is 119877 independent

Advances in High Energy Physics 3

To describe the supergravity interactions among 119872 the-ory objects with no light-conemomentum exchange and onemay define Γ

119892

(119884) through

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (11)

with 119865 being the zero mode of the 11119889 supergravity along119909minus and 119878cla the classical action of 11119889 supergravity The

integrating out of F induces the effective action for the 119872

theory object 119884 with the supergravity interaction (withouttransferring the light-cone momentum) taken into account

Γ119892

(119884) = sum

119899

1

119899int 119889

10

1199091

sdot sdot sdot int 11988910

119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(12)

where 119866Fc(1199091 119909119899) is the connected Greenrsquos function ofsupergravity in light-cone gauge with the zero light-conemomentum and 119881

119865(119884)

(119909) is the current density of the con-figuration 119884 coupling with the supergravity field 119865(119909) Undera Legendre transformation Γ

119892

(119884) could be written as

Γ119892

(119884) = 119878eff (119877 119865)

+ int11988910

119909119865 (119909)119881119877119865(119884)

(119909) sim 119878eff (119877 119865) + 119878119877119865

(119884)

(13)


minus120575119878eff (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119877119865(119884)

(119909) =120575119878

119877119865

(119884)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865

(14)

So if119882(119877 119865) = 119878eff(119877 119865) Γ(119884) = Γ119892

(119884)Although M theoryMatrix theory correspondence and

AdSCFT correspondence are very different it is possibleto construct the connection between the two In [19ndash21] itwas shown that PWMM expanded around the certain 12

BPS states gives SYM119877times119878

2 SYM119877times119878

3119885

119896

and SYM119877times119878

3 whilethe backreaction of the corresponding 12 BPS states onpp-wave produces the gravity dual We will investigate thecorrespondence in more detail

In (7) 119865 is the generic 11119889 supergravity field with the 119909minus

isometry 119882(119877 119865) if is indeed equal to 119878eff(119877 119865) gives theeffective action of the field 119865 On the other hand in (2) onlythe 4119889 field 120601

0

(119909) is given from which the 5119889 field 120601(119909 119903) isobtained from the equation of motion or from the RG flow119882[120601(120601

0

)] is the action of 120601(119909 119903) This is the holography ofAdSCFT One may want to turn on the arbitrary 120601(119909 119903) onAdS

5

and try to find the corresponding 4119889 gauge dual Thedual gauge theorymay not be SYM

4

since 1206010

can only encodea subset of 5119889 fields which are in one-to-one correspondencewith the 4119889 fields In fact since the transverse space ofAdS

5

times1198785 is 1198785 other than 119877

6 the gauge theory dual may havethe scalar fields 119883

119868 other than 119883119868 Suppose the coordinate

of AdS5

times 1198785 is (119909 119903 Ω) for a 10119889 scalar 119867

119899

(119909 119903 Ω) =

ℎ119899

(119909 119903)119884119899

(Ω) with 119884119899

(Ω) = 119862119886

1sdotsdotsdot119886

119899

(119910119886

1 sdot sdot sdot 119910119886

119899)|119910|119899 the

spherical harmonic of 1198785 the operator counterpart is

119867119899

(119909 119903 119883) = ℎ119899

(119909 119903) 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) (15)

ℎ119899

(119909 119903) can be the arbitrary 5119889 function In SYM4

weonly have ℎ

119899

(119909)119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) to represent such fieldsNevertheless for SYM

4

with the scalar field 119883 and the4119889 background 119865

0

(119909) a 119883 rarr 119883 1198650

(119909) rarr 119865(119909 119903)

transformation can be made under which the partitionfunction remains invariant If the SYM

4

with the scalar field119883

and the 5119889 background119865(119909 119903) is the gauge theory descriptionof the string theory onAdS

5

times1198785 with the background119865(119909 119903)

its partition function will then equal 119890119882[119865(119909119903)] with 119882 beingthe 10119889 supergravity action So we arrive at (2) 119883 rarr 119883

is a Weyl transformation under which 1198650

(119909) must evolve as119865(119909 119903) to preserve the partition function We will show thatfor such 119865(119909 119903) 120575119882[119865(119909 119903)]120575119865(119909 119903) = 0 so if 119882[119865(119909 119903)]

is the action of the supergravity 119865(119909 119903) will be the on-shell solution The discussion can also be extended to SCFT

3

and SCFT6

With no source term added under the 119883 rarr

119883 transformation the induced fields give the near horizongeometry of 1198722 and 1198725 respectively The holography inAdSCFT is very similar to the holography in noncriticalstring coupling with 2119889 being the gravity and with 119892 rarr 119892

replaced by119883 rarr 119883This paper is organized as follows In Section 2 we

consider the free energy of the Matrix theory on super-gravity background 119865 that is translation invariant alongthe 119909

minus direction and its relation with the 11119889 effectiveaction of 119865 In Section 3 we consider Matrix theory on theconfiguration representing branes and its gravity dual Thediscussion will then be specified to the PWMM from whichSYM

119877times119878

2 SYM119877times119878

3119885

119896


3 can be obtained [19ndash21] In Section 4 we give a possible way to induce the radialdependent 5119889 fields from the 4119889 background fields in SYM

4

2 Free Energy and the EffectiveAction of Supergravity

In this section we will consider 119882(119877 119865) the free energyof the Matrix theory on a generic 11119889 supergravity back-ground 119865 that is translation invariant along 119909

minus Since thesupermembrane action in light-cone gauge only containsone free parameter 119901

+ as the discrete regularization of thesupermembrane actionMatrix theory action 119878

119877119865

(119884) also hasone free parameter which could be taken as 119877 the radiusof 119909minus 119901+ = 119873119877 and 119873 rarr infin is fixed The concrete 119877

dependence of 119878119877119865

(119884) is 119878119877119865

(119884) = 119878119865

0119877119865

minus119877

2119865

minusminus

(119884) where 1198650

119865minus

119865minusminus

represent fields with zero one and two 119909minus indices

As a result 119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus

) On the otherhand due to the coordinate invariance the 119877 dependenceof the effective action of the supergravity field 119865 is also119878eff(119877 119865) = 119878eff(1198650 119877119865

minus

1198772

119865minusminus

) From 119882(119877 119865) we candefine Γ(119884) = 119878

119877119865

(119884) + 119882(119877 119865) with 119865 solved through120575[119878

119877119865

(119884) + 119882(119877 119865)]120575119865 = 0 Γ(119884) is 119877 or equivalently 119901+independent Γ(119884) could be taken as the effective action of thematrix configuration119884 In fact at the one-loop level Γ(119884) andthe standard effective action of the Matrix theory coincideIf 119882(119877 119865) = 119878eff(119877 119865) we will have Γ(119884) = Γ

119892

(119884) withΓ119892

(119884) being the effective action describing the supergravity


interactions among theM theory objects with the zero light-cone momentum exchange

21 The Action of the MatrixTheory on a Generic BackgroundTheMatrix theory action in flat spacetime is

119878119877

= 119877int119889119909+

times Tr( 1

21198772119863+

119883119868

119863+

119883119868

+1

41198976119901

[119883119868

119883119869

]2

minus119894

119877120579119863

+

120579 +1

1198973119901

120579120574119868

[119883119868

120579])

(16)

where119883119868 120579 and1198600

are119873times119873 hermitianmatrices with119873 rarr

infin and 119868 = 1 sdot sdot sdot 9 119877 rarr infin 119901+ = 119873119877 119897119901

is the Plancklength

119883119868 119909

+ and 119877 have the dimension of length so eachcommutator is multiplied by a factor 11198973

119901

to make the actiondimensionless With the replacement 119883

119868

rarr 119897119901

119883119868 119909+ rarr

119897119901

119909+ 119877 rarr 119897

119901

119877 we get the action

119878119877

= 119877int119889119909+

times Tr( 1

21198772119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894

119877120579119863

+

120579 + 120579120574119868

[119883119868

120579])

(17)

in which 119897119901

is cancelled and 119883119868 120579 119909+ and 119877 are all dimen-

sionless In the following we will still adopt this conventionso 119897

119901

will not appear explicitlyThe 11119889 supergravity field after the gauge fixing has the

nonzero components (119892+minus

119892++

119892119868+

119892119868119869

) (119862119868119869+

119862119868119869119870

) and(120595

+

120595119868

) [22] Based on the Hamiltonian in [22] one canwrite down the action of the bosonic membrane on suchsupergravity background

119878119887

119865

0

= int119889119909+

1198892

120590119892+minus

119875+

times [119875+2

119863+

119883119868

119863+

119883119869

21198922+minus

119892119868119869

+119875+

119863+

119883119868

119892+minus

(119862119869

+119875+

119892119869

+

119892+minus

)119892119868119869

+119875+2

119892++

21198922+minus

minus 119883119868

119883119869

119883119870

119883119871

119892119868119870

119892119869119871

+119875+

119862+

119892+minus

]

(18)

where 119862119868

= 119883119869

119883119870

119862119868

119869119870

and 119862+

= 119883119869

119883119870

119862+119869119870

Note thatit is 119892

+minus

119875+ that appears The Matrix theory version is

119878119887

119865

0

= int119889119909+

times Tr(119877119892+minus

119863+

119883119868

119863+

119883119869

2(119877119892+minus

)2

119892119868119869

+119863+

119883119868

119877119892+minus

(119862119869

+119892119869

+

119877119892+minus

)119892119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892119868119870

119892119869119871

+119892++

2(119877119892+minus

)2

+119862+

119877119892+minus

)

(19)

where 119862119868

= minus(1198942)[119883119869

119883119870

]119862119868

119869119870

and 119862+

= minus(1198942)[119883119869

119883119870

]

119862+119869119870

Without the gauge fixing 119892119868minus

119892minusminus

119862minus119868119869

119862+minus119868

120595minus

= 0so terms involving them can also be added

119878119887

119892

119868minus

= int119889119909+


119863+

119883119868

119863+

119883119869

119863+

119883119870

2(119877119892+minus

)3

119892119869119870

minus119863+

119883119868

4119877119892+minus

[119883119870

119883119871

] [119883119872

119883119873

]

times 119892119870119872

119892119871119873

+119863+

119883119871

119877119892+minus

[119883119869

119883119872

] [119883119868

119883119873

]

times119892119871119869

119892119872119873

times 119877119892119868minus

)

(20)

and similarly for 119878119887119892

minusminus

which is the sum of the 4th order terms(119863

+

119883)4 (119863

+

119883)2

[119883119883]2 and [119883119883]

4 The current densitiesfor119892

minusminus

119892119868minus

119862minus119868119869

119862+minus119868

120595minus

were derived in [3 23ndash25] throughthe calculation of the one-loop effective action Otherwisein the light-cone quantization of the membrane one mayadd 119863

+

119883minus

= (12)119863+

119883119868

119863+

119883119868

+ (1]2)119883119868

119883119869

119883119868

119883119869

and119863119886

119883minus

= 119863+

119883119868

119863119886

119883119868

into the action which could couple withthe 119909

minus indexed fieldsIn all these terms it is 119877119892

+minus

119877119892119868minus

1198772119892minusminus

119877119862minus119868119869

119877119862+minus119868

119877120595

minus

that are involved (Similarly in supermembrane actionit is 119892

+minus

119875+ 119892

119868minus

119875+ 119892

minusminus

119875+2 119862

minus119868119869

119875+ 119862

+minus119868

119875+ 120595

minus

119875+ that


will appear)119877 is the radius of 119909minus Under the parameterizationtransformation 119909

+

= 119891(1199091015840+

) 119909minus = 1205721199091015840minus

hArr 119877 = 1205721198771015840

1198921015840

++

= 11989110158402

(1199091015840+

) 119892++

1198921015840

+minus

= 1205721198911015840

(1199091015840+

) 119892+minus

1198921015840

+119868

= 1198911015840

(1199091015840+

) 119892+119868

1198921015840

minus119868

= 120572119892minus119868

1198921015840

minusminus

= 1205722

119892minusminus

1198621015840

+minus119869

= 1205721198911015840

(1199091015840+

) 119862+minus119869

1198621015840

+119868119869

= 1198911015840

(1199091015840+

) 119862+119868119869

1198621015840

minus119868119869

= 120572119862minus119868119869

(21)

the action is invariant This is consistent with the coordinatetransformation

1198891199042

= 119892++

119889119891 (1199091015840+

) 119889119891 (1199091015840+

) + 2119892119868+

119889119909119868

119889119891 (1199091015840+

)

+ 119892119868119869

119889119909119868

119889119909119869

+ 2120572119892+minus

119889119891 (1199091015840+

) 1198891199091015840minus

+ 2120572119892119868minus

119889119909119868

1198891199091015840minus

+ 1205722

119892minusminus

1198891199091015840minus

1198891199091015840minus

= 119892++

11989110158402

(1199091015840+

) 1198891199091015840+

1198891199091015840+

+ 21198911015840

(1199091015840+

) 119892119868+

119889119909119868

1199091015840+

+ 119892119868119869

119889119909119868

119889119909119869

+ 2120572119892+minus

1198911015840

(1199091015840+

) 1198891199091015840+

1198891199091015840minus

+ 2120572119892119868minus

119889119909119868

1198891199091015840minus

+ 1205722

119892minusminus

1198891199091015840minus

1198891199091015840minus

119862 = 119862119868119869119870

119889119909119868

and 119889119909119869

and 119889119909119870

+ 119862119868119869+

119889119909119868

and 119889119909119869

and 119889119891 (1199091015840+

)

+ 120572119862119868119869minus

119889119909119868

and 119889119909119869

and 1198891199091015840minus

+ 120572119862119868+minus

119889119909119868

and 119889119891 (1199091015840+

) and 1198891199091015840minus

= 119862119868119869119870

119889119909119868

and 119889119909119869

and 119889119909119870

+ 119862119868119869+

1198911015840

(1199091015840+

) 119889119909119868

and 119889119909119869

and 1198891199091015840+

+ 120572119862119868119869minus

119889119909119868

and 119889119909119869

and 119889119909minus

+ 120572119862119868+minus

1198911015840

(1199091015840+

) 119889119909119868

and 1198891199091015840+

and 119889119909minus

(22)

119878119877119865

the supersymmetric extension of (19) and (20) is notconstructed yetWith 119878

119877119865

given the current density for119865 canbe defined as

119881119865(119884)

(119909) =120575119878

119877119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(23)

In particular when 119865 is the flat background for which theonly nonvanishing fields are 119892

+minus

= minus1 and 119892119868119869

= 120575119868119869

119881119865(119884)

(119909)

reduces to the localized vertex operator of the supergravityfield 119865 In [3 4 23ndash25] the vertex operators for varioussupergravity fields are constructed Although the exact 119878

119877119865

is unknown with the vertex operators at hand one can write

down theMatrix theory action onweakly curved backgroundin linear gravity approximation [3 4 23 24]

119878119877119865

= 119878119877

minus 119877int119889119909+

times STr [119881ℎ

119868119869(119883)

ℎ119868119869

(119883) + 119881ℎ

119868+(119883)

ℎ119868+

(119883)

+ 119881ℎ

++(119883)

ℎ++

(119883) + 119881ℎ

119868minus(119883)

ℎ119868minus

(119883)

+ 119881ℎ

+minus(119883)

ℎ+minus

(119883) + 119881ℎ

minusminus(119883)

ℎminusminus

(119883)

+ 119881119862

119868119869119870(119883)

119862119868119869119870

(119883) + 119881119862

+119869119870(119883)

119862+119869119870

(119883)

+ 119881119862

119868119869minus(119883)

119862119868119869minus

(119883) + 119881119862

+minus119870(119883)

119862+minus119870

(119883)

+ 119881120595(119883)

120595 (119883) + 119881120595

+(119883)

120595+

(119883)

+ 119881120595

minus(119883)

120595minus

(119883) ]

(24)

where for example

119881ℎ

119868+(119883)

ℎ119868+

(119883) = (119863+

119883119868

119877minus

1

4120579120574

119868119869

120579120597

120597119883119869

)ℎ119868+

(119883)

119877 (25)

ℎ119868119869

(119883) 120595minus

(119883) are 11119889 background fields with the 119888-number coordinate 119909 replaced by the matrix coordi-nate 119883 The background fields are only the functions of(119909+

1199091

1199099

) they are the zeromodes of the 11119889 supergrav-ity along 119909

minusThe vertex operator can take three different forms First

it can be the operator defined in a 1119889 SYM theory just as thatin AdSCFTThen the background fields like ℎ

119868+

(119883) shouldbe expanded as the Taylor series

ℎ119868+

(119909+

1198831

1198839

)

=

infin

sum

119899=0

1

119899(120597119896

1

sdot sdot sdot 120597119896

119899

ℎ119868+

) (119909+

0 0)119883119896


119899

(26)

with (120597119896

1


119899

ℎ119868+

)(119909+

0 0) the derivative of ℎ119868+

(119909+

)

at (119909+ 0) [3 23ndash25]

STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0)

times STr[ 1

119877(119863+

119883119868

119877minus

1

4120579120574

119868119869

120579120597

120597119883119869

)119883119896


119899]

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0) 119868119896

1sdotsdotsdot119896

119899

ℎ

119868+

(119909+

)

(27)

Since all background fields enter into the action in the formof119891(119909

+

0 0) the 119880(119873) SYM1

lives at (119909+ 0 0) When

119883119894

997888rarr 119883119894

minus 119886119894

119891 (119909+

0 0) 997888rarr 119891(119909+

1198861

1198869

)

(28)


SYM1

undergoes a translation Nothing specifies where theSYM

1

should be so one may put it at any point in 1198779

Similar to AdSCFT there is a one-to-one correspondencebetween operators and fields However no holography ispresent here Fields living on SYM

1

are Taylor series coef-ficients which uniquely determines the 11119889 backgroundThe background fields are arbitrary and are not necessarilyon shell On the other hand in AdSCFT fields living onCFT are boundary values from which the full backgroundis solved through the equations of motion or the RG flow Incontrast to the chiral primary operators Φ119896

1sdotsdotsdot119896

119899 in AdSCFTthe moment operators 1198681198961 sdotsdotsdot119896119899 do not need to be traceless As aresult 1 1198681198961 1198681198961 sdotsdotsdot119896119899 couples with the 10119889 field while1 Φ

119896

1 Φ119896

1sdotsdotsdot119896

119899 only couples with the 9119889 fieldIn the second form the vertex operator is defined in 10119889

spacetime

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

) = 120575 (1199090

minus 119909+

) 1205759

(119909119868

minus 119883119868

)

= 12057510

(119909120583

minus 119883120583

)

(29)

where 119883119868 are 119873 times 119873 matrices and for uniformity we have

set 1198830

= 119909+1

119873times119873

This is a matrix generalization of the 120575-function In special situations when all of the119883119868 are diagonalthat is119883119868

= diag1199091198681

119909119868

2

119909119868

119873

(29) becomes

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

)

= diag 12057510

(119909120583

minus 119909120583

1

) 12057510

(119909120583

minus 119909120583

2

) 12057510

(119909120583

minus 119909120583

119873

)

(30)

With the generalized 120575-function it is straightforward to writedown the current densities for various fields For ℎ

119868+

we have

119881ℎ

119868+(119909) =

120575

120575ℎ119868+

(119909)minus119877int119889119909

+STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

12057510

(119909 minus 119883)]

(31)

It is not convenient to deal with the 120575-function One maywant to do a Fourier transformation which gives the thirdrepresentation of the vertex operator

119881ℎ

119868+

(119901) = int11988910

119909119881ℎ

119868+(119909) 119890

119894119901119909

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

119890119894119901119883

]

= minus119877int119889119909+STr[minus

119863+

119883119868

119877+

119894

4120579120574

119868119869

120579119901119869

] 119890119894119901119883

(32)

22 Partition Function of Matrix Theory on Curved Back-ground Suppose the exact form of 119878

119865

is given and the par-tition function of Matrix theory on supergravity backgroundis

119885 (119877 119865) = 119890119882(119877119865)

= int [119889119883119889120579 1198891198600

] 119890minus119878

119877119865(119883120579119860

0)

= int [119889119884] 119890minus119878

119877119865(119884)

(33)

where 119865 is the 11119889 background field (119892+minus

119892++

119892119868+

119892119868119869

119892minusminus

119892119868minus

) (119862119868119869+

119862119868119869119870

119862119868119869minus

119862119868+minus

) and (120595+

120595119868

120595minus

) mentionedbefore and 119884 collectively represents (119883119868

120579 1198600

)In gaugestring correspondence partition function is an

important quantity the value of which should be equal onboth sides For AdS

5

CFT4

correspondence it is expectedthat

119885SYM4

= exp minus119865 (34)

should hold where 119865 = sum119899

ℎ=0

119865119899

ℎ

is the free energy of thestrings on AdS

5

times 1198785 119885SYM

4

and expminus119865 are the partitionfunction of SYM

4

and the partition function of the secondquantized type IIB string theory on AdS

5

times 1198785 respectively

For the present situation the comparison is relatively trivialOn one side we have a gauge theory with the partitionfunction given by (33) on the other side the119872-theory sectorwith the light-conemomentum 119901

+ is described by theMatrixmodel with 119901

+

= 119873119877 for which the partition function isagain (33)

Suppose 119878119872119875

+119865

(119884) is the membrane action on background119865 119878119872

119875

+119865

(119884) should be general covariant so for 119875+

= 1198751015840+

120572119909+

= 119891(1199091015840+

) and 119883 = 119892(1198831015840

) there is 119878119872

119875

+119865

(119884) =

119878119872

119875

1015840+119865

1015840(1198841015840

) where 1198651015840 is the field coming from the coordinate

transformation

119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (35)

For the bosonic action we can see this is indeed the case If[119889119884] = [119889119884

1015840

] that is the path integral measure is coordinateindependent we will have

int [119889119884] 119890minus119878

119872

119875+119865(119884)

= int [1198891198841015840

] 119890minus119878

119872

1198751015840+1198651015840 (1198841015840)

(36)

After thematrix regularization themembrane configurations119884(119909

+

1205901

1205902

) and 1198841015840

(119909+

1205901

1205902

) become the matrix configu-rations 119884(119909

+

) and 1198841015840

(119909+

) and there is

119885 (119877 119865) = 119890119882(119877119865)

= int [119889119884] 119890minus119878

119877119865(119884)

= int [1198891198841015840

] 119890minus119878

11987710158401198651015840 (1198841015840)

= 119890119882(119877

1015840119865

1015840)

= 119885 (1198771015840

1198651015840

)

(37)

119882(119877 119865) = 119882(1198771015840

1198651015840

) At least restricted to (35) 119882(119877 119865) isthe diffeomorphism invariant functional of 119865


Let 1198650

119865minus

119865minusminus

represent fields with zero one and two 119909minus

indices Since

119878119877119865

(119884) = 119878119865

0119877119865

minus119877

2119865

minusminus(119884) (38)

119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus

) (39)

For the 11119889 supergravity fields (1198650

119865minus

119865minusminus

) which are trans-lation invariant along the 119909

minus direction 119909minus isin [0 2120587119877) the11119889 supergravity effective action is 119878eff(119877 119865

0

119865minus

119865minusminus

) 119878eff isinvariant under the coordinate transformation

119909minus

997888rarr119909minus

119877 119865

0

997888rarr 1198650

119865minus

997888rarr 119877119865minus

119865minusminus

997888rarr 1198772

119865minusminus

(40)

so

119878eff (119877 1198650

119865minus

119865minusminus

) = 119878eff (1 1198650 119877119865minus

1198772

119865minusminus

) (41)

The radius of 119909minus is absorbed in 119865minus

119865minusminus

as is in (39) In thisrespect119882 is consistent with 119878eff

Let 119865 = 119865 + 119865 where 119865 is the flat background with 119892+minus

=

minus1 119892119868119869

= 120575119868119869

and the rest fields being zero119882(119877 119865) = 0

119882(119877 119865) = int11988910

119909Γ119888

119877119865

(119909) 119865 (119909)

+1

2int 119889

10

1199091

times int11988910

1199092

Γ119888

119877119865

(1199091

1199092

) 119865 (1199091

) 119865 (1199092

) + sdot sdot sdot

= sum

119899

1

119899int 119889

10

1199091


119909119899

Γ119888

119877119865

(1199091

119909119899

)

times 119865 (1199091

) sdot sdot sdot 119865 (119909119899

)

(42)

where (119865 collectively represents supergravity fields Forexample Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) = 1205752

119882(119877119865)120575119892++

(1199091

)120575119892minusminus

(1199092

)|119865=119865

)

Γ119888

119877119865

(1199091

119909119899

) =120575119899

119882(119877 119865)

120575119865(1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(43)

For the same119865 the 11119889 supergravity effective action 119878eff(119877 119865)

is

119878eff (119877 119865) = sum

119899

1

119899int

2120587119877

0

119889119909minus

1

times int119889119909+

1

1198899x1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

int119889119909+

119899

1198899x119899

times Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

times 119865 (119909+

1

x1

) sdot sdot sdot 119865 (119909+

119899

x119899

)

(44)

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

) is the vertex function of the 11119889

supergravity If 119878eff(119877 119865) = 119882(119877 119865) there will be

Γ119888

119877119865

(1199091

119909119899

)

= Γ119888

119877119865

(119909+

1

x1

119909+

119899

x119899

)

= int

2120587119877

0

119889119909minus

1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

(45)

In linear gravity approximation 120575119881119865

(119909)120575119865(119910) = 0where 119881

119865

is the current density coupling with 119865(119909) as isdefined in (23)

Γ119888

119877119865

(1199091

119909119899

) = (minus1)119899

⟨119881119865

(1199091

) sdot sdot sdot 119881119865

(119909119899

)⟩119888

(46)

⟨119881119865

(1199091


(119909119899

)⟩119888

is the connected correlation function ofthe current density in contrast to

⟨119881119865

(1199091


(119909119899

)⟩

= int [119889119883119889120579 1198891198600

] 119890minus119878

0119881119865

(1199091


(119909119899

)

=(minus1)

119899

120575119899

119885 (119877 119865)

120575119865 (1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(47)

which is the correlation function of the current density In(42)119882(119877 119865) is expanded around the flat background119865 Onecan of course expand 119882(119877 119865) on a different backgroundgiving rise to the different Γ119888

119877119865

(1199091

119909119899

)In terms of 119881

119865

(119901)

⟨119881119865

(1199091


(119909119899

)⟩119888

= int11988910

1199011


119901119899

119890minus119894(119901

1119909

1+sdotsdotsdot+119901

119899119909

119899)

times ⟨119881119865

(1199011


(119901119899

)⟩119888

(48)

Note that 119881119865(119883)

(119909+

119909119868

minus 119886119868

) = 119881119865(119883+119886)

(119909+

119909119868

) 1198780

(119883 + 119886) =

1198780

(119883) so

⟨119881119865

(1199091


(119909119899

)⟩119888

= ⟨119881119865

(1199091

+ 1198861


(119909119899

+ 119886119899

)⟩119888

(49)

The correlation function is translation invariant⟨119881

119865

(1199011


(119901119899

)⟩119888

= 0 if 1199011

+ sdot sdot sdot + 119901119899

= 0Let us first consider the one point function

⟨119881119865

(119909)⟩ = minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865

= V119865

(50)

V119865

is the vacuum expectation value of the current densitywhen the background field is 119865 and is a constant inspacetime 119878

119877

is 119878119874(9) invariant so V119865

should also be 119878119874(9)

invariant As a result

V119892

119868+

= V119892

119868minus

= V119862

119868119869119870

= V119862

119868119869+

= V119862

119868119869minus

= V119862

119868+minus

= V120595

119868

= 0

(51)


V119892

119868119869

= 0 if the traceless condition is imposed V120595

+

=

V120595

minus

= 0 due to the supersymmetryThe nonvanishing currentdensities are V

ℎ

++

Vℎ

minusminus

and Vℎ

+minus

In particular Vℎ

++

= 119901+

1198719 is

the vacuum expectation value of the light-cone momentumdensity For the generic value of 119865

⟨119881119865

(119909)⟩119865

= minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865+

119865

= V119865

(119909)

= sum

119899

minus1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Γ119888

119877119865

(119909 1199091

119909119899minus1

)

times 119865 (1199091

) sdot sdot sdot 119865 (119909119899minus1

)

(52)

is the vacuum expectation value of the current densityin presence of the background field In string theory thevanishing of the one point function the tadpole for vertexoperators gives the equations of motion for backgroundfields Similarly here if 119882(119877 119865) is the effective action of thesupergravity fields on SUGRA solution background therewill be ⟨119881

119865

(119909)⟩119865

= 0 except for ℎ++

whose vertex operatoris the same as the tachyon in bosonic stringWe will return tothis problem later

23 Another Effective Action of Matrix Theory For the given119865(119909) V

119865

(119909) is uniquely determined Conversely different119865(119909) may result in the same V

119865

(119909) This is quite like thesource-gravity coupled system For the given gravity field thedensity of the source can be obtained through 120575119878

119892

120575119892120583]

=

minus119879120583] On the other hand with the given source the gravity

solution is not unique Nevertheless with the proper bound-ary condition imposed there is always a privileged solutionWewill choose the boundary condition so that for V

119865

(119909) = 0119865(119909) = 0 119865(119909) could be interpreted as the field generatedby the current density V

119865

(119909) Other boundary conditionscorrespond to adding the external supergravity backgroundfor example the plane wave background in addition to fieldsgenerated by source We will discuss this situation later

Then there is a one-to-one correspondence between V119865

and 119865 and so a Legendre transformation is possible Beforethat we will first define Γ(119884)

Γ (119884) = 119878119877119865

(119884) + 119882 (119877 119865) (53)

119890Γ(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

(54)

119865 is solved from the equation

int [119889]120575

120575119865 (119909)119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

[minus

120575119878119877119865

()

120575119865 (119909)+

120575119878119877119865

(119884)

120575119865 (119909)] = 0

(55)

or equivalently

int [119889] 119890minus119878

119877119865(

119884)

120575119878119877119865

()

120575119865 (119909)=

120575119878119877119865

(119884)

120575119865 (119909)= minus

120575119882 (119877 119865)

120575119865 (119909) (56)

In some sense 119865(119909) is the field generated by 119884 Take aderivative of (54) with respect to 119884 using (55) we get

120575Γ (119884)

120575119884=

120575119878119877119865

(119884)

120575119884 (57)

where 119865 is solved from (55) The variation on the right-handside of (57) only acts on 119884 with 119865 being fixed

Recall that the 119877 dependence of 119878119877119865

(119884) only comes from119877119865

minus

and 1198772

119865minusminus

so if the solution of (56) is (1198650

119865minus

119865minusminus

) with119877 replaced by120572119877 the solutionwill become (119865

0

119865minus

120572 119865minusminus

1205722

)119878119877119865

(119884) 119882(119877 119865) and Γ(119884) remain invariant Γ(119884) is 119877-independent In supermembrane picture suppose there aretwo supermembrane theories with the light-conemomentum119875+ and 119875

1015840+ and consider the fields generated by the sameconfiguration 119884(119909

+

1205901

1205902

) Let 1199091015840minus rarr 1198751015840+

1199091015840minus

119875+ 1198751015840+ rarr

119875+ in this frame the generated fields are the same changing

back we get the relation 1198650

= 1198650

119865minus

119875+

= 1198651015840

minus

1198751015840+ 119865

minusminus

119875+2

=

1198651015840

minusminus

1198751015840+2 When plugged into 119878

119872

119875

+119865

(119884) the 119877 dependence iscanceled so the obtained Γ(119884) is the same

With these properties collected we may consider thepossible interpretation of Γ(119884) and119882(119877 119865) If119882(119877 119865) is theeffective action of supergravity since 119878

119877119865

(119884) is the matrixtheory action on background 119865 Γ(119884)will be the action of thesource-gravity coupled system and is on shell with respect tosupergravity and thus could be taken as the effective actionof the configuration 119884 120575Γ(119884)120575119884 = 120575119878

119877119865

(119884)120575119884 = 0 isthe quantum corrected equation of motion which differsfrom the classical equation of motion 120575119878

0

(119884)120575119884 = 0 inthat the background fields in former are generated by theconfiguration 119884 itself while the background fields in latterare given For time-independent 119884 the stationary point ofthe effective action gives the vacuum configuration In thiscase 120575Γ(119884)120575119884 = 120575119878

119877119865

(119884)120575119884 = 0 is equivalent to therequirement that the branes should not exert force to eachother which is the no force condition for BPS configurations[26] All of the above statements are based on the assumptionthat 119882(119877 119865) is the effective action of supergravity whichhowever is unproved

Let us continue to explore the properties of Γ(119884) and119882(119877 119865) For simplicity let 119877 = 1 or in other words let the119909minus indexed fields absorb 119877 In a weakly curved background

119878119865

(119884) = 1198780

(119884) + int 11988910

119909119865(119909)119881119865(119884)

(119909) Γ(119884) = 1198780

(119884) + Γ(V119865

)where

Γ (V119865

) = int11988910

119909119865 (119909) V119865

(119909) + 119882(119865 + 119865) (58)

is the Legendre transformation of119882(119865)

minus120575119882 (119865)

120575119865 (119909)

100381610038161003816100381610038161003816100381610038161003816 119865

= V119865

= 119881119865(119884)

(119909)

120575Γ (V119865

)

120575V119865

(119909)

1003816100381610038161003816100381610038161003816100381610038161003816V119865

= 119865 (119909)

(59)


Let

Λ119865

(1199091

119909119899

) =120575119899

Γ (V119865

)

120575V119865

(1199091

) sdot sdot sdot 120575V119865

(119909119899

)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

(60)

then

Γ (119884) = 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times V119865

(1199091

) sdot sdot sdot V119865

(119909119899

)

= 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(61)

For the given V119865

(119909)

119865 (119909) = sum

119899

1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Λ119865

(119909 1199091

119909119899minus1

)

times V119865

(1199091


(119909119899minus1

)

(62)

Equation (52) gives the current density V119865

(119909) generated by thefield 119865(119909) (62) gives the field 119865(119909) generated by the currentdensity V

119865

(119909) If Λ119865

is the connected Greenrsquos function ofsupergravity119865(119909)will be the vacuumexpectation value of thesupergravity field 119865 in presence of the source V

119865

In classicallevel 119865(119909) could be calculated by 120575119878

119892

120575119865(119909) = minusV119865

(119909) with119878119892

the classical action of supergravity

Γ119888

119865

(119909 minus 119911) = minus120575V

119865

(119909)

120575119865 (119911)

100381610038161003816100381610038161003816100381610038161003816 119865=0

Λ119865

(119911 minus 119910) =1205752

Γ (V119865

)

120575V119865

(119911) 120575V119865

(119910)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

=120575119865 (119911)

120575V119865

(119910)

100381610038161003816100381610038161003816100381610038161003816V119865=0

(63)

so

minusint11988910

119911Γ119888

119865

(119909 minus 119911) Λ119865

(119911 minus 119910) = 12057510

(119909 minus 119910) (64)

minusΛ119865

(1199091

minus 1199092

) is the inverse of Γ119888119865

(1199091

minus 1199092

) Subsequently

Λ119865

(1199091

1199092

1199093

) = int11988910

1199101

int11988910

1199102

times int11988910

1199103

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Γ119888

119865

(1199101

1199102

1199103

)

Γ119888

119865

(1199091

1199092

1199093

) = minusint11988910

1199101

int11988910

1199102

times int11988910

1199103

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Λ119865

(1199101

1199102

1199103

)

Λ119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Λ119865

(1199104

minus 1199094

)

times [Γ119888

119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Λ119865

(1199111

minus 1199112

)

times (Γ119888

119865

(1199111

1199101

1199104

)

times Γ119888

119865

(1199112

1199102

1199103

)

+ Γ119888

119865

(1199111

1199102

1199104

)

times Γ119888

119865

(1199112

1199101

1199103

)

+ Γ119888

119865

(1199111

1199103

1199104

)

times Γ119888

119865

(1199112

1199101

1199102

)) ]

Γ119888

119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Γ119888

119865

(1199104

minus 1199094

)

times [Λ119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Γ119888

119865

(1199111

minus 1199112

)

times (Λ119865

(1199111

1199101

1199104

)

times Λ119865

(1199112

1199102

1199103

)

+ Λ119865

(1199111

1199102

1199104

)

times Λ119865

(1199112

1199101

1199103

)

+ Λ119865

(1199111

1199103

1199104

)

times Λ119865

(1199112

1199101

1199102

)) ]

(65)


The relation betweenΛ119865

and Γ119888

119865

shows thatΛ119865

and Γ119888

119865

are theconnected Greenrsquos function and the vertex function of a par-ticular quantumfield theoryΛ

119865

(119911minus119910) = (120575119865(119911)120575V119865

(119910))|V119865=0

gives the change of the supergravity field with respect to thecurrent density so it is natural to take it as the propagator ofthe supergraviton We will see some evidence for it

In (54) let = 119884 + 120578

119890Γ(119884)

= int [119889120578] 119890minus119878

119865(119884+120578)+119878

119865(119884)

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minusint119889120591120575119878

119865

(119884)

120575119884 (120591)120578 (120591)

minus1

2int119889120591 119889120591

1015840

1205752

119878119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)

times 120578 (120591) 120578 (1205911015840

) + sdot sdot sdot

= minus119878(1)

119865

(119884 120578) minus 119878(2)

119865

(119884 120578) + sdot sdot sdot

(66)

119878119865

could be written as (We only write 1205972

120591

but one should beaware that the fermionic kinetic term 120597

120591

also exists)

119878119865

(119884) = int119889120591 [1

2119884120597

2

120591

119884 + 1198810

(119884) + 119881 (119865 119884)]

= 1198780

(119884) + int119889120591119881 (119865 119884)

(67)

with 119881(119865 119884) taking the form as that in (27) For the given 119884solve 119865

0

from 120575119878119865

0

(119884)120575119884 = 0 and expand 119878(119899)

119865

(119884 120578) around1198650

119865 = 1198650

+ 119891 Consider

119878(1)

119865

(119884 120578) =120575119878

(1)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650

119891 + sdot sdot sdot

119878(2)

119865

(119884 120578) = 119878(2)

119865

0

(119884 120578) +120575119878

(2)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


(68)

119891 = 119891(119884) is solved as the functional of 119884 Let Γ(119884) = Γ1

(119884) +

Γ2

(119884) + sdot sdot sdot since

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minus119878(2)

119865

0

(119884 120578) + sdot sdot sdot

Γ1

(119884) = ln int [119889120578] 119890minus119878

(2)

1198650(119884120578)

= ln (det119860119865

0

)12

(69)

with

119860119865

0

(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842+

1205972

119881 (1198650

119884)

1205971198842] 120575 (120591 minus 120591

1015840

)

(70)

In particular when 1205751198780

(119884)120575119884 = 0 1198650

= 0 119860119865

0

reduces to

119860(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842] 120575 (120591 minus 120591

1015840

) (71)

The corresponding Γ1

(119884) is the same as the one-loop con-tribution Γ

1

eff(119884) of the standard effective action Γeff(119884) inbackground gauge

119890Γeff(119884) = int [119889120578] 119890

minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

(72)

subject to the condition

int [119889120578]120575

120575119869 (120591)119890minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

= 0 (73)

Γ1

(119884) = Γ1

eff(119884) has been calculated for the arbitrary 119884

satisfying 1205751198780

(119884)120575119884 = 0 [3 25]

Γ1

(119884) =1

2int119889

10

1199091

times int11988910

1199092

1198660

Fc (1199091 1199092) 119881119865(119884) (1199091) 119881119865(119884) (1199092)

(74)

with 1198660

Fc being the free supergraviton propagator Comparedwith (61) Λ

1198651

(1199091

1199092

) = 1198660

Fc(1199091 1199092) If the result can beextended to the 119899-point Greenrsquos functions and to the full-loopcalculation there will be

Λ119865

(1199091

119909119899

) = 119866Fc (1199091 119909119899) (75)

where 119866Fc is the connected full Greenrsquos function of super-gravity Γ

119888

119865

(1199091

119909119899

) and 119882(119865) will then become the 119899-point vertex function and the effective action of supergravityrespectively

In quantum field theory unlike the 119878-matrix the effectiveaction is not the observable and thus is not uniquely definedDifferent gauges and the parameterization give the differenteffective actions Γ(119884) and Γeff(119884) differ from a parameter-ization transformation Γ(119884) = Γeff(119884) However to comparethe Matrix theory with supergravity we do need a privilegedeffective action On supergravity side in light-cone gauge theexpected effective action is

Γ119892

[119884 (120591)] = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(76)

The effective action defined in (61) is also the Taylor seriesof 119881

119865(119884)

and thus could be compared with (76) directly Thestandard effective action Γeff(119884) is expanded as the Taylorseries of119884 A careful reorganization is needed to get119881

119865(119884)

butit is unclear whether it is always possible to do so Moreoverunder a Legendre transformation (76) becomes

Γ119892

[119884 (120591)] = 119878eff (119865)

+ int11988910

119909119865 (119909)119881119865(119884)

(119909) sim 119878eff (119865) + 119878119865

[119884 (120591)]

(77)

with

minus120575119878eff(119865)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119865(119884)

(119909) =120575119878

119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(78)


119878eff(119865) is the effective action of supergravity Equation (77) isalmost the same as (53)The only problem is whether119882(119865) =

119878eff(119865) or not

24 Free Energy of String and the Effective Action of theBackground Fields In string theory there is a similar storyIn [17 18] it was shown that the effective action of thesupergravity could be taken as the renormalized free energyof the strings on background 119865

119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)

where 119865 satisfies the free field equationEquation (79) looks consistent with our philosophy the

free energy of the stringsmembranes on a given backgroundgives the effective action of the background fields Howeverthere is a difference119865 is the renormalized field other than thebare field In fact in (79) a particularWeyl gauge119892

119886119887

= 1198902120590

119892119886119887

is always imposed and so 119882119904

(119865 120590) also has the dependenceon 120590

120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904

(119865) although 120573(119865) = 0 is notthe necessary condition

Suppose 119878eff(119865) is the effective action of the stringmodes

119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650

) = 0119865will evolve alongthe RG flow that is 119882

119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840

] A specialproperty of 119865

0

is that it will finally reach an IR fixed point 119865119882119904

(1198650

120590) = 119882119904

(119865infin) In fact since1198650

satisfies the first-order120573 equation the RG flow may bring it to an IR fixed pointSince 120573(119865) = 0 119882

119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650

)119865 is calculated from 119865

0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895

)120573(119865) = 0 is equivalent to 120575119878eff(119865)120575119865(119909)|

119865=

119865

= 0 [18] In[18] it was shown that 119882

119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904

(119865) = 119878eff(119865) There is a one-to-one corre-spondence between the solution space of 120575119878eff(119865)120575119865(119909) =

0 and the solution space of Δ119865 = 0 So for all 119865 with120575119878eff(119865)120575119865(119909) = 0 = 120573(119865) 119882

119904

(119865 120590) = 119878eff(119865) In 120590-modelapproach only for such 119865 the conformal factor decouplesthus the calculated string free energy is physical In thissubspace 119882

119904

(119865) = 119878eff(119865)

119882119904

(119865 120590) with 120573(119865) = 0 usually depends on 120590 Howeverthe 120590-dependence drops out for 119865

0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)

is the 119878-matrix functional [27] For the on-shell 1199011

119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)

is the connected scattering amplitude It is only in Polyakovapproach canwe construct the 119878-matrix functional in thiswaysince it intrinsically involves some kind of renormalizationwhich is equivalent to the subtraction of the massless poleexchange contribution [18 27]

To conclude for 119865 satisfying 120575119878eff(119865)120575119865(119909) = 0119882119904

(119865 120590) = 119882119904

(119865) = 119878eff(119865) However the off-shell extensionof the effective action and the free energy are both ambiguousin Sigma-model approach

Another support for the identification of the string theoryfree energy on a particular background and the effectiveaction of the background fields comes from AdSCFT SYM

4

is a nonperturbative description of the type IIB string theoryon AdS

5

times 1198785 It is expected that the Hilbert spaces of

both sides are isomorphic to each other Similar with theChern-Simonstopological string correspondence 119885SYM

4

=

119890119882

119904 where 119882119904

is the free energy of the string theory onAdS

5

times 1198785 The question is what will be the string dual of

SYM4

if the background metric of SYM4

is 120578120583] + 120575119892

120583](119909)

other than 120578120583] for very small 120575119892

120583](119909) A natural expectationis that such SYM

4

is dual to the type IIB string theoryon AdS

5

times 1198785 with a little modification of the background

metric 120575119866120583](119909 120590) that is entirely determined by 120575119892

120583](119909)Correspondingly 119885SYM

4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904

[120575119866120583](119909 120590)] is the free energy of the string theory on

AdS5

times1198785 with the background perturbation 120575119866

120583](119909 120590) beingturned on Note that for the stringy explanation of theSYM

4

partition function to be possible the type IIB dualmust have the definite background since the string partitionfunction is always defined on a given background Alsofor the state correspondence to be valid the dual type IIBstring theory should have the definite background otherwiseit is impossible to determine the string spectrum Nowreturn to the original topic Gauge theory calculation gives


119885SYM4

[120575119892120583](119909)] = 119890

119878eff[120575119866120583](119909120590)] where 119878eff[120575119866120583](119909 120590)] is thesupergravity effective action for the modified backgroundfields so 119878eff[120575119866120583](119909 120590)] = 119882

119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904

holdsThere is a naive way to interpret 119882

119904

(119865) = 119878eff(119865)Suppose the classical supergravity action is 119878

119888

(119865) from thefield theoryrsquos point of view

119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)

where 119865 is the fluctuation on 119865 or in other words super-gravitons living on background 119865 The effective action is thesum of the connected 1PI vacuum-vacuum diagrams of thesupergravitons on background 119865 The elementary propagatorand the vertices can be read from 119878

119888

(119865 + 119865) Now considerthe string theory on background 119865 we may have

119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904

(119865) is the sum of the irreducible vacuum-vacuumstring diagrams on background 119865 Note that in stringdiagram there is nither a concept of 1PI nor 1PR Also thereis no classical action like 119878

119888

(119865 + 119865) to determine the basicconstitution of the diagram The integration simply coversall possible string configurations Equation (86) can be takenas the stringy refined version of (85) In (86) we secretlyassumed that the unphysical worldsheet conformal factor isdecoupled In Polyakov approach this is possible only when119865 is the solution of 120575119878eff(119865)120575119865 = 0 The cancelation of theWeyl anomaly gives the eom for the effective action of thesupergravity including the 120572

1015840 corrections so 119878eff(119865) shouldbe the effective action with the 120572

1015840 corrections taken intoaccount

A natural119872-theory extension is

119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865

(119884) is the supermembrane action on 11119889 super-gravity background 119865 The membrane is already the secondquantized object so the left-hand side of (87) is 119890

119878eff(119865)

other than 119878eff(119865) For the generic 11119889 background 119865 119878119872119865

(119884)

should be covariantm and so the worldvolume metric mustbe introduced and integrated making the supermembranetheory nonrenormalizable For 119865 which is translation invari-ant along 119909

minus the light-cone gauge can be imposed Theconfigurations are then truncated to those with the light-conemomentum119901

+The integration out of such configurations onbackground 119865 gives the effective action of 119865 with the radiusof 119909minus sim 1119901

+

3 Matrix Theory on a Particular Vacuum

If119882(119877 119865) in (42) is the effective action of the 11119889 supergrav-ity field 119865(119909) that is translation invariant along 119909

minus for theon-shell 119865(119909) there will be 120575119882(119877 119865)120575119865(119909) = minusV

119865

(119909) = 0However for 119865 = 119865 which is obviously on shell the currentdensities V

119892

++

V119892

minusminus

and V119892

+minus

are nonvanishing (Note that the

vertex operators for 119892++

and 119892+minus

are quite like the vertexoperators for tachyon and dilaton in bosonic string theoryin which there is also a tadpole in flat spacetime [28])

119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)

With 120572 120573 = + minus one may try to solve 119892120572120573

(119909) from120575119882(119877 119865)120575119892

120572120573

(119909) = 0 and then take this value other than119892+minus

(119909) = minus1 119892minusminus

(119909) = 119892++

(119909) = 0 as the backgroundto do the expansion V

119892

120572120573

are constants so the generated119892120572120573

(119909) are also constants (although the constant 119892120572120573

(119909)

does not really solve the equation) and do not representthe substantial change of the background We will simplyneglect these tadpoles V

119892

++

should be distinguished from120575119878

119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510

(119909) which is the light-cone momentum density of a supergraviton localized at(120591 0 0) and of course will produce the nontrivial119892

minusminus

(119909)In (54) 119865 = 119865(119877 119884) is the functional of 119884 solved through

(55) For the arbitrary 119865 we may define

119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865

(119884) is the action of the source-gravity coupled system and is not necessarily on shell withrespect to gravity If 119878

119877119865

(120578) gives the Matrix theory descrip-tion of 119872 theory on background 119865 119878

119877119865

(119884 120578) will describe119872 theory on background 119865 in presence of the brane 119884 If 119884satisfies the equations of motion that is 120575119878

119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865

(119884 120578) starts from the quadratic term For 119865 satisfying120575Ω(119877 119884 119865)120575119865(119909) = 0 119878

119877119865

(119884 120578) then gives a description of


119872 theory on a background 119865 generated by the brane 119884 Inthis case Ω[119877 119884 119865(119884)] = Γ(119884)

0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)

The expectation values of all current densities vanish Thereis no tadpole

The 119880(infin) symmetry of Matrix theory is destroyed by 119884In particular if

119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)

the preserved gauge symmetry is 119880(1198731

) times sdot sdot sdot times 119880(119873119898

)Nevertheless the original119880(infin) symmetry still has its mani-festation For all 119906 isin 119880(infin)

120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731


) 120578 transforms like a gaugefield

31 Applied to PWMM In the following we will focus on aspecial example the plane wave matrix model [2] PWMMis a Matrix theory description of 119872-theory on 11119889 pp-wavebackground

1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)

The background preserves 32 supersymmetries while therest 11119889 supergravity solutions with 32 supersymmetries areflat spacetime AdS

4

times 1198787 and AdS

7

times 1198784 [29] In pp-wave

the dynamics of the 119872-theory sector with the light-conemomentum 119901

+

= 119873119877 is described by the 119880(119873) matrixmodel

119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)

119873 rarr infin 119877 rarr infin PWMM also preserves 32 supersymme-tries and has the same symmetry group as that of the pp-wavebackground

The classical supersymmetric solutions of the action are[2]

[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833

form the 119873 dimensional representation of119878119880(2) Suppose (3119877120583)119883

119894

= 119871119894 119894 = 1 2 3 119871

119894 can bedecomposed as

119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904

(119904 = 1 119905) stands for the spin 119895119904

irreduciblerepresentation of 119878119880(2)

(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)

The solution can be taken as a set of 119873119904

coincident fuzzyspheres with the radius (1205833119877)radic119895

119904

(119895119904

+ 1) [2] The 119880(119873)

gauge symmetry is broken into119880(1198731

) times119880(1198732

) times sdot sdot sdot times119880(119873119905

)All fuzzy spheres are concentric When 119895

119904

rarr infin thefuzzy spheres become a set of membrane spheres given bysum3

119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904

+ 1)1205836119877 Each sphericalmembrane has 119901

minus

= 0 119901+ = (2119895 + 1)119877 so they are alsocalled the giant gravitons

The solution can also be interpreted as the spherical 1198725

branes with a dual assignment of the light-cone momentum[30] Any partition of 119873 may be represented by a Youngdiagram whose column lengths are the elements in the parti-tion In the 1198722 interpretation such a diagram correspondsto a state with one membrane for each column with thenumber of boxes in the column being the number of unitsof momentum In the dual 1198725 interpretation it is the rowsof the Young diagram that correspond to the individual1198725 with the row lengths giving the number of units ofmomentum carried by each1198725 In both cases the total light-cone momentum is always 119873119877 If 119873

119904

are finite they arefuzzy 1198725 branes When 119873

119904

rarr infin the fuzzy 1198725 becomethe spherical 1198725 wrapping 119878

5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903

Inpatricular the trivial vacuum 119883

119894

= 0 represents a single 1198725

braneAll of the supersymmetric solutions preserve 16 non-

linearly realized supersymmetries They are the 12 BPSstates on pp-wave background Although the backgroundand the Lagrangian are both maximally supersymmetricthere is no state in matrix model preserving all of the 32

supersymmetries The reason is that all states have the samenonzero light-cone momentum 119873119877 which itself woulddestroy the linearly realized supersymmetriesThe situation isdifferent in SYM

4

in which we do have a vacuum preserving


32 supersymmetries representing the ground state of thestring theory on AdS

5

times 1198785 (Brane like) 12 BPS states are

giant gravitons on AdS5

times 1198785 [31ndash33]

Similar to the PWMM ldquotiny graviton matrix modelrdquo(TGMT) is proposed as the nonperturbative description ofthe type IIB string theory on pp-wave background [34ndash36] TGMT can also be taken as the DLCQ of the typeIIB string theory on AdS

5

times 1198785 capturing the physics seen

from the infinite momentum frame (IMF) since in IMFAdS

5

times1198785 is viewed as the pp-wave [34ndash36] Although TGMT

preserves 32 supersymmetries the vacuum configurationscarrying the definite light-cone momentum are all 12 BPSmatching exactly with the 12 BPS states on type IIB pp-wave background which are giant gravitons (spherical 1198633

branes) and type IIB strings Lifted to 11119889 the light-conedimension is replaced by 119879

2 while the TGMT which is adiscrete regularization of the 1198633 branes in type IIB picturebecomes the regularization of the 1198725 branes [35] Notethat the 4-form field in type IIB pp-wave when lifted to11119889 becomes the 6-form field coupling electrically with the1198725 branes so it is natural to construct the matrix modelvia the discrete regularization of 1198725 branes other than theusual 1198722 branes On type IIB pp-wave 1198633 branes are 1198725

braneswrapping1198792 while the type IIB strings aremembraneswrapping one of 1198781 In contrast to the PWMM in TGMT thetrivial vacuum 119883 = 0 represents type IIB string (with 1198722

origin) while the nontrivial vacuum represents 1198633 branes(with1198725 origin) which is probably because the PWMMandthe TGMT are constructed from 1198722 and 1198725 respectivelyRecall that in PWMM the fuzzy configuration may have the1198722 and1198725dual interpretations [30] it is interesting to figureout whether a similar 1198651-1198633 dual interpretation also existsfor configurations in TGMT

With denoting the supersymmetric vacuum in (96)119872theory on pp-wave background in presence of the brane isdescribed by the Matrix model with the action

119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877

( 120578) starts from the quad-ratic term

For simplicity in the following we will assume 119877 is ab-sorbed into the fields If the backreaction of the brane

on pp-wave background is turned on the field 119865() will begenerated

119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)

pp-wave with added respects the 119878119880(2 | 4) symmetry sothe generated 119865() as well as the corresponding 119878

PW119865(

119884)

(120578) willhave the same symmetry

With the fields 119865() given in principle one can writedown 119878

PW119865(

119884)

(120578) explicitly The classical supersymmetric solu-tions of 119878

PW119865(

119884)

(120578) are still (96) To see this note that thespherical giant gravitons are 12 BPS states on pp-wave sothey should not exert force to each other [26] or in otherwords giant gravitons are still stable even if the backreactionof the other giant gravitons on pp-wave is turned on Indeedin [37] the giant graviton on the backreacted geometry isanalyzed The stable configuration is still the same as thatin the pp-wave case In [38] the quantum effective action ofPWMM around was calculated at the one-loop level isalso the stationary point of the effective action We may have120575Γ

PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884

= 0 With the pp-wavebackground replaced by the backreacted geometry 119878PW

119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)

( 120578) still starts from the quadratic term since120575119878

PW119865(

119884)

(119884)120575119884|119884=

119884

= 0Notice that although any configuration

1015840 in (96) maybe the classical vacuum of 119878PW

119865(

119884)

(120578) is special because thevacuum expectation values of current densities vanish onlyfor 119878PW

119865(

119884)

( 120578) but not for the generic 119878PW119865(

119884)

(1015840

120578)The 11119889 geometry produced by 12 BPS states of PWMM

was constructed in [19 39] The geometry has a bubblestructure containing noncontractible 7 cycles and 4 cyclessupporting 119865

4

and 1198657

fluxesThe geometry is smooth withoutsingularity and then sourcelessThis is an explicit realizationof the geometric transition [40] The backreaction makes thethe worldvolume of the 119872 branes shrink and the transversesphere blowupAs a result althoughwe start from the source-gravity coupled action the obtained supergravity solution issmooth and satisfies the sourceless equations too The braneaction as well as the current density is zero on the generatedsupergravity background

Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)

is the momentum eigenstate in 119909minus direction so the

generated 11119889 background119865() is translation invariant along119909minus In large 119903 limit the local structure of the giant gravitons is

not important while the asymptotic geometry is just the pp-wave with the perturbation roughly given by [39]

1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)


which is the field produced by the supergraviton which isstatic in 9119889 space carrying the definite light-conemomentum[41] 119876 = 30120587

3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901

isabsorbed into 119877 and 119903) With the coordinate transformation119909 rarr 119892

minus13

119904

119909 119909minus rarr 11989223

119904

119909minus (104) becomes

1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)

while the radius of 119909minus is now 119892minus23

119904

119877Under the 119909

minus reduction in string frame

1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)

In particular for (105)

1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)

is the near-horizon geometry of the1198630 branes [41]The type IIA solution coming from the 119909

minus reduction ofthe 11119889 field119865()was constructed in [19 37]When 119903 rarr infinthe perturbation which is the reduction of (104) along 119909

minus isthe near-horizon geometry of 119873 coincident 1198630-branes Theappearance of the near-horizon geometry is because of thenull reduction The reduction of (104) along 119909

+

minus 119909minus gives

the 1198630 brane solution [41] Different from AdSCFT hereno near-horizon limit is taken and the brane solution itselfbecomes the near-horizon geometry when reduced along 119909

minus

32 The Gauge Theory Dual from the Matrix Model Accord-ing to the previous proposal the Matrix theory descriptionof the 119872 theory on background (104) in presence of thesupergravitonwith119901

+

= 119873119877 is 119878119877119865

(119884 120578) where119865 is the fieldin (104) 119884 = 0 (if 119865 is the field in (105) 119878

119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877

is the BFSS action in (17) On the other hand accordingto AdSCFT the gauge theory description of the type IIAstring theory on background (104) is SYM

1

with the action119878119877=1

(120578) Since (104) becomes (107) under the 119909minus reduction it

is desirable to find a limit to make (108) reduce to 119878119877=1

(120578)

Consider the coordinate transformation 119909119868

rarr 119909119868

119877119909minus

rarr 119877119909minus under which

119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)

997888rarr 119865lowast

(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)

Correspondingly in Matrix theory action 119878119877119865

with a fieldredefinition 119883

119868

rarr 119883119868

119877 and also a 119909minus

rarr 119877119909minus rescaling

to make the radius of 119909minus become 1 the background fields

appearing in action are just 119865lowast

(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)

With afurther rescaling 120579 rarr 120579119877

32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)

that will appearReturn to (108) For any119877 119878

119877119865

is equivalent to 1198781119865

lowast(119877)

1198773

in which 119892lowast

minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865

lArrrArr lim119877rarr0

1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1

arrives Note that for finite 119909 lim119877rarr0

119909119877 = infin so bytaking the 119877 rarr 0 limit the background field that matterslives at the 119903 rarr infin region far away from the source

For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+

(119909) all fields are marginal so for the arbitrary 119877 [5]

119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)

Now consider the background 119865() coming from thebackreaction of the giant graviton on pp-wave Suppose

is a vacuum in (96) with

2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)

The radii of the spherical membranes are 119903119904

sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909

minus gives the smoothtype IIA solution that is constructed in [19] The genericsolution of type IIA supergravity with 119878119880(2 | 4) symmetryis characterized by a function 119881(120588 120578) and is given as [19]

1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)

where the dot and the prime represent the derivatives withrespect to log 120588 and 120578 respectively 119881 can be taken asan electrostatic potential for an axially symmetric systemwith conducting disks and a background potential 120588 is thedistance from the center axis and 120578 is the coordinate in thedirection along the center axis 119881(120588 120578) = 119881

119887

(120588 120578) + V120578

(120588 120578)where 119881

119887

is the background potential and V120578

is determinedby a configuration of conducting disks Each 119878119880(2 | 4)

symmetric theory is specified by 119881119887

each vacuum of thetheory is specified by a configuration of conducting disks

From (115) one can also get the uplifted 11119889 solution Forexample

1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)

Only the region 120578 ge 0 is meaningful There is an infinitelylarge conducting disk sitting at 120578 = 0 Vacuum (97)corresponds to 119905 disks located at 120578

1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905

+1)The electric chargeson these disks are equal to (120587

2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904

is the radius of the spherical membranes Thecorrespondence between the spacetime coordinates and thePWMM fields is (119883

1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+

With the given disk configuration the correspondingpotential is

119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)

which when plugged into (115) and (116) gives the 10119889

solution 11986510

(119909) as well as the 11119889 solution 119865(119909) 119909 standsfor the 9 space coordinates since the solution is translationinvariant along 119909

+ and 119909minus

Under the coordinate transformation 119909 rarr 119909119877 119909minus rarr

119877119909minus the disks are now located at 120578

119904

= (1205872)(2119895119904

+ 1) whilethe fields will transform as 119865(119909) rarr 119865

lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast

(119877 119909) can be obtained by plugging

119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)

into (115) and (116) Except for 1198781119865

lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)

into (115) and (116) one can get 119865lowast

10

(119877 119909) and the corre-sponding 119865

lowast

(119877 119909) In particular for PWMM 119881 = 119881PW119887


1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast

(119877 119909) = 119865(119909) as we haveseen before Both 119881(120588119877 120578119877) and 119877

3

119881(120588119877 120578119877) satisfycylindrically symmetric Laplace equation so 119865

lowast

(119877 119909) and119865lowast

(119877 119909) are also the type IIA supergravity solutions In fact119865(119909) 119865

lowast

(119877 119909) and 119865lowast

(119877 119909) are related to each other by thecoordinate transformation 119878

119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773

The limit of interest is 119899 rarr infin 119877 rarr infin 1198773119899 = 1205722 and

120583 fixed In this limit the concentric spherical membranes allhave the infinite radii with the difference (120572

119894

minus120572119895

)119877 rarr 0 Forthe finite 120588 120578 120588119877 rarr 0 120578119877 rarr 0 we need to consider thebehavior of V

120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877

can be approximated as the potential generated by dipoleslocated at plusmn120578

119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882

120578 minus (23)1205783 Indeed with

1198773

119881(120588119877 120578119877) plugged into (116) 119899 rarr infin 119877 rarr infin1198773

119899 being fixed the 11119889 solution finally approaches thepp-wave As a result lim 119878

119877119865

hArr lim 119878PW119877=1

1198773 Even if the

background field is 119865() in the limit taken we still getPWMM

On the other hand plug 1198773

119881(120588119877 120578119877) into the reduced10119889 solution (115) minus 2 = 119877

3

(V120578119877

minus 2V120578119877

) the V120578119877

partnow matters Let 119908(120588

2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)

The 10119889 solution has the dependence on the disk configura-tion Equation (122) is lim119865

lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623

1198773 we will get lim 119865

lowast10

(119877 119909) By takingthe 119877 rarr infin limit the relevant region is 120588 rarr 0 120578 rarr 0which is again far from the source The 11119889 field approachesthe pp-wave but the reduced 10119889 field still depends on thedisk configuration Similarly for (104) suppose 119876 = 119873119896

7then when 119896 rarr 0 the 11119889 background is fat while theassociated 10119889 background (107) is still the near-horizongeometry but becomes singular now Nevertheless let 119906 =

119903119896 119903 rarr 0 119906 fixed (107) can be written as

1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)

This is similar to the limit taken in AdSCFT [7]Now consider the exact form of 119878

1119865

lowast(119877)

For the generic11119889 background 119865(119909) the bosonic part of 119878

1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+

times Tr(119892lowast+minus

119863+

119883119868

119863+

119883119869

21198922lowast+minus

119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)

The relation between 119865lowast

(119877 119909) and the 119909minus reduced 10119889

background 119865lowast10

(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899

119892lowastminusminus

= 1198904120601

lowast3

119892lowast119898minus

= 1198904120601

lowast3

119860lowast119898

(125)

In the limit with 119899 rarr infin 119877 rarr infin 1198773119899 fixed if 119892lowastminusminus

rarr 0119892lowast

119868minus

= 0 the rest 119865lowast10

(119877 119909) is finite there will be

1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869

radicminus119866lowast++

997888rarr 119890120601

lowast119860lowast+

(126)


With these relations

lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909

+ Trradicminus119866lowast++

times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)

So in the limit taken 1198781198871119865

lowast(119877)

could be identifiedwith 119878119887

1119865

lowast10(119877)

the action of SYM

1

on the 10119889 background 119865lowast10

(119877 119909) Thediscussion can also be extended to the full 119878

1119865

lowast(119877)

with thefermionic part included

Specified to the background generated by on pp-wave(126) is satisfied for the solution in (122) so lim 119878

1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)

Indeed one can verify thevalidity of (127) by directly plugging (122) in it In the limittaken PWMM can be taken as the SYM

1

living on 10119889

background (122) Equation (104) also satisfies the abovecriteria Actually

int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+

times Trradicminus119866

++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++

and so forth are the near-horizon geometry ofthe1198630 branes in (107) with 119892

119904

= 1In the limit with 119899 rarr infin119877 rarr infin1198773119899 = 120572

2 fixed (122)gives the solution in the region of finite 120588 and 120578 To study theregion near the spherical membrane shells with 120588 120578minus119899 finitea change of variables 120578 rarr 120578 + 119899 can be made As is shown in[42] in this limit

1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887

is the background potential for SYM119877times119878

2 Alternatively we may directly plug

119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)

into (116) and (115) taking the limit at the end LetV120578119877

(120588119877 (120578 + 119899)119877) = 119906(1205882

120578) the 11119889 background 119865lowast

(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)

which has the topology of 119877 times 1198777

times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2

now have the same prefactor Equations(122) and (133) can be taken as the background for 1119889 and3119889 gauge theories respectively

On gauge theory side to study the fluctuation around thespherical membranes we should expand 119878

PW119877=1

1198773 around

In [20] it was shown that in the limit of 119899 rarr infin 1198773119899fixed (i) PWMM around a certain vacuum is equivalent toSYM

119877times119878

2 around each vacuum and (ii) SYM119877times119878

2 around acertain vacuum with a periodicity imposed is equivalent toSYM

119877times119878

3119885

119896

around each vacuum In particular SYM119877times119878

3 canbe realized as the PWMM around a certain vacuum with aperiodicity condition imposed

Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2

is the action of the 119880(119873) SYM119877times119878

2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)

1120583 is the radius of 1198782 which is parameterized by (120579 120601)

1198922

119877times119878

2 = 1198773

1205833

119899 119910 represents the vacuum

Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)

Region I and II correspond to 0 le 120579 lt (1205872) + 120576 and(1205872) minus 120576 lt 120579 le 120587 respectively 120572

119904

is identified withthe 120572

119904

in (113) The 119880(119873) gauge group is spontaneouslybroken to 119880(119873

1

) times 119880(1198732


) Equation (135)corresponds to 119905 disks located at 120578

1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905

with the electric charges on eachequal to (120587

2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

The previous 120578119904

=

(1205872)(119899 + 120572119904

) in PWMM now becomes 120578119904

= (1205872)120572119904

due tothe 120578 rarr 120578 + 119899 redefinition

The correspondence between the spacetime coordinatesand the SYM

119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2

) Equation (134) is the SYM119877times119878

2 on flatbackground Plug (133) into (134) similar to (127) for bosonicpart we have

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588

2119906119899)14 The action remains the

same so (134) is also the action of SYM119877times119878

2 on background(133) The conclusion holds for fermionic part except for arescaling of 120582

Indeed as we will show in Appendix A such phe-nomenon is very common Generically the action of SYM

119901+1

on flat background is equal to the action of the SYM119901+1

onthe near-horizon geometry of the 119863119901 branes (For 119901 ge 6 theworldvolume theories of119863119901 branes do not decouple from thebulk as is discussed in [7]) For SCFT

3

and SCFT6

on 1198722

and 1198725 such requirement can even offer some clues for thestructure of the field theory


119906 = 119906(1205882

120578) is the function of the radial directions Forthe given field configuration [Φ119883

4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)

Compared with (134) with 119883119898

rarr 119883119898 Φ rarr Φ the 3119889

background fields on 119877 times 1198782 will get the radial dependence

119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)

This is some kind of realization of the holography on whichwe will discuss more in the next section One special featurehere is that the background fields depend on two radialdirections 120588 and 120578 Let 119903 = (120588

2

+ 1205782

)12 1198891199042

10

in (133) can bewritten as

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)

with SYM119877times119878

2 living at 119903 = infin However with the energy 119864

indentified with 119903 the RG flow cannot give 119906(1205882

120578) which isnot just the function of 119903 Instead we will make the 119883

119898

rarr

119883119898Φ rarr Φ transformation to recover 120588120578 dependence of the

3119889 background fields in 119878119884119872119877times119878

2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW

(119865) is the 119909minus reduction of the 11119889 supergravity effective

action of the field generated by the brane on pp-wave Inthe limit with 119899 rarr infin 119877 rarr infin 1198773119899 = 120572

2 fixed under thechange of variables 120578 rarr 120578 + 119899

119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2

(119865) is the type IIA action for the above-mentioned supergravity solution dual to a vacuum ofSYM

119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)

As is demonstrated in [19 20] from SYM119877times119878

2 and thecorresponding type IIA solution it is also possible to getthe SYM

119877times119878

3 and the associated type IIB solution In (135)suppose

Φ =120583

2

times diag[ 119904 minus 1 119904 minus 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]

(144)where 119904 runs from minusinfin to infin Expanding the action (134)around this vacuum and imposing the condition 119910

(119904+1119905+1)

=

119910(119904119905) on all of the field fluctuations one will get [20]

119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3

is the action of the 119880(119872) SYM on 119877 times 1198783


This is a special example of Taylorrsquos prescription for thecompactification (the 119879-duality) in matrix models [43] Thenew ingredient is the nontrivial gauge field which makes anontrivial fibration of 1198781 over 1198782 rather than a direct productas a result it is 1198783 other than 119878

2

times 1198781 that is obtained [20]

On gravity side start from the trivial vacuumof SYM119877times119878

2 for which there is only a single disk with the electric charge(1205872

8)119872 compactify the 120578 direction the disk configurationin covering space will contain the infinite copies of disks withthe period 1205872 corresponding to the vacuum (144) The typeIIA geometry generated by (144) after a 119879-duality transfor-mation becomes the type IIB geometry AdS

5

times 1198785 [19] On

field theory side since 120578 is compactified the field fluctuationsshould respect the periodicity condition 119910

(119904+1119905+1)

= 119910(119904119905)

Taylorrsquos prescription for the compactification also involvesthe 119879-duality transformation making SYM

119877times119878

2 in type IIAbecome 119878

119877times119878

3

in type IIB Equation (143) turns into

119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)

For the trivial vacuum of 119878119877times119878

3

119865 represents AdS5

times 1198785

backgroundFinally the geometry arising from the backreaction of the

1198633 giant gravitons with the definite light-cone momentumon type IIB pp-wave background was also constructed in[39] It is tempting to find the corresponding gauge theorydual one attempt is to expand the TGMT [34ndash36] aroundthe corresponding 12 BPS configuration and then take thecertain limit TGMT is the discrete regularization of the 1198633

branes so the resulted gauge theory should be a 4119889 gaugetheory as is required since the geometry is generated byspherical1198633 branes on pp-wave

4 Holography in AdSCFT

SYM4

is the gauge dual of the string theory on AdS5

times 1198785 A

natural question is what will be the gauge dual of the stringtheory if the field perturbation is added toAdS

5

times1198785 For BFSS

matrix model we have matrices 119883119894 119894 = 1 9 representingnine transverse coordinates so for a scalar field

120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1

sdot sdot sdot (1199099

)119899

9

(148)

we may get the operator realization

Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)

with 119909119894 replaced bymatrix119883

119894 AddingΦ(119905) to the Lagrangiangives a matrix model on a background with the scalar field120601(119905 119909

1

1199099

) turned on The situation for SYM4

is a littledifferent In BFSSmodel the original background is flat all ofthe fields could be expanded as the Taylor series in Cartesiancoordinates For SYM

4

the original background is AdS5

times1198785

fields are expanded in terms of the spherical harmonics on 1198785

For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899

(119910|119910|) is a spherical harmonic of rank 119899 ℎ119899

(119909 |119910|)

is a scalar field on AdS5

transforming in the (0 119899 0) irrepof 119878119874(6) It is necessary to find the operator correspondenceof 119884119899(119910|119910|) In SYM

4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) whichhowever is the operator realization of 119862

119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898

) 119877and Σ may be the operators corresponding to 119903 = |119910| = 119890

120590

and 120590 respectively For the 5119889 field 119865(119909 119903) for all 1205900

= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900

) are coupling constants of the 4119889 gaugetheory living in (119909 120590

0

) Similar to (28) when the gauge theorymoves along 120590 Σ rarr Σminus 119886119868 119891(119899)(119909 120590

0

) rarr 119891(119899)

(119909 1205900

+ 119886) Σcould be decomposed into the 119880(1) part 120590 and the tracelesspart Σ

119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904

The configuration (1198835

11988310

) is equivalent to(119883

5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898


One may take 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904

) as the operatorcorresponding to 119884

119899

(119910|119910|) and consider the gauge theorywith the vertex operator perturbation realized as (in factwe will choose 119883 instead of 119883

119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899

(119909 120590) can be the arbitrary 5119889 function Gauge theory likethis is difficult to approach directly We still prefer SYM

4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) added which after a suitabletransition will become a gauge theory with the operatorℎ119899

(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899

(119909 120590) is notarbitrary anymore but is determined by ℎ

119899

(119909)This is quite similar to the noncritical string couplingwith

2119889 gravity For critical string with 26 coordinates any 26119889

background fields can be represented by the vertex operatorperturbations on the stringworldsheet action For noncriticalstring with for example 25 coordinates only the vertexoperator perturbations corresponding to 25119889 fields can beconstructed However if the noncritical string is coupled tothe conformalmode of the 2119889 gravity so that the total numberof degrees of freedom is 26 after a property transformationwith the partition function kept invariant the theory willbecome the critical string coupling with the 26119889 backgroundfields The 26119889 fields are induced from the 25119889 fields with theconformalmode acting as the 26th dimension For SYM

4

therole of the conformal mode is played by 120590

41 Noncritical String Coupling with 2119889 Gravity Let us havea simple review of the noncritical string [44ndash49] Considerthe bosonic string living in 119889 dimensional spacetime Thecorresponding nonlinear sigma model action is

119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)

120583 ] = 1 119889 The partition function is defined as

119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)

The theory is conformal invariant since all 119892 are integratedThe integration over the small 119892 gives the divergence tocure which a cutoff should be introduced destroying theconformal invariance Under the conformal gauge fixing 119892 =

119890120593

119892120593 ge 0 119892 is a small metric giving the cut-off scale

119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871

(120593 119892) is the Liouville action With 119863119892

119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892

120593 the partition function becomes

119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)

The original 119889 dimensional background field 119865(119883) after thegravitational dressing becomes the 119889 + 1 dimensional field119865(119883 120590) 119865(119883 0) = 119865(119883) One can also make a change of thevariables to move the boundary from 120593 = 0 to 120593 = minusinfin

119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus

119878(120593minusinfin+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)

With the cutoff being introduced the conformal transforma-tion should be accompanied by a change of the boundary

119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)

With the cutoff being removed 119885(119892) = 119885(119890minus120596

119892)


119878(120593119883 119892) is the action of the bosonic string couplingwith the background fields119865(119883 120590)The conformal invarianceindicates that 119865(119883 120590) should be the on-shell solution ofgravity To arrive at this result it is important that in (157)the change of the 2119889 metric is always compensated by theadjustment of the background fields keeping the form of thesigma model invariant

42 Inducing the Radial Dependent Fields Now considerSYM

4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894

119909radicminus119892

times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)

Select 119883 so that 119883 can be expressed as 119883 = 119890120590

119883 with 120590 isin

(minusinfin 0] 119883 is the infinite matrix representing the maximum119883 The partition function is

119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)

Due to the Weyl invariance

119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)

with no counterterm added SYM4

is conformal invariantso at least for finite 120596 and 119892

120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596

119883) However this may not be the case when120596 = infin or when 119892

120583] rarr 0 119878119888[120596 119892] will still be kept for theinfinite conformal transformation

Nevertheless with 119892120583] = 120578

120583] we always have 119878119888

[120590 119892] = 0With119883 parameterized as119883 = 119890

120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)

From (166) to (167) the integration over120590 in119883 is converted tothe integration over120590 in119892The partition function of SYM

4

onflat background is equal to the partition function of a gaugetheory on a curved background From (167) one may readthe background metric

1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033

= +infin and 120590 isin (minusinfin 0] 1199033

=

+infin because we take 119883 as the standard matrix For finite 1199033

120590 isin (minusinfin +infin] With 119903

3

being the radius of AdS5

times 1198785 (167)

could be taken as the partition function of a gauge theory onAdS

5

times 1198785

Equations (166) and (167) have the direct extension toSCFT

3

and SCFT6

for 1198722 and 1198725 For SCFT3

119883 has theweight 12 and the Weyl transformation is (119890

120590

119883 119892120583]) rarr

(119883 1198904120590

119892120583]) The induced metric is

1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6

119883 has the weight 2 and the Weyltransformation is (119890

120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035

Equations (169) and (170) are the near-horizongeometries of1198722 and1198725 respectively

More generically as we will show in Appendix A forSYM

119901+1

SCFT3

and SCFT6

on 119863119901 1198722 and 1198725 the actionon flat background and the action on the near-horizongeometry of 119863119901 1198722 and 1198725 are the same As a result for1198722 and 1198725 with the near-horizon geometry given by (169)and (170) 119883 rarr (119903

2

119903)119883 and 119883 rarr (1199035

119903)119883 are always


accompanied by 1198891199092

3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6

For119863119901 with the near-horizon geometry

1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)

minus(7minus119901)2

(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4

119883 transformation will then make119889119909

2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901

)(7minus119901)(119901minus3)4 The coupling constant as well as the

metric now gets the radial dependence ((7 minus 119901)2)((7 minus

119901)2) = 1 and the weight of119883 is 1In the above situations the transformations are all made

for 119883 This is not always the case Consider the 2119889119873 = (4 4)

SCFT and the 1198601198891198783

times 1198783

times 1198794 geometry [50 51] The scalars

119883119894 119894 = 1 2 3 4 are related to the transverse 1198794 for which the

metric is 11988911990924

There is no radial dependent prefactor for119883 toabsorb so 119883 has the weight 0 In fact the chiral primariesare constructed from the fermions Ψ that have the weight12 and are representations of the 119877-symmetry group 119878119874(4)

associated with 1198783

Return to SYM4

with the operator perturbation added

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889

metric The partition function is (here the path integralmeasure depends on both 119892

120583]0 and 1206010

However except for119892120583]0 no other field will enter into the path integral measure

directly Even though the transformation of the path integralmeasure still has the dependence on other external fieldsAn explicit example is the chiral anomaly in gauge theoryWe will discuss the path integral measure in more detail inAppendix B)

119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)

We still want to do a119883 rarr 119883 transformation

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)

When 120590 = minusinfin 119878119888(120590 119892120583]0 1206010) may not be zero Instead of

1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010

we can find the suitably adjusted 119892120583](120590)

and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)

In (176) the integration over 119883 119860 and 120582 has already beencarried out both sides only depend on the 4119889 function 120590(119909)Obviously 119892

120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)

The 5119889 functions 119892120583](119909 120590) and 120601(119909 120590) are induced from the

4119889 fields 119892120583]0(119909) and 120601(119909) For finite 120590 119892

120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010

More generically with the 4119889 background field 119865

0

turnedon for the given 4119889 function 120590(119909) since 120590 le 0 119890120590119883 rarr 119883 is aUV to IR transformation the effect of which can be cancelledby the adjustment of the coupling constants keeping thepartition function invariant 119890120590119883 rarr 119883 119865

0

rarr 119865(120590) Withall possible operators included the change of the scale can


always be compensated by the change of the backgroundfields leaving the form of the Lagrangian invariant

119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)

In (178) with 119865 specified to 120601 the induced action119868[119892

120583](120590) 119865(120590) 119883 119860 120582] contains the term int1198894

119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886


119899) |119910| = infin since 119883 is theinfinite matrix The corresponding 10119889 field is Φ(119909 120590 Ω) =

|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899

(Ω) is the spherical harmonic on 1198785 Equation (178) is the

partition function of the gauge theory on AdS5

times 1198785 with the

background fields Φ(119909 120590 Ω) and g120583](119909 120590) turned on

On gravity side the asymptotic expansion of the gravitysolution g(120590) is [52]

g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)

where 120590 isin (minusinfin +infin) Let 120590 rarr 120590 + 120590infin

with 120590 isin (minusinfin 0]120590infin

= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590

infin119890minus3120590

)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920

Similarly for the solution of thescalar field Φ(120590) [53]

Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012

+ sdot sdot sdot + 119890(4minus2119899)120590

1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)

with 120590 isin (minusinfin +infin) 120601(120590) = 119890(4minus119899)120590

infinΦ(120590 + 120590infin

)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)

with 120590 isin (minusinfin 0] When 119899 gt 2 120601(120590) = 119890(119899minus4)120590

1206010

for finite 120590For g-Φ coupled system the subleading terms are determinedby both 119892

0

and 1206010

Φ(119909 120590) and g120583](119909 120590) are fields on AdS

5

with 1199033

finite It is expected that the 119883 rarr 119883 transformationwill give 119892

120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596

119883 the induced fields become 119892120583](120590 + 120596) and

120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596

(120590) are fields induced from 1198902120596

119892120583]0 and

119890(119899minus4)120596

1206010

respectively There will be 119892120596

120583](120590) = 119892120583](120590 + 120596)

120601120596

(120590) = 120601(120590 + 120596) Indeed in (182) and (180) with 1206010

and1198920

replaced by 119890(119899minus4)120596

1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010

120590 rarr 120590 + 120596 is adiffeomorphism transformation of the gravity solution g

120583](120590)

and Φ(120590) [54]Equation (184) is valid for [119892

120583](119909 120590) 120601(119909 120590)] inducedfrom [119892

120583]0(119909) 1206010(119909)] For the generic 5119889 functions[119892120583](119909 120590) 120601(119909 120590)] we only have

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)


for some 5119889 functions [119892120596

120583](119909 120590) 120601120596

(119909 120590)] For constant 120596under the scale transformation 119909 rarr 119890

120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)

In special cases if 119890minus2120596

119892120596

120583](120590) = 119892120583](120590) 120601

120596

(120590) = 120601(120590)the background is ldquoconformalrdquo For [119892

120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0

120601 is the marginal fieldFor fields which are ldquoconformalrdquo

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)

120601(119909 120590) is on shellRestricted to [119892

120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +

120596) 120601(120590 + 120596)] = [119892120583](120590) 120601(120590)] [119892120583](120590) 120601(120590)] is not ldquoconfor-

malrdquo Instead of (191) if we require

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596

radicminus119892(1199090

1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell


120601(119909 120590) is the functional of 1206010

(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)

since nothing could guarantee that the one-point function ofthe SYM

4

would vanish in presence of the arbitrary source1206010

(119909) The physical field is Φ = 119890(119899minus4)120590

infin120601 so there is also

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)

The difference between 119868[119892120583]0(119909) 1206010(119909) 119883 119860 120582] and

119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582] is that 119883 is replaced by 119883 while

the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]

has the natural interpretation as a gauge theory on AdS5

times 1198785

with the external field

radicminus119892 (119909 120590)119865 (119909 120590 Ω) = radicminus119892 (119909 120590)119865 (119909 120590) 119899

= 119890119899120590

infinradicminus119892 (119909 120590)119865 (119909 120590) 119884119899

= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)

turned on where 119865(119909 120590) and 119892120583](119909 120590) can be the arbitrary

5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590

infin119865 are physicalfields on AdS

5

According to the previous philosophy

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892

[119892120583](119909 120590) 120601(119909 120590) 119883 0] is the effective action of the type

IIB supergravity on AdS5

times 1198785

For a special subset of fields 120601(119909 120590) that can be derivedfrom the 4119889 fields 120601

0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)

that isΦ(119909 120590) is the on-shell solution of the supergravity Forthe given 120601

0

(119909) there are two ways to get the induced 120601(119909 120590)one is through (176) and the other is to find the solutionof 120575119878

119892

120575Φ(1199090

1205900

) = 0 imposing lim120590rarr0

119890(4minus119899)120590

infinΦ(119909 120590) =

1206010

(119909) as the boundary conditionFinally we arrive at

119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)

Starting from 119868[119892120583]0(119909) 1206010(119909) 119883 119860 120582] SYM

4

with the sourceterm added after a 119883 rarr 119883 transformation we get119868[119892

120583](119909 120590) 120601(119909 120590) 119883 119860 120582] the action of the gauge the-ory on AdS

5

times 1198785 whose free energy may be equal to

119878119892

[119892120583](119909 120590) 120601(119909 120590) 119883 0] During the transition 119865(119909 120590)

are carefully adjusted to make the partition functionof 119868[119892

120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]

remain the same then we arrive at (201) where 119865(119909 120590) isinduced from 119865

0

(119909) and is on shell with respect to 119878119892

43 Imposing the Cutoff

119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)

where 119871 is the Lagrangian of the supergravity 119878119892

is divergentOne may impose a cutoff 120590 120590 = minusinfin

119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)

Ongauge theory side the one-loop effective action119882(119892120583]0) =

minus ln det(119863)2 is also divergent With a cutoff 120598being intro-duced [55]

119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin

119889120590Tr (exp minus119890minus2(120590+120590

infin)

119863)

= int

120590

minusinfin

119889120590Tr (exp minus119890minus2120590D)

= 119882(g120583] (0) 120590)

(204)

where 119890minus2(120590+120590infin) = 120588 g120583](0) = 119890

2120590

infin119892120583]0D is the Laplace oper-

ator with the metric g120583](0) Obviously 119882(119890

2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g

120583](0) 120590) For smallbut finite 120588 [56]

Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)


The renormalized 119882ren(119892120583]0) is finite This is the realizationof the holographic renormalization [52] on gauge theory sideWith 119892

120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604

is the conformal anomaly [57] On the other hand

119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))

119882nonloc (g120583] (0)) minus 119882ren (119892120583]0) = 120590

infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2

2 and 119882nonloc(g120583](0)) couldbe compared with the local part and the nonlocal part ofthe gravity action in [58] The nonlocal part contains thelogarithmical divergence [59] which is just 120590

infin

1198604

On gravity side the corresponding gravity solution has

the near boundary expansion

119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920

120598) The on-shell gravity actionwith the cutoff 120588 = 120598 is

119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)

which is entirely determined by the boundary value 119866(1198920

120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)

The renormalized 119878ren(119892120583]0) is finite [52] Since119866(119890

2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)

In both gauge theory and gravity subtracting of the infinityintroduces the conformal anomaly

In (204) 120598 is the UV cutoff in gauge theory It is desirableto find a direct and exact way to impose it In the followingwe will consider a cutoff imposed on 119883 which although hassome relevance with 120598 is still not it

Take119883 = 119890120590

119883 other than119883 as the upper bound of119883 thepartition function is

119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)

since the field induced from 1198902120596

119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596

119883) the induced fields have thesame boundary value 119892

120583](120590 + 120596)For the infinitesimal 4119889 function 120575120590(119909)

119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)

(119909) we may have

119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)

Also we may define 119871119887

(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)

(119909) = 0 For infinite 120575120590 forexample 119882(119892

120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)

Although (219) and (204) look similar the two kinds ofcutoffs are different In particular

ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)

because the 119885(119892120583]0 119883) minus 119885(119892

120583]0 119883) part also has the contri-bution to 119878

119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920

119883) From (206) or (212)one can see

120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)

For finite119883 and 120598 (226) cannot be directly identified with 119878119886

and 119878119887 However when119883 rarr 119883 120598 rarr 0

119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604

is the conformal anomaly which with theinfinite part subtracted becomes 119860

4

In the previous subsection the radial dependent function

119865(120590) is induced via (176) Nevertheless it can also be inducedfrom 119882(119892

120583]0 119883) with 119883 being the upper bound of theintegration Take a particular finite matrix119883

lowast as the standardso that the arbitrary configuration of119883 can be represented by119883 = 119890

120590

119883lowast with 120590 isin (minusinfin +infin)

119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)

Starting from 120590 = 0 with 120590 increasing 119882(119892120583]0 119890

120590

119883lowast

) willalso increase so 119892

120583]0 should change accordingly to make thepartition function invariant Namely we have

119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892

120583](120590) rarr 0the simple scaling relation may not be valid For finite 119892

120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892

120583]0 fixed and 119883lowast replaced by

119890minus120596

119883lowast the induced 119892

120583](120590) remains the same Also with 119883lowast

fixed and 119892120583](0) replaced by 119892

120583](120596) the induced field will be119892120583](120596 + 120590) 119892

120583](120590) is the unique function trajectory that doesnot depend on the119883

lowast chosenWe can compare it with the previously mentioned non-

critical string coupling with 2119889 gravity

119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)

where 119865(119883) is the 119889 dimensional background field 119892 is thecutoff metric With 119892 replaced by 119892

120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596

119892 is a conformal transforma-tion which should be accompanied by 119865 rarr 119865(120596) to makethe partition function invariant 119885(119865 119892) = 119885(119865(120596) 119890

120596

119892)119865(120596) is a function trajectory that does not depend on the 119892

chosen A further 119890120593119892120596 rarr 119892120596 transformation gives

119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596

(120593) = 119865(120593 + 120596) is inducedJust as (224) ln 119885(119865(120596) 119890

120596

119892) cannot be identified withthe 119889 + 1 dimensional gravity actionwith a cutoffThe gravityaction interpretation is possible onlywhen119892 rarr 0 that is thecutoff is removed

Here 120593 is directly related to the worldsheet metric 119892 =

119890120593

119892 119865(120593) gives the RG flow of the 119889 dimensional fields Ingauge theory case 120590 represents the radial direction alongwhich all fields including the metric 119892

120583] will evolve 119865(120590)gives the radial evolution of the fields which cannot bedirectly identified with the RG flow To discuss the RG flowwe should consider the renormalized 119882ren = 119878ren for which(see eg [60])

int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)

where119860 represents the local anomalies For the renormaliza-tion scale 120583

120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)

120573119894 is the anomalous dimension 120573119894 = 120573

119894

119865119894

0

is the 120573 function

5 Conclusion

With 119873 rarr infinbeing fixed and 119877 varying the free energyof the Matrix theory on a 119909

minus-translation invariant 11119889

supergravity background 119865 is the functional of 119877 and 119865 thatis119882 = 119882(119877 119865) Under the coordinate transformation

119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840

)119882(119877 119865) preserves part of the 11119889 diffeomorphisminvariance 119882(119877 119865) can be compared with 119878eff(119877 119865) theeffective action of the supergravity for the same field 119865 Onfield theory side naively 119878eff(119865) is calculated from

119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)

with 119878cla being the classical action of the 11119889 supergrav-ity 119890

119878eff(119865) can be taken as the sum of the 1PI graphs ofthe vacuum-vacuum amplitude for supergravitons livingon background 119865(Of course (239) is not well-defined)In 119872 theory the basic objects are membranes other thanparticles 119890119882(119865) is the sum of the membrane configurationson background 119865 In some sense 119882(119865) gives an 119872 theoryrefined version of the 119878eff(119865) in (239)

In Matrix theory the problem is that we only considerthe 119872 theory sector with the definite light-cone momentum119901+ and the background 119865 that is translation invariant along

119909minus For the same 119865 but different 119901

+ 119882(119877 119865) also has the119877 or equivalently 119901+ dependence Nevertheless 119882(119877 119865) =

119882(1198650

119877119865minus


) with 119877 being the radius of the 119909minus In fact

for 119878eff on the same 11119889 background there is also 119878eff(119877 119865) =

119878eff(1198650 119877119865minus


) In 119882(119901+

119865) 119901+ is only a parameter

encoding the scale of 119909minus so it is enough to consider the sectorwith the definite 119901

+ 119873 rarr infin in Matrix theory is kept fixedin order to maintain the consistency with membrane theorywhich is only characterized by one parameter 119901+

On field theory side to describe the supergravity interac-tions among119872 theory objects with no light-conemomentumexchange one may consider Γ

119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)

where 119865 is the zero mode of the 11119889 supergravity along119909minus The integration out of 119865 gives the effective action for

the 119872 theory object 119884 with the supergravity interactions(without transferring the light-cone momentum) all takeninto account Under the Legendre transformation we mayget Γ

119892

(119884) = 119878eff(119877 119865) + 119878119877119865

(119884) where 119878eff is given by(239) 120575[119878eff(119877 119865) + 119878

119877119865

(119884)]120575119865 = 0 With 119878eff(119877 119865) beingreplaced by 119882(119877 119865) one can define another effective actionΓ(119884) = 119882(119877 119865) + 119878

119877119865

(119884) which is totally constructed fromthe Matrix theory with no input like 119878cla(119877 119865) added Γ(119884)

is 119901+ independent which is expected since the scattering

amplitudes with the zero light-cone momentum transfershould not depend on 119901

+ due to the Lorentz invariance [61]When 120575119878

0

(119884)120575119884 = 0 at the one-loop order Γ(119884) and thestandard Matrix theory effective action Γeff(119884) are the same

A special property of Matrix theory is that various braneconfigurations have the natural matrix realization so it ispossible to construct some AdSCFT type gaugegravitycorrespondences from it In [20] SYM

119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878

3 are all obtained by expanding the PWMM aroundthe particular 12 BPS states while the backreaction of the12 BPS states on pp-wave after the 119909

minus reduction producesthe dual 10119889 geometry [19] A special type IIA limit is takenin both gauge theory and gravity side In this limit thebackreacted 11119889 geometry approaches the pp-wave so theMatrix theory dual is still PWMM On the other hand the 119909minusreduced 10119889 geometry (122) is a1198630-type solutionThe actionof the PWMM equals the action of the SYM

119877

on background(122) To study the fluctuations around the 12 BPS states the120578 rarr 120578 + 119899 and the 119884 rarr 119884 + redefinition should be madeon gravity and the Matrix theory side The 10119889 geometrythen becomes (133) a 1198632-type solution Correspondingly


119878PW

(119884 + ) becomes the action of SYM119877times119878

2 which withthe background (133) plugged in remains invariant With theSYM

119877times119878

2 and the gravity dual at hand a further119879-duality-liketransformation gives SYM

119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4

is the nonpertubative definition of the type IIBstring theory on AdS

5

times 1198785 It is unlikely such gaugestring

correspondence will suddenly vanish just because the metricin SYM

4

deviates 120578120583] a little If the correspondence still exists

the gravity dual will be the type IIB string theory on AdS5

times

1198785 with a little background perturbation turned on Thena one-to-one correspondence should exist between the 4119889

field 1198650

(119909) in SYM4

and the 5119889 field F(119909 120590) on AdS5

Thenatural candidate of F(119909 120590) is the gravity solution on AdS

5

with 1198650

(119909) the boundary condition For the correspondenceto be valid it is necessary to deriveF(119909 120590)merely fromSYM

4

In the simplest situation when the background metric ingauge theory is the standard 120578

120583] the near-horizon geometryof 1198633 1198722 and 1198725 could be induced from SYM

4

SCFT3

and SCFT

6

by a 119883 rarr 119883 transformation It is expected thatthe same method when applied to SYM

4

with the arbitrary1198650

(119909) turned on will give the corresponding F(119909 120590) With119883 rarr 119883 119865

0

(119909) rarr 119865(119909 120590) SYM4

becomes a gauge theoryliving in 5119889 background F(119909 120590) times the transverse 119878

5 Thefree energy can be expressed as the functional of F(119909 120590) thatis119882

0

(1198650

) = 119882(F) 120575119882(F)120575F(119909 120590) = 0 so if119882(F) is the 10119889gravity action as is in Matrix theory case F(119909 120590) will be theon-shell solution

Appendices

A The Action on the Flat Backgroundand the Near-Horizon Geometry

The action of the (119901 + 1)-dimensional SYM theory on 11987791

background could be written as

1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899

119892 = det119892120583] The near-horizon

geometry of the119863119901 branes is

119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

11986001sdotsdotsdot119901

= (119903

119903119901

)

7minus119901


(A2)

With 119903 replaced by the matrix 119877 = (119883119898

119883119898

)12 the gauge

theory on the near-horizon geometry background has theaction

119868119865

= minus1

4120587int119889

119901+1

119909 Tr radicminus119892119890minusΦ

times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904

= 1 except for a rescaling of120582 119868

0

= 119868119865

The SYM action in 11987791 and the SYM action in

the near-horizon geometry are the same Before and after thebackreaction the action remains invariant

Similarly the near-horizon geometries of 1198722 and 1198725

branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)

respectively For both1198722 and1198725

radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)

Instead for 1198722 (the 3-algebra was proposed in [62ndash64] toconstruct the model for two coincident1198722 branes)

radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)

We see another necessity of the sextic potential For 3119889 SCFTthe dimension of the scalar field is 12 so the potential


term should contain six scalars For 1198725 (3-algebra like thisappeared in [65])

radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)

The scalar dimension is 2 so effectively the potentialshould contain two-dimension 2 scalars and two-dimension1 scalars

B Path Integral Measureand the External Fields

For simplicity consider the scalar field with the action

119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =

119868[119892120583]0 1206010 119883] Following [66] wewill define the scalar density

field 119883 to eliminate the dependence of the path integralmeasure on 119892

120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)

The partition function is then

119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)

with no 119892120583]0 entering into the path integral measure

119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596

being the conformal anomaly Note that to arrive at(B6) we potentially assumed

int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)

On gravity side this is equivalent to assume thatlim

120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

120598) minus lim120598rarr0

119878 (119892120583]0 1206010 120598)

(B8)

119860120596

in (B8) could be compared with (227) From (B5)

120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)

indicating that 119860120596

will depend on both 119892120583]0 and 120601

0

Indeedthe direct calculation of the conformal anomaly gives 119860

120596

=

119860120596

(119892120583]0 1206010) composed by the gravity part (119892

120583]0) and thematter part (120601

0

) Correspondingly the transformation ofthe path integral measure will also depend on 119892

120583]0 and 1206010

although neither of them enters into119863119883 explicitly

Similarly when the theory is coupled to the external119878119880(4)

119877

gauge field 119860119886

120583

although 119860119886

120583

does not enter intothe path integral measure the Jacobian of the path integralmeasure under the 119877-symmetry transformation gives the 119877-symmetry anomaly which is the function of 119860119886

120583

Finally on gravity side one can read the 119877-symmetry

anomaly from the type IIB supergravity action directly butshould make the regularization of the action first to getthe conformal anomaly Correspondingly on gauge theoryside the 119877-symmetry anomaly exists originally while theconformal anomaly is introduced by regularization

Acknowledgments

Theworkwas supported in part by theMitchell-Heep chair inHighEnergy Physics (CMC) and by theDOEGrantDEFG03-95-Er-40917

References

[1] T BanksW Fischler S H Shenker and L Susskind ldquoM theoryas a matrix modela conjecturerdquo Physical Review D vol 55 no8 pp 5112ndash5128 1997

[2] D Berenstein J Maldacena and H Nastase ldquoStrings in flatspace and pp waves from 119873 = 4 super Yang Millsrdquo Journal ofHigh Energy Physics vol 2002 no 4 article 13 2002

[3] W Taylor and M Van Raamsdonk ldquoSupergravity currents andlinearized interactions for matrix theory configurations withfermionic backgroundsrdquo Journal of High Energy Physics vol1999 no 4 article 013 1999


[4] A Dasgupta H Nicolai and J Plefka ldquoVertex operators forthe supermembranerdquoThe Journal of High Energy Physics no 5article 7 28 2000

[5] K Dasgupta M M Sheikh-Jabbari and M Van RaamsdonkldquoMatrix perturbation theory for M-theory on a pp-waverdquoJournal of High Energy Physics vol 2002 no 5 article 56 2002

[6] J Maldacena ldquoThe large 119873 limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 no 2 pp 231ndash252 1998

[7] N Itzhaki J M Maldacena J Sonnenschein and S Yankielow-icz ldquoSupergravity and the large119873 limit of theories with sixteensuperchargesrdquo Physical Review D vol 58 no 4 Article ID046004 1998

[8] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters Bvol 428 no 1-2 pp 105ndash114 1998

[9] E Witten ldquoAnti de Sitter space and holographyrdquo Advances inTheoretical and Mathematical Physics vol 2 no 2 pp 253ndash2911998

[10] S Sethi and L Susskind ldquoRotational invariance in the M(atrix)formulation of type IIB theoryrdquo Physics Letters B vol 400 no3-4 pp 265ndash268 1997

[11] L Susskind ldquoT duality in M(atrix) theory and S duality in fieldtheoryrdquo httpxxxlanlgovabshep-th9611164

[12] O J Ganor S Ramgoolam and W Taylor ldquoBranes fluxes andduality in M(atrix)-theoryrdquo Nuclear Physics B vol 492 no 1-2pp 191ndash204 1997

[13] B deWit JHoppe andHNicolai ldquoOn the quantummechanicsof supermembranesrdquoNuclear Physics B vol 305 no 4 pp 545ndash581 1988

[14] L Motl ldquoProposals on non-perturbative superstring interac-tionsrdquo httplanlarxivorgabshep-th9701025

[15] R Dijkgraaf E Verlinde and H Verlinde ldquoMatrix stringtheoryrdquo Nuclear Physics B vol 500 no 1ndash3 pp 43ndash61 1997

[16] Y Sekino and T Yoneya ldquoFrom supermembrane to matrixstringrdquo Nuclear Physics B vol 619 no 1ndash3 pp 22ndash50 2001

[17] E S Fradkin and A A Tseytlin ldquoEffective field theory fromquantized stringsrdquo Physics Letters B vol 158 no 4 pp 316ndash3221985

[18] A A Tseytlin ldquoSigmamodel approach to string theory effectiveactions with tachyonsrdquo Journal of Mathematical Physics vol 42no 7 pp 2854ndash2871 2001

[19] H Lin and J Maldacena ldquoFivebranes from gauge theoryrdquoPhysical Review D vol 74 no 8 Article ID 084014 40 pages2006

[20] G Ishiki S Shimasaki Y Takayama and A Tsuchiya ldquoEmbed-ding of theories with 119878119880(2|4) symmetry into the plane wavematrix modelrdquo Journal of High Energy Physics vol 20069 no11 article 089 2006

[21] T Ishii G Ishiki S Shimasaki and A Tsuchiya ldquo119873 = 4 superYang-Mills theory from the plane wave matrix modelrdquo PhysicalReview D vol 78 no 10 Article ID 106001 21 pages 2008

[22] B deWit K Peeters and J Plefka ldquoSuperspace geometryfor supermembrane backgroundsrdquo Nuclear Physics B vol 532Article ID 980320 pp 99ndash123 1998

[23] W Taylor and M Van Raamsdonk ldquoMultiple D0-branes inweakly curved backgroundsrdquoNuclear Physics B vol 558 no 1-2pp 63ndash95 1999

[24] W Taylor and M Van Raamsdonk ldquoMultiple D119901-branes inweak background fieldsrdquo Nuclear Physics B vol 573 no 3 pp703ndash734 2000

[25] D Kabat and W Taylor ldquoLinearized supergravity from Matrixtheoryrdquo Physics Letters B vol 426 no 3-4 pp 297ndash305 1998

[26] A A Tseytlin ldquolsquoNo-forcersquo condition and BPS combinations of119901-branes in 12 and (2 0) dimensionsrdquo Nuclear Physics B vol487 no 1-2 pp 141ndash154 1997

[27] A Jevicki and C Lee ldquoS-matrix generating functional andeffective actionrdquo Physical Review D vol 37 no 6 pp 1485ndash14911988

[28] E S Fradkin and A A Tseytlin ldquoQuantum string theoryeffective actionrdquoNuclear Physics B vol 261 no 1 pp 1ndash27 1985

[29] J Figueroa-OrsquoFarrill and G Papadopoulos ldquoHomogeneousfluxes branes and a maximally supersymmetric solution of M-theoryrdquo Journal of High Energy Physics vol 2001 no 08 article036 2001

[30] J Maldacena M M Sheikh-Jabbari and M Van RaamsdonkldquoTranverse fivebranes in Matrix theoryrdquo Journal of High EnergyPhysics vol 2003 no 1 article 038 2003

[31] V Balasubramanian M Berkooz A Naqvi and M J StrasslerldquoGiant gravitons in conformal field theoryrdquo Journal of HighEnergy Physics vol 2002 no 4 article 34 2002

[32] A Hashimoto S Hirano and N Itzhaki ldquoLarge branes in AdSand their field theory dualrdquoThe Journal of High Energy Physicsvol 2000 no 8 article 51 2000

[33] J McGreevy L Susskind and N Toumbas ldquoInvasion of thegiant gravitons from anti-de Sitter spacerdquo The Journal of HighEnergy Physics vol 2000 no 6 article 8 2000

[34] M M Sheikh-Jabbari ldquoTiny graviton matrix theory DLCQof IIB plane-wave string theory a conjecturerdquo Journal of HighEnergy Physics vol 2004 no 9 article 17 2004

[35] M M Sheikh-Jabbari andM Torabian ldquoClassification of all 12BPS solutions of the tiny gravitonmatrix theoryrdquo Journal ofHighEnergy Physics vol 2005 no 4 article 001 2005

[36] M Ali-Akbari M M Sheikh-Jabbari and M Torabian ldquoTinygraviton matrix theorySYM correspondence analysis of BPSstatesrdquo Physical Review D vol 74 no 6 Article ID 0660052006

[37] H Lin ldquoThe supergravity dual of the BMN matrix modelrdquoJournal of High Energy Physics vol 2004 no 12 article 0012004

[38] K Sugiyama and K Yoshida ldquoGiant graviton and quantumstability in the Matrix model on a pp-wave backgroundrdquoPhysical Review D vol 66 no 8 Article ID 085022 9 pages2002

[39] H Lin O Lunin and J Maldacena ldquoBubbling AdS space and12 BPS geometriesrdquo Journal of High Energy Physics vol 2004no 10 article 025 2004

[40] F Cachazo K A Intriligator and C Vafa ldquoA large N duality viaa geometric transitionrdquo Nuclear Physics B vol 603 Article ID010306 3 pages 2001

[41] K Becker M Becker J Polchinski and A Tseytlin ldquoHigherorder graviton scattering in M(atrix) theoryrdquo Physical ReviewD vol 56 no 6 pp R3174ndashR3178 1997

[42] H Ling A R Mohazab H-H Shieh G van Anders and MVan Raamsdonk ldquoLittle string theory from a double-scaledmatrix modelrdquo Journal of High Energy Physics vol 2006 no 10article 018 2006

[43] W Taylor ldquoD-brane field theory on compact spacesrdquo PhysicsLetters B vol 394 no 3-4 pp 283ndash287 1997

[44] I Antoniadis C Bachas J Ellis and D V NanopoulosldquoCosmological string theories and discrete inflationrdquo PhysicsLetters B vol 211 no 4 pp 393ndash399 1988


[45] I Antoniadis C Bachas J Ellis and D V Nanopoulos ldquoAnexpanding universe in string theoryrdquoNuclear Physics B vol 328no 1 pp 117ndash139 1989

[46] I Antoniadis C Bachas J Ellis and D V NanopoulosldquoComments on cosmological string solutionsrdquo Physics Letters Bvol 257 no 3-4 pp 278ndash284 1991

[47] J Distler and H Kawai ldquoConformal field theory and 2Dquantum gravityrdquoNuclear Physics B vol 321 no 2 pp 509ndash5271989

[48] J Polchinski ldquoA two-dimensional model for quantum gravityrdquoNuclear Physics B vol 324 no 1 pp 123ndash140 1989

[49] A Dhar and S R Wadia ldquoNoncritical strings RG flows andholographyrdquo Nuclear Physics B vol 590 no 1-2 pp 261ndash2722000

[50] A Strominger and C Vafa ldquoMicroscopic origin of theBekenstein-Hawking entropyrdquo Physics Letters B vol 379 no 1ndash4 pp 99ndash104 1996

[51] J R David G Mandal and S R Wadia ldquoMicroscopic formu-lation of black holes in string theoryrdquo Physics Reports vol 369no 6 pp 549ndash686 2002

[52] M Henningson and K Skenderis ldquoThe holographic Weylanomalyrdquo The Journal of High Energy Physics no 7 article 231998

[53] S de Haro K Skenderis and S N Solodukhin ldquoHolo-graphic reconstruction of spacetime and renormalization in theAdSCFT correspondencerdquo Communications in MathematicalPhysics vol 217 no 3 pp 595ndash622 2001

[54] C Imbimbo A Schwimmer S Theisen and S YankielowiczldquoDiffeomorphisms and holographic anomaliesrdquo Classical andQuantum Gravity vol 17 no 5 pp 1129ndash1138 2000

[55] N D Birrell and P C W Davies Quantum Fields in CurvedSpace vol 7 ofCambridgeMonographs onMathematical PhysicsCambridge University Press Cambridge UK 1982

[56] B S DeWitt Dynamical Theory of Groups and Fields Gordonand Breach Science Publishers New York NY USA 1965

[57] M J Duff ldquoTwenty years of the Weyl anomalyrdquo Classical andQuantum Gravity vol 11 no 6 pp 1387ndash1403 1994

[58] J De Boer E Verlinde and H Verlinde ldquoOn the holographicrenormalization grouprdquo Journal of High Energy Physics vol 4no 8 article 003 pp 1ndash15 2000

[59] M Fukuma S Matsuura and T Sakai ldquoA note on the Weylanomaly in the holographic renormalization grouprdquo Progress ofTheoretical Physics vol 104 no 5 pp 1089ndash1108 2000

[60] J Erdmenger ldquoField-theoretical interpretation of the holo-graphic renormalization grouprdquo Physical Review D vol 64 no8 Article ID 085012 12 pages 2001

[61] T Banks ldquoTasi lectures on matrix theoryrdquo httparxivorgabshep-th9911068

[62] J Bagger and N Lambert ldquoModeling multiple M2-branesrdquoPhysical Review D vol 75 no 4 Article ID 045020 7 pages2007

[63] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D vol 77 no 6Article ID 065008 6 pages 2008

[64] A Gustavsson ldquoAlgebraic structures on parallel M2-branesrdquoNuclear Physics B vol 811 no 1-2 pp 66ndash76 2009

[65] N Lambert and C Papageorgakis ldquoNonabelian (20) tensormultiplets and 3-algebrasrdquo Journal of High Energy Physics vol2010 no 8 article 083 2010

[66] K Fujikawa ldquoComment on chiral and conformal anomaliesrdquoPhysical Review Letters vol 44 no 26 pp 1733ndash1736 1980

Submit your manuscripts athttpwwwhindawicom

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limit (the radius of 1198781 approaches 0) matrices commute (KKmodes could be dropped) and Matrix string theory reducesto the second quantized type IIA string theory in light-cone gaugeThe configurations aremultistring configurationswith the transition amplitude given by the integration of allintermediate string joining and splitting processes [14 15]

Since the state the spectrum and the transition ampli-tude are all in one-to-one correspondence the partitionfunction of the gauge theory equals the partition function oftheMstring theory For SYM

4

we have [8 9]

119885SYM4

(120573 1206010

) = 119885AdS5times119878

5 (120573 1206010

) (1)

where 120573 = 1119879 is the radius of the time direction The zerotemperature partition function is only the functional of 120601

0

for which [8 9]

119885SYM4

(1206010

) = 119890119882[120601(120601

0)]

(2)

120601(1206010

) is the on-shell supergravity solution with the boundaryvalue 120601

0

119882 is the type IIB supergravity action on AdS5

times 1198785

If the gravity dual of SYM4

with the source 1206010

added is thetype IIB string theory on AdS

5

times 1198785 with the background

field 120601(1206010

) turned on in zero temperature limit the partitionfunction will only contain the contribution of the groundstate geometry so 119885SYM

4

(1206010

) = 119890119882[120601(120601

0)]

Except for (2) 119885SYM4

(1206010

) also has another expression onAdS

5

times 1198785 Let 119882

119904

be the free energy of the type IIB stringtheory on AdS

5

times 1198785 and 119885SYM

4

the partition function ofSYM

4

it is expected that

119885SYM4

= 119890119882

119904 (3)

where 119882 = sum119899

ℎ=0

119882119899

ℎ

When the background field 1206010

varies SYM4

undergoes a change of the coupling constantsCorrespondingly there is also a change of the couplingconstants for strings living in AdS

5

times 1198785 which is in fact a

modification of the 5119889 background For (3) to be valid a one-to-one correspondence 120601

0

harr 120601(1206010

) should exist

119885SYM4

(1206010

) = 119890119882

119904[120601(120601

0)]

(4)

On AdS5

side the string free energy 119882119904

[120601(1206010

)] cannot bedefined without a definite 5119889 background 120601(120601

0

) Also forthe state (spectrum) correspondence to be valid the definite5119889 background is necessary otherwise it is impossible todetermine the string spectrum If (2) and (4) both hold

119882[120601 (1206010

)] = 119882119904

[120601 (1206010

)] (5)

The free energy of the string theory on a given backgroundequals the effective action of the background fields

For Matrix theory similarly

119885Matrix (120573 119901+

119865) = 119885119872

(120573 119901+

119865) (6)

119865 is the 11119889 supergravity background and is the translationinvariant along the 119909

minus direction 119885119872

(120573 119901+

119865) is the par-tition function of the M theory sector with the light-conemomentum 119901

+ which is supposed to be the supermembrane

Equation (6) is trivially satisfied sinceMatrix theory is just theregularization of the supermembrane with the definite light-cone momentum When 120573 = infin

119885Matrix (119901+

119865) = 119890119882(119901

+119865)

(7)

119882(119901+

119865) is the 119901+-parameterized functional of 119865 with the

10119889 covarianceIn string theory one can also calculate the free energy of

the strings on a given background 119865

119882119904

(119865) =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 (8)

In [17 18] it was shown that for 119865 satisfying the free fieldequation 119882

119904

(119865) could be taken as the effective action ofthe renormalized background field 119865(119865) that is 119882

119904

(119865) =

119878eff[119865(119865)]It is tempting to establish a relation between119882(119901

+

119865) andthe effective action of the 11119889 supergravity However Matrixtheory no matter if it was taken as the DLCQ formulationof the M theory or as the discrete regularization of thesupermembrane theory in light-cone gauge only describesthe sub-Hilbert space of theM theory with the definite light-cone momentum 119901

+ without capturing all the information ofthe covariant theory For different 119901+ 119882(119901

+

119865) is differentNevertheless let119865 = 119865

0

119865minus

119865minusminus

where1198650

119865minus

119865minusminus

representfields with zero one and two 119909

minus indices respectively one canfind that 119882(119901

+

119865) = 119882(1198650

119865minus

119901+

119865minusminus

119901+2

) Let 119901+ = 119873119877

with 119877 being the radius of 119909minus 119873 rarr infin 119877 rarr infin and

119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus

) On the other hand for 11119889

supergravity fields 119865 that are translation invariant along 119909minus

suppose 119878eff(119877 119865) is the effective action of 119865 and there isalso 119878eff(119877 119865) = 119878eff(1198650 119877119865

minus

1198772

119865minusminus

) The 119877 dependence of119882(119877 119865) is consistent with 119878eff(119877 119865)

One may want to consider the complete partition func-tionwith all light-conemomentum taken into account whichis roughly 119890

119882(119865)

= int119889119901+

119890119882(119901

+119865) Each 119882(119901

+

119865) only differsby a rescaling of (119865

minus

119865minusminus

) so the summation does not givemore information It is enough to consider 119882(119901

+

119865) withthe definite 119901

+ In fact 119865 is the translation invariant along119909minus as a result a sector with the definite 119901

+ has the enoughdegrees of freedom to produce 119878eff(119877 119865) The complete Mtheory degrees of freedom including sectors with all light-cone momentum is necessary only when 119865 is the field withthe 11119889 spacetime dependence

For the arbitrary matrix configuration 119884 we may defineΓ(119884) via

119890Γ(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= 119890119882(119877119865)+119878

119877119865(119884)

(9)


minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865

=120575119878

119877119865

(119884)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865

(10)

It is easy to see that Γ(119884) is 119877 independent



119892

(119884) through

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (11)




Γ119892

(119884) = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(12)


119865(119884)


119892


Γ119892

(119884) = 119878eff (119877 119865)

+ int11988910

119909119865 (119909)119881119877119865(119884)

(119909) sim 119878eff (119877 119865) + 119878119877119865

(119884)

(13)


minus120575119878eff (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119877119865(119884)

(119909) =120575119878

119877119865

(119884)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865

(14)

So if119882(119877 119865) = 119878eff(119877 119865) Γ(119884) = Γ119892




2 SYM119877times119878

3119885

119896





0


0


5


4

since 1206010


5




of AdS5


119899

(119909 119903 Ω) =

ℎ119899

(119909 119903)119884119899

(Ω) with 119884119899

(Ω) = 119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)|119910|119899 the


119867119899

(119909 119903 119883) = ℎ119899

(119909 119903) 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) (15)

ℎ119899


weonly have ℎ

119899

(119909)119862119886

1sdotsdotsdot119886

119899

(119883119886



4


0

(119909) a 119883 rarr 119883 1198650

(119909) rarr 119865(119909 119903)


4



5






3

and SCFT6






119877times119878

2 SYM119877times119878

3119885

119896



4





119877119865


dependence of 119878119877119865

(119884) is 119878119877119865

(119884) = 119878119865

0119877119865

minus119877

2119865

minusminus

(119884) where 1198650

119865minus

119865minusminus


As a result 119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus


minus

1198772

119865minusminus


119877119865


119877119865


119892

(119884) withΓ119892





119878119877

= 119877int119889119909+

times Tr( 1

21198772119863+

119883119868

119863+

119883119868

+1

41198976119901

[119883119868

119883119869

]2

minus119894

119877120579119863

+

120579 +1

1198973119901

120579120574119868

[119883119868

120579])

(16)

where119883119868 120579 and1198600



is the Plancklength

119883119868 119909


119901


119868

rarr 119897119901

119883119868 119909+ rarr

119897119901

119909+ 119877 rarr 119897

119901


119878119877

= 119877int119889119909+

times Tr( 1

21198772119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894

119877120579119863

+

120579 + 120579120574119868

[119883119868

120579])

(17)

in which 119897119901



119901



119892++

119892119868+

119892119868119869

) (119862119868119869+

119862119868119869119870

) and(120595

+

120595119868


119878119887

119865

0

= int119889119909+

1198892

120590119892+minus

119875+

times [119875+2

119863+

119883119868

119863+

119883119869

21198922+minus

119892119868119869

+119875+

119863+

119883119868

119892+minus

(119862119869

+119875+

119892119869

+

119892+minus

)119892119868119869

+119875+2

119892++

21198922+minus

minus 119883119868

119883119869

119883119870

119883119871

119892119868119870

119892119869119871

+119875+

119862+

119892+minus

]

(18)

where 119862119868

= 119883119869

119883119870

119862119868

119869119870

and 119862+

= 119883119869

119883119870

119862+119869119870


+minus


119878119887

119865

0

= int119889119909+


119863+

119883119868

119863+

119883119869

2(119877119892+minus

)2

119892119868119869

+119863+

119883119868

119877119892+minus

(119862119869

+119892119869

+

119877119892+minus

)119892119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892119868119870

119892119869119871

+119892++

2(119877119892+minus

)2

+119862+

119877119892+minus

)

(19)

where 119862119868

= minus(1198942)[119883119869

119883119870

]119862119868

119869119870

and 119862+

= minus(1198942)[119883119869

119883119870

]

119862+119869119870


119892minusminus

119862minus119868119869

119862+minus119868

120595minus


119878119887

119892

119868minus

= int119889119909+


119863+

119883119868

119863+

119883119869

119863+

119883119870

2(119877119892+minus

)3

119892119869119870

minus119863+

119883119868

4119877119892+minus

[119883119870

119883119871

] [119883119872

119883119873

]

times 119892119870119872

119892119871119873

+119863+

119883119871

119877119892+minus

[119883119869

119883119872

] [119883119868

119883119873

]

times119892119871119869

119892119872119873

times 119877119892119868minus

)

(20)


minusminus


+

119883)4 (119863

+

119883)2

[119883119883]2 and [119883119883]


minusminus

119892119868minus

119862minus119868119869

119862+minus119868

120595minus


+

119883minus

= (12)119863+

119883119868

119863+

119883119868

+ (1]2)119883119868

119883119869

119883119868

119883119869

and119863119886

119883minus

= 119863+

119883119868

119863119886

119883119868



+minus

119877119892119868minus


119877119862minus119868119869

119877119862+minus119868

119877120595

minus


+minus

119875+ 119892

119868minus

119875+ 119892

minusminus

119875+2 119862

minus119868119869

119875+ 119862

+minus119868

119875+ 120595

minus

119875+ that



+

= 119891(1199091015840+

) 119909minus = 1205721199091015840minus

hArr 119877 = 1205721198771015840

1198921015840

++

= 11989110158402

(1199091015840+

) 119892++

1198921015840

+minus

= 1205721198911015840

(1199091015840+

) 119892+minus

1198921015840

+119868

= 1198911015840

(1199091015840+

) 119892+119868

1198921015840

minus119868

= 120572119892minus119868

1198921015840

minusminus

= 1205722

119892minusminus

1198621015840

+minus119869

= 1205721198911015840

(1199091015840+

) 119862+minus119869

1198621015840

+119868119869

= 1198911015840

(1199091015840+

) 119862+119868119869

1198621015840

minus119868119869

= 120572119862minus119868119869

(21)


1198891199042

= 119892++

119889119891 (1199091015840+

) 119889119891 (1199091015840+

) + 2119892119868+

119889119909119868

119889119891 (1199091015840+

)

+ 119892119868119869

119889119909119868

119889119909119869

+ 2120572119892+minus

119889119891 (1199091015840+

) 1198891199091015840minus

+ 2120572119892119868minus

119889119909119868

1198891199091015840minus

+ 1205722

119892minusminus

1198891199091015840minus

1198891199091015840minus

= 119892++

11989110158402

(1199091015840+

) 1198891199091015840+

1198891199091015840+

+ 21198911015840

(1199091015840+

) 119892119868+

119889119909119868

1199091015840+

+ 119892119868119869

119889119909119868

119889119909119869

+ 2120572119892+minus

1198911015840

(1199091015840+

) 1198891199091015840+

1198891199091015840minus

+ 2120572119892119868minus

119889119909119868

1198891199091015840minus

+ 1205722

119892minusminus

1198891199091015840minus

1198891199091015840minus

119862 = 119862119868119869119870

119889119909119868

and 119889119909119869

and 119889119909119870

+ 119862119868119869+

119889119909119868

and 119889119909119869

and 119889119891 (1199091015840+

)

+ 120572119862119868119869minus

119889119909119868

and 119889119909119869

and 1198891199091015840minus

+ 120572119862119868+minus

119889119909119868

and 119889119891 (1199091015840+

) and 1198891199091015840minus

= 119862119868119869119870

119889119909119868

and 119889119909119869

and 119889119909119870

+ 119862119868119869+

1198911015840

(1199091015840+

) 119889119909119868

and 119889119909119869

and 1198891199091015840+

+ 120572119862119868119869minus

119889119909119868

and 119889119909119869

and 119889119909minus

+ 120572119862119868+minus

1198911015840

(1199091015840+

) 119889119909119868

and 1198891199091015840+

and 119889119909minus

(22)

119878119877119865


119877119865


119881119865(119884)

(119909) =120575119878

119877119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(23)


+minus

= minus1 and 119892119868119869

= 120575119868119869

119881119865(119884)

(119909)


119877119865



119878119877119865

= 119878119877

minus 119877int119889119909+


119868119869(119883)

ℎ119868119869

(119883) + 119881ℎ

119868+(119883)

ℎ119868+

(119883)

+ 119881ℎ

++(119883)

ℎ++

(119883) + 119881ℎ

119868minus(119883)

ℎ119868minus

(119883)

+ 119881ℎ

+minus(119883)

ℎ+minus

(119883) + 119881ℎ

minusminus(119883)

ℎminusminus

(119883)

+ 119881119862

119868119869119870(119883)

119862119868119869119870

(119883) + 119881119862

+119869119870(119883)

119862+119869119870

(119883)

+ 119881119862

119868119869minus(119883)

119862119868119869minus

(119883) + 119881119862

+minus119870(119883)

119862+minus119870

(119883)

+ 119881120595(119883)

120595 (119883) + 119881120595

+(119883)

120595+

(119883)

+ 119881120595

minus(119883)

120595minus

(119883) ]

(24)

where for example

119881ℎ

119868+(119883)

ℎ119868+

(119883) = (119863+

119883119868

119877minus

1

4120579120574

119868119869

120579120597

120597119883119869

)ℎ119868+

(119883)

119877 (25)

ℎ119868119869

(119883) 120595minus


1199091

1199099




119868+


ℎ119868+

(119909+

1198831

1198839

)

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0)119883119896


119899

(26)

with (120597119896

1


119899

ℎ119868+

)(119909+


(119909+

)

at (119909+ 0) [3 23ndash25]

STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0)

times STr[ 1

119877(119863+

119883119868

119877minus

1

4120579120574

119868119869

120579120597

120597119883119869

)119883119896


119899]

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0) 119868119896

1sdotsdotsdot119896

119899

ℎ

119868+

(119909+

)

(27)


+

0 0) the 119880(119873) SYM1


119883119894

997888rarr 119883119894

minus 119886119894

119891 (119909+

0 0) 997888rarr 119891(119909+

1198861

1198869

)

(28)


SYM1


1



1


1sdotsdotsdot119896


119896

1 Φ119896

1sdotsdotsdot119896


spacetime

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

) = 120575 (1199090

minus 119909+

) 1205759

(119909119868

minus 119883119868

)

= 12057510

(119909120583

minus 119883120583

)

(29)


set 1198830

= 119909+1

119873times119873


= diag1199091198681

119909119868

2

119909119868

119873

(29) becomes

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

)

= diag 12057510

(119909120583

minus 119909120583

1

) 12057510

(119909120583

minus 119909120583

2

) 12057510

(119909120583

minus 119909120583

119873

)

(30)


119868+

we have

119881ℎ

119868+(119909) =

120575

120575ℎ119868+

(119909)minus119877int119889119909

+STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

12057510

(119909 minus 119883)]

(31)


119881ℎ

119868+

(119901) = int11988910

119909119881ℎ

119868+(119909) 119890

119894119901119909

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

119890119894119901119883

]


119863+

119883119868

119877+

119894

4120579120574

119868119869

120579119901119869

] 119890119894119901119883

(32)


119865


119885 (119877 119865) = 119890119882(119877119865)

= int [119889119883119889120579 1198891198600

] 119890minus119878

119877119865(119883120579119860

0)

= int [119889119884] 119890minus119878

119877119865(119884)

(33)


119892++

119892119868+

119892119868119869

119892minusminus

119892119868minus

) (119862119868119869+

119862119868119869119870

119862119868119869minus

119862119868+minus

) and (120595+

120595119868

120595minus


120579 1198600



5

CFT4


119885SYM4

= exp minus119865 (34)


ℎ=0

119865119899

ℎ


5

times 1198785 119885SYM

4


4


5




+


Suppose 119878119872119875

+119865


119875

+119865


= 1198751015840+

120572119909+

= 119891(1199091015840+

) and 119883 = 119892(1198831015840

) there is 119878119872

119875

+119865

(119884) =

119878119872

119875

1015840+119865

1015840(1198841015840


transformation

119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (35)


1015840


int [119889119884] 119890minus119878

119872

119875+119865(119884)

= int [1198891198841015840

] 119890minus119878

119872

1198751015840+1198651015840 (1198841015840)

(36)


+

1205901

1205902

) and 1198841015840

(119909+

1205901

1205902


+

) and 1198841015840

(119909+

) and there is

119885 (119877 119865) = 119890119882(119877119865)

= int [119889119884] 119890minus119878

119877119865(119884)

= int [1198891198841015840

] 119890minus119878

11987710158401198651015840 (1198841015840)

= 119890119882(119877

1015840119865

1015840)

= 119885 (1198771015840

1198651015840

)

(37)

119882(119877 119865) = 119882(1198771015840

1198651015840



Let 1198650

119865minus

119865minusminus


indices Since

119878119877119865

(119884) = 119878119865

0119877119865

minus119877

2119865


119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus

) (39)


119865minus

119865minusminus



0

119865minus

119865minusminus


119909minus


119877 119865

0

997888rarr 1198650

119865minus

997888rarr 119877119865minus

119865minusminus

997888rarr 1198772

119865minusminus

(40)

so

119878eff (119877 1198650

119865minus

119865minusminus

) = 119878eff (1 1198650 119877119865minus

1198772

119865minusminus

) (41)


119865minusminus



=

minus1 119892119868119869

= 120575119868119869


119882(119877 119865) = int11988910

119909Γ119888

119877119865

(119909) 119865 (119909)

+1

2int 119889

10

1199091

times int11988910

1199092

Γ119888

119877119865

(1199091

1199092

) 119865 (1199091

) 119865 (1199092

) + sdot sdot sdot

= sum

119899

1

119899int 119889

10

1199091


119909119899

Γ119888

119877119865

(1199091

119909119899

)

times 119865 (1199091

) sdot sdot sdot 119865 (119909119899

)

(42)


119877119892

++119892

minusminus

(1199091

1199092

) = 1205752

119882(119877119865)120575119892++

(1199091


(1199092

)|119865=119865

)

Γ119888

119877119865

(1199091

119909119899

) =120575119899

119882(119877 119865)

120575119865(1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(43)


is

119878eff (119877 119865) = sum

119899

1

119899int

2120587119877

0

119889119909minus

1

times int119889119909+

1

1198899x1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

int119889119909+

119899

1198899x119899

times Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

times 119865 (119909+

1

x1

) sdot sdot sdot 119865 (119909+

119899

x119899

)

(44)

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899



Γ119888

119877119865

(1199091

119909119899

)

= Γ119888

119877119865

(119909+

1

x1

119909+

119899

x119899

)

= int

2120587119877

0

119889119909minus

1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

(45)


(119909)120575119865(119910) = 0where 119881

119865


Γ119888

119877119865

(1199091

119909119899

) = (minus1)119899

⟨119881119865

(1199091


(119909119899

)⟩119888

(46)

⟨119881119865

(1199091


(119909119899

)⟩119888


⟨119881119865

(1199091


(119909119899

)⟩

= int [119889119883119889120579 1198891198600

] 119890minus119878

0119881119865

(1199091


(119909119899

)

=(minus1)

119899

120575119899

119885 (119877 119865)

120575119865 (1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(47)


119877119865

(1199091

119909119899

)In terms of 119881

119865

(119901)

⟨119881119865

(1199091


(119909119899

)⟩119888

= int11988910

1199011


119901119899

119890minus119894(119901

1119909


119899119909

119899)

times ⟨119881119865

(1199011


(119901119899

)⟩119888

(48)

Note that 119881119865(119883)

(119909+

119909119868

minus 119886119868

) = 119881119865(119883+119886)

(119909+

119909119868

) 1198780

(119883 + 119886) =

1198780

(119883) so

⟨119881119865

(1199091


(119909119899

)⟩119888

= ⟨119881119865

(1199091

+ 1198861


(119909119899

+ 119886119899

)⟩119888

(49)


119865

(1199011


(119901119899

)⟩119888

= 0 if 1199011

+ sdot sdot sdot + 119901119899


⟨119881119865

(119909)⟩ = minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865

= V119865

(50)

V119865


119877

is 119878119874(9) invariant so V119865



V119892

119868+

= V119892

119868minus

= V119862

119868119869119870

= V119862

119868119869+

= V119862

119868119869minus

= V119862

119868+minus

= V120595

119868

= 0

(51)


V119892

119868119869


+

=

V120595

minus


ℎ

++

Vℎ

minusminus

and Vℎ

+minus

In particular Vℎ

++

= 119901+

1198719 is


⟨119881119865

(119909)⟩119865

= minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865+

119865

= V119865

(119909)

= sum

119899

minus1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Γ119888

119877119865

(119909 1199091

119909119899minus1

)

times 119865 (1199091


)

(52)


119865

(119909)⟩119865




119865


119865


119892

120575119892120583]

=



119865


119865




Γ (119884) = 119878119877119865

(119884) + 119882 (119877 119865) (53)

119890Γ(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

(54)


int [119889]120575

120575119865 (119909)119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

[minus

120575119878119877119865

()

120575119865 (119909)+

120575119878119877119865

(119884)

120575119865 (119909)] = 0

(55)

or equivalently

int [119889] 119890minus119878

119877119865(

119884)

120575119878119877119865

()

120575119865 (119909)=

120575119878119877119865

(119884)

120575119865 (119909)= minus

120575119882 (119877 119865)

120575119865 (119909) (56)


120575Γ (119884)

120575119884=

120575119878119877119865

(119884)

120575119884 (57)



(119884) only comes from119877119865

minus

and 1198772

119865minusminus


119865minus

119865minusminus


0

119865minus


1205722

)119878119877119865



+

1205901

1205902

) Let 1199091015840minus rarr 1198751015840+

1199091015840minus

119875+ 1198751015840+ rarr



= 1198650

119865minus

119875+

= 1198651015840

minus

1198751015840+ 119865

minusminus

119875+2

=

1198651015840

minusminus


119872

119875

+119865



119877119865


119877119865


0


119877119865



119878119865

(119884) = 1198780

(119884) + int 11988910

119909119865(119909)119881119865(119884)

(119909) Γ(119884) = 1198780

(119884) + Γ(V119865

)where

Γ (V119865

) = int11988910

119909119865 (119909) V119865

(119909) + 119882(119865 + 119865) (58)


minus120575119882 (119865)

120575119865 (119909)

100381610038161003816100381610038161003816100381610038161003816 119865

= V119865

= 119881119865(119884)

(119909)

120575Γ (V119865

)

120575V119865

(119909)

1003816100381610038161003816100381610038161003816100381610038161003816V119865

= 119865 (119909)

(59)


Let

Λ119865

(1199091

119909119899

) =120575119899

Γ (V119865

)

120575V119865

(1199091


(119909119899

)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

(60)

then

Γ (119884) = 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times V119865

(1199091


(119909119899

)

= 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(61)


(119909)

119865 (119909) = sum

119899

1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Λ119865

(119909 1199091

119909119899minus1

)

times V119865

(1199091


(119909119899minus1

)

(62)



119865

(119909) If Λ119865


119865


119892

120575119865(119909) = minusV119865

(119909) with119878119892


Γ119888

119865

(119909 minus 119911) = minus120575V

119865

(119909)

120575119865 (119911)

100381610038161003816100381610038161003816100381610038161003816 119865=0

Λ119865

(119911 minus 119910) =1205752

Γ (V119865

)

120575V119865

(119911) 120575V119865

(119910)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

=120575119865 (119911)

120575V119865

(119910)

100381610038161003816100381610038161003816100381610038161003816V119865=0

(63)

so

minusint11988910

119911Γ119888

119865

(119909 minus 119911) Λ119865

(119911 minus 119910) = 12057510

(119909 minus 119910) (64)

minusΛ119865

(1199091

minus 1199092


(1199091

minus 1199092

) Subsequently

Λ119865

(1199091

1199092

1199093

) = int11988910

1199101

int11988910

1199102

times int11988910

1199103

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Γ119888

119865

(1199101

1199102

1199103

)

Γ119888

119865

(1199091

1199092

1199093


1199101

int11988910

1199102

times int11988910

1199103

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Λ119865

(1199101

1199102

1199103

)

Λ119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Λ119865

(1199104

minus 1199094

)

times [Γ119888

119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Λ119865

(1199111

minus 1199112

)

times (Γ119888

119865

(1199111

1199101

1199104

)

times Γ119888

119865

(1199112

1199102

1199103

)

+ Γ119888

119865

(1199111

1199102

1199104

)

times Γ119888

119865

(1199112

1199101

1199103

)

+ Γ119888

119865

(1199111

1199103

1199104

)

times Γ119888

119865

(1199112

1199101

1199102

)) ]

Γ119888

119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Γ119888

119865

(1199104

minus 1199094

)

times [Λ119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Γ119888

119865

(1199111

minus 1199112

)

times (Λ119865

(1199111

1199101

1199104

)

times Λ119865

(1199112

1199102

1199103

)

+ Λ119865

(1199111

1199102

1199104

)

times Λ119865

(1199112

1199101

1199103

)

+ Λ119865

(1199111

1199103

1199104

)

times Λ119865

(1199112

1199101

1199102

)) ]

(65)



and Γ119888

119865

shows thatΛ119865

and Γ119888

119865


119865

(119911minus119910) = (120575119865(119911)120575V119865

(119910))|V119865=0


In (54) let = 119884 + 120578

119890Γ(119884)

= int [119889120578] 119890minus119878

119865(119884+120578)+119878

119865(119884)

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minusint119889120591120575119878

119865

(119884)

120575119884 (120591)120578 (120591)

minus1

2int119889120591 119889120591

1015840

1205752

119878119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)

times 120578 (120591) 120578 (1205911015840

) + sdot sdot sdot

= minus119878(1)

119865

(119884 120578) minus 119878(2)

119865

(119884 120578) + sdot sdot sdot

(66)

119878119865


120591


120591

also exists)

119878119865

(119884) = int119889120591 [1

2119884120597

2

120591

119884 + 1198810

(119884) + 119881 (119865 119884)]

= 1198780

(119884) + int119889120591119881 (119865 119884)

(67)


0

from 120575119878119865

0

(119884)120575119884 = 0 and expand 119878(119899)

119865

(119884 120578) around1198650

119865 = 1198650

+ 119891 Consider

119878(1)

119865

(119884 120578) =120575119878

(1)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


119878(2)

119865

(119884 120578) = 119878(2)

119865

0

(119884 120578) +120575119878

(2)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


(68)


(119884) +

Γ2


minus119878119865

(119884 + 120578) + 119878119865

(119884) = minus119878(2)

119865

0

(119884 120578) + sdot sdot sdot

Γ1

(119884) = ln int [119889120578] 119890minus119878

(2)

1198650(119884120578)

= ln (det119860119865

0

)12

(69)

with

119860119865

0

(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842+

1205972

119881 (1198650

119884)

1205971198842] 120575 (120591 minus 120591

1015840

)

(70)


(119884)120575119884 = 0 1198650

= 0 119860119865

0

reduces to

119860(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842] 120575 (120591 minus 120591

1015840

) (71)



1


119890Γeff(119884) = int [119889120578] 119890

minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

(72)


int [119889120578]120575

120575119869 (120591)119890minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

= 0 (73)

Γ1

(119884) = Γ1



(119884)120575119884 = 0 [3 25]

Γ1

(119884) =1

2int119889

10

1199091

times int11988910

1199092

1198660

Fc (1199091 1199092) 119881119865(119884) (1199091) 119881119865(119884) (1199092)

(74)

with 1198660


1198651

(1199091

1199092

) = 1198660


Λ119865

(1199091

119909119899

) = 119866Fc (1199091 119909119899) (75)


119888

119865

(1199091

119909119899



Γ119892

[119884 (120591)] = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(76)


119865(119884)


119865(119884)


Γ119892

[119884 (120591)] = 119878eff (119865)

+ int11988910

119909119865 (119909)119881119865(119884)

(119909) sim 119878eff (119865) + 119878119865

[119884 (120591)]

(77)

with

minus120575119878eff(119865)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119865(119884)

(119909) =120575119878

119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(78)



119878eff(119865) or not


119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)



119886119887

= 1198902120590

119892119886119887



120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904



119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650


119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840


0


(1198650

120590) = 119882119904



119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650


0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895


119865=

119865


119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904



119904


119904

(119865) = 119878eff(119865)

119882119904


0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)


119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)



(119865 120590) = 119882119904



4


5



4

=

119890119882

119904 where 119882119904


5


SYM4


is 120578120583] + 120575119892

120583](119909)



4


5




4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904


AdS5



4



119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119892

(119884) through

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (11)




Γ119892

(119884) = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(12)


119865(119884)


119892


Γ119892

(119884) = 119878eff (119877 119865)

+ int11988910

119909119865 (119909)119881119877119865(119884)

(119909) sim 119878eff (119877 119865) + 119878119877119865

(119884)

(13)


minus120575119878eff (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119877119865(119884)

(119909) =120575119878

119877119865

(119884)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865

(14)

So if119882(119877 119865) = 119878eff(119877 119865) Γ(119884) = Γ119892




2 SYM119877times119878

3119885

119896





0


0


5


4

since 1206010


5




of AdS5


119899

(119909 119903 Ω) =

ℎ119899

(119909 119903)119884119899

(Ω) with 119884119899

(Ω) = 119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)|119910|119899 the


119867119899

(119909 119903 119883) = ℎ119899

(119909 119903) 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) (15)

ℎ119899


weonly have ℎ

119899

(119909)119862119886

1sdotsdotsdot119886

119899

(119883119886



4


0

(119909) a 119883 rarr 119883 1198650

(119909) rarr 119865(119909 119903)


4



5






3

and SCFT6






119877times119878

2 SYM119877times119878

3119885

119896



4





119877119865


dependence of 119878119877119865

(119884) is 119878119877119865

(119884) = 119878119865

0119877119865

minus119877

2119865

minusminus

(119884) where 1198650

119865minus

119865minusminus


As a result 119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus


minus

1198772

119865minusminus


119877119865


119877119865


119892

(119884) withΓ119892





119878119877

= 119877int119889119909+

times Tr( 1

21198772119863+

119883119868

119863+

119883119868

+1

41198976119901

[119883119868

119883119869

]2

minus119894

119877120579119863

+

120579 +1

1198973119901

120579120574119868

[119883119868

120579])

(16)

where119883119868 120579 and1198600



is the Plancklength

119883119868 119909


119901


119868

rarr 119897119901

119883119868 119909+ rarr

119897119901

119909+ 119877 rarr 119897

119901


119878119877

= 119877int119889119909+

times Tr( 1

21198772119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894

119877120579119863

+

120579 + 120579120574119868

[119883119868

120579])

(17)

in which 119897119901



119901



119892++

119892119868+

119892119868119869

) (119862119868119869+

119862119868119869119870

) and(120595

+

120595119868


119878119887

119865

0

= int119889119909+

1198892

120590119892+minus

119875+

times [119875+2

119863+

119883119868

119863+

119883119869

21198922+minus

119892119868119869

+119875+

119863+

119883119868

119892+minus

(119862119869

+119875+

119892119869

+

119892+minus

)119892119868119869

+119875+2

119892++

21198922+minus

minus 119883119868

119883119869

119883119870

119883119871

119892119868119870

119892119869119871

+119875+

119862+

119892+minus

]

(18)

where 119862119868

= 119883119869

119883119870

119862119868

119869119870

and 119862+

= 119883119869

119883119870

119862+119869119870


+minus


119878119887

119865

0

= int119889119909+


119863+

119883119868

119863+

119883119869

2(119877119892+minus

)2

119892119868119869

+119863+

119883119868

119877119892+minus

(119862119869

+119892119869

+

119877119892+minus

)119892119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892119868119870

119892119869119871

+119892++

2(119877119892+minus

)2

+119862+

119877119892+minus

)

(19)

where 119862119868

= minus(1198942)[119883119869

119883119870

]119862119868

119869119870

and 119862+

= minus(1198942)[119883119869

119883119870

]

119862+119869119870


119892minusminus

119862minus119868119869

119862+minus119868

120595minus


119878119887

119892

119868minus

= int119889119909+


119863+

119883119868

119863+

119883119869

119863+

119883119870

2(119877119892+minus

)3

119892119869119870

minus119863+

119883119868

4119877119892+minus

[119883119870

119883119871

] [119883119872

119883119873

]

times 119892119870119872

119892119871119873

+119863+

119883119871

119877119892+minus

[119883119869

119883119872

] [119883119868

119883119873

]

times119892119871119869

119892119872119873

times 119877119892119868minus

)

(20)


minusminus


+

119883)4 (119863

+

119883)2

[119883119883]2 and [119883119883]


minusminus

119892119868minus

119862minus119868119869

119862+minus119868

120595minus


+

119883minus

= (12)119863+

119883119868

119863+

119883119868

+ (1]2)119883119868

119883119869

119883119868

119883119869

and119863119886

119883minus

= 119863+

119883119868

119863119886

119883119868



+minus

119877119892119868minus


119877119862minus119868119869

119877119862+minus119868

119877120595

minus


+minus

119875+ 119892

119868minus

119875+ 119892

minusminus

119875+2 119862

minus119868119869

119875+ 119862

+minus119868

119875+ 120595

minus

119875+ that



+

= 119891(1199091015840+

) 119909minus = 1205721199091015840minus

hArr 119877 = 1205721198771015840

1198921015840

++

= 11989110158402

(1199091015840+

) 119892++

1198921015840

+minus

= 1205721198911015840

(1199091015840+

) 119892+minus

1198921015840

+119868

= 1198911015840

(1199091015840+

) 119892+119868

1198921015840

minus119868

= 120572119892minus119868

1198921015840

minusminus

= 1205722

119892minusminus

1198621015840

+minus119869

= 1205721198911015840

(1199091015840+

) 119862+minus119869

1198621015840

+119868119869

= 1198911015840

(1199091015840+

) 119862+119868119869

1198621015840

minus119868119869

= 120572119862minus119868119869

(21)


1198891199042

= 119892++

119889119891 (1199091015840+

) 119889119891 (1199091015840+

) + 2119892119868+

119889119909119868

119889119891 (1199091015840+

)

+ 119892119868119869

119889119909119868

119889119909119869

+ 2120572119892+minus

119889119891 (1199091015840+

) 1198891199091015840minus

+ 2120572119892119868minus

119889119909119868

1198891199091015840minus

+ 1205722

119892minusminus

1198891199091015840minus

1198891199091015840minus

= 119892++

11989110158402

(1199091015840+

) 1198891199091015840+

1198891199091015840+

+ 21198911015840

(1199091015840+

) 119892119868+

119889119909119868

1199091015840+

+ 119892119868119869

119889119909119868

119889119909119869

+ 2120572119892+minus

1198911015840

(1199091015840+

) 1198891199091015840+

1198891199091015840minus

+ 2120572119892119868minus

119889119909119868

1198891199091015840minus

+ 1205722

119892minusminus

1198891199091015840minus

1198891199091015840minus

119862 = 119862119868119869119870

119889119909119868

and 119889119909119869

and 119889119909119870

+ 119862119868119869+

119889119909119868

and 119889119909119869

and 119889119891 (1199091015840+

)

+ 120572119862119868119869minus

119889119909119868

and 119889119909119869

and 1198891199091015840minus

+ 120572119862119868+minus

119889119909119868

and 119889119891 (1199091015840+

) and 1198891199091015840minus

= 119862119868119869119870

119889119909119868

and 119889119909119869

and 119889119909119870

+ 119862119868119869+

1198911015840

(1199091015840+

) 119889119909119868

and 119889119909119869

and 1198891199091015840+

+ 120572119862119868119869minus

119889119909119868

and 119889119909119869

and 119889119909minus

+ 120572119862119868+minus

1198911015840

(1199091015840+

) 119889119909119868

and 1198891199091015840+

and 119889119909minus

(22)

119878119877119865


119877119865


119881119865(119884)

(119909) =120575119878

119877119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(23)


+minus

= minus1 and 119892119868119869

= 120575119868119869

119881119865(119884)

(119909)


119877119865



119878119877119865

= 119878119877

minus 119877int119889119909+


119868119869(119883)

ℎ119868119869

(119883) + 119881ℎ

119868+(119883)

ℎ119868+

(119883)

+ 119881ℎ

++(119883)

ℎ++

(119883) + 119881ℎ

119868minus(119883)

ℎ119868minus

(119883)

+ 119881ℎ

+minus(119883)

ℎ+minus

(119883) + 119881ℎ

minusminus(119883)

ℎminusminus

(119883)

+ 119881119862

119868119869119870(119883)

119862119868119869119870

(119883) + 119881119862

+119869119870(119883)

119862+119869119870

(119883)

+ 119881119862

119868119869minus(119883)

119862119868119869minus

(119883) + 119881119862

+minus119870(119883)

119862+minus119870

(119883)

+ 119881120595(119883)

120595 (119883) + 119881120595

+(119883)

120595+

(119883)

+ 119881120595

minus(119883)

120595minus

(119883) ]

(24)

where for example

119881ℎ

119868+(119883)

ℎ119868+

(119883) = (119863+

119883119868

119877minus

1

4120579120574

119868119869

120579120597

120597119883119869

)ℎ119868+

(119883)

119877 (25)

ℎ119868119869

(119883) 120595minus


1199091

1199099




119868+


ℎ119868+

(119909+

1198831

1198839

)

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0)119883119896


119899

(26)

with (120597119896

1


119899

ℎ119868+

)(119909+


(119909+

)

at (119909+ 0) [3 23ndash25]

STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0)

times STr[ 1

119877(119863+

119883119868

119877minus

1

4120579120574

119868119869

120579120597

120597119883119869

)119883119896


119899]

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0) 119868119896

1sdotsdotsdot119896

119899

ℎ

119868+

(119909+

)

(27)


+

0 0) the 119880(119873) SYM1


119883119894

997888rarr 119883119894

minus 119886119894

119891 (119909+

0 0) 997888rarr 119891(119909+

1198861

1198869

)

(28)


SYM1


1



1


1sdotsdotsdot119896


119896

1 Φ119896

1sdotsdotsdot119896


spacetime

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

) = 120575 (1199090

minus 119909+

) 1205759

(119909119868

minus 119883119868

)

= 12057510

(119909120583

minus 119883120583

)

(29)


set 1198830

= 119909+1

119873times119873


= diag1199091198681

119909119868

2

119909119868

119873

(29) becomes

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

)

= diag 12057510

(119909120583

minus 119909120583

1

) 12057510

(119909120583

minus 119909120583

2

) 12057510

(119909120583

minus 119909120583

119873

)

(30)


119868+

we have

119881ℎ

119868+(119909) =

120575

120575ℎ119868+

(119909)minus119877int119889119909

+STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

12057510

(119909 minus 119883)]

(31)


119881ℎ

119868+

(119901) = int11988910

119909119881ℎ

119868+(119909) 119890

119894119901119909

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

119890119894119901119883

]


119863+

119883119868

119877+

119894

4120579120574

119868119869

120579119901119869

] 119890119894119901119883

(32)


119865


119885 (119877 119865) = 119890119882(119877119865)

= int [119889119883119889120579 1198891198600

] 119890minus119878

119877119865(119883120579119860

0)

= int [119889119884] 119890minus119878

119877119865(119884)

(33)


119892++

119892119868+

119892119868119869

119892minusminus

119892119868minus

) (119862119868119869+

119862119868119869119870

119862119868119869minus

119862119868+minus

) and (120595+

120595119868

120595minus


120579 1198600



5

CFT4


119885SYM4

= exp minus119865 (34)


ℎ=0

119865119899

ℎ


5

times 1198785 119885SYM

4


4


5




+


Suppose 119878119872119875

+119865


119875

+119865


= 1198751015840+

120572119909+

= 119891(1199091015840+

) and 119883 = 119892(1198831015840

) there is 119878119872

119875

+119865

(119884) =

119878119872

119875

1015840+119865

1015840(1198841015840


transformation

119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (35)


1015840


int [119889119884] 119890minus119878

119872

119875+119865(119884)

= int [1198891198841015840

] 119890minus119878

119872

1198751015840+1198651015840 (1198841015840)

(36)


+

1205901

1205902

) and 1198841015840

(119909+

1205901

1205902


+

) and 1198841015840

(119909+

) and there is

119885 (119877 119865) = 119890119882(119877119865)

= int [119889119884] 119890minus119878

119877119865(119884)

= int [1198891198841015840

] 119890minus119878

11987710158401198651015840 (1198841015840)

= 119890119882(119877

1015840119865

1015840)

= 119885 (1198771015840

1198651015840

)

(37)

119882(119877 119865) = 119882(1198771015840

1198651015840



Let 1198650

119865minus

119865minusminus


indices Since

119878119877119865

(119884) = 119878119865

0119877119865

minus119877

2119865


119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus

) (39)


119865minus

119865minusminus



0

119865minus

119865minusminus


119909minus


119877 119865

0

997888rarr 1198650

119865minus

997888rarr 119877119865minus

119865minusminus

997888rarr 1198772

119865minusminus

(40)

so

119878eff (119877 1198650

119865minus

119865minusminus

) = 119878eff (1 1198650 119877119865minus

1198772

119865minusminus

) (41)


119865minusminus



=

minus1 119892119868119869

= 120575119868119869


119882(119877 119865) = int11988910

119909Γ119888

119877119865

(119909) 119865 (119909)

+1

2int 119889

10

1199091

times int11988910

1199092

Γ119888

119877119865

(1199091

1199092

) 119865 (1199091

) 119865 (1199092

) + sdot sdot sdot

= sum

119899

1

119899int 119889

10

1199091


119909119899

Γ119888

119877119865

(1199091

119909119899

)

times 119865 (1199091

) sdot sdot sdot 119865 (119909119899

)

(42)


119877119892

++119892

minusminus

(1199091

1199092

) = 1205752

119882(119877119865)120575119892++

(1199091


(1199092

)|119865=119865

)

Γ119888

119877119865

(1199091

119909119899

) =120575119899

119882(119877 119865)

120575119865(1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(43)


is

119878eff (119877 119865) = sum

119899

1

119899int

2120587119877

0

119889119909minus

1

times int119889119909+

1

1198899x1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

int119889119909+

119899

1198899x119899

times Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

times 119865 (119909+

1

x1

) sdot sdot sdot 119865 (119909+

119899

x119899

)

(44)

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899



Γ119888

119877119865

(1199091

119909119899

)

= Γ119888

119877119865

(119909+

1

x1

119909+

119899

x119899

)

= int

2120587119877

0

119889119909minus

1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

(45)


(119909)120575119865(119910) = 0where 119881

119865


Γ119888

119877119865

(1199091

119909119899

) = (minus1)119899

⟨119881119865

(1199091


(119909119899

)⟩119888

(46)

⟨119881119865

(1199091


(119909119899

)⟩119888


⟨119881119865

(1199091


(119909119899

)⟩

= int [119889119883119889120579 1198891198600

] 119890minus119878

0119881119865

(1199091


(119909119899

)

=(minus1)

119899

120575119899

119885 (119877 119865)

120575119865 (1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(47)


119877119865

(1199091

119909119899

)In terms of 119881

119865

(119901)

⟨119881119865

(1199091


(119909119899

)⟩119888

= int11988910

1199011


119901119899

119890minus119894(119901

1119909


119899119909

119899)

times ⟨119881119865

(1199011


(119901119899

)⟩119888

(48)

Note that 119881119865(119883)

(119909+

119909119868

minus 119886119868

) = 119881119865(119883+119886)

(119909+

119909119868

) 1198780

(119883 + 119886) =

1198780

(119883) so

⟨119881119865

(1199091


(119909119899

)⟩119888

= ⟨119881119865

(1199091

+ 1198861


(119909119899

+ 119886119899

)⟩119888

(49)


119865

(1199011


(119901119899

)⟩119888

= 0 if 1199011

+ sdot sdot sdot + 119901119899


⟨119881119865

(119909)⟩ = minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865

= V119865

(50)

V119865


119877

is 119878119874(9) invariant so V119865



V119892

119868+

= V119892

119868minus

= V119862

119868119869119870

= V119862

119868119869+

= V119862

119868119869minus

= V119862

119868+minus

= V120595

119868

= 0

(51)


V119892

119868119869


+

=

V120595

minus


ℎ

++

Vℎ

minusminus

and Vℎ

+minus

In particular Vℎ

++

= 119901+

1198719 is


⟨119881119865

(119909)⟩119865

= minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865+

119865

= V119865

(119909)

= sum

119899

minus1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Γ119888

119877119865

(119909 1199091

119909119899minus1

)

times 119865 (1199091


)

(52)


119865

(119909)⟩119865




119865


119865


119892

120575119892120583]

=



119865


119865




Γ (119884) = 119878119877119865

(119884) + 119882 (119877 119865) (53)

119890Γ(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

(54)


int [119889]120575

120575119865 (119909)119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

[minus

120575119878119877119865

()

120575119865 (119909)+

120575119878119877119865

(119884)

120575119865 (119909)] = 0

(55)

or equivalently

int [119889] 119890minus119878

119877119865(

119884)

120575119878119877119865

()

120575119865 (119909)=

120575119878119877119865

(119884)

120575119865 (119909)= minus

120575119882 (119877 119865)

120575119865 (119909) (56)


120575Γ (119884)

120575119884=

120575119878119877119865

(119884)

120575119884 (57)



(119884) only comes from119877119865

minus

and 1198772

119865minusminus


119865minus

119865minusminus


0

119865minus


1205722

)119878119877119865



+

1205901

1205902

) Let 1199091015840minus rarr 1198751015840+

1199091015840minus

119875+ 1198751015840+ rarr



= 1198650

119865minus

119875+

= 1198651015840

minus

1198751015840+ 119865

minusminus

119875+2

=

1198651015840

minusminus


119872

119875

+119865



119877119865


119877119865


0


119877119865



119878119865

(119884) = 1198780

(119884) + int 11988910

119909119865(119909)119881119865(119884)

(119909) Γ(119884) = 1198780

(119884) + Γ(V119865

)where

Γ (V119865

) = int11988910

119909119865 (119909) V119865

(119909) + 119882(119865 + 119865) (58)


minus120575119882 (119865)

120575119865 (119909)

100381610038161003816100381610038161003816100381610038161003816 119865

= V119865

= 119881119865(119884)

(119909)

120575Γ (V119865

)

120575V119865

(119909)

1003816100381610038161003816100381610038161003816100381610038161003816V119865

= 119865 (119909)

(59)


Let

Λ119865

(1199091

119909119899

) =120575119899

Γ (V119865

)

120575V119865

(1199091


(119909119899

)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

(60)

then

Γ (119884) = 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times V119865

(1199091


(119909119899

)

= 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(61)


(119909)

119865 (119909) = sum

119899

1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Λ119865

(119909 1199091

119909119899minus1

)

times V119865

(1199091


(119909119899minus1

)

(62)



119865

(119909) If Λ119865


119865


119892

120575119865(119909) = minusV119865

(119909) with119878119892


Γ119888

119865

(119909 minus 119911) = minus120575V

119865

(119909)

120575119865 (119911)

100381610038161003816100381610038161003816100381610038161003816 119865=0

Λ119865

(119911 minus 119910) =1205752

Γ (V119865

)

120575V119865

(119911) 120575V119865

(119910)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

=120575119865 (119911)

120575V119865

(119910)

100381610038161003816100381610038161003816100381610038161003816V119865=0

(63)

so

minusint11988910

119911Γ119888

119865

(119909 minus 119911) Λ119865

(119911 minus 119910) = 12057510

(119909 minus 119910) (64)

minusΛ119865

(1199091

minus 1199092


(1199091

minus 1199092

) Subsequently

Λ119865

(1199091

1199092

1199093

) = int11988910

1199101

int11988910

1199102

times int11988910

1199103

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Γ119888

119865

(1199101

1199102

1199103

)

Γ119888

119865

(1199091

1199092

1199093


1199101

int11988910

1199102

times int11988910

1199103

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Λ119865

(1199101

1199102

1199103

)

Λ119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Λ119865

(1199104

minus 1199094

)

times [Γ119888

119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Λ119865

(1199111

minus 1199112

)

times (Γ119888

119865

(1199111

1199101

1199104

)

times Γ119888

119865

(1199112

1199102

1199103

)

+ Γ119888

119865

(1199111

1199102

1199104

)

times Γ119888

119865

(1199112

1199101

1199103

)

+ Γ119888

119865

(1199111

1199103

1199104

)

times Γ119888

119865

(1199112

1199101

1199102

)) ]

Γ119888

119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Γ119888

119865

(1199104

minus 1199094

)

times [Λ119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Γ119888

119865

(1199111

minus 1199112

)

times (Λ119865

(1199111

1199101

1199104

)

times Λ119865

(1199112

1199102

1199103

)

+ Λ119865

(1199111

1199102

1199104

)

times Λ119865

(1199112

1199101

1199103

)

+ Λ119865

(1199111

1199103

1199104

)

times Λ119865

(1199112

1199101

1199102

)) ]

(65)



and Γ119888

119865

shows thatΛ119865

and Γ119888

119865


119865

(119911minus119910) = (120575119865(119911)120575V119865

(119910))|V119865=0


In (54) let = 119884 + 120578

119890Γ(119884)

= int [119889120578] 119890minus119878

119865(119884+120578)+119878

119865(119884)

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minusint119889120591120575119878

119865

(119884)

120575119884 (120591)120578 (120591)

minus1

2int119889120591 119889120591

1015840

1205752

119878119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)

times 120578 (120591) 120578 (1205911015840

) + sdot sdot sdot

= minus119878(1)

119865

(119884 120578) minus 119878(2)

119865

(119884 120578) + sdot sdot sdot

(66)

119878119865


120591


120591

also exists)

119878119865

(119884) = int119889120591 [1

2119884120597

2

120591

119884 + 1198810

(119884) + 119881 (119865 119884)]

= 1198780

(119884) + int119889120591119881 (119865 119884)

(67)


0

from 120575119878119865

0

(119884)120575119884 = 0 and expand 119878(119899)

119865

(119884 120578) around1198650

119865 = 1198650

+ 119891 Consider

119878(1)

119865

(119884 120578) =120575119878

(1)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


119878(2)

119865

(119884 120578) = 119878(2)

119865

0

(119884 120578) +120575119878

(2)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


(68)


(119884) +

Γ2


minus119878119865

(119884 + 120578) + 119878119865

(119884) = minus119878(2)

119865

0

(119884 120578) + sdot sdot sdot

Γ1

(119884) = ln int [119889120578] 119890minus119878

(2)

1198650(119884120578)

= ln (det119860119865

0

)12

(69)

with

119860119865

0

(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842+

1205972

119881 (1198650

119884)

1205971198842] 120575 (120591 minus 120591

1015840

)

(70)


(119884)120575119884 = 0 1198650

= 0 119860119865

0

reduces to

119860(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842] 120575 (120591 minus 120591

1015840

) (71)



1


119890Γeff(119884) = int [119889120578] 119890

minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

(72)


int [119889120578]120575

120575119869 (120591)119890minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

= 0 (73)

Γ1

(119884) = Γ1



(119884)120575119884 = 0 [3 25]

Γ1

(119884) =1

2int119889

10

1199091

times int11988910

1199092

1198660

Fc (1199091 1199092) 119881119865(119884) (1199091) 119881119865(119884) (1199092)

(74)

with 1198660


1198651

(1199091

1199092

) = 1198660


Λ119865

(1199091

119909119899

) = 119866Fc (1199091 119909119899) (75)


119888

119865

(1199091

119909119899



Γ119892

[119884 (120591)] = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(76)


119865(119884)


119865(119884)


Γ119892

[119884 (120591)] = 119878eff (119865)

+ int11988910

119909119865 (119909)119881119865(119884)

(119909) sim 119878eff (119865) + 119878119865

[119884 (120591)]

(77)

with

minus120575119878eff(119865)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119865(119884)

(119909) =120575119878

119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(78)



119878eff(119865) or not


119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)



119886119887

= 1198902120590

119892119886119887



120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904



119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650


119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840


0


(1198650

120590) = 119882119904



119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650


0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895


119865=

119865


119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904



119904


119904

(119865) = 119878eff(119865)

119882119904


0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)


119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)



(119865 120590) = 119882119904



4


5



4

=

119890119882

119904 where 119882119904


5


SYM4


is 120578120583] + 120575119892

120583](119909)



4


5




4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904


AdS5



4



119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119878119877

= 119877int119889119909+

times Tr( 1

21198772119863+

119883119868

119863+

119883119868

+1

41198976119901

[119883119868

119883119869

]2

minus119894

119877120579119863

+

120579 +1

1198973119901

120579120574119868

[119883119868

120579])

(16)

where119883119868 120579 and1198600



is the Plancklength

119883119868 119909


119901


119868

rarr 119897119901

119883119868 119909+ rarr

119897119901

119909+ 119877 rarr 119897

119901


119878119877

= 119877int119889119909+

times Tr( 1

21198772119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894

119877120579119863

+

120579 + 120579120574119868

[119883119868

120579])

(17)

in which 119897119901



119901



119892++

119892119868+

119892119868119869

) (119862119868119869+

119862119868119869119870

) and(120595

+

120595119868


119878119887

119865

0

= int119889119909+

1198892

120590119892+minus

119875+

times [119875+2

119863+

119883119868

119863+

119883119869

21198922+minus

119892119868119869

+119875+

119863+

119883119868

119892+minus

(119862119869

+119875+

119892119869

+

119892+minus

)119892119868119869

+119875+2

119892++

21198922+minus

minus 119883119868

119883119869

119883119870

119883119871

119892119868119870

119892119869119871

+119875+

119862+

119892+minus

]

(18)

where 119862119868

= 119883119869

119883119870

119862119868

119869119870

and 119862+

= 119883119869

119883119870

119862+119869119870


+minus


119878119887

119865

0

= int119889119909+


119863+

119883119868

119863+

119883119869

2(119877119892+minus

)2

119892119868119869

+119863+

119883119868

119877119892+minus

(119862119869

+119892119869

+

119877119892+minus

)119892119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892119868119870

119892119869119871

+119892++

2(119877119892+minus

)2

+119862+

119877119892+minus

)

(19)

where 119862119868

= minus(1198942)[119883119869

119883119870

]119862119868

119869119870

and 119862+

= minus(1198942)[119883119869

119883119870

]

119862+119869119870


119892minusminus

119862minus119868119869

119862+minus119868

120595minus


119878119887

119892

119868minus

= int119889119909+


119863+

119883119868

119863+

119883119869

119863+

119883119870

2(119877119892+minus

)3

119892119869119870

minus119863+

119883119868

4119877119892+minus

[119883119870

119883119871

] [119883119872

119883119873

]

times 119892119870119872

119892119871119873

+119863+

119883119871

119877119892+minus

[119883119869

119883119872

] [119883119868

119883119873

]

times119892119871119869

119892119872119873

times 119877119892119868minus

)

(20)


minusminus


+

119883)4 (119863

+

119883)2

[119883119883]2 and [119883119883]


minusminus

119892119868minus

119862minus119868119869

119862+minus119868

120595minus


+

119883minus

= (12)119863+

119883119868

119863+

119883119868

+ (1]2)119883119868

119883119869

119883119868

119883119869

and119863119886

119883minus

= 119863+

119883119868

119863119886

119883119868



+minus

119877119892119868minus


119877119862minus119868119869

119877119862+minus119868

119877120595

minus


+minus

119875+ 119892

119868minus

119875+ 119892

minusminus

119875+2 119862

minus119868119869

119875+ 119862

+minus119868

119875+ 120595

minus

119875+ that



+

= 119891(1199091015840+

) 119909minus = 1205721199091015840minus

hArr 119877 = 1205721198771015840

1198921015840

++

= 11989110158402

(1199091015840+

) 119892++

1198921015840

+minus

= 1205721198911015840

(1199091015840+

) 119892+minus

1198921015840

+119868

= 1198911015840

(1199091015840+

) 119892+119868

1198921015840

minus119868

= 120572119892minus119868

1198921015840

minusminus

= 1205722

119892minusminus

1198621015840

+minus119869

= 1205721198911015840

(1199091015840+

) 119862+minus119869

1198621015840

+119868119869

= 1198911015840

(1199091015840+

) 119862+119868119869

1198621015840

minus119868119869

= 120572119862minus119868119869

(21)


1198891199042

= 119892++

119889119891 (1199091015840+

) 119889119891 (1199091015840+

) + 2119892119868+

119889119909119868

119889119891 (1199091015840+

)

+ 119892119868119869

119889119909119868

119889119909119869

+ 2120572119892+minus

119889119891 (1199091015840+

) 1198891199091015840minus

+ 2120572119892119868minus

119889119909119868

1198891199091015840minus

+ 1205722

119892minusminus

1198891199091015840minus

1198891199091015840minus

= 119892++

11989110158402

(1199091015840+

) 1198891199091015840+

1198891199091015840+

+ 21198911015840

(1199091015840+

) 119892119868+

119889119909119868

1199091015840+

+ 119892119868119869

119889119909119868

119889119909119869

+ 2120572119892+minus

1198911015840

(1199091015840+

) 1198891199091015840+

1198891199091015840minus

+ 2120572119892119868minus

119889119909119868

1198891199091015840minus

+ 1205722

119892minusminus

1198891199091015840minus

1198891199091015840minus

119862 = 119862119868119869119870

119889119909119868

and 119889119909119869

and 119889119909119870

+ 119862119868119869+

119889119909119868

and 119889119909119869

and 119889119891 (1199091015840+

)

+ 120572119862119868119869minus

119889119909119868

and 119889119909119869

and 1198891199091015840minus

+ 120572119862119868+minus

119889119909119868

and 119889119891 (1199091015840+

) and 1198891199091015840minus

= 119862119868119869119870

119889119909119868

and 119889119909119869

and 119889119909119870

+ 119862119868119869+

1198911015840

(1199091015840+

) 119889119909119868

and 119889119909119869

and 1198891199091015840+

+ 120572119862119868119869minus

119889119909119868

and 119889119909119869

and 119889119909minus

+ 120572119862119868+minus

1198911015840

(1199091015840+

) 119889119909119868

and 1198891199091015840+

and 119889119909minus

(22)

119878119877119865


119877119865


119881119865(119884)

(119909) =120575119878

119877119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(23)


+minus

= minus1 and 119892119868119869

= 120575119868119869

119881119865(119884)

(119909)


119877119865



119878119877119865

= 119878119877

minus 119877int119889119909+


119868119869(119883)

ℎ119868119869

(119883) + 119881ℎ

119868+(119883)

ℎ119868+

(119883)

+ 119881ℎ

++(119883)

ℎ++

(119883) + 119881ℎ

119868minus(119883)

ℎ119868minus

(119883)

+ 119881ℎ

+minus(119883)

ℎ+minus

(119883) + 119881ℎ

minusminus(119883)

ℎminusminus

(119883)

+ 119881119862

119868119869119870(119883)

119862119868119869119870

(119883) + 119881119862

+119869119870(119883)

119862+119869119870

(119883)

+ 119881119862

119868119869minus(119883)

119862119868119869minus

(119883) + 119881119862

+minus119870(119883)

119862+minus119870

(119883)

+ 119881120595(119883)

120595 (119883) + 119881120595

+(119883)

120595+

(119883)

+ 119881120595

minus(119883)

120595minus

(119883) ]

(24)

where for example

119881ℎ

119868+(119883)

ℎ119868+

(119883) = (119863+

119883119868

119877minus

1

4120579120574

119868119869

120579120597

120597119883119869

)ℎ119868+

(119883)

119877 (25)

ℎ119868119869

(119883) 120595minus


1199091

1199099




119868+


ℎ119868+

(119909+

1198831

1198839

)

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0)119883119896


119899

(26)

with (120597119896

1


119899

ℎ119868+

)(119909+


(119909+

)

at (119909+ 0) [3 23ndash25]

STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0)

times STr[ 1

119877(119863+

119883119868

119877minus

1

4120579120574

119868119869

120579120597

120597119883119869

)119883119896


119899]

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0) 119868119896

1sdotsdotsdot119896

119899

ℎ

119868+

(119909+

)

(27)


+

0 0) the 119880(119873) SYM1


119883119894

997888rarr 119883119894

minus 119886119894

119891 (119909+

0 0) 997888rarr 119891(119909+

1198861

1198869

)

(28)


SYM1


1



1


1sdotsdotsdot119896


119896

1 Φ119896

1sdotsdotsdot119896


spacetime

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

) = 120575 (1199090

minus 119909+

) 1205759

(119909119868

minus 119883119868

)

= 12057510

(119909120583

minus 119883120583

)

(29)


set 1198830

= 119909+1

119873times119873


= diag1199091198681

119909119868

2

119909119868

119873

(29) becomes

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

)

= diag 12057510

(119909120583

minus 119909120583

1

) 12057510

(119909120583

minus 119909120583

2

) 12057510

(119909120583

minus 119909120583

119873

)

(30)


119868+

we have

119881ℎ

119868+(119909) =

120575

120575ℎ119868+

(119909)minus119877int119889119909

+STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

12057510

(119909 minus 119883)]

(31)


119881ℎ

119868+

(119901) = int11988910

119909119881ℎ

119868+(119909) 119890

119894119901119909

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

119890119894119901119883

]


119863+

119883119868

119877+

119894

4120579120574

119868119869

120579119901119869

] 119890119894119901119883

(32)


119865


119885 (119877 119865) = 119890119882(119877119865)

= int [119889119883119889120579 1198891198600

] 119890minus119878

119877119865(119883120579119860

0)

= int [119889119884] 119890minus119878

119877119865(119884)

(33)


119892++

119892119868+

119892119868119869

119892minusminus

119892119868minus

) (119862119868119869+

119862119868119869119870

119862119868119869minus

119862119868+minus

) and (120595+

120595119868

120595minus


120579 1198600



5

CFT4


119885SYM4

= exp minus119865 (34)


ℎ=0

119865119899

ℎ


5

times 1198785 119885SYM

4


4


5




+


Suppose 119878119872119875

+119865


119875

+119865


= 1198751015840+

120572119909+

= 119891(1199091015840+

) and 119883 = 119892(1198831015840

) there is 119878119872

119875

+119865

(119884) =

119878119872

119875

1015840+119865

1015840(1198841015840


transformation

119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (35)


1015840


int [119889119884] 119890minus119878

119872

119875+119865(119884)

= int [1198891198841015840

] 119890minus119878

119872

1198751015840+1198651015840 (1198841015840)

(36)


+

1205901

1205902

) and 1198841015840

(119909+

1205901

1205902


+

) and 1198841015840

(119909+

) and there is

119885 (119877 119865) = 119890119882(119877119865)

= int [119889119884] 119890minus119878

119877119865(119884)

= int [1198891198841015840

] 119890minus119878

11987710158401198651015840 (1198841015840)

= 119890119882(119877

1015840119865

1015840)

= 119885 (1198771015840

1198651015840

)

(37)

119882(119877 119865) = 119882(1198771015840

1198651015840



Let 1198650

119865minus

119865minusminus


indices Since

119878119877119865

(119884) = 119878119865

0119877119865

minus119877

2119865


119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus

) (39)


119865minus

119865minusminus



0

119865minus

119865minusminus


119909minus


119877 119865

0

997888rarr 1198650

119865minus

997888rarr 119877119865minus

119865minusminus

997888rarr 1198772

119865minusminus

(40)

so

119878eff (119877 1198650

119865minus

119865minusminus

) = 119878eff (1 1198650 119877119865minus

1198772

119865minusminus

) (41)


119865minusminus



=

minus1 119892119868119869

= 120575119868119869


119882(119877 119865) = int11988910

119909Γ119888

119877119865

(119909) 119865 (119909)

+1

2int 119889

10

1199091

times int11988910

1199092

Γ119888

119877119865

(1199091

1199092

) 119865 (1199091

) 119865 (1199092

) + sdot sdot sdot

= sum

119899

1

119899int 119889

10

1199091


119909119899

Γ119888

119877119865

(1199091

119909119899

)

times 119865 (1199091

) sdot sdot sdot 119865 (119909119899

)

(42)


119877119892

++119892

minusminus

(1199091

1199092

) = 1205752

119882(119877119865)120575119892++

(1199091


(1199092

)|119865=119865

)

Γ119888

119877119865

(1199091

119909119899

) =120575119899

119882(119877 119865)

120575119865(1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(43)


is

119878eff (119877 119865) = sum

119899

1

119899int

2120587119877

0

119889119909minus

1

times int119889119909+

1

1198899x1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

int119889119909+

119899

1198899x119899

times Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

times 119865 (119909+

1

x1

) sdot sdot sdot 119865 (119909+

119899

x119899

)

(44)

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899



Γ119888

119877119865

(1199091

119909119899

)

= Γ119888

119877119865

(119909+

1

x1

119909+

119899

x119899

)

= int

2120587119877

0

119889119909minus

1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

(45)


(119909)120575119865(119910) = 0where 119881

119865


Γ119888

119877119865

(1199091

119909119899

) = (minus1)119899

⟨119881119865

(1199091


(119909119899

)⟩119888

(46)

⟨119881119865

(1199091


(119909119899

)⟩119888


⟨119881119865

(1199091


(119909119899

)⟩

= int [119889119883119889120579 1198891198600

] 119890minus119878

0119881119865

(1199091


(119909119899

)

=(minus1)

119899

120575119899

119885 (119877 119865)

120575119865 (1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(47)


119877119865

(1199091

119909119899

)In terms of 119881

119865

(119901)

⟨119881119865

(1199091


(119909119899

)⟩119888

= int11988910

1199011


119901119899

119890minus119894(119901

1119909


119899119909

119899)

times ⟨119881119865

(1199011


(119901119899

)⟩119888

(48)

Note that 119881119865(119883)

(119909+

119909119868

minus 119886119868

) = 119881119865(119883+119886)

(119909+

119909119868

) 1198780

(119883 + 119886) =

1198780

(119883) so

⟨119881119865

(1199091


(119909119899

)⟩119888

= ⟨119881119865

(1199091

+ 1198861


(119909119899

+ 119886119899

)⟩119888

(49)


119865

(1199011


(119901119899

)⟩119888

= 0 if 1199011

+ sdot sdot sdot + 119901119899


⟨119881119865

(119909)⟩ = minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865

= V119865

(50)

V119865


119877

is 119878119874(9) invariant so V119865



V119892

119868+

= V119892

119868minus

= V119862

119868119869119870

= V119862

119868119869+

= V119862

119868119869minus

= V119862

119868+minus

= V120595

119868

= 0

(51)


V119892

119868119869


+

=

V120595

minus


ℎ

++

Vℎ

minusminus

and Vℎ

+minus

In particular Vℎ

++

= 119901+

1198719 is


⟨119881119865

(119909)⟩119865

= minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865+

119865

= V119865

(119909)

= sum

119899

minus1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Γ119888

119877119865

(119909 1199091

119909119899minus1

)

times 119865 (1199091


)

(52)


119865

(119909)⟩119865




119865


119865


119892

120575119892120583]

=



119865


119865




Γ (119884) = 119878119877119865

(119884) + 119882 (119877 119865) (53)

119890Γ(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

(54)


int [119889]120575

120575119865 (119909)119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

[minus

120575119878119877119865

()

120575119865 (119909)+

120575119878119877119865

(119884)

120575119865 (119909)] = 0

(55)

or equivalently

int [119889] 119890minus119878

119877119865(

119884)

120575119878119877119865

()

120575119865 (119909)=

120575119878119877119865

(119884)

120575119865 (119909)= minus

120575119882 (119877 119865)

120575119865 (119909) (56)


120575Γ (119884)

120575119884=

120575119878119877119865

(119884)

120575119884 (57)



(119884) only comes from119877119865

minus

and 1198772

119865minusminus


119865minus

119865minusminus


0

119865minus


1205722

)119878119877119865



+

1205901

1205902

) Let 1199091015840minus rarr 1198751015840+

1199091015840minus

119875+ 1198751015840+ rarr



= 1198650

119865minus

119875+

= 1198651015840

minus

1198751015840+ 119865

minusminus

119875+2

=

1198651015840

minusminus


119872

119875

+119865



119877119865


119877119865


0


119877119865



119878119865

(119884) = 1198780

(119884) + int 11988910

119909119865(119909)119881119865(119884)

(119909) Γ(119884) = 1198780

(119884) + Γ(V119865

)where

Γ (V119865

) = int11988910

119909119865 (119909) V119865

(119909) + 119882(119865 + 119865) (58)


minus120575119882 (119865)

120575119865 (119909)

100381610038161003816100381610038161003816100381610038161003816 119865

= V119865

= 119881119865(119884)

(119909)

120575Γ (V119865

)

120575V119865

(119909)

1003816100381610038161003816100381610038161003816100381610038161003816V119865

= 119865 (119909)

(59)


Let

Λ119865

(1199091

119909119899

) =120575119899

Γ (V119865

)

120575V119865

(1199091


(119909119899

)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

(60)

then

Γ (119884) = 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times V119865

(1199091


(119909119899

)

= 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(61)


(119909)

119865 (119909) = sum

119899

1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Λ119865

(119909 1199091

119909119899minus1

)

times V119865

(1199091


(119909119899minus1

)

(62)



119865

(119909) If Λ119865


119865


119892

120575119865(119909) = minusV119865

(119909) with119878119892


Γ119888

119865

(119909 minus 119911) = minus120575V

119865

(119909)

120575119865 (119911)

100381610038161003816100381610038161003816100381610038161003816 119865=0

Λ119865

(119911 minus 119910) =1205752

Γ (V119865

)

120575V119865

(119911) 120575V119865

(119910)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

=120575119865 (119911)

120575V119865

(119910)

100381610038161003816100381610038161003816100381610038161003816V119865=0

(63)

so

minusint11988910

119911Γ119888

119865

(119909 minus 119911) Λ119865

(119911 minus 119910) = 12057510

(119909 minus 119910) (64)

minusΛ119865

(1199091

minus 1199092


(1199091

minus 1199092

) Subsequently

Λ119865

(1199091

1199092

1199093

) = int11988910

1199101

int11988910

1199102

times int11988910

1199103

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Γ119888

119865

(1199101

1199102

1199103

)

Γ119888

119865

(1199091

1199092

1199093


1199101

int11988910

1199102

times int11988910

1199103

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Λ119865

(1199101

1199102

1199103

)

Λ119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Λ119865

(1199104

minus 1199094

)

times [Γ119888

119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Λ119865

(1199111

minus 1199112

)

times (Γ119888

119865

(1199111

1199101

1199104

)

times Γ119888

119865

(1199112

1199102

1199103

)

+ Γ119888

119865

(1199111

1199102

1199104

)

times Γ119888

119865

(1199112

1199101

1199103

)

+ Γ119888

119865

(1199111

1199103

1199104

)

times Γ119888

119865

(1199112

1199101

1199102

)) ]

Γ119888

119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Γ119888

119865

(1199104

minus 1199094

)

times [Λ119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Γ119888

119865

(1199111

minus 1199112

)

times (Λ119865

(1199111

1199101

1199104

)

times Λ119865

(1199112

1199102

1199103

)

+ Λ119865

(1199111

1199102

1199104

)

times Λ119865

(1199112

1199101

1199103

)

+ Λ119865

(1199111

1199103

1199104

)

times Λ119865

(1199112

1199101

1199102

)) ]

(65)



and Γ119888

119865

shows thatΛ119865

and Γ119888

119865


119865

(119911minus119910) = (120575119865(119911)120575V119865

(119910))|V119865=0


In (54) let = 119884 + 120578

119890Γ(119884)

= int [119889120578] 119890minus119878

119865(119884+120578)+119878

119865(119884)

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minusint119889120591120575119878

119865

(119884)

120575119884 (120591)120578 (120591)

minus1

2int119889120591 119889120591

1015840

1205752

119878119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)

times 120578 (120591) 120578 (1205911015840

) + sdot sdot sdot

= minus119878(1)

119865

(119884 120578) minus 119878(2)

119865

(119884 120578) + sdot sdot sdot

(66)

119878119865


120591


120591

also exists)

119878119865

(119884) = int119889120591 [1

2119884120597

2

120591

119884 + 1198810

(119884) + 119881 (119865 119884)]

= 1198780

(119884) + int119889120591119881 (119865 119884)

(67)


0

from 120575119878119865

0

(119884)120575119884 = 0 and expand 119878(119899)

119865

(119884 120578) around1198650

119865 = 1198650

+ 119891 Consider

119878(1)

119865

(119884 120578) =120575119878

(1)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


119878(2)

119865

(119884 120578) = 119878(2)

119865

0

(119884 120578) +120575119878

(2)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


(68)


(119884) +

Γ2


minus119878119865

(119884 + 120578) + 119878119865

(119884) = minus119878(2)

119865

0

(119884 120578) + sdot sdot sdot

Γ1

(119884) = ln int [119889120578] 119890minus119878

(2)

1198650(119884120578)

= ln (det119860119865

0

)12

(69)

with

119860119865

0

(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842+

1205972

119881 (1198650

119884)

1205971198842] 120575 (120591 minus 120591

1015840

)

(70)


(119884)120575119884 = 0 1198650

= 0 119860119865

0

reduces to

119860(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842] 120575 (120591 minus 120591

1015840

) (71)



1


119890Γeff(119884) = int [119889120578] 119890

minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

(72)


int [119889120578]120575

120575119869 (120591)119890minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

= 0 (73)

Γ1

(119884) = Γ1



(119884)120575119884 = 0 [3 25]

Γ1

(119884) =1

2int119889

10

1199091

times int11988910

1199092

1198660

Fc (1199091 1199092) 119881119865(119884) (1199091) 119881119865(119884) (1199092)

(74)

with 1198660


1198651

(1199091

1199092

) = 1198660


Λ119865

(1199091

119909119899

) = 119866Fc (1199091 119909119899) (75)


119888

119865

(1199091

119909119899



Γ119892

[119884 (120591)] = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(76)


119865(119884)


119865(119884)


Γ119892

[119884 (120591)] = 119878eff (119865)

+ int11988910

119909119865 (119909)119881119865(119884)

(119909) sim 119878eff (119865) + 119878119865

[119884 (120591)]

(77)

with

minus120575119878eff(119865)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119865(119884)

(119909) =120575119878

119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(78)



119878eff(119865) or not


119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)



119886119887

= 1198902120590

119892119886119887



120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904



119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650


119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840


0


(1198650

120590) = 119882119904



119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650


0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895


119865=

119865


119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904



119904


119904

(119865) = 119878eff(119865)

119882119904


0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)


119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)



(119865 120590) = 119882119904



4


5



4

=

119890119882

119904 where 119882119904


5


SYM4


is 120578120583] + 120575119892

120583](119909)



4


5




4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904


AdS5



4



119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





+

= 119891(1199091015840+

) 119909minus = 1205721199091015840minus

hArr 119877 = 1205721198771015840

1198921015840

++

= 11989110158402

(1199091015840+

) 119892++

1198921015840

+minus

= 1205721198911015840

(1199091015840+

) 119892+minus

1198921015840

+119868

= 1198911015840

(1199091015840+

) 119892+119868

1198921015840

minus119868

= 120572119892minus119868

1198921015840

minusminus

= 1205722

119892minusminus

1198621015840

+minus119869

= 1205721198911015840

(1199091015840+

) 119862+minus119869

1198621015840

+119868119869

= 1198911015840

(1199091015840+

) 119862+119868119869

1198621015840

minus119868119869

= 120572119862minus119868119869

(21)


1198891199042

= 119892++

119889119891 (1199091015840+

) 119889119891 (1199091015840+

) + 2119892119868+

119889119909119868

119889119891 (1199091015840+

)

+ 119892119868119869

119889119909119868

119889119909119869

+ 2120572119892+minus

119889119891 (1199091015840+

) 1198891199091015840minus

+ 2120572119892119868minus

119889119909119868

1198891199091015840minus

+ 1205722

119892minusminus

1198891199091015840minus

1198891199091015840minus

= 119892++

11989110158402

(1199091015840+

) 1198891199091015840+

1198891199091015840+

+ 21198911015840

(1199091015840+

) 119892119868+

119889119909119868

1199091015840+

+ 119892119868119869

119889119909119868

119889119909119869

+ 2120572119892+minus

1198911015840

(1199091015840+

) 1198891199091015840+

1198891199091015840minus

+ 2120572119892119868minus

119889119909119868

1198891199091015840minus

+ 1205722

119892minusminus

1198891199091015840minus

1198891199091015840minus

119862 = 119862119868119869119870

119889119909119868

and 119889119909119869

and 119889119909119870

+ 119862119868119869+

119889119909119868

and 119889119909119869

and 119889119891 (1199091015840+

)

+ 120572119862119868119869minus

119889119909119868

and 119889119909119869

and 1198891199091015840minus

+ 120572119862119868+minus

119889119909119868

and 119889119891 (1199091015840+

) and 1198891199091015840minus

= 119862119868119869119870

119889119909119868

and 119889119909119869

and 119889119909119870

+ 119862119868119869+

1198911015840

(1199091015840+

) 119889119909119868

and 119889119909119869

and 1198891199091015840+

+ 120572119862119868119869minus

119889119909119868

and 119889119909119869

and 119889119909minus

+ 120572119862119868+minus

1198911015840

(1199091015840+

) 119889119909119868

and 1198891199091015840+

and 119889119909minus

(22)

119878119877119865


119877119865


119881119865(119884)

(119909) =120575119878

119877119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(23)


+minus

= minus1 and 119892119868119869

= 120575119868119869

119881119865(119884)

(119909)


119877119865



119878119877119865

= 119878119877

minus 119877int119889119909+


119868119869(119883)

ℎ119868119869

(119883) + 119881ℎ

119868+(119883)

ℎ119868+

(119883)

+ 119881ℎ

++(119883)

ℎ++

(119883) + 119881ℎ

119868minus(119883)

ℎ119868minus

(119883)

+ 119881ℎ

+minus(119883)

ℎ+minus

(119883) + 119881ℎ

minusminus(119883)

ℎminusminus

(119883)

+ 119881119862

119868119869119870(119883)

119862119868119869119870

(119883) + 119881119862

+119869119870(119883)

119862+119869119870

(119883)

+ 119881119862

119868119869minus(119883)

119862119868119869minus

(119883) + 119881119862

+minus119870(119883)

119862+minus119870

(119883)

+ 119881120595(119883)

120595 (119883) + 119881120595

+(119883)

120595+

(119883)

+ 119881120595

minus(119883)

120595minus

(119883) ]

(24)

where for example

119881ℎ

119868+(119883)

ℎ119868+

(119883) = (119863+

119883119868

119877minus

1

4120579120574

119868119869

120579120597

120597119883119869

)ℎ119868+

(119883)

119877 (25)

ℎ119868119869

(119883) 120595minus


1199091

1199099




119868+


ℎ119868+

(119909+

1198831

1198839

)

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0)119883119896


119899

(26)

with (120597119896

1


119899

ℎ119868+

)(119909+


(119909+

)

at (119909+ 0) [3 23ndash25]

STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0)

times STr[ 1

119877(119863+

119883119868

119877minus

1

4120579120574

119868119869

120579120597

120597119883119869

)119883119896


119899]

=

infin

sum

119899=0

1

119899(120597119896

1


119899

ℎ119868+

) (119909+

0 0) 119868119896

1sdotsdotsdot119896

119899

ℎ

119868+

(119909+

)

(27)


+

0 0) the 119880(119873) SYM1


119883119894

997888rarr 119883119894

minus 119886119894

119891 (119909+

0 0) 997888rarr 119891(119909+

1198861

1198869

)

(28)


SYM1


1



1


1sdotsdotsdot119896


119896

1 Φ119896

1sdotsdotsdot119896


spacetime

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

) = 120575 (1199090

minus 119909+

) 1205759

(119909119868

minus 119883119868

)

= 12057510

(119909120583

minus 119883120583

)

(29)


set 1198830

= 119909+1

119873times119873


= diag1199091198681

119909119868

2

119909119868

119873

(29) becomes

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

)

= diag 12057510

(119909120583

minus 119909120583

1

) 12057510

(119909120583

minus 119909120583

2

) 12057510

(119909120583

minus 119909120583

119873

)

(30)


119868+

we have

119881ℎ

119868+(119909) =

120575

120575ℎ119868+

(119909)minus119877int119889119909

+STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

12057510

(119909 minus 119883)]

(31)


119881ℎ

119868+

(119901) = int11988910

119909119881ℎ

119868+(119909) 119890

119894119901119909

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

119890119894119901119883

]


119863+

119883119868

119877+

119894

4120579120574

119868119869

120579119901119869

] 119890119894119901119883

(32)


119865


119885 (119877 119865) = 119890119882(119877119865)

= int [119889119883119889120579 1198891198600

] 119890minus119878

119877119865(119883120579119860

0)

= int [119889119884] 119890minus119878

119877119865(119884)

(33)


119892++

119892119868+

119892119868119869

119892minusminus

119892119868minus

) (119862119868119869+

119862119868119869119870

119862119868119869minus

119862119868+minus

) and (120595+

120595119868

120595minus


120579 1198600



5

CFT4


119885SYM4

= exp minus119865 (34)


ℎ=0

119865119899

ℎ


5

times 1198785 119885SYM

4


4


5




+


Suppose 119878119872119875

+119865


119875

+119865


= 1198751015840+

120572119909+

= 119891(1199091015840+

) and 119883 = 119892(1198831015840

) there is 119878119872

119875

+119865

(119884) =

119878119872

119875

1015840+119865

1015840(1198841015840


transformation

119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (35)


1015840


int [119889119884] 119890minus119878

119872

119875+119865(119884)

= int [1198891198841015840

] 119890minus119878

119872

1198751015840+1198651015840 (1198841015840)

(36)


+

1205901

1205902

) and 1198841015840

(119909+

1205901

1205902


+

) and 1198841015840

(119909+

) and there is

119885 (119877 119865) = 119890119882(119877119865)

= int [119889119884] 119890minus119878

119877119865(119884)

= int [1198891198841015840

] 119890minus119878

11987710158401198651015840 (1198841015840)

= 119890119882(119877

1015840119865

1015840)

= 119885 (1198771015840

1198651015840

)

(37)

119882(119877 119865) = 119882(1198771015840

1198651015840



Let 1198650

119865minus

119865minusminus


indices Since

119878119877119865

(119884) = 119878119865

0119877119865

minus119877

2119865


119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus

) (39)


119865minus

119865minusminus



0

119865minus

119865minusminus


119909minus


119877 119865

0

997888rarr 1198650

119865minus

997888rarr 119877119865minus

119865minusminus

997888rarr 1198772

119865minusminus

(40)

so

119878eff (119877 1198650

119865minus

119865minusminus

) = 119878eff (1 1198650 119877119865minus

1198772

119865minusminus

) (41)


119865minusminus



=

minus1 119892119868119869

= 120575119868119869


119882(119877 119865) = int11988910

119909Γ119888

119877119865

(119909) 119865 (119909)

+1

2int 119889

10

1199091

times int11988910

1199092

Γ119888

119877119865

(1199091

1199092

) 119865 (1199091

) 119865 (1199092

) + sdot sdot sdot

= sum

119899

1

119899int 119889

10

1199091


119909119899

Γ119888

119877119865

(1199091

119909119899

)

times 119865 (1199091

) sdot sdot sdot 119865 (119909119899

)

(42)


119877119892

++119892

minusminus

(1199091

1199092

) = 1205752

119882(119877119865)120575119892++

(1199091


(1199092

)|119865=119865

)

Γ119888

119877119865

(1199091

119909119899

) =120575119899

119882(119877 119865)

120575119865(1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(43)


is

119878eff (119877 119865) = sum

119899

1

119899int

2120587119877

0

119889119909minus

1

times int119889119909+

1

1198899x1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

int119889119909+

119899

1198899x119899

times Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

times 119865 (119909+

1

x1

) sdot sdot sdot 119865 (119909+

119899

x119899

)

(44)

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899



Γ119888

119877119865

(1199091

119909119899

)

= Γ119888

119877119865

(119909+

1

x1

119909+

119899

x119899

)

= int

2120587119877

0

119889119909minus

1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

(45)


(119909)120575119865(119910) = 0where 119881

119865


Γ119888

119877119865

(1199091

119909119899

) = (minus1)119899

⟨119881119865

(1199091


(119909119899

)⟩119888

(46)

⟨119881119865

(1199091


(119909119899

)⟩119888


⟨119881119865

(1199091


(119909119899

)⟩

= int [119889119883119889120579 1198891198600

] 119890minus119878

0119881119865

(1199091


(119909119899

)

=(minus1)

119899

120575119899

119885 (119877 119865)

120575119865 (1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(47)


119877119865

(1199091

119909119899

)In terms of 119881

119865

(119901)

⟨119881119865

(1199091


(119909119899

)⟩119888

= int11988910

1199011


119901119899

119890minus119894(119901

1119909


119899119909

119899)

times ⟨119881119865

(1199011


(119901119899

)⟩119888

(48)

Note that 119881119865(119883)

(119909+

119909119868

minus 119886119868

) = 119881119865(119883+119886)

(119909+

119909119868

) 1198780

(119883 + 119886) =

1198780

(119883) so

⟨119881119865

(1199091


(119909119899

)⟩119888

= ⟨119881119865

(1199091

+ 1198861


(119909119899

+ 119886119899

)⟩119888

(49)


119865

(1199011


(119901119899

)⟩119888

= 0 if 1199011

+ sdot sdot sdot + 119901119899


⟨119881119865

(119909)⟩ = minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865

= V119865

(50)

V119865


119877

is 119878119874(9) invariant so V119865



V119892

119868+

= V119892

119868minus

= V119862

119868119869119870

= V119862

119868119869+

= V119862

119868119869minus

= V119862

119868+minus

= V120595

119868

= 0

(51)


V119892

119868119869


+

=

V120595

minus


ℎ

++

Vℎ

minusminus

and Vℎ

+minus

In particular Vℎ

++

= 119901+

1198719 is


⟨119881119865

(119909)⟩119865

= minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865+

119865

= V119865

(119909)

= sum

119899

minus1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Γ119888

119877119865

(119909 1199091

119909119899minus1

)

times 119865 (1199091


)

(52)


119865

(119909)⟩119865




119865


119865


119892

120575119892120583]

=



119865


119865




Γ (119884) = 119878119877119865

(119884) + 119882 (119877 119865) (53)

119890Γ(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

(54)


int [119889]120575

120575119865 (119909)119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

[minus

120575119878119877119865

()

120575119865 (119909)+

120575119878119877119865

(119884)

120575119865 (119909)] = 0

(55)

or equivalently

int [119889] 119890minus119878

119877119865(

119884)

120575119878119877119865

()

120575119865 (119909)=

120575119878119877119865

(119884)

120575119865 (119909)= minus

120575119882 (119877 119865)

120575119865 (119909) (56)


120575Γ (119884)

120575119884=

120575119878119877119865

(119884)

120575119884 (57)



(119884) only comes from119877119865

minus

and 1198772

119865minusminus


119865minus

119865minusminus


0

119865minus


1205722

)119878119877119865



+

1205901

1205902

) Let 1199091015840minus rarr 1198751015840+

1199091015840minus

119875+ 1198751015840+ rarr



= 1198650

119865minus

119875+

= 1198651015840

minus

1198751015840+ 119865

minusminus

119875+2

=

1198651015840

minusminus


119872

119875

+119865



119877119865


119877119865


0


119877119865



119878119865

(119884) = 1198780

(119884) + int 11988910

119909119865(119909)119881119865(119884)

(119909) Γ(119884) = 1198780

(119884) + Γ(V119865

)where

Γ (V119865

) = int11988910

119909119865 (119909) V119865

(119909) + 119882(119865 + 119865) (58)


minus120575119882 (119865)

120575119865 (119909)

100381610038161003816100381610038161003816100381610038161003816 119865

= V119865

= 119881119865(119884)

(119909)

120575Γ (V119865

)

120575V119865

(119909)

1003816100381610038161003816100381610038161003816100381610038161003816V119865

= 119865 (119909)

(59)


Let

Λ119865

(1199091

119909119899

) =120575119899

Γ (V119865

)

120575V119865

(1199091


(119909119899

)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

(60)

then

Γ (119884) = 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times V119865

(1199091


(119909119899

)

= 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(61)


(119909)

119865 (119909) = sum

119899

1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Λ119865

(119909 1199091

119909119899minus1

)

times V119865

(1199091


(119909119899minus1

)

(62)



119865

(119909) If Λ119865


119865


119892

120575119865(119909) = minusV119865

(119909) with119878119892


Γ119888

119865

(119909 minus 119911) = minus120575V

119865

(119909)

120575119865 (119911)

100381610038161003816100381610038161003816100381610038161003816 119865=0

Λ119865

(119911 minus 119910) =1205752

Γ (V119865

)

120575V119865

(119911) 120575V119865

(119910)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

=120575119865 (119911)

120575V119865

(119910)

100381610038161003816100381610038161003816100381610038161003816V119865=0

(63)

so

minusint11988910

119911Γ119888

119865

(119909 minus 119911) Λ119865

(119911 minus 119910) = 12057510

(119909 minus 119910) (64)

minusΛ119865

(1199091

minus 1199092


(1199091

minus 1199092

) Subsequently

Λ119865

(1199091

1199092

1199093

) = int11988910

1199101

int11988910

1199102

times int11988910

1199103

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Γ119888

119865

(1199101

1199102

1199103

)

Γ119888

119865

(1199091

1199092

1199093


1199101

int11988910

1199102

times int11988910

1199103

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Λ119865

(1199101

1199102

1199103

)

Λ119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Λ119865

(1199104

minus 1199094

)

times [Γ119888

119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Λ119865

(1199111

minus 1199112

)

times (Γ119888

119865

(1199111

1199101

1199104

)

times Γ119888

119865

(1199112

1199102

1199103

)

+ Γ119888

119865

(1199111

1199102

1199104

)

times Γ119888

119865

(1199112

1199101

1199103

)

+ Γ119888

119865

(1199111

1199103

1199104

)

times Γ119888

119865

(1199112

1199101

1199102

)) ]

Γ119888

119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Γ119888

119865

(1199104

minus 1199094

)

times [Λ119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Γ119888

119865

(1199111

minus 1199112

)

times (Λ119865

(1199111

1199101

1199104

)

times Λ119865

(1199112

1199102

1199103

)

+ Λ119865

(1199111

1199102

1199104

)

times Λ119865

(1199112

1199101

1199103

)

+ Λ119865

(1199111

1199103

1199104

)

times Λ119865

(1199112

1199101

1199102

)) ]

(65)



and Γ119888

119865

shows thatΛ119865

and Γ119888

119865


119865

(119911minus119910) = (120575119865(119911)120575V119865

(119910))|V119865=0


In (54) let = 119884 + 120578

119890Γ(119884)

= int [119889120578] 119890minus119878

119865(119884+120578)+119878

119865(119884)

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minusint119889120591120575119878

119865

(119884)

120575119884 (120591)120578 (120591)

minus1

2int119889120591 119889120591

1015840

1205752

119878119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)

times 120578 (120591) 120578 (1205911015840

) + sdot sdot sdot

= minus119878(1)

119865

(119884 120578) minus 119878(2)

119865

(119884 120578) + sdot sdot sdot

(66)

119878119865


120591


120591

also exists)

119878119865

(119884) = int119889120591 [1

2119884120597

2

120591

119884 + 1198810

(119884) + 119881 (119865 119884)]

= 1198780

(119884) + int119889120591119881 (119865 119884)

(67)


0

from 120575119878119865

0

(119884)120575119884 = 0 and expand 119878(119899)

119865

(119884 120578) around1198650

119865 = 1198650

+ 119891 Consider

119878(1)

119865

(119884 120578) =120575119878

(1)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


119878(2)

119865

(119884 120578) = 119878(2)

119865

0

(119884 120578) +120575119878

(2)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


(68)


(119884) +

Γ2


minus119878119865

(119884 + 120578) + 119878119865

(119884) = minus119878(2)

119865

0

(119884 120578) + sdot sdot sdot

Γ1

(119884) = ln int [119889120578] 119890minus119878

(2)

1198650(119884120578)

= ln (det119860119865

0

)12

(69)

with

119860119865

0

(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842+

1205972

119881 (1198650

119884)

1205971198842] 120575 (120591 minus 120591

1015840

)

(70)


(119884)120575119884 = 0 1198650

= 0 119860119865

0

reduces to

119860(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842] 120575 (120591 minus 120591

1015840

) (71)



1


119890Γeff(119884) = int [119889120578] 119890

minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

(72)


int [119889120578]120575

120575119869 (120591)119890minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

= 0 (73)

Γ1

(119884) = Γ1



(119884)120575119884 = 0 [3 25]

Γ1

(119884) =1

2int119889

10

1199091

times int11988910

1199092

1198660

Fc (1199091 1199092) 119881119865(119884) (1199091) 119881119865(119884) (1199092)

(74)

with 1198660


1198651

(1199091

1199092

) = 1198660


Λ119865

(1199091

119909119899

) = 119866Fc (1199091 119909119899) (75)


119888

119865

(1199091

119909119899



Γ119892

[119884 (120591)] = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(76)


119865(119884)


119865(119884)


Γ119892

[119884 (120591)] = 119878eff (119865)

+ int11988910

119909119865 (119909)119881119865(119884)

(119909) sim 119878eff (119865) + 119878119865

[119884 (120591)]

(77)

with

minus120575119878eff(119865)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119865(119884)

(119909) =120575119878

119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(78)



119878eff(119865) or not


119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)



119886119887

= 1198902120590

119892119886119887



120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904



119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650


119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840


0


(1198650

120590) = 119882119904



119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650


0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895


119865=

119865


119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904



119904


119904

(119865) = 119878eff(119865)

119882119904


0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)


119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)



(119865 120590) = 119882119904



4


5



4

=

119890119882

119904 where 119882119904


5


SYM4


is 120578120583] + 120575119892

120583](119909)



4


5




4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904


AdS5



4



119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




SYM1


1



1


1sdotsdotsdot119896


119896

1 Φ119896

1sdotsdotsdot119896


spacetime

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

) = 120575 (1199090

minus 119909+

) 1205759

(119909119868

minus 119883119868

)

= 12057510

(119909120583

minus 119883120583

)

(29)


set 1198830

= 119909+1

119873times119873


= diag1199091198681

119909119868

2

119909119868

119873

(29) becomes

120575

120575119865 (119909)119865 (119909

+

1198831

1198839

)

= diag 12057510

(119909120583

minus 119909120583

1

) 12057510

(119909120583

minus 119909120583

2

) 12057510

(119909120583

minus 119909120583

119873

)

(30)


119868+

we have

119881ℎ

119868+(119909) =

120575

120575ℎ119868+

(119909)minus119877int119889119909

+STr [119881ℎ

119868+(119883)

ℎ119868+

(119883)]

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

12057510

(119909 minus 119883)]

(31)


119881ℎ

119868+

(119901) = int11988910

119909119881ℎ

119868+(119909) 119890

119894119901119909

= minus119877int119889119909+STr [119881

ℎ

119868+(119883)

119890119894119901119883

]


119863+

119883119868

119877+

119894

4120579120574

119868119869

120579119901119869

] 119890119894119901119883

(32)


119865


119885 (119877 119865) = 119890119882(119877119865)

= int [119889119883119889120579 1198891198600

] 119890minus119878

119877119865(119883120579119860

0)

= int [119889119884] 119890minus119878

119877119865(119884)

(33)


119892++

119892119868+

119892119868119869

119892minusminus

119892119868minus

) (119862119868119869+

119862119868119869119870

119862119868119869minus

119862119868+minus

) and (120595+

120595119868

120595minus


120579 1198600



5

CFT4


119885SYM4

= exp minus119865 (34)


ℎ=0

119865119899

ℎ


5

times 1198785 119885SYM

4


4


5




+


Suppose 119878119872119875

+119865


119875

+119865


= 1198751015840+

120572119909+

= 119891(1199091015840+

) and 119883 = 119892(1198831015840

) there is 119878119872

119875

+119865

(119884) =

119878119872

119875

1015840+119865

1015840(1198841015840


transformation

119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (35)


1015840


int [119889119884] 119890minus119878

119872

119875+119865(119884)

= int [1198891198841015840

] 119890minus119878

119872

1198751015840+1198651015840 (1198841015840)

(36)


+

1205901

1205902

) and 1198841015840

(119909+

1205901

1205902


+

) and 1198841015840

(119909+

) and there is

119885 (119877 119865) = 119890119882(119877119865)

= int [119889119884] 119890minus119878

119877119865(119884)

= int [1198891198841015840

] 119890minus119878

11987710158401198651015840 (1198841015840)

= 119890119882(119877

1015840119865

1015840)

= 119885 (1198771015840

1198651015840

)

(37)

119882(119877 119865) = 119882(1198771015840

1198651015840



Let 1198650

119865minus

119865minusminus


indices Since

119878119877119865

(119884) = 119878119865

0119877119865

minus119877

2119865


119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus

) (39)


119865minus

119865minusminus



0

119865minus

119865minusminus


119909minus


119877 119865

0

997888rarr 1198650

119865minus

997888rarr 119877119865minus

119865minusminus

997888rarr 1198772

119865minusminus

(40)

so

119878eff (119877 1198650

119865minus

119865minusminus

) = 119878eff (1 1198650 119877119865minus

1198772

119865minusminus

) (41)


119865minusminus



=

minus1 119892119868119869

= 120575119868119869


119882(119877 119865) = int11988910

119909Γ119888

119877119865

(119909) 119865 (119909)

+1

2int 119889

10

1199091

times int11988910

1199092

Γ119888

119877119865

(1199091

1199092

) 119865 (1199091

) 119865 (1199092

) + sdot sdot sdot

= sum

119899

1

119899int 119889

10

1199091


119909119899

Γ119888

119877119865

(1199091

119909119899

)

times 119865 (1199091

) sdot sdot sdot 119865 (119909119899

)

(42)


119877119892

++119892

minusminus

(1199091

1199092

) = 1205752

119882(119877119865)120575119892++

(1199091


(1199092

)|119865=119865

)

Γ119888

119877119865

(1199091

119909119899

) =120575119899

119882(119877 119865)

120575119865(1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(43)


is

119878eff (119877 119865) = sum

119899

1

119899int

2120587119877

0

119889119909minus

1

times int119889119909+

1

1198899x1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

int119889119909+

119899

1198899x119899

times Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

times 119865 (119909+

1

x1

) sdot sdot sdot 119865 (119909+

119899

x119899

)

(44)

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899



Γ119888

119877119865

(1199091

119909119899

)

= Γ119888

119877119865

(119909+

1

x1

119909+

119899

x119899

)

= int

2120587119877

0

119889119909minus

1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

(45)


(119909)120575119865(119910) = 0where 119881

119865


Γ119888

119877119865

(1199091

119909119899

) = (minus1)119899

⟨119881119865

(1199091


(119909119899

)⟩119888

(46)

⟨119881119865

(1199091


(119909119899

)⟩119888


⟨119881119865

(1199091


(119909119899

)⟩

= int [119889119883119889120579 1198891198600

] 119890minus119878

0119881119865

(1199091


(119909119899

)

=(minus1)

119899

120575119899

119885 (119877 119865)

120575119865 (1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(47)


119877119865

(1199091

119909119899

)In terms of 119881

119865

(119901)

⟨119881119865

(1199091


(119909119899

)⟩119888

= int11988910

1199011


119901119899

119890minus119894(119901

1119909


119899119909

119899)

times ⟨119881119865

(1199011


(119901119899

)⟩119888

(48)

Note that 119881119865(119883)

(119909+

119909119868

minus 119886119868

) = 119881119865(119883+119886)

(119909+

119909119868

) 1198780

(119883 + 119886) =

1198780

(119883) so

⟨119881119865

(1199091


(119909119899

)⟩119888

= ⟨119881119865

(1199091

+ 1198861


(119909119899

+ 119886119899

)⟩119888

(49)


119865

(1199011


(119901119899

)⟩119888

= 0 if 1199011

+ sdot sdot sdot + 119901119899


⟨119881119865

(119909)⟩ = minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865

= V119865

(50)

V119865


119877

is 119878119874(9) invariant so V119865



V119892

119868+

= V119892

119868minus

= V119862

119868119869119870

= V119862

119868119869+

= V119862

119868119869minus

= V119862

119868+minus

= V120595

119868

= 0

(51)


V119892

119868119869


+

=

V120595

minus


ℎ

++

Vℎ

minusminus

and Vℎ

+minus

In particular Vℎ

++

= 119901+

1198719 is


⟨119881119865

(119909)⟩119865

= minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865+

119865

= V119865

(119909)

= sum

119899

minus1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Γ119888

119877119865

(119909 1199091

119909119899minus1

)

times 119865 (1199091


)

(52)


119865

(119909)⟩119865




119865


119865


119892

120575119892120583]

=



119865


119865




Γ (119884) = 119878119877119865

(119884) + 119882 (119877 119865) (53)

119890Γ(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

(54)


int [119889]120575

120575119865 (119909)119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

[minus

120575119878119877119865

()

120575119865 (119909)+

120575119878119877119865

(119884)

120575119865 (119909)] = 0

(55)

or equivalently

int [119889] 119890minus119878

119877119865(

119884)

120575119878119877119865

()

120575119865 (119909)=

120575119878119877119865

(119884)

120575119865 (119909)= minus

120575119882 (119877 119865)

120575119865 (119909) (56)


120575Γ (119884)

120575119884=

120575119878119877119865

(119884)

120575119884 (57)



(119884) only comes from119877119865

minus

and 1198772

119865minusminus


119865minus

119865minusminus


0

119865minus


1205722

)119878119877119865



+

1205901

1205902

) Let 1199091015840minus rarr 1198751015840+

1199091015840minus

119875+ 1198751015840+ rarr



= 1198650

119865minus

119875+

= 1198651015840

minus

1198751015840+ 119865

minusminus

119875+2

=

1198651015840

minusminus


119872

119875

+119865



119877119865


119877119865


0


119877119865



119878119865

(119884) = 1198780

(119884) + int 11988910

119909119865(119909)119881119865(119884)

(119909) Γ(119884) = 1198780

(119884) + Γ(V119865

)where

Γ (V119865

) = int11988910

119909119865 (119909) V119865

(119909) + 119882(119865 + 119865) (58)


minus120575119882 (119865)

120575119865 (119909)

100381610038161003816100381610038161003816100381610038161003816 119865

= V119865

= 119881119865(119884)

(119909)

120575Γ (V119865

)

120575V119865

(119909)

1003816100381610038161003816100381610038161003816100381610038161003816V119865

= 119865 (119909)

(59)


Let

Λ119865

(1199091

119909119899

) =120575119899

Γ (V119865

)

120575V119865

(1199091


(119909119899

)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

(60)

then

Γ (119884) = 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times V119865

(1199091


(119909119899

)

= 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(61)


(119909)

119865 (119909) = sum

119899

1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Λ119865

(119909 1199091

119909119899minus1

)

times V119865

(1199091


(119909119899minus1

)

(62)



119865

(119909) If Λ119865


119865


119892

120575119865(119909) = minusV119865

(119909) with119878119892


Γ119888

119865

(119909 minus 119911) = minus120575V

119865

(119909)

120575119865 (119911)

100381610038161003816100381610038161003816100381610038161003816 119865=0

Λ119865

(119911 minus 119910) =1205752

Γ (V119865

)

120575V119865

(119911) 120575V119865

(119910)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

=120575119865 (119911)

120575V119865

(119910)

100381610038161003816100381610038161003816100381610038161003816V119865=0

(63)

so

minusint11988910

119911Γ119888

119865

(119909 minus 119911) Λ119865

(119911 minus 119910) = 12057510

(119909 minus 119910) (64)

minusΛ119865

(1199091

minus 1199092


(1199091

minus 1199092

) Subsequently

Λ119865

(1199091

1199092

1199093

) = int11988910

1199101

int11988910

1199102

times int11988910

1199103

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Γ119888

119865

(1199101

1199102

1199103

)

Γ119888

119865

(1199091

1199092

1199093


1199101

int11988910

1199102

times int11988910

1199103

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Λ119865

(1199101

1199102

1199103

)

Λ119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Λ119865

(1199104

minus 1199094

)

times [Γ119888

119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Λ119865

(1199111

minus 1199112

)

times (Γ119888

119865

(1199111

1199101

1199104

)

times Γ119888

119865

(1199112

1199102

1199103

)

+ Γ119888

119865

(1199111

1199102

1199104

)

times Γ119888

119865

(1199112

1199101

1199103

)

+ Γ119888

119865

(1199111

1199103

1199104

)

times Γ119888

119865

(1199112

1199101

1199102

)) ]

Γ119888

119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Γ119888

119865

(1199104

minus 1199094

)

times [Λ119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Γ119888

119865

(1199111

minus 1199112

)

times (Λ119865

(1199111

1199101

1199104

)

times Λ119865

(1199112

1199102

1199103

)

+ Λ119865

(1199111

1199102

1199104

)

times Λ119865

(1199112

1199101

1199103

)

+ Λ119865

(1199111

1199103

1199104

)

times Λ119865

(1199112

1199101

1199102

)) ]

(65)



and Γ119888

119865

shows thatΛ119865

and Γ119888

119865


119865

(119911minus119910) = (120575119865(119911)120575V119865

(119910))|V119865=0


In (54) let = 119884 + 120578

119890Γ(119884)

= int [119889120578] 119890minus119878

119865(119884+120578)+119878

119865(119884)

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minusint119889120591120575119878

119865

(119884)

120575119884 (120591)120578 (120591)

minus1

2int119889120591 119889120591

1015840

1205752

119878119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)

times 120578 (120591) 120578 (1205911015840

) + sdot sdot sdot

= minus119878(1)

119865

(119884 120578) minus 119878(2)

119865

(119884 120578) + sdot sdot sdot

(66)

119878119865


120591


120591

also exists)

119878119865

(119884) = int119889120591 [1

2119884120597

2

120591

119884 + 1198810

(119884) + 119881 (119865 119884)]

= 1198780

(119884) + int119889120591119881 (119865 119884)

(67)


0

from 120575119878119865

0

(119884)120575119884 = 0 and expand 119878(119899)

119865

(119884 120578) around1198650

119865 = 1198650

+ 119891 Consider

119878(1)

119865

(119884 120578) =120575119878

(1)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


119878(2)

119865

(119884 120578) = 119878(2)

119865

0

(119884 120578) +120575119878

(2)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


(68)


(119884) +

Γ2


minus119878119865

(119884 + 120578) + 119878119865

(119884) = minus119878(2)

119865

0

(119884 120578) + sdot sdot sdot

Γ1

(119884) = ln int [119889120578] 119890minus119878

(2)

1198650(119884120578)

= ln (det119860119865

0

)12

(69)

with

119860119865

0

(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842+

1205972

119881 (1198650

119884)

1205971198842] 120575 (120591 minus 120591

1015840

)

(70)


(119884)120575119884 = 0 1198650

= 0 119860119865

0

reduces to

119860(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842] 120575 (120591 minus 120591

1015840

) (71)



1


119890Γeff(119884) = int [119889120578] 119890

minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

(72)


int [119889120578]120575

120575119869 (120591)119890minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

= 0 (73)

Γ1

(119884) = Γ1



(119884)120575119884 = 0 [3 25]

Γ1

(119884) =1

2int119889

10

1199091

times int11988910

1199092

1198660

Fc (1199091 1199092) 119881119865(119884) (1199091) 119881119865(119884) (1199092)

(74)

with 1198660


1198651

(1199091

1199092

) = 1198660


Λ119865

(1199091

119909119899

) = 119866Fc (1199091 119909119899) (75)


119888

119865

(1199091

119909119899



Γ119892

[119884 (120591)] = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(76)


119865(119884)


119865(119884)


Γ119892

[119884 (120591)] = 119878eff (119865)

+ int11988910

119909119865 (119909)119881119865(119884)

(119909) sim 119878eff (119865) + 119878119865

[119884 (120591)]

(77)

with

minus120575119878eff(119865)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119865(119884)

(119909) =120575119878

119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(78)



119878eff(119865) or not


119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)



119886119887

= 1198902120590

119892119886119887



120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904



119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650


119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840


0


(1198650

120590) = 119882119904



119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650


0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895


119865=

119865


119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904



119904


119904

(119865) = 119878eff(119865)

119882119904


0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)


119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)



(119865 120590) = 119882119904



4


5



4

=

119890119882

119904 where 119882119904


5


SYM4


is 120578120583] + 120575119892

120583](119909)



4


5




4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904


AdS5



4



119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Let 1198650

119865minus

119865minusminus


indices Since

119878119877119865

(119884) = 119878119865

0119877119865

minus119877

2119865


119882(119877 119865) = 119882(1198650

119877119865minus

1198772

119865minusminus

) (39)


119865minus

119865minusminus



0

119865minus

119865minusminus


119909minus


119877 119865

0

997888rarr 1198650

119865minus

997888rarr 119877119865minus

119865minusminus

997888rarr 1198772

119865minusminus

(40)

so

119878eff (119877 1198650

119865minus

119865minusminus

) = 119878eff (1 1198650 119877119865minus

1198772

119865minusminus

) (41)


119865minusminus



=

minus1 119892119868119869

= 120575119868119869


119882(119877 119865) = int11988910

119909Γ119888

119877119865

(119909) 119865 (119909)

+1

2int 119889

10

1199091

times int11988910

1199092

Γ119888

119877119865

(1199091

1199092

) 119865 (1199091

) 119865 (1199092

) + sdot sdot sdot

= sum

119899

1

119899int 119889

10

1199091


119909119899

Γ119888

119877119865

(1199091

119909119899

)

times 119865 (1199091

) sdot sdot sdot 119865 (119909119899

)

(42)


119877119892

++119892

minusminus

(1199091

1199092

) = 1205752

119882(119877119865)120575119892++

(1199091


(1199092

)|119865=119865

)

Γ119888

119877119865

(1199091

119909119899

) =120575119899

119882(119877 119865)

120575119865(1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(43)


is

119878eff (119877 119865) = sum

119899

1

119899int

2120587119877

0

119889119909minus

1

times int119889119909+

1

1198899x1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

int119889119909+

119899

1198899x119899

times Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

times 119865 (119909+

1

x1

) sdot sdot sdot 119865 (119909+

119899

x119899

)

(44)

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899



Γ119888

119877119865

(1199091

119909119899

)

= Γ119888

119877119865

(119909+

1

x1

119909+

119899

x119899

)

= int

2120587119877

0

119889119909minus

1

sdot sdot sdot int

2120587119877

0

119889119909minus

119899

Γ119888

119865

(119909minus

1

119909+

1

x1

119909minus

119899

119909+

119899

x119899

)

(45)


(119909)120575119865(119910) = 0where 119881

119865


Γ119888

119877119865

(1199091

119909119899

) = (minus1)119899

⟨119881119865

(1199091


(119909119899

)⟩119888

(46)

⟨119881119865

(1199091


(119909119899

)⟩119888


⟨119881119865

(1199091


(119909119899

)⟩

= int [119889119883119889120579 1198891198600

] 119890minus119878

0119881119865

(1199091


(119909119899

)

=(minus1)

119899

120575119899

119885 (119877 119865)

120575119865 (1199091

) sdot sdot sdot 120575119865 (119909119899

)

100381610038161003816100381610038161003816100381610038161003816119865=119865

(47)


119877119865

(1199091

119909119899

)In terms of 119881

119865

(119901)

⟨119881119865

(1199091


(119909119899

)⟩119888

= int11988910

1199011


119901119899

119890minus119894(119901

1119909


119899119909

119899)

times ⟨119881119865

(1199011


(119901119899

)⟩119888

(48)

Note that 119881119865(119883)

(119909+

119909119868

minus 119886119868

) = 119881119865(119883+119886)

(119909+

119909119868

) 1198780

(119883 + 119886) =

1198780

(119883) so

⟨119881119865

(1199091


(119909119899

)⟩119888

= ⟨119881119865

(1199091

+ 1198861


(119909119899

+ 119886119899

)⟩119888

(49)


119865

(1199011


(119901119899

)⟩119888

= 0 if 1199011

+ sdot sdot sdot + 119901119899


⟨119881119865

(119909)⟩ = minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865

= V119865

(50)

V119865


119877

is 119878119874(9) invariant so V119865



V119892

119868+

= V119892

119868minus

= V119862

119868119869119870

= V119862

119868119869+

= V119862

119868119869minus

= V119862

119868+minus

= V120595

119868

= 0

(51)


V119892

119868119869


+

=

V120595

minus


ℎ

++

Vℎ

minusminus

and Vℎ

+minus

In particular Vℎ

++

= 119901+

1198719 is


⟨119881119865

(119909)⟩119865

= minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865+

119865

= V119865

(119909)

= sum

119899

minus1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Γ119888

119877119865

(119909 1199091

119909119899minus1

)

times 119865 (1199091


)

(52)


119865

(119909)⟩119865




119865


119865


119892

120575119892120583]

=



119865


119865




Γ (119884) = 119878119877119865

(119884) + 119882 (119877 119865) (53)

119890Γ(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

(54)


int [119889]120575

120575119865 (119909)119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

[minus

120575119878119877119865

()

120575119865 (119909)+

120575119878119877119865

(119884)

120575119865 (119909)] = 0

(55)

or equivalently

int [119889] 119890minus119878

119877119865(

119884)

120575119878119877119865

()

120575119865 (119909)=

120575119878119877119865

(119884)

120575119865 (119909)= minus

120575119882 (119877 119865)

120575119865 (119909) (56)


120575Γ (119884)

120575119884=

120575119878119877119865

(119884)

120575119884 (57)



(119884) only comes from119877119865

minus

and 1198772

119865minusminus


119865minus

119865minusminus


0

119865minus


1205722

)119878119877119865



+

1205901

1205902

) Let 1199091015840minus rarr 1198751015840+

1199091015840minus

119875+ 1198751015840+ rarr



= 1198650

119865minus

119875+

= 1198651015840

minus

1198751015840+ 119865

minusminus

119875+2

=

1198651015840

minusminus


119872

119875

+119865



119877119865


119877119865


0


119877119865



119878119865

(119884) = 1198780

(119884) + int 11988910

119909119865(119909)119881119865(119884)

(119909) Γ(119884) = 1198780

(119884) + Γ(V119865

)where

Γ (V119865

) = int11988910

119909119865 (119909) V119865

(119909) + 119882(119865 + 119865) (58)


minus120575119882 (119865)

120575119865 (119909)

100381610038161003816100381610038161003816100381610038161003816 119865

= V119865

= 119881119865(119884)

(119909)

120575Γ (V119865

)

120575V119865

(119909)

1003816100381610038161003816100381610038161003816100381610038161003816V119865

= 119865 (119909)

(59)


Let

Λ119865

(1199091

119909119899

) =120575119899

Γ (V119865

)

120575V119865

(1199091


(119909119899

)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

(60)

then

Γ (119884) = 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times V119865

(1199091


(119909119899

)

= 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(61)


(119909)

119865 (119909) = sum

119899

1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Λ119865

(119909 1199091

119909119899minus1

)

times V119865

(1199091


(119909119899minus1

)

(62)



119865

(119909) If Λ119865


119865


119892

120575119865(119909) = minusV119865

(119909) with119878119892


Γ119888

119865

(119909 minus 119911) = minus120575V

119865

(119909)

120575119865 (119911)

100381610038161003816100381610038161003816100381610038161003816 119865=0

Λ119865

(119911 minus 119910) =1205752

Γ (V119865

)

120575V119865

(119911) 120575V119865

(119910)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

=120575119865 (119911)

120575V119865

(119910)

100381610038161003816100381610038161003816100381610038161003816V119865=0

(63)

so

minusint11988910

119911Γ119888

119865

(119909 minus 119911) Λ119865

(119911 minus 119910) = 12057510

(119909 minus 119910) (64)

minusΛ119865

(1199091

minus 1199092


(1199091

minus 1199092

) Subsequently

Λ119865

(1199091

1199092

1199093

) = int11988910

1199101

int11988910

1199102

times int11988910

1199103

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Γ119888

119865

(1199101

1199102

1199103

)

Γ119888

119865

(1199091

1199092

1199093


1199101

int11988910

1199102

times int11988910

1199103

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Λ119865

(1199101

1199102

1199103

)

Λ119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Λ119865

(1199104

minus 1199094

)

times [Γ119888

119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Λ119865

(1199111

minus 1199112

)

times (Γ119888

119865

(1199111

1199101

1199104

)

times Γ119888

119865

(1199112

1199102

1199103

)

+ Γ119888

119865

(1199111

1199102

1199104

)

times Γ119888

119865

(1199112

1199101

1199103

)

+ Γ119888

119865

(1199111

1199103

1199104

)

times Γ119888

119865

(1199112

1199101

1199102

)) ]

Γ119888

119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Γ119888

119865

(1199104

minus 1199094

)

times [Λ119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Γ119888

119865

(1199111

minus 1199112

)

times (Λ119865

(1199111

1199101

1199104

)

times Λ119865

(1199112

1199102

1199103

)

+ Λ119865

(1199111

1199102

1199104

)

times Λ119865

(1199112

1199101

1199103

)

+ Λ119865

(1199111

1199103

1199104

)

times Λ119865

(1199112

1199101

1199102

)) ]

(65)



and Γ119888

119865

shows thatΛ119865

and Γ119888

119865


119865

(119911minus119910) = (120575119865(119911)120575V119865

(119910))|V119865=0


In (54) let = 119884 + 120578

119890Γ(119884)

= int [119889120578] 119890minus119878

119865(119884+120578)+119878

119865(119884)

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minusint119889120591120575119878

119865

(119884)

120575119884 (120591)120578 (120591)

minus1

2int119889120591 119889120591

1015840

1205752

119878119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)

times 120578 (120591) 120578 (1205911015840

) + sdot sdot sdot

= minus119878(1)

119865

(119884 120578) minus 119878(2)

119865

(119884 120578) + sdot sdot sdot

(66)

119878119865


120591


120591

also exists)

119878119865

(119884) = int119889120591 [1

2119884120597

2

120591

119884 + 1198810

(119884) + 119881 (119865 119884)]

= 1198780

(119884) + int119889120591119881 (119865 119884)

(67)


0

from 120575119878119865

0

(119884)120575119884 = 0 and expand 119878(119899)

119865

(119884 120578) around1198650

119865 = 1198650

+ 119891 Consider

119878(1)

119865

(119884 120578) =120575119878

(1)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


119878(2)

119865

(119884 120578) = 119878(2)

119865

0

(119884 120578) +120575119878

(2)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


(68)


(119884) +

Γ2


minus119878119865

(119884 + 120578) + 119878119865

(119884) = minus119878(2)

119865

0

(119884 120578) + sdot sdot sdot

Γ1

(119884) = ln int [119889120578] 119890minus119878

(2)

1198650(119884120578)

= ln (det119860119865

0

)12

(69)

with

119860119865

0

(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842+

1205972

119881 (1198650

119884)

1205971198842] 120575 (120591 minus 120591

1015840

)

(70)


(119884)120575119884 = 0 1198650

= 0 119860119865

0

reduces to

119860(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842] 120575 (120591 minus 120591

1015840

) (71)



1


119890Γeff(119884) = int [119889120578] 119890

minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

(72)


int [119889120578]120575

120575119869 (120591)119890minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

= 0 (73)

Γ1

(119884) = Γ1



(119884)120575119884 = 0 [3 25]

Γ1

(119884) =1

2int119889

10

1199091

times int11988910

1199092

1198660

Fc (1199091 1199092) 119881119865(119884) (1199091) 119881119865(119884) (1199092)

(74)

with 1198660


1198651

(1199091

1199092

) = 1198660


Λ119865

(1199091

119909119899

) = 119866Fc (1199091 119909119899) (75)


119888

119865

(1199091

119909119899



Γ119892

[119884 (120591)] = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(76)


119865(119884)


119865(119884)


Γ119892

[119884 (120591)] = 119878eff (119865)

+ int11988910

119909119865 (119909)119881119865(119884)

(119909) sim 119878eff (119865) + 119878119865

[119884 (120591)]

(77)

with

minus120575119878eff(119865)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119865(119884)

(119909) =120575119878

119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(78)



119878eff(119865) or not


119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)



119886119887

= 1198902120590

119892119886119887



120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904



119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650


119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840


0


(1198650

120590) = 119882119904



119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650


0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895


119865=

119865


119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904



119904


119904

(119865) = 119878eff(119865)

119882119904


0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)


119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)



(119865 120590) = 119882119904



4


5



4

=

119890119882

119904 where 119882119904


5


SYM4


is 120578120583] + 120575119892

120583](119909)



4


5




4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904


AdS5



4



119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




V119892

119868119869


+

=

V120595

minus


ℎ

++

Vℎ

minusminus

and Vℎ

+minus

In particular Vℎ

++

= 119901+

1198719 is


⟨119881119865

(119909)⟩119865

= minus120575119882 (119877 119865)

120575119865 (119909)

10038161003816100381610038161003816100381610038161003816119865=119865+

119865

= V119865

(119909)

= sum

119899

minus1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Γ119888

119877119865

(119909 1199091

119909119899minus1

)

times 119865 (1199091


)

(52)


119865

(119909)⟩119865




119865


119865


119892

120575119892120583]

=



119865


119865




Γ (119884) = 119878119877119865

(119884) + 119882 (119877 119865) (53)

119890Γ(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

(54)


int [119889]120575

120575119865 (119909)119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

[minus

120575119878119877119865

()

120575119865 (119909)+

120575119878119877119865

(119884)

120575119865 (119909)] = 0

(55)

or equivalently

int [119889] 119890minus119878

119877119865(

119884)

120575119878119877119865

()

120575119865 (119909)=

120575119878119877119865

(119884)

120575119865 (119909)= minus

120575119882 (119877 119865)

120575119865 (119909) (56)


120575Γ (119884)

120575119884=

120575119878119877119865

(119884)

120575119884 (57)



(119884) only comes from119877119865

minus

and 1198772

119865minusminus


119865minus

119865minusminus


0

119865minus


1205722

)119878119877119865



+

1205901

1205902

) Let 1199091015840minus rarr 1198751015840+

1199091015840minus

119875+ 1198751015840+ rarr



= 1198650

119865minus

119875+

= 1198651015840

minus

1198751015840+ 119865

minusminus

119875+2

=

1198651015840

minusminus


119872

119875

+119865



119877119865


119877119865


0


119877119865



119878119865

(119884) = 1198780

(119884) + int 11988910

119909119865(119909)119881119865(119884)

(119909) Γ(119884) = 1198780

(119884) + Γ(V119865

)where

Γ (V119865

) = int11988910

119909119865 (119909) V119865

(119909) + 119882(119865 + 119865) (58)


minus120575119882 (119865)

120575119865 (119909)

100381610038161003816100381610038161003816100381610038161003816 119865

= V119865

= 119881119865(119884)

(119909)

120575Γ (V119865

)

120575V119865

(119909)

1003816100381610038161003816100381610038161003816100381610038161003816V119865

= 119865 (119909)

(59)


Let

Λ119865

(1199091

119909119899

) =120575119899

Γ (V119865

)

120575V119865

(1199091


(119909119899

)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

(60)

then

Γ (119884) = 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times V119865

(1199091


(119909119899

)

= 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(61)


(119909)

119865 (119909) = sum

119899

1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Λ119865

(119909 1199091

119909119899minus1

)

times V119865

(1199091


(119909119899minus1

)

(62)



119865

(119909) If Λ119865


119865


119892

120575119865(119909) = minusV119865

(119909) with119878119892


Γ119888

119865

(119909 minus 119911) = minus120575V

119865

(119909)

120575119865 (119911)

100381610038161003816100381610038161003816100381610038161003816 119865=0

Λ119865

(119911 minus 119910) =1205752

Γ (V119865

)

120575V119865

(119911) 120575V119865

(119910)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

=120575119865 (119911)

120575V119865

(119910)

100381610038161003816100381610038161003816100381610038161003816V119865=0

(63)

so

minusint11988910

119911Γ119888

119865

(119909 minus 119911) Λ119865

(119911 minus 119910) = 12057510

(119909 minus 119910) (64)

minusΛ119865

(1199091

minus 1199092


(1199091

minus 1199092

) Subsequently

Λ119865

(1199091

1199092

1199093

) = int11988910

1199101

int11988910

1199102

times int11988910

1199103

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Γ119888

119865

(1199101

1199102

1199103

)

Γ119888

119865

(1199091

1199092

1199093


1199101

int11988910

1199102

times int11988910

1199103

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Λ119865

(1199101

1199102

1199103

)

Λ119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Λ119865

(1199104

minus 1199094

)

times [Γ119888

119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Λ119865

(1199111

minus 1199112

)

times (Γ119888

119865

(1199111

1199101

1199104

)

times Γ119888

119865

(1199112

1199102

1199103

)

+ Γ119888

119865

(1199111

1199102

1199104

)

times Γ119888

119865

(1199112

1199101

1199103

)

+ Γ119888

119865

(1199111

1199103

1199104

)

times Γ119888

119865

(1199112

1199101

1199102

)) ]

Γ119888

119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Γ119888

119865

(1199104

minus 1199094

)

times [Λ119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Γ119888

119865

(1199111

minus 1199112

)

times (Λ119865

(1199111

1199101

1199104

)

times Λ119865

(1199112

1199102

1199103

)

+ Λ119865

(1199111

1199102

1199104

)

times Λ119865

(1199112

1199101

1199103

)

+ Λ119865

(1199111

1199103

1199104

)

times Λ119865

(1199112

1199101

1199102

)) ]

(65)



and Γ119888

119865

shows thatΛ119865

and Γ119888

119865


119865

(119911minus119910) = (120575119865(119911)120575V119865

(119910))|V119865=0


In (54) let = 119884 + 120578

119890Γ(119884)

= int [119889120578] 119890minus119878

119865(119884+120578)+119878

119865(119884)

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minusint119889120591120575119878

119865

(119884)

120575119884 (120591)120578 (120591)

minus1

2int119889120591 119889120591

1015840

1205752

119878119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)

times 120578 (120591) 120578 (1205911015840

) + sdot sdot sdot

= minus119878(1)

119865

(119884 120578) minus 119878(2)

119865

(119884 120578) + sdot sdot sdot

(66)

119878119865


120591


120591

also exists)

119878119865

(119884) = int119889120591 [1

2119884120597

2

120591

119884 + 1198810

(119884) + 119881 (119865 119884)]

= 1198780

(119884) + int119889120591119881 (119865 119884)

(67)


0

from 120575119878119865

0

(119884)120575119884 = 0 and expand 119878(119899)

119865

(119884 120578) around1198650

119865 = 1198650

+ 119891 Consider

119878(1)

119865

(119884 120578) =120575119878

(1)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


119878(2)

119865

(119884 120578) = 119878(2)

119865

0

(119884 120578) +120575119878

(2)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


(68)


(119884) +

Γ2


minus119878119865

(119884 + 120578) + 119878119865

(119884) = minus119878(2)

119865

0

(119884 120578) + sdot sdot sdot

Γ1

(119884) = ln int [119889120578] 119890minus119878

(2)

1198650(119884120578)

= ln (det119860119865

0

)12

(69)

with

119860119865

0

(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842+

1205972

119881 (1198650

119884)

1205971198842] 120575 (120591 minus 120591

1015840

)

(70)


(119884)120575119884 = 0 1198650

= 0 119860119865

0

reduces to

119860(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842] 120575 (120591 minus 120591

1015840

) (71)



1


119890Γeff(119884) = int [119889120578] 119890

minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

(72)


int [119889120578]120575

120575119869 (120591)119890minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

= 0 (73)

Γ1

(119884) = Γ1



(119884)120575119884 = 0 [3 25]

Γ1

(119884) =1

2int119889

10

1199091

times int11988910

1199092

1198660

Fc (1199091 1199092) 119881119865(119884) (1199091) 119881119865(119884) (1199092)

(74)

with 1198660


1198651

(1199091

1199092

) = 1198660


Λ119865

(1199091

119909119899

) = 119866Fc (1199091 119909119899) (75)


119888

119865

(1199091

119909119899



Γ119892

[119884 (120591)] = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(76)


119865(119884)


119865(119884)


Γ119892

[119884 (120591)] = 119878eff (119865)

+ int11988910

119909119865 (119909)119881119865(119884)

(119909) sim 119878eff (119865) + 119878119865

[119884 (120591)]

(77)

with

minus120575119878eff(119865)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119865(119884)

(119909) =120575119878

119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(78)



119878eff(119865) or not


119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)



119886119887

= 1198902120590

119892119886119887



120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904



119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650


119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840


0


(1198650

120590) = 119882119904



119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650


0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895


119865=

119865


119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904



119904


119904

(119865) = 119878eff(119865)

119882119904


0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)


119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)



(119865 120590) = 119882119904



4


5



4

=

119890119882

119904 where 119882119904


5


SYM4


is 120578120583] + 120575119892

120583](119909)



4


5




4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904


AdS5



4



119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Let

Λ119865

(1199091

119909119899

) =120575119899

Γ (V119865

)

120575V119865

(1199091


(119909119899

)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

(60)

then

Γ (119884) = 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times V119865

(1199091


(119909119899

)

= 1198780

(119884) + sum

119899

1

119899

times int11988910

1199091


119909119899

Λ119865

(1199091

119909119899

)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(61)


(119909)

119865 (119909) = sum

119899

1

(119899 minus 1)

times int11988910

1199091


119909119899minus1

Λ119865

(119909 1199091

119909119899minus1

)

times V119865

(1199091


(119909119899minus1

)

(62)



119865

(119909) If Λ119865


119865


119892

120575119865(119909) = minusV119865

(119909) with119878119892


Γ119888

119865

(119909 minus 119911) = minus120575V

119865

(119909)

120575119865 (119911)

100381610038161003816100381610038161003816100381610038161003816 119865=0

Λ119865

(119911 minus 119910) =1205752

Γ (V119865

)

120575V119865

(119911) 120575V119865

(119910)

1003816100381610038161003816100381610038161003816100381610038161003816V119865=0

=120575119865 (119911)

120575V119865

(119910)

100381610038161003816100381610038161003816100381610038161003816V119865=0

(63)

so

minusint11988910

119911Γ119888

119865

(119909 minus 119911) Λ119865

(119911 minus 119910) = 12057510

(119909 minus 119910) (64)

minusΛ119865

(1199091

minus 1199092


(1199091

minus 1199092

) Subsequently

Λ119865

(1199091

1199092

1199093

) = int11988910

1199101

int11988910

1199102

times int11988910

1199103

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Γ119888

119865

(1199101

1199102

1199103

)

Γ119888

119865

(1199091

1199092

1199093


1199101

int11988910

1199102

times int11988910

1199103

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Λ119865

(1199101

1199102

1199103

)

Λ119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Λ119865

(1199101

minus 1199091

) Λ119865

(1199102

minus 1199092

)

times Λ119865

(1199103

minus 1199093

) Λ119865

(1199104

minus 1199094

)

times [Γ119888

119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Λ119865

(1199111

minus 1199112

)

times (Γ119888

119865

(1199111

1199101

1199104

)

times Γ119888

119865

(1199112

1199102

1199103

)

+ Γ119888

119865

(1199111

1199102

1199104

)

times Γ119888

119865

(1199112

1199101

1199103

)

+ Γ119888

119865

(1199111

1199103

1199104

)

times Γ119888

119865

(1199112

1199101

1199102

)) ]

Γ119888

119865

(1199091

1199092

1199093

1199094

) = int11988910

1199101

int11988910

1199102

int11988910

1199103

times int11988910

1199104

Γ119888

119865

(1199101

minus 1199091

) Γ119888

119865

(1199102

minus 1199092

)

times Γ119888

119865

(1199103

minus 1199093

) Γ119888

119865

(1199104

minus 1199094

)

times [Λ119865

(1199101

1199102

1199103

1199104

)

+ int11988910

1199111

times int11988910

1199112

Γ119888

119865

(1199111

minus 1199112

)

times (Λ119865

(1199111

1199101

1199104

)

times Λ119865

(1199112

1199102

1199103

)

+ Λ119865

(1199111

1199102

1199104

)

times Λ119865

(1199112

1199101

1199103

)

+ Λ119865

(1199111

1199103

1199104

)

times Λ119865

(1199112

1199101

1199102

)) ]

(65)



and Γ119888

119865

shows thatΛ119865

and Γ119888

119865


119865

(119911minus119910) = (120575119865(119911)120575V119865

(119910))|V119865=0


In (54) let = 119884 + 120578

119890Γ(119884)

= int [119889120578] 119890minus119878

119865(119884+120578)+119878

119865(119884)

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minusint119889120591120575119878

119865

(119884)

120575119884 (120591)120578 (120591)

minus1

2int119889120591 119889120591

1015840

1205752

119878119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)

times 120578 (120591) 120578 (1205911015840

) + sdot sdot sdot

= minus119878(1)

119865

(119884 120578) minus 119878(2)

119865

(119884 120578) + sdot sdot sdot

(66)

119878119865


120591


120591

also exists)

119878119865

(119884) = int119889120591 [1

2119884120597

2

120591

119884 + 1198810

(119884) + 119881 (119865 119884)]

= 1198780

(119884) + int119889120591119881 (119865 119884)

(67)


0

from 120575119878119865

0

(119884)120575119884 = 0 and expand 119878(119899)

119865

(119884 120578) around1198650

119865 = 1198650

+ 119891 Consider

119878(1)

119865

(119884 120578) =120575119878

(1)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


119878(2)

119865

(119884 120578) = 119878(2)

119865

0

(119884 120578) +120575119878

(2)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


(68)


(119884) +

Γ2


minus119878119865

(119884 + 120578) + 119878119865

(119884) = minus119878(2)

119865

0

(119884 120578) + sdot sdot sdot

Γ1

(119884) = ln int [119889120578] 119890minus119878

(2)

1198650(119884120578)

= ln (det119860119865

0

)12

(69)

with

119860119865

0

(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842+

1205972

119881 (1198650

119884)

1205971198842] 120575 (120591 minus 120591

1015840

)

(70)


(119884)120575119884 = 0 1198650

= 0 119860119865

0

reduces to

119860(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842] 120575 (120591 minus 120591

1015840

) (71)



1


119890Γeff(119884) = int [119889120578] 119890

minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

(72)


int [119889120578]120575

120575119869 (120591)119890minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

= 0 (73)

Γ1

(119884) = Γ1



(119884)120575119884 = 0 [3 25]

Γ1

(119884) =1

2int119889

10

1199091

times int11988910

1199092

1198660

Fc (1199091 1199092) 119881119865(119884) (1199091) 119881119865(119884) (1199092)

(74)

with 1198660


1198651

(1199091

1199092

) = 1198660


Λ119865

(1199091

119909119899

) = 119866Fc (1199091 119909119899) (75)


119888

119865

(1199091

119909119899



Γ119892

[119884 (120591)] = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(76)


119865(119884)


119865(119884)


Γ119892

[119884 (120591)] = 119878eff (119865)

+ int11988910

119909119865 (119909)119881119865(119884)

(119909) sim 119878eff (119865) + 119878119865

[119884 (120591)]

(77)

with

minus120575119878eff(119865)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119865(119884)

(119909) =120575119878

119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(78)



119878eff(119865) or not


119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)



119886119887

= 1198902120590

119892119886119887



120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904



119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650


119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840


0


(1198650

120590) = 119882119904



119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650


0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895


119865=

119865


119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904



119904


119904

(119865) = 119878eff(119865)

119882119904


0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)


119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)



(119865 120590) = 119882119904



4


5



4

=

119890119882

119904 where 119882119904


5


SYM4


is 120578120583] + 120575119892

120583](119909)



4


5




4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904


AdS5



4



119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





and Γ119888

119865

shows thatΛ119865

and Γ119888

119865


119865

(119911minus119910) = (120575119865(119911)120575V119865

(119910))|V119865=0


In (54) let = 119884 + 120578

119890Γ(119884)

= int [119889120578] 119890minus119878

119865(119884+120578)+119878

119865(119884)

minus119878119865

(119884 + 120578) + 119878119865

(119884) = minusint119889120591120575119878

119865

(119884)

120575119884 (120591)120578 (120591)

minus1

2int119889120591 119889120591

1015840

1205752

119878119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)

times 120578 (120591) 120578 (1205911015840

) + sdot sdot sdot

= minus119878(1)

119865

(119884 120578) minus 119878(2)

119865

(119884 120578) + sdot sdot sdot

(66)

119878119865


120591


120591

also exists)

119878119865

(119884) = int119889120591 [1

2119884120597

2

120591

119884 + 1198810

(119884) + 119881 (119865 119884)]

= 1198780

(119884) + int119889120591119881 (119865 119884)

(67)


0

from 120575119878119865

0

(119884)120575119884 = 0 and expand 119878(119899)

119865

(119884 120578) around1198650

119865 = 1198650

+ 119891 Consider

119878(1)

119865

(119884 120578) =120575119878

(1)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


119878(2)

119865

(119884 120578) = 119878(2)

119865

0

(119884 120578) +120575119878

(2)

119865

(119884 120578)

120575119865

10038161003816100381610038161003816100381610038161003816100381610038161198650


(68)


(119884) +

Γ2


minus119878119865

(119884 + 120578) + 119878119865

(119884) = minus119878(2)

119865

0

(119884 120578) + sdot sdot sdot

Γ1

(119884) = ln int [119889120578] 119890minus119878

(2)

1198650(119884120578)

= ln (det119860119865

0

)12

(69)

with

119860119865

0

(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842+

1205972

119881 (1198650

119884)

1205971198842] 120575 (120591 minus 120591

1015840

)

(70)


(119884)120575119884 = 0 1198650

= 0 119860119865

0

reduces to

119860(120591 1205911015840

) = [1205972

120591

+1205972

1198810

(119884)

1205971198842] 120575 (120591 minus 120591

1015840

) (71)



1


119890Γeff(119884) = int [119889120578] 119890

minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

(72)


int [119889120578]120575

120575119869 (120591)119890minus119878

0(119884+120578)+int 119889120591Tr[119869(120591)120578(120591)]

= 0 (73)

Γ1

(119884) = Γ1



(119884)120575119884 = 0 [3 25]

Γ1

(119884) =1

2int119889

10

1199091

times int11988910

1199092

1198660

Fc (1199091 1199092) 119881119865(119884) (1199091) 119881119865(119884) (1199092)

(74)

with 1198660


1198651

(1199091

1199092

) = 1198660


Λ119865

(1199091

119909119899

) = 119866Fc (1199091 119909119899) (75)


119888

119865

(1199091

119909119899



Γ119892

[119884 (120591)] = sum

119899

1

119899int 119889

10

1199091


119909119899

119866Fc (1199091 119909119899)

times 119881119865(119884)

(1199091

) sdot sdot sdot 119881119865(119884)

(119909119899

)

(76)


119865(119884)


119865(119884)


Γ119892

[119884 (120591)] = 119878eff (119865)

+ int11988910

119909119865 (119909)119881119865(119884)

(119909) sim 119878eff (119865) + 119878119865

[119884 (120591)]

(77)

with

minus120575119878eff(119865)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

= 119881119865(119884)

(119909) =120575119878

119865

(119884)

120575119865(119909)

10038161003816100381610038161003816100381610038161003816119865

(78)



119878eff(119865) or not


119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)



119886119887

= 1198902120590

119892119886119887



120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904



119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650


119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840


0


(1198650

120590) = 119882119904



119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650


0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895


119865=

119865


119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904



119904


119904

(119865) = 119878eff(119865)

119882119904


0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)


119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)



(119865 120590) = 119882119904



4


5



4

=

119890119882

119904 where 119882119904


5


SYM4


is 120578120583] + 120575119892

120583](119909)



4


5




4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904


AdS5



4



119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119878eff(119865) or not


119878eff [119865 (119865)] =

infin

sum

119898=0

119890120601(2minus2119898)

int119872

2minus2119898

[119889119883] 119890minus119878

119865 = 119882119904

(119865) (79)



119886119887

= 1198902120590

119892119886119887



120575119882119904

(119865 120590)

120575120590= minus

120575119882119904

(119865 120590)

120575119865120573 (119865) (80)

When 120573(119865) = 0 119882119904

(119865 120590) = 119882119904



119878eff (119865) = minus1

2int119889

119863

119909119865Δ119865 + 119878int (119865) (81)

Consider1198650

withΔ1198650

= 0 Since120573(1198650


119904

(1198650

120590) = 119882119904

[119865(1205901015840

) 1205901015840


0


(1198650

120590) = 119882119904



119904

(119865infin) = 119882119904

(119865 120590) = 119882119904

(119865) = 119882119904

(1198650

120590) =

119882119904

(1198650


0

via

119865 = 1198650

+ Δminus1

120575119878int (119865)

120575119865

= 1198650

(119909) + int119889119863

119910119863 (119909 minus 119910 119865)

120575119878int (119865)

120575119865 (119910)

(82)

120575119878eff(119865)120575119865(119909)|119865=

119865

= 0 Since 120575119878eff(119865)120575119865119894

= 120581119894119895

120573(119865119895


119865=

119865


119904

(1198650

120590) = 119878eff(119865) which means119882119904

(119865 120590) = 119882119904



119904


119904

(119865) = 119878eff(119865)

119882119904


0

In fact

119882119904

(1198650

120590) = 119882119904

(1198650

) = 119878eff (119865)

= 119878eff [1198650 + Δminus1

120575119878int (1198650)

1205751198650

+ sdot sdot sdot ]

(83)


119901119899

119882119904

(1198650

120590) = sum

119899

1

119899int 119889

9

1199011


119901119899

119878119888

119865

(1199011

119901119899

)

times 1198650

(1199011


(119901119899

)

1198650

(119909) = int1198899

1199011198650

(119901)

119878119888

119865

(1199011

119901119899

)

=1

1198730

infin

sum

119898=0

119890120601(2minus2119898)

times int119872

2minus2119898

[119889119883] 119890minus119878

0119881119865

(1199011


(119901119899

)

(84)



(119865 120590) = 119882119904



4


5



4

=

119890119882

119904 where 119882119904


5


SYM4


is 120578120583] + 120575119892

120583](119909)



4


5




4

[120575119892120583](119909)] = 119890

119882

119904[120575119866

120583](119909120590)] where119882119904


AdS5



4



119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119885SYM4

[120575119892120583](119909)] = 119890


119904

[120575119866120583](119909 120590)] if 119885SYM

4

= 119890119882

119904


119904


119888


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878

119888(119865+

119865)

(85)


119888


119890119878eff(119865) = 119885

119904

(119865) = 119890119882

119904(119865)

(86)

where 119882119904


119888





119890119878eff(119865) = 119885

119872

(119865) = int [119889119884] 119890minus119878

119872

119865(119884)

(87)

where 119878119872

119865


119878eff(119865)


(119884)




+




119865


119892

++

V119892

minusminus

and V119892

+minus



and 119892+minus


119882(119877 119865)

= minusint11988910

119909 [V119892

++

119892++

(119909) + V119892

minusminus

119892minusminus

(119909) + V119892

+minus

119892+minus

(119909)]

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

++119892

minusminus

(1199091

1199092

) 119892++

(1199091

) 119892minusminus

(1199092

)

+1

2int119889

10

1199091

times int11988910

1199092

Γ119888

119877119892

+minus119892

+minus

(1199091

1199092

) 119892+minus

(1199091

) 119892+minus

(1199092

) + sdot sdot sdot

(88)


(119909) from120575119882(119877 119865)120575119892

120572120573



(119909) = 119892++


119892

120572120573



(119909)


119892

++


119877119865

(119884)120575119892++

(119909)|119884=0

= (119873119877)12057510


minusminus



119890Ω(119877119884119865)

= int [119889] 119890minus119878

119877119865(

119884)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884+120578)+119878

119877119865(119884)

= int [119889120578] 119890minus119878

119877119865(119884120578)

(89)

Ω(119877 119884 119865) = 119882(119877 119865) + 119878119877119865


119877119865


119877119865


119877119865

(119884)120575119884 = 0

119878119877119865

(119884 120578) = int119889120591120575119878

119877119865

(119884)

120575119884 (120591)120578 (120591)

+1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

=1

2int119889120591 119889120591

1015840

1205752

119878119877119865

(119884)

120575119884 (120591) 120575119884 (1205911015840)120578 (120591) 120578 (120591

1015840

) + sdot sdot sdot

(90)

119878119877119865


119877119865




0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





0 = int [119889120578] 119890minus119878

119877119865(119884120578)

120575119878119877119865

(119884 120578)

120575119865 (119909)= ⟨

120575119878119877119865

(119884 120578)

120575119865 (119909)⟩ (91)



119884 = diag[

[

1198721

1198721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119872119898

119872119898

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119898

]

]

(92)




120578 997888rarr 119906120578119906+

+ (119906119884119906+

minus 119884) 120578 997888rarr 120578 + 119894 [120572 120578] + 119894 [120572 119884]

(93)

so for 119906 notin 119880(1198731




1198891199042

= minus2119889119909+

119889119909minus

+

9

sum

119894=1

119889119909119894

119889119909119894

minus [

3

sum

119886=1

1205832

9(119909

119886

)2

+

9

sum

119886

1015840=4

1205832

36(119909

119886

1015840

)

2

]119889119909+

119889119909+

119865123+

= 120583

(94)


4


7



+


119878PW119877

= 119877int119889119909+

times Tr 1

41198772(119863

+

119883119894

)2

minus119894

119877120579119863

+

120579 +1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6119877)

2

(119883119886

)2

minus (120583

12119877)

2

(119883119886

1015840

)

2

minus120583

6119877119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8119877120579120574123

120579

(95)



[119883119894

119883119895

] = 119894120583

3119877120598119894119895119896

119883119896

119894 119895 119896 = 1 2 3

119883119894

= 0 119894 = 4 9

119883

119894

= 0 119894 = 1 9

(96)

1198831

1198832

1198833


119894

= 119871119894 119894 = 1 2 3 119871


119871119894

= diag[[

[

119871119894

119895

1


119895

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

119871119894

119895

2


119895

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2


119895

119905


119895

119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]]

]

(97)

where 119871119894

119895

119904



(21198951

+ 1)1198731

+ (21198952

+ 1)1198732

+ sdot sdot sdot + (2119895119905

+ 1)119873119905

= 119873 (98)



119904

(119895119904

+ 1) [2] The 119880(119873)


) times119880(1198732



119904


119894=1

(119909119894

119904

)2

= 1199032

119904

with 119903119904

prop (2119895119904


minus




119904


119904


5 given by sum9

119894=4

(119909119894

119903

)2

= 1199032

119903


119894





4




5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





5





5



5








119878PW119877

( 120578) = 119878PW119877

( + 120578) minus 119878PW119877

() (99)

Since (120575119878PW119877

(119884)120575119884)|119884=

119884

= 0 119878PW119877




119890Γ

PW(

119884)

= int [119889120578] 119890minus119878

PW119865()

(120578)+119878

PW119865()

(

119884)

= 119890119882

PW[119865(

119884)]+119878

PW119865()

(

119884)

(100)

where

120575119878PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= minus119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

(101)


PW119865(

119884)



PW119865(

119884)


PW119865(

119884)


PW(119884)120575119884|

119884=

119884

= 120575119878PW119865(

119884)

(119884)120575119884|119884=

119884


119877

( 120578)

is modified to

119878PW119865(119884)

( 120578) = 119878PW119865(119884)

( + 120578) minus 119878PW119865(119884)

() (102)

119878PW119865(

119884)


PW119865(

119884)

(119884)120575119884|119884=

119884



119865(

119884)


119865(

119884)


119884)

(1015840



4

and 1198657


Return to (100)

119878PW119865(119884)

() = 0120575119878

PW119865

()

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

ΓPW

() = 119882PW

[119865 ()] 120575119882

PW(119865)

120575119865(119909)

100381610038161003816100381610038161003816100381610038161003816119865=119865(119884)

= 0

119890119882

PW[119865(

119884)]

= int [119889120578] 119890minus119878

PW119865()

(

119884120578)

(103)




1198891199042

11

= 2119889119909+

119889119909minus

+ 119892minusminus

119889119909minus

119889119909minus

+ 119889119909119894

119889119909119894

119892minusminus

=119876

1199037

(104)



3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





3

1198731198772 (It is not 119876 = 30120587

3

1198979

119901

1198731198772 because 119897

119901


minus13

119904

119909 119909minus rarr 11989223

119904


1198891199042

11

= 211989213

119904

119889119909+

119889119909minus

+11989243

119904

119876

1199037119889119909

minus

119889119909minus

+ 119892minus23

119904

119889119909119894

119889119909119894

(105)


119904

119877Under the 119909


1198891199042

= 119890minus21206013

119866119898119899

119889119909119898

119889119909119899

+ 11989041206013

(119889119909minus

+ 119860119898

119889119909119898

)2

119892119898119899

= 119890minus21206013

119866119898119899

+ 11989041206013

119860119898

119860119899

119892minusminus

= 11989041206013

119892119898minus

= 11989041206013

119860119898

(106)


1198891199042

10

= minus11990372

11987612

119889119909+

119889119909+

+11987612

11990372119889119909

119868

119889119909119868

119890120601

= 119892119904

(119876

1199037)

34

119860+

=1199037

119892119904

119876

(107)




+



minus


+

= 119873119877 is 119878119877119865


119877119865

rarr (1119892119904

) 119878119877119865

)Cosider

119878119877119865

(0 120578) = 119878119877

(120578) + int119889119909+ Tr (119892

minusminus

119879minusminus

) (120578) (108)

119878119877


1




(120578)


rarr 119909119868

119877119909minus


119865 (119909) = (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

119892++

119892119868+

119892minusminus

119892119868minus

119862119868119869+

119862119868119869minus

120595+

120595minus

)


(119877 119909) = 119877119902

119865(119909

119877)

= (119877119892+minus

119892119868119869

1198772 119862

119868+minus

119862119868119869119870

1198773120595119868

119877 119892++

119892119868+

119877 119877

2

119892minusminus

119892119868minus

119862119868119869+

1198772119862119868119869minus

119877 120595

+

119877120595minus

)

(109)



119868

rarr 119883119868





(119877119883) 119878119877119865

rarr 1198781119865

lowast(119877)


32 119878119877119865

rarr 1198781119865

lowast(119877)

1198773 In 119878

1119865

lowast(119877)

it is

119865lowast

(119877119883) = 119877119901

119865(119883

119877)

= (119892+minus

119892119868119869

119862119868+minus

119862119868119869119870

120595119868

1198772

119892++

119877119892119868+

119892minusminus

1198772119892119868minus

119877 119877119862

119868119869+

119862119868119869minus

119877 119877120595

+

120595minus

119877)

(110)


119877119865


lowast(119877)

1198773


minusminus

(119877 119909) = 119892minusminus

(119903119877)1198772

sim 1198773

1199037 In 119877 rarr 0 limit

119892lowast

minusminus

(119877 119909) rarr 0

lim119877rarr0

119878119877119865


1

1198773119878119877=1

= lim119877rarr0

1

1198773int119889119909

+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

(111)

SYM1



For PWMM 1198772119892++

(119909119877) = 119892++

(119909) and 119877119862119868119869+

(119909119877) =

119862119868119869+


119878PW119877

lArrrArr1

1198773119878PW119877=1

=1

1198773int119889119909

+

times Tr 1

4(119863

+

119883119894

)2

minus 119894120579119863+

120579


+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




+1

2[119883

119894

119883119895

]2

+ 2120579120574119894

[119883119894

120579]

minus (120583

6)

2

(119883119886

)2

minus (120583

12)

2

(119883119886

1015840

)

2

minus120583

6119894120598119886119887119888

119883119886

119883119887

119883119888

minus120583

8120579120574123

120579

(112)



2119895119904

+ 1 = 119899 + 120572119904

(113)

119899 997888rarr infin 120572119904

finite (114)


sim (119899 + 120572119904

)119877119904 = 1 119905 119865() reduced along 119909


1198891199042

10

= ( minus 2

minus11988110158401015840

)

12

times minus4

minus 2119889119909

+

119889119909+

+minus2119881

10158401015840

(119889120588

2

+ 1198891205782

) + 4119889Ω2

5

+ 211988110158401015840

Δ119889Ω

2

2

1198904120601

=

4( minus 2)3

minus119881101584010158402Δ2

1198621

= minus2

1015840

minus 2119889119909

+

1198654

= 1198891198623

1198623

= minus42

11988110158401015840

Δ119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (

1015840

Δ+ 120578)119889

2

Ω

Δ = ( minus 2)11988110158401015840

minus (1015840

)2

(115)


119887

(120588 120578) + V120578

(120588 120578)where 119881

119887






1198891199042

11

=

223

( minus 2)

(minus119881101584010158402Δ2)13

119889119909minus

119889119909minus

minus253

1015840

(minus119881101584010158402Δ2)13

119889119909+

119889119909minus

+

253

13

[(1015840

)2

minus 11988110158401015840

]

(minus11988110158401015840Δ2)13

119889119909+

119889119909+

+

223

(Δ)13

(minus11988110158401015840)13

minus119881

10158401015840

(119889120588

2

+ 1198891205782

)

+2119889Ω2

5

+11988110158401015840

Δ119889Ω

2

2

(116)

For PWMM

119881PW119887

(120588 120578) = 1205882

120578 minus2

31205783

(117)


1

= (1205872119877)(21198951

+ 1) 1205782

=

(1205872119877)(21198952

+1) 120578119905

= (1205872119877)(2119895119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905

respectively 120578

119904


1

1198832

1198833

) harr (120578Ω2

) (1198834

1198839

) harr

(120588Ω5

) 119909+ harr 119909+


119881 (120588 120578) = 1205882

120578 minus2

31205783

+ V120578

(120588 120578) (118)


solution 11986510


+ and 119909minus



119904

= (1205872)(2119895119904


lowast

(119877 119909) = 119877119902

119865(119909119877)11986510

(119909) rarr 119865lowast10

(119877 119909) As a result 119878119877119865

rarr 1198781119865

lowast(119877)

119865lowast10

(119877 119909)

and 119865lowast


119881(120588

119877120578

119877) =

1

1198773(120588

2

120578 minus2

31205783

) + V120578119877

(120588

119877120578

119877) (119)


lowast(119877)

we also have 119878119877119865

sim

1198781119865

lowast(119877)

1198773 with 119865

lowast

(119877 119909) = 119877119901

119865(119909119877) given by (110) Plug

1198773

119881(120588

119877120578

119877) = 120588

2

120578 minus2

31205783

+ 1198773

V120578119877

(120588

119877120578

119877) (120)


10


lowast



1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1198773

119881(120588119877 120578119877) = 119881(120588 120578) so 119865lowast


3


lowast

(119877 119909) and119865lowast


lowast

(119877 119909) and 119865lowast


119877119865

hArr 1198781119865

lowast(119877)

hArr 1198781119865

lowast(119877)

1198773



119894

minus120572119895


120578

(120588 120578) around 0 Since 120578119904

119877 rarr infin near 0 V120578119877


119904

119877

1198773

V120578119877

(120588

119877120578

119877) sim

119905

sum

119904=1

119873119904

1198773

[(120578119904

minus 120578

119877)

2

+ (120588

119877)

2

]

minus12

minus [(120578119904

+ 120578

119877)

2

+ (120588

119877)

2

]

minus12

sim

119905

sum

119904=1

21198774

119873119904

120578

1205782

119904

997888rarr 0

(121)

lim1198773

119881(120588119877 120578119877) = 1205882


1198773



119877119865

hArr lim 119878PW119877=1

1198773 Even if the




3

(V120578119877

minus 2V120578119877

) the V120578119877


2

120578) = 1198773V120578119877

(120588119877 120578119877)

1198891199042

10

= 4

minus(

1205972

120588

2119908

120578)

minus12

119889119909+

119889119909+

+(

1205972

120588

2119908

120578)

12

[1198891205882

+ 1205882

119889Ω2

5

+ 1198891205782

+ 1205782

119889Ω2

2

]

1198904120601

= (

1205972

120588

2119908

120578)

3

1198621

= minus2(

1205972

120588

2119908

120578)

minus1

119889119909+

1198654

= 1198891198623

1198623

= minus161205783

119889119909+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (41205782

1205972

120588

2119908 minus 120597120588

2119908 + 120578120597120588

2120597120578

119908)1198892

Ω

(122)


lowast

10

(119877 119909) With 119890120601

rarr 1198773

119890120601 119862

1

rarr

1198621

1198773 119862

3

rarr 1198623


lowast10




1198891199042

10

= 1198962

(minus11990672

11987312

119889119906+

119889119906+

+11987312

11990672119889119906

119868

119889119906119868

)

11989041206013

=119873

1199067 119860

+

=1199067

119873

(123)


1119865

lowast(119877)


1119865

lowast(119877)

is

119878119887

1119865

lowast(119877)

= int119889119909+


119863+

119883119868

119863+

119883119869


119892lowast119868119869

+119863+

119883119868

119892lowast+minus

(119888119869

lowast

+119892119869

lowast+

119892lowast+minus

)119892lowast119868119869

+1

4[119883

119868

119883119869

] [119883119870

119883119871

] 119892lowast119868119870

119892lowast119869119871

+119892lowast++


+119888lowast+

119892lowast+minus

) + 119878119892

lowast119868minus

+ 119878119892

lowastminusminus

(124)




(119877 119909) is

119892lowast119898119899

= 119890minus2120601

lowast3

119866lowast119898119899

+ 1198904120601

lowast3

119860lowast119898

119860lowast119899


= 1198904120601

lowast3


= 1198904120601

lowast3

119860lowast119898

(125)


rarr 0119892lowast

119868minus



1198773

119878119892

119868minus

= 0 1198773

119878119892

minusminus

997888rarr 0

119866lowast++

997888rarr minus1198902120601

lowast119860lowast+

119860lowast+

119866lowast119868119869

= 1198902120601

lowast3

119892lowast119868119869


997888rarr 119890120601

lowast119860lowast+

(126)



lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





lim 119878119887

1119865

lowast(119877)

= lim 1

119890120601lowastint119889119909


times (minus1

2119866lowast119868119869

119866++

lowast

119863+

119883119868

119863+

119883119869

minus 119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

+ (minus119866++

lowast

)12

119866lowast119868119869

119862119869

lowast119870119871

times 119863+

119883119868

[119883119870

119883119871

]

+1

4119866lowast119868119869

119866lowast119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus1

21198902120601

lowast3

119866++

lowast

119892lowast++

+ (minus119866++

lowast

)12

119862lowast+119869119870

[119883119869

119883119870

] )

= lim 119878119887

1119865

lowast10(119877)

(127)

For 119892lowast++

= 120583++

119883119868

119883119869

119892lowast119868119869

119892119869lowast+

= ]+

119883119869

1198902120601

lowast3

119866++

lowast

119892lowast++

= 120583++

119866++

lowast

119866lowast119868119869

119883119868

119883119869

119866lowast119868119869

119866++

lowast

119863+

119883119868

119892119869

lowast+

= ]+

119866lowast119868119869

119866++

lowast

119863+

119883119868

119883119869

(128)


lowast(119877)


1119865

lowast10(119877)

the action of SYM

1



1119865

lowast(119877)



1119865

lowast(119877)

=

lim 119878PW119877=1

1198773

= lim 1198781119865

lowast10(119877)


1

living on 10119889


int119889119909+ Tr(1

2119863+

119883119868

119863+

119883119868

+1

4[119883

119868

119883119869

]2

minus 119894120579119863+

120579 + 120579120574119868

[119883119868

120579] )

= int119889119909+


++

119890120601(minus

1

2119866119868119869

119866++

119863+

119883119868

119863+

119883119869

+1

4119866119868119869

119866119870119871

[119883119868

119883119870

] [119883119869

119883119871

]

minus 119894119890120601

(minus119866++

)12

120579119863+

120579

+ 119890120601

11986612

119868119868

120579120574119868

[119883119868

120579] )

(129)

where 120601 119866++


119904



1

1198773119881

PW119887

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] 997888rarr119899

1198773(120588

2

minus 21205782

)

=1

1205722119881

SYM119877times1198782

119887

(130)

where 119881SYM119877times1198782

119887



119881(120588

119877120578 + 119899

119877)

=1

1198773[120588

2

(120578 + 119899) minus2

3(120578 + 119899)

3

] + V120578119877

(120588

119877120578 + 119899

119877)

(131)


(120588119877 (120578 + 119899)119877) = 119906(1205882


(119877 119909)

is

1198891199042

11

= (2

120572211989921205972

120588

2119906)

23

times [

1205724

1198991205972

120588

2119906

4119889119909

minus

119889119909minus

minus 21205722

119899 (1 + 120597120588

2120597120578

119906) 119889119909+

119889119909minus

+ 1198992

(4119889119909+

119889119909+

minus 119889Ω2

2

)

minus4 (1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

) ]

(132)


times 1198782

times 119877 The 10119889


(119877 119909) is

1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

(1198891205782

+ 1198891205882

+ 1205882

119889Ω2

5

)

119890120601

=1205722

2(

1205972

120588

2119906

119899)

14


1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1198621

= minus

2 (1 + 120597120588

2120597120578

119906)

12057221205972

120588

2119906

119889119909+

1198654

= 1198891198623

1198623

= minus4119899

12057221205972

120588

2119906119889119909

+

and 1198892

Ω

1198673

= 1198891198612

1198612

= (minus

1 + 120597120588

2120597120578

119906

41205972

120588

2119906

+ 120578)1198892

Ω

(133)

119889119909+

119889119909+ and 119889Ω

2

2



PW119877=1

1198773 around


119877times119878



119877times119878

3119885

119896



Concretely (1198991198773) 119878PW119877=1

( 120578) = (1198991198773

)119878PW119877=1

( + 120578) =

119878119877times119878

2

(119910 + 119910) where 119878119877times119878

2


2

with119873 = 1198731

+ sdot sdot sdot + 119873119905

119878119877times119878

2

=1

1198922

119877times119878

2

int119889119909+

119889Ω2

1205832

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

Φ119863119886

Φ

minus1205832

2Φ2

+ 12058311986512

Φ minus1

2119863119886

119883119898

119863119886

119883119898

minus1205832

81198832

119898

+1

4[119883

119898

119883119899

]2

+1

2[Φ119883

119898

]2

minus119894

2120582Γ

119886

119863119886

120582 +119894120583

8120582Γ

123

120582

minus1

2120582Γ

3

[Φ 120582] minus1

2120582Γ

119898

[119883119898

120582]

(134)


1198922

119877times119878

2 = 1198773

1205833


Φ =120583

2diag [

[

1205721

1205721

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

1

1205722

1205722

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

2

120572119905

120572119905

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119873

119905

]

]

1198601

= 0

1198602

=

tan 120579

2Φ in region I

minuscot 1205792

Φ in region II

(135)


119904


119904


1

) times 119880(1198732



1

= (1205872)1205721

1205782

=

(1205872)1205722

120578119905

= (1205872)120572119905


2

8)1198731

(1205872

8)1198732

(1205872

8)119873119905


=

(1205872)(119899 + 120572119904


= (1205872)120572119904



119877times119878

2 fields is Φ harr 120578 (1198834

1198839

) harr (120588Ω5

)(119909+

Ω2

) harr (119909+

Ω2



119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119866120578120578

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2119866120578120578

Φ2

+ 120583(minus119866)minus12

11986612

120578120578

11986512

Φ

minus1

2119866119886119887

119866119898119899

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119866119898119899

119883119898

119883119899

+1

4119866119898119901

119866119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

+1

2119866120578120578

119866119898119899

[Φ119883119898

] [Φ119883119899

] )

(136)

where 119866119886119887

sim 119866119898119899

sim 119866120578120578

sim 119866Ω

2 sim (1205972

120588

2119906119899)12 119866 = det119866

119886119887

sim

minus(1205972

120588

2119906119899)minus32 119890120601 sim 120572

2

(1205972

120588





119901+1



3

and SCFT6

on 1198722



119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119906 = 119906(1205882


4

1198839

] let

[Φ 1198834

1198839

]

= [

[

(

1205972

120588

2119906

119899)

14

Φ(

1205972

120588

2119906

119899)

14

1198834

(

1205972

120588

2119906

119899)

14

1198839]

]

(137)

then

119878119877times119878

2

119887

= 1205833

int119889119909+

119889Ω2

1205832


119890120601

times (minus1

4119866119886119888

119866119887119889

119865119886119887

119865119888119889

minus1

2119866119886119887

119863119886

Φ119863119887

Φ minus1205832

2119866Ω

2Φ2

+ 120583(minus119866)minus12

11986512

Φ minus1

2119866119886119887

119863119886

119883119898

119863119887

119883119899

minus1205832

8119866Ω

2119883119898

119883119899

+1

4[119883

119898

119883119899

] [119883119901

119883119902

]

+1

2[Φ 119883

119898

] [Φ 119883119899

])

(138)


rarr 119883119898 Φ rarr Φ the 3119889


119890120601

= 1 997888rarr 119890120601

=1205722

2(

1205972

120588

2119906

119899)

14

(1205882

120578)

1198891199042

3

= minus4119889119909+

119889119909+

+ 119889Ω2

2

997888rarr 1198891199042

3

= (

1205972

120588

2119906

119899)

minus12

(1205882

120578) (minus4119889119909+

119889119909+

+ 119889Ω2

2

)

(139)


2

+ 1205782

)12 1198891199042

10


1198891199042

10

= (

1205972

120588

2119906

119899)

minus12

(minus4119889119909+

119889119909+

+ 119889Ω2

2

)

+ 4(

1205972

120588

2119906

119899)

12

1198891199032

+ 41199032

(

1205972

120588

2119906

119899)

12

119889Ω2

6

(140)





119898

rarr



2 For PWMM we have

119890119882

PW(119865)

= int [119889120578] 119890minus119878

PW(

119884120578)

(141)

119882PW




119878PW

( 120578) 997888rarr 119878119877times119878

2

(119910 119910)

119882PW

(119865) 997888rarr 119882119877times119878

2

(119865)

(142)

where 119882119877times119878

2


119877times119878

2 Then

119890119882

119877times1198782

(119865)

= int [119889119910] 119890minus119878

119877times1198782

(119910 119910)

(143)



119877times119878


Φ =120583

2


119872

119904 119904⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

119904 + 1 119904 + 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119872

]


(119904+1119905+1)

=


119878119877times119878

2

(119910 + 119910)100381610038161003816100381610038161003816[119910(119904+1119905+1)=119910(119904119905)]

= 119878119877times119878

3

(119911)

=1

1198922

119877times119878

3

int119889119905119889Ω

3

(1205832)3

times Tr minus1

4119865119886119887

119865119886119887

minus1

2119863119886

119883119898

119863119886

119883119898

minus1

12119883

2

119898

minus119894

2120582Γ

119886

119863119886

120582 minus1

2120582Γ

119898

[119883119898

120582]

+1

4[119883

119898

119883119899

]2

(145)where

1198922

119877times119878

3 =4120587

1205831198922

119877times119878

2 (146)

119878119877times119878

3




2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2





5

times 1198785 [19] On


(119904+1119905+1)

= 119910(119904119905)


119877times119878


119877times119878

3


119890119882

119877times1198783

(119865)

= int [119889119911] 119890minus119878

119877times1198782

(119911)

(147)


3


times 1198785





SYM4


times 1198785 A


5



120601 (119905 1199091

1199099

) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times (1199091

)119899

1


)119899

9

(148)


Φ (119905) =

infin

sum

119899

1=0

sdot sdot sdot

infin

sum

119899

9=0

1

1198991


times (120597119899

1

1


9

9

120601) (119905 0 0)

times STr [(1198831

)119899

1


)119899

9

]

(149)



1

1199099



4


times1198785


For scalar ℎ

ℎ (119909 119910) =

infin

sum

119899=2

ℎ119899

(1199091003816100381610038161003816119910

1003816100381610038161003816) 119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

)

119884119899

(119910

10038161003816100381610038161199101003816100381610038161003816

) =

119862119886

1sdotsdotsdot119886

119899

(119910119886


119899)

10038161003816100381610038161199101003816100381610038161003816

119899

(150)

where 119884119899


(119909 |119910|)



4

we have 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



119886

1sdotsdotsdot119886

119899

(119910119886


119899)Naively let 1198772 = 119890

2Σ

= 119883119898

119883119898 Σ = (12) ln (119883

119898

119883119898


120590


= ln 1199030

formally

119865 (119909 119877) = 119891 (119909 Σ)

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) (Σ minus 1205900

119868)

+1

2119891(2)

(119909 1205900

) (Σ minus 1205900

119868)2

+ sdot sdot sdot

= 119891(0)

(119909 1205900

) + 119891(1)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

+1

2119891(2)

(119909 1205900

) [1

2ln(

119883119898

119883119898

1199032

0

)]

2

+ sdot sdot sdot

(151)

Functions 119891(119899)(119909 1205900


0


0

) rarr 119891(119899)

(119909 1205900


119904

= Σ minus 120590119868

120590 =1

119873TrΣ =

1

2119873Tr ln (119883

119898

119883119898

)

=1

2119873ln det (119883119898

119883119898

)

Σ119904

=1

2ln (119883

119898

119904

119883119898

119904

) det (119883119898

119904

119883119898

119904

) = 1

(152)

119883119898

= 119890120590

119883119898

119904


11988310


5

119904

11988310

119904

120590) 120590 rarr 120590 + 119886119883119898

rarr 119890119886

119883119898



1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1sdotsdotsdot119886

119899

Tr(119883119886

1

119904


119899

119904


119899


119904

with det(119883119898

119883119898

) = infin119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) sim 119884119899

(119910) where |119910| = infin)

119867119899

(119909 120590) = ℎ119899

(119909 120590) 119862119886

1sdotsdotsdot119886

119899

Tr (119883119886

1

119904


119899

119904

) (153)

ℎ119899


4

with ℎ119899

(119909)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886



(119909 120590)119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) The 5119889 field ℎ119899


119899




4



119878 (119892119883) =1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883)

+1205721015840

119877Φ (119883) + 119879 (119883) + sdot sdot sdot ]

(154)


119885 = int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

int

119863119892

119892119863119892

119883

vol (diff2

)119890minus119878(119892119883)

= int

119863119890

120596119892

(119890120596

119892)119863119890

120596119892

119883

vol (diff2

)119890minus119878(119890

120596119892119883)

(155)


119890120593


119863119892

119892

vol (diff2

)=

119863119892

120593

vol (conf2

)119890minus119878

119871(120593119892)

(156)

where 119878119871


119883 rarr 119863119892

119883119863119892

120593 rarr 119863119892


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus119878(119892119883)minus119878

119871(120593119892)

= int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

(157)

where

119878 (120593119883 119892)

=1

41205871205721015840int119889

2

120585radic119892 [119892119886119887

120597119886

120593120597119887

120593

+ 119892119886119887

120597119886

119883120583

120597119887

119883]119866120583] (119883 120593)

+ 120598119886119887

120597119886

119883120583

120597119887

119883]119861120583] (119883 120593)

+1205721015840

Φ (119883 120593) + (119883 120593) + sdot sdot sdot ]

(158)


119885 (119892 0) = int

infin

120593=0

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593119883119892)

= int

infin

120593=0

119863119892

(120593 minus infin)119863119892

119883

vol (conf2

)119890minus


= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878(120593+infin119883119892)

= int

infin

120593=minusinfin

119863119892

120593119863119892

119883

vol (conf2

)119890minus

119878

1015840(120593119883119892)

(159)

In 1198781015840 1198651015840(120593) = 119865(120593 + infin)Since 119892 = 119890

120593+120596

119890minus120596

119892 for 120596 ge 0

119885 (119892 0) = int

infin

120593=0

119863119890

minus120596119892

(120593 + 120596)119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593+120596119883119890

minus120596119892)

= int

infin

120593=120596

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 120596)

(160)


119885 (119890minus120596

119892 0) minus 119885 (119892 0) = int

120596

120593=0

119863119890

minus120596119892

120593119863119890

minus120596119892

119883

vol (conf2

)119890minus

119878(120593119883119890

minus120596119892)

= 119885 (119890minus120596

119892 0) minus 119885 (119890minus120596

119892 120596)

(161)


119892)




4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






4

with the action

119868 [119892120583] 119883 119860 120582] = minus

1

1198922

119884119872

int1198894


times Tr(1

2119892120583]119892120582120590

119865120583120582

119865]120590

+ 119892120583]119863120583

119883119898

119863]119883119898

+1

6119883

2

119898

minus 119894120582Γ120583

119863120583

120582

minus 120582Γ119898

[119883119898

120582]

minus1

2[119883

119898

119883119899

]2

)

(162)


119883 with 120590 isin


119885(119892120583] 119883) = int

119883

119863119892

119883119863119892

119860119863119892

120582 exp minus119868 [119892120583] 119883 119860 120582]

= 119890119882(119892

120583]

119883)

(163)


119868 [119892120583] 119883 119860 120582] = 119868 [119890

2120596

119892120583] 119890

minus120596

119883119860 119890minus(32)120596

120582] (164)

Suppose

119863119892

119883119863119892

119860119863119892

120582 = 119863119890

2120596119892

(119890minus120596

119883)119863119890

2120596119892

times 119860119863119890

2120596119892

(119890minus(32)120596

120582) 119890119878

119888[120596119892]

(165)



120583] 119878119888

[120596 119892] = 0 119885(119892120583] 119883) =

119885(119890minus2120596

119892120583] 119890

120596




120583] we always have 119878119888


120590

119883

119885(119892120583] 119883) = int

0

120590=minusinfin

119863119892

(119890120590

119883)119863119892

119860119863119892

120582

times exp minus119868 [119892120583] 119890

120590

119883119860 120582]

(166)

= int

0

120590=minusinfin

119863119890

2120590119892

119883119863119890

2120590119892

119860119863119890

2120590119892

120582

times exp minus119868 [1198902120590

119892120583] 119883 119860 120582]

(167)


4


1198891199042

= 1198902120590

1198891199092

4

+ 1199032

3

(1198891205902

+ 119889Ω2

5

)

= (119903

1199033

)

2

1198891199092

4

+ (1199033

119903)

2

(1198891199032

+ 1199032

119889Ω2

5

)

(168)

119903 = 119890120590

1199033

119903 isin [0 +infin) 1199033


=



3


times 1198785 (167)


5

times 1198785


3

and SCFT6

for 1198722 and 1198725 For SCFT3


120590

119883 119892120583]) rarr

(119883 1198904120590


1198891199042

= 1198904120590

1198891199092

3

+ 1199032

2

(1198891205902

+ 119889Ω2

7

)

= (119903

1199032

)

4

1198891199092

3

+ (1199032

119903)

2

(1198891199032

+ 1199032

119889Ω2

7

)

(169)

with 119903 = 119890120590

1199032

For SCFT6


120590

119883 119892120583]) rarr (119883 119890

120590

119892120583]) The induced

metric is

1198891199042

= 119890120590

1198891199092

6

+ 1199032

5

(1198891205902

+ 119889Ω2

4

)

= (119903

1199035

)1198891199092

6

+ (1199035

119903)

2

(1198891199032

+ 1199032

119889Ω2

4

)

(170)

with 119903 = 119890120590

1199035



119901+1

SCFT3

and SCFT6


2

119903)119883 and 119883 rarr (1199035

119903)119883 are always



3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





3

rarr (1199031199032

)4

1198891199092

3

and 1198891199092

6

rarr (1199031199035

)1198891199092

6


1198891199042

= (119903

119903119901

)

(7minus119901)2

1198891199092

119901+1

+ (119903

119903119901

)


(1198891199032

+ 1199032

119889Ω2

8minus119901

)

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4

(171)

the 119883 rarr (119903119901

119903)(7minus119901)4


2

119901+1

rarr (119903119903119901

)(7minus119901)2

1198891199092

119901+1

119890Φ

= 119892119904

rarr 119890Φ

=

119892119904

(119903119903119901





SCFT and the 1198601198891198783

times 1198783



metric is 11988911990924



Return to SYM4


119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [119892120583]0 119883 119860 120582] + int119889

4

119909radicminus119892120583]0 120601

0

(119909)119874 (119909)

(172)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899) 119892120583]0 is the arbitrary 4119889


120583]0 and 1206010



119885(119892120583]0 1206010 119883) = int

119883

119863(119892

0120601

0)

119883119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119883 119860 120582]

119868 [119892120583]0 1206010 119883 119860 120582]

= 119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883119860 119890minus(32)120596

120582]

(173)

For finite 120596 119892120583]0 1206010

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (174)


119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

119883119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

times 119860119863(119890

2120590119892

0119890

(119899minus4)120590120601

0)

(119890minus(32)120590

120582)

times exp minus119868 [1198902120590

119892120583]0 119890

(119899minus4)120590

1206010

119883 119860 119890minus(32)120590

120582]

+ 119878119888

(120590 119892120583]0 1206010)

(175)


1198902120590

119892120583]0 and 119890

(119899minus4)120590

1206010


and 120601(120590) so that

119863(119892

0120601

0)

(119890120590

119883)119863(119892

0120601

0)

119860119863(119892

0120601

0)

120582

times exp minus119868 [119892120583]0 1206010 119890

120590

119883119860 120582]

= 119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

(176)


120583]0 = 119892120583](0) 1206010 = 120601(0) We also require

119863[119892(120590

1)120601(120590

1)]

(119890120590

2119883)119863[119892(120590

1)120601(120590

1)]

119860119863[119892(120590

1)120601(120590

1)]

120582

times exp minus119868 [119892120583] (1205901) 120601 (120590

1

) 119890120590

2119883119860 120582]

= 119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

119883119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

times 119860119863[119892(120590

1+120590

2)120601(120590

1+120590

2)]

120582

times exp minus119868 [119892120583] (1205901 + 120590

2

) 120601 (1205901

+ 1205902

) 119883 119860 120582]

(177)



120583](120590) = 1198902120590

119892120583]0

120601(120590) = 119890(119899minus4)120590

1206010


0


0




119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119885(119892120583]0 1198650 119883)

= int

0

120590=minusinfin

119863[119892(120590)119865(120590)]

119883119863[119892(120590)119865(120590)]

119860119863[119892(120590)119865(120590)]

120582

times exp minus119868 [119892120583] (120590) 119865 (120590) 119883 119860 120582]

= 119885 (119892120583] (120590) 119865 (120590) 119883 0)

(178)



119909radicminus119892(120590)120601

(119909 120590)119874(119909) with 119874(119909) given by 119874 = 119862119886

1sdotsdotsdot119886

119899

Tr(119883119886


119899)119874(119909) sim 119862

119886

1sdotsdotsdot119886

119899

(119910119886



|119910|119899minus4

120601(119909 120590)119884119899

(Ω) and g120583](119909 120590) = |119910|

4

119892120583](119909 120590) where

119884119899






g (120590) = 1198902120590

[1198920

+ 119890minus2120590

1198922

+ 119890minus4120590

1198924

minus 2120590119890minus4120590

ℎ4

] + 119874 (119890minus3120590

)

(179)



= +infin and 119892(120590) = 119890minus2120590

infing(120590)

119892 (120590) = 1198902120590

[1198920

+ 119890minus2(120590+120590

infin)

1198922

+ 119890minus4(120590+120590

infin)

1198924

minus2 (120590 + 120590infin

) 119890minus4(120590+120590

infin)

ℎ4

] + 119874 (119890minus5120590


)

(180)

For finite 120590 119892(120590) = 1198902120590

1198920


Φ (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2120590

1206012


1206012119899minus4

minus2120590119890(4minus2119899)120590

1205932119899minus4

] + 119874 [119890minus(119899+1)120590

]

(181)



)

120601 (120590) = 119890(119899minus4)120590

[1206010

+ 119890minus2(120590+120590

infin)

1206012

+ sdot sdot sdot

+ 119890(4minus2119899)(120590+120590

infin)

1206012119899minus4

minus2 (120590 + 120590infin

) 119890(4minus2119899)(120590+120590

infin)

1205932119899minus4

]

+ 119874 [119890(3minus2119899)120590

infin119890minus(119899+1)120590

]

(182)


1206010


0

and 1206010


5

with 1199033


120583](119909 120590) and 120601(119909 120590) in (180) and (182)

From (177)

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

= 119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

(183)

As a result

119885(119892120583]0 1206010 119883)

= 119885 (119892120583] (120590) 120601 (120590) 119883 0)

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883 0)

= int

120596

120590=minusinfin

119863[119892(120590)120601(120590)]

(119890minus120596

119883)119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119890

minus120596

119883119860 120582]

= 119885 (119892120583] (120590) 120601 (120590) 119890

minus120596

119883120596)

(184)

With 119883 rarr 119890minus120596


120601(120590 + 120596) Also since

119885(119892120583]0 1206010 119883) = 119885 (119890

2120596

119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883) (185)

we have

119885(119892120583] (120590) 120601 (120590) 119883 0) = 119885 (119892

120596

120583] (120590) 120601120596

(120590) 119890minus120596

119883 0)

(186)

where 119892120596

120583](120590) and 120601120596


119892120583]0 and

119890(119899minus4)120596

1206010


120583](120590) = 119892120583](120590 + 120596)

120601120596


and1198920


1206010

and 1198902120596

1198920

one may get 120601(120590 + 120596) and119892(120590 + 120596) 119892

0

rarr 1198902120596

1198920

1206010

rarr 119890(119899minus4)120596

1206010


120583](120590)


120583](119909 120590) 120601(119909 120590)] inducedfrom [119892


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120596

120583] (119909 120590) 120601120596

(119909 120590) 119890minus120596

119883 0)

(187)



120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120583](119909 120590) 120601120596


120596

119909

119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119890minus2120596

119892120596

120583] (119890120596

119909 120590) 120601120596

(119890120596

119909 120590) 119890minus120596

119883 0)

(188)


119892120596

120583](120590) = 119892120583](120590) 120601

120596


120583](119909 120590) 120601(119909 120590)] thisrequires 119892

120583](120590 + 120596) = 1198902120596

119892120583](120590) 120601(120590 + 120596) = 120601(120590) and thus

119892120583](120590) = 119890

2120590

119892120583]0 120601(120590) = 120601

0


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= 119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

(189)

and so

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

=

120575119885 (119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883 0)

120575120601 (1198901205961199090

1205900

)

(190)

As a result

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨1198904120596

radicminus119892(1198901205961199090

1205900

)119874120596

(119890120596

1199090

)120575(120590 minus 1205900

)⟩120596

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

(119890minus120596

119883)119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119890

120596

119909 120590) 120601 (119890120596

119909 120590) 119890minus120596

119883119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= int

0

120590=minusinfin

119863[119892(120590)

120601(120590)]

119883119863[119892(120590)

120601(120590)]

119860119863[119892(120590)

120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]

times 119890(4minus119899)120596

radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)

= 119890(4minus119899)120596

⟨radicminus119892 (1198901205961199090

1205900

)119874 (119890120596

1199090

) 120575 (120590 minus 1205900

)⟩

(191)

and then

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(192)


120583](119909 120590) 120601(119909 120590)] usually [119890minus2120596

119892120583](120590 +



⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (119909 120590) 120601 (119909 120590) 119883 119860 120582]


1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)

= ⟨119890(4minus119899)120596


1205900

)119874(1199090

)119890119894119901(120590+120596)

⟩120596

= int

0

120590=minusinfin

119863[119892(120590+120596)120601(120590+120596)]

(119890minus120596

119883)119863[119892(120590+120596)120601(120590+120596)]

times 119860119863[119892(120590+120596)120601(120590+120596)]

120582

times exp minus119868 [119892120583] (120590 + 120596) 120601 (120590 + 120596) 119890

minus120596

119883119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= int

0

120590=minusinfin

119863[119892(120590)120601(120590)]

119883119863[119892(120590)120601(120590)]

119860119863[119892(120590)120601(120590)]

120582

times exp minus119868 [119892120583] (120590) 120601 (120590) 119883 119860 120582]

times 119890(4minus119899)120596


1205900

)119874 (1199090

) 119890119894119901(120590+120596)

= 119890(119894119901minus119899+4)120596

⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

(193)

still

0 = ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 119890119894119901120590

⟩

= ⟨radicminus119892 (1199090

1205900

)119874 (1199090

) 120575 (120590 minus 1205900

)⟩

=

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)

(194)

120601(119909 120590) is also on shell



(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





(119909) In general

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

1205751206010

(1199090

)

=

120575119885 (119892120583]0 (119909) 1206010 (119909) 119883)

1205751206010

(1199090

)= 0

(195)


4




120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575Φ (1199090

1205900

)= 0 (196)



the field 1198650

(119909) becomes 119865(119909 120590) 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


times 1198785



= 119890119899120590


= radicminusg (119909 120590)F (119909 120590) 119884119899

(197)


5119889 functions g120583] = 119890

4120590

infin119892120583] and F = 119890

(119899minus4)120590


5


119885(119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

= exp 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(198)

119878119892



times 1198785


0

(119909)

120575119885 (119892120583] (119909 120590) 120601 (119909 120590) 119883 0)

120575120601 (1199090

1205900

)= 0 (199)

and then

120575119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

120575Φ (1199090

1205900

)= 0 (200)


0


119892

120575Φ(1199090

1205900


119890(4minus119899)120590

infinΦ(119909 120590) =

1206010


119882(119892120583]0 1206010 119883) = ln119885(119892

120583]0 1206010 119883)

= 119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

(201)


4



5


119878119892



120583]0(119909) 1206010(119909) 119883 119860 120582] and 119868[119892120583](119909 120590) 120601(119909 120590) 119883 119860 120582]


0



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 0]

= int

0

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(202)



119878119892

[119892120583] (119909 120590) 120601 (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590) 120601 (119909 120590)]

(203)



119882(119892120583]0 120598) =

1

2int

infin

120598

119889120588

120588Tr (119890minus120588119863)

= int

120590

minusinfin


infin)

119863)

= int

120590

minusinfin


= 119882(g120583] (0) 120590)

(204)


2120590



2120596

119892120583]0 119890

2120596

120598) =

119882(119892120583]0 120598) = 119882(119890

2120596g120583](0) 120590 minus 120596) = 119882(g


Tr (119890minus120588119863) =

infin

sum

119899=0

1198602119899

120588119899minus2

1198602119899

= int1198894

119909radicminus119892 1198862119899

(119892)

(205)

so

119882(119892120583]0) = lim

120598rarr0

119882(119892120583]0 120598)

= lim120598rarr0

[1198600

41205982+

1198602

2120598minus

1198604

ln 120598

2] + 119882ren (119892

120583]0)

(206)



120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120583]0 rarr 1198902120596

119892120583]0 120598 rarr 119890

2120596

120598 [52]

119882ren (1198902120596

119892120583]0) minus 119882ren (119892

120583]0) = 1205961198604

(207)

1198604


119882(g120583] (0) 120590)

=A0

4119890minus4120590+

A2

2119890minus2120590+ A

4

120590 + 119882nonloc (g120583] (0)) (208)

so

119882(g120583] (0) 0) =

A0

4+A2

2+ 119882nonloc (g120583] (0))

= 119882loc (g120583] (0)) + 119882nonloc (g120583] (0))


infin

1198604

(209)

119882loc(g120583](0)) = A0

4 + A2


infin

1198604



119866 (1198920

120588) =1198920

120588+ 119892

2

+ 120588 (1198924

+ ln 120588ℎ4

) + 119874 (1205882

) (210)

At 120588 = 120598 the metric is 119866(1198920


119878 (119892120583]0 120598) = 119878 [119866 (119892

0

120598)] = int

infin

120598

119889120588

2120588119871 (120588) (211)


120598)

119878 (119892120583]0) = lim

120598rarr0

119878 (119892120583]0 120598)

= lim120598rarr0

119897

16120587119866119873

int1198894


times [1198860

(119892)

1205982+

1198862

(119892)

120598

minus 1198864

(119892) ln 120598]

+ 119878ren (119892120583]0)

(212)


2120596

1198920

1198902120596

120588) = 119866(1198920

120588) 119866(1198902120596

1198920

1198902120596

120598) = 119866(1198920

120598)119878(119890

2120596

119892120583]0 119890

2120596

120598) = 119878(119892120583]0 120598) [52]

119878ren (1198902120596

119892120583]0) minus 119878ren (119892

120583]0)

=119897

8120587119866119873

int1198894

119909radicminus1198921205961198864

(119892) = 1205961198604

(213)



Take119883 = 119890120590


119885(119892120583]0 119883) = int

119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= 119885 (119892120583] (120590) 119883 120590)

(214)

For finite 119892120583]0 and 120596 119885(119892

120583]0 119890120596

119883) = 119885(1198902120596

119892120583]0 119883) There is

also119885(119892

120583]0 119890120596

119883) = 119885 (119892120583] (120590) 119883 120590 + 120596)

= 119885 (119892120583] (120590 + 120596) 119883 120590) = 119885 (119890

2120596

119892120583]0 119883)

(215)


119892120583]0 is 119892

120583](120590 + 120596) From119885(119890

2120596

119892120583]0 119883) and 119885(119892

120583]0 119890120596



119885(119892120583]0 119890

120575120590

119883)

= int

119890

120575120590119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582 exp minus119868 [119892120583]0 119883 119860 120582]

= exp int1198894

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)119885 (119892

120583]0 119883)

(216)

where

119871119886

(1198920119883)(119909) = lim

120575120590rarr0

[1

120575120590 (119909)

times (int

119890

120575120590(119909)119883

119863119892

0

119883119863119892

0

119860119863119892

0

120582

times exp minus119868 [119892120583]0 119883 119860 120582]

minus119885 (119892120583]0 119883))]

times 119885minus1

(119892120583]0 119883)

(217)

With 119878119886

(1198920

119883) = int 1198894

119909119871119886

(119892

0119883)


119882(119892120583]0)

= int

infin

0

119889120590119878119886

(1198920

119890120590

119883) + 119882(119892120583]0 119883)

= int

infin

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

0

119889120588

120588119878119886

(1198920

119883

120588)

(218)


where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where 120588 = 119890minus120590

119882(119892120583]0 119883) = int

0

minusinfin

119889120590119878119886

(1198920

119890120590

119883) = int

infin

1

119889120588

120588119878119886

(1198920

119883

120588)

(219)

119882(119892120583]0 119890

120575120590

119883) minus 119882(119892120583]0 119883) = int119889

4

119909119871119886

(1198920 119883)(119909) 120575120590 (119909)

(220)


(119892

0119883)

with

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119883) = int1198894

119909119871119887

(1198920119883)(119909) 120575120590 (119909)

(221)

if

119882(1198902120575120590

119892120583]0 119883) minus 119882(119892

120583]0 119890120575120590

119883)

= int1198894

119909119871119888

(1198920119883)(119909) 120575120590 (119909)

119871119888

(119892

0119883)

(119909) = 119871119887

(119892

0119883)

(119909) minus 119871119886

(119892

0119883)

(119909)

(222)

For finite 120575120590 and 119892120583]0 119871

119888

(119892

0119883)


120583]0 119890120596

119883) and119882(1198902120596

119892120583]0 119883) with 120596 = minusinfin

119882(119892120583]0 119890

120596

119883) = ln119885(119892120583] (120590 + 120596) 119883 0)

119882 (1198902120596

119892120583]0 119883) = ln119885(119892

120596

120583] (120590) 119883 0)

(223)

119892120583](120596) = 119890

2120596

119892120583]0 so 119892

120596

120583](120590) = 119892120583](120590 + 120596)


ln119885(119892120583]0 119883) = ln119885(119892

120583] (120590) 119883 120590) = 119878119892

[119892120583] (119909 120590) 119883 120590]

= int

120590

minusinfin

119889120590int1198894

119909119871 [119892120583] (119909 120590)]

(224)



119892

[119892120583](119909 120590) 119883 120590]

Let

119878119887

(1198920

119883) = int1198894

119909119871119887

(1198920119883)(119909)

119878119888

(1198920

119883) = int1198894

119909119871119888

(1198920119883)(119909)

(225)

then 119878119888(1198920

119883) = 119878119887

(1198920

119883) minus 119878119886

(1198920


120598120597

120597120598119882(119892

120583]0 120598) = minus1198600

21205982minus

1198602

2120598minus

1198604

2+ 119874 (120598)

= minus119892120583]0

120597

120597119892120583]0

119882(119892120583]0 120598)

(226)



119878119887

(1198920

119883) = lim120598rarr0

[1198600

1205982+

1198602

120598+ 119860

4

] = 119878119886

(1198920

119883) (227)

1198600

1205982

+1198602

120598+1198604


4





120590


119882(119892120583]0 119883) = 119882(119892

120583]0 119890120590

119883lowast

) = ln119885(119892120583]0 119890

120590

119883lowast

) (228)


120590

119883lowast



119882(119892120583]0 119883

lowast

) = 119882(119892120583] (120590) 119890

120590

119883lowast

) (229)

with 119892120583](0) = 119892

120583]0

120575119882

120575119892120583] (120590)

120575119892120583] (120590)

120575120590+

120575119882

120575120590= 0 (230)

For finite 120590 119892120583](120590) = 119890

minus2120590

119892120583]0 When 120590 rarr infin 119892


120583]0

and 120596119882(119892120583]0 119890

minus120596

119883lowast

) = 119882(119890minus2120596

119892120583]0 119883

lowast

) If

119882(119890minus2120596

119892120583]0 119883

lowast

) = 119882[119892120583] (120596 + 120590) 119890

120590

119883lowast

]

= 119882(119892120583] (120596) 119883

lowast

)

(231)

there will be

119882(119892120583]0 119890

minus120596

119883lowast

) = 119882[119892120583] (120590) 119890

120590

(119890minus120596

119883lowast

)] (232)

So in (229) with 119892120583](0) = 119892


119890minus120596








119885 (119865 119892) = int

infin

120593=0

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(233)


120596

= 119890120596

119892

119885 (119865 119890120596

119892) = int

infin

120593=0

119863(119890

120593119892

120596)

120593119863(119890

120593119892

120596)

119883

vol (conf2

)119890minus119878(119865119890

120593119892

120596119883)minus119878

119871(120593119892

120596)

= int

infin

120593=120596

119863(119890

120593119892)

120593119863(119890

120593119892)

119883

vol (conf2

)119890minus119878(119865119890

120593119892119883)minus119878

119871(120593119892)

(234)


119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119885(119865 119890120596

119892) = 119885(119865 119892) 119892 rarr 119890120596


120596



119885 (119865 (120596) 119890120596

119892) = int

infin

120593=0

119863119892

120596120593119863119892

120596119883

vol (conf2

)119890minus

119878(120593119883119865

120596(120593)119892

120596)

(235)

in which 119865120596


120596



119890120593



int1198894

119909radic1198920

120596 (119909) [minus2119892120583]

0

120575

120575119892120583]

0

+ (Δ119894

minus 4 + 120573119894

) 119865119894

0

120575

120575119865119894

0

]119882ren

= int1198894

119909 [120596 (119909)119860 + 120597120583

120596 (119909)119885120583

]

(236)


120583120597119882ren120597120583

+ int1198894

119909radic1198920

times [2119892120583]

0

120575

120575119892120583]

0

minus (Δ119894

minus 4) 119865119894

0

120575

120575119865119894

0

]119882ren = 0

120583120597119882ren120597120583

+ int1198894

119909radic1198920

[120573119894

120575119882ren120575119865

119894

0

] = int1198894

119909119860

(237)


119894

119865119894

0


5 Conclusion




119909minus

= 1205721199091015840minus

119909+

= 119891 (1199091015840+

) 119909 = 119892 (1199091015840

) (238)

with 119909 = (1199091

1199099

) if 119877 rarr 1198771015840 119865 rarr 119865

1015840 119882(119877 119865) =

119882(1198771015840

1198651015840


119890119878eff(119865) = int

1PI[119889119865] 119890

minus119878cla(119865+119865) (239)






119882(1198650

119877119865minus




119878eff(1198650 119877119865minus


) In 119882(119901+





119892

(119884) given by

119890Γ

119892(119884)

= int [119889119865] 119890119878

119877119865(119884)minus119878cla(119877119865) (240)



119892

(119884) = 119878eff(119877 119865) + 119878119877119865


119877119865


119877119865





0



119877times119878

2 SYM119877times119878

3119885

119896

andSYM

119877times119878



119877



119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119878PW



119877times119878


119877times119878

3 and AdS5

times 1198785 [19 21]

SYM4


5



4



times


field 1198650

(119909) in SYM4



5

with 1198650


4



4

SCFT3

and SCFT

6


4



0

(119909) rarr 119865(119909 120590) SYM4



0

(1198650


Appendices




1198680

= minus1

1198922119901

int119889119901+1


times Tr(1

2119892120583120582

119892]120590119865120583]119865120582120590 + 119892

120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

])

(A1)

where 119892120583] = 120578

120583] 119892119898119899 = 120575119898119899



119892120583] = (

119903

119903119901

)

(7minus119901)2

120578120583] 119892

119898119899

= (119903

119903119901

)


120575119898119899

119890Φ

= 119892119904

(119903

119903119901

)

(7minus119901)(119901minus3)4


= (119903

119903119901

)

7minus119901


(A2)


119883119898

)12 the gauge


119868119865

= minus1

4120587int119889

119901+1


times (1

2119892120583120582

119892]120590119865120583]119865120582120590

+ 119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

minus 119894120582Γ120583

119863120583

120582 minus 120582Γ119898

[119883119898

120582]

minus1

2119892119898119901

119892119899119902

times [119883119898

119883119899

] [119883119901

119883119902

] )

(A3)

where 119892 = det119892120583] With 4120587119892

119904


0

= 119868119865




branes are

119892120583] = (

119903

1199032

)

4

120578120583] 119892

119898119899

= (119903

1199032

)

minus2

120575119898119899

119860012

= (119903

1199032

)

6

120576012

119892120583] = (

119903

1199035

) 120578120583] 119892

119898119899

= (119903

1199035

)

minus2

120575119898119899

119860012

= (119903

1199035

)

3

120576012

(A4)


radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

= radicminus119892119892120583]119892119898119899

119863120583

119883119898

119863]119883119899

(A5)

But

radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

=radicminus119892119892119898119901

119892119899119902

[119883119898

119883119899

] [119883119901

119883119902

]

(A6)


radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

= radicminus119892119892119897119901

119892119898119902

119892119899119903

[119883119897

119883119898

119883119899

] [119883119901

119883119902

119883119903

]

(A7)




radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

= radicminus119892119892120583]119892119898119902119892119899119903 [119862

120583

119883119898

119883119899

] [119862] 119883

119902

119883119903

]

(A8)




119868 [119892120583]0 1206010 119883]

= int1198894


Tr(119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ 1206010

119874)

(B1)

where 119874 = 119862119886

1sdotsdotsdot119886

119899

(119883119886


119899) 119868[1198902120596119892120583]0 119890

(119899minus4)120596

1206010

119890minus120596

119883] =



120583]0

119883119898

= (minus1198920

)14

119883119898

(B2)

In terms of119883

119868 [119892120583]0 1206010 119883] = 119868 [119892

120583]0 1206010 119883]

= int1198894

119909Tr [119892120583]0

120597120583

119883119898

120597]119883119898

+1

61198770

1198832

119898

+ (minus1198920

)(2minus119899)4

1206010

119874]

119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119890120596

119883] = 119868 [119892120583]0 1206010 119883]

(B3)


119885(119892120583]0 1206010) = int119863119883 exp minus119868 [119892

120583]0 1206010 119883] (B4)


119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

)

= int119863119883 exp minus119868 [1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

119883]

= int119863119883 exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

= int119863(119890120596

1198831015840

) exp minus119868 [119892120583]0 1206010 119883

1015840

]

(B5)

where1198831015840

= 119890minus120596

119883 If119863(119890120596

1198831015840

) = 119890119860

1205961198631198831015840 then

ln119885(1198902120596

119892120583]0 119890

(119899minus4)120596

1206010

) = 119860120596

+ ln 119885(119892120583]0 1206010) (B6)

with 119860120596


int119863119883 exp minus119868 [119892120583]0 1206010 119883]

= int119863(119890minus120596

119883) exp minus119868 [119892120583]0 1206010 119890

minus120596

119883]

(B7)


120598rarr0

119878(119892120583]0 1206010 120598) = lim

120598rarr0

119878(119892120583]0 1206010 119890

2120596

120598) so

119860120596

= lim120598rarr0

119878 (1198902120596

119892120583]0 119890

(119899minus4)120596

1206010


119878 (119892120583]0 1206010 120598)

(B8)

119860120596


120575119860120596

120575120596

10038161003816100381610038161003816100381610038161003816120596=0

=1

119885int119863119883 exp minus119868 [119892

120583]0 1206010 119883]120575119868

120575119883

119883

= ⟨120575119868

120575119883

119883⟩

(B9)



0


120596

=

119860120596



0


120583]0 and 1206010



119877


120583

although 119860119886

120583


120583



Acknowledgments


References










































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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