logarithmic operators in ads3/cft2

11 May 2000

Ž .Physics Letters B 480 2000 348–354

Logarithmic operators in AdS rCFT3 2

Alex Lewis 1

Department of Mathematical Physics, National UniÕersity of Ireland, Maynooth, Ireland

Received 21 February 2000; accepted 27 March 2000Editor: P.V. Landshoff

Abstract

We discuss the relation between singletons in AdS and logarithmic operators in the CFT on the boundary. In 23

dimensions there can be more logarithmic operators apart from those which correspond to singletons in AdS, becauselogarithmic operators can occur when the dimensions of primary fields differ by an integer instead of being equal. Theseoperators may be needed to account for the greybody factor for gauge bosons in the bulk. q 2000 Published by ElsevierScience B.V. All rights reserved.

One particularly interesting example of thew xAdSrCFT correspondence 1–3 is the AdS rCFT3 2

correspondence, which relates supergravity on AdS3

=S3 to a 2-dimensional CFT. One advantage of thisis that 2-dimensional conformal field theories arevery well understood, and that makes AdS espe-3

cially suitable for studying the relation between sin-gletons on AdS and logarithmic conformal field the-

Ž .ories LCFT , since almost all previous work onLCFT has concentrated on the 2-dimensional case.

w xAccording to 2,3 , at the boundary of AdSDq 1Ž .we have a coupling between bulk fields F x, z andi

Ž . Dboundary fields O x , Hd xF O , where the bound-i i i

ary fields are subject to the boundary condition

1 E-mail: [email protected]

Ž . Ž .F x, z sl x, R , with zsR the boundary ofi i

AdS . The relation between correlation functionsdq1

in CFT and the bulk supergravity action isD

d D x l OH i iÝi yS w�F 4xi² :e se 1Ž .

w xThis relation was used in Refs. 4,5 to show that, ifthere are singletons in AdS , the theory on thedq1

boundary is in fact an LCFT. A theory of freesingletons is formulated in terms of a dipole-ghost

w xpair of fields A and B which satisfy 6

E mE qm2 AqBs0, E mE qm2 Bs0 2Ž .Ž . Ž .m m

these fields have the bulk AdS action

1Dq 1 mn 2 2'Ss d x g g E AE Bym ABy B 3Ž .Ž .H m n 2

The fields A and B couple to boundary fields C andŽ .D and using Eq. 1 the two-point functions of C

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00390-7

( )A. LewisrPhysics Letters B 480 2000 348–354 349

Ž w xand D are found to be see Ref. 5 for details of the.calculation

² :C x C y s0,Ž . Ž .c

² : ² :C x D y s D x C y s ,Ž . Ž . Ž . Ž . 2< <xyy D

1² : < <D x D y s dy2cln xyy 4Ž . Ž . Ž .Ž .2< <xyy D

Ž . 2with the dimension D given by D DyD sm ,Ž .csD 2 DyD and ds2 DyD. these are the usual

two-point functions for logarithmic operators in CFTw x7,8 . These correlation functions occur if the Hamil-

Žtonian in two dimensions, the Virasoro generator.L is non-diagonalizable, and has the Jordan form0

< : < : < : < : < :L C sh C , L D sh D q C 5Ž .0 0

and similarly for L , where for singletons we have0

hsh and so Ds2h. The theories with this type ofoperators are called Logarithmic CFTs and their

w xproperties have been studied extensively 8 sincew xthey were introduced in Ref. 7 . Applications of

LCFT to strings and D-brane scattering were devel-w xoped in Refs. 9,10 . A recent paper relevant to AdS3

w xis 11One way to see if fields of this type exist in a

theory is to look at the four-point functions of ordi-nary fields. If there are no logarithmic operators, theoperator product expansion for primary fields has theform

f ki j

O x O x ; O x q PPPŽ . Ž . Ž .Ýi 1 j 2 k 1D qD yDi j k< <xi 12

6Ž .² Ž . Ž .: < <y2 D iand O x O x s x d , which leads toi 1 j 2 12 i j

an expansion for four-point functions of the form² :O x O x O x O xŽ . Ž . Ž . Ž .i 1 j 2 j 3 i 4

f k f li j i j

s F x 7Ž . Ž .Ý D qD D qDi j i j< < < <x xkl 12 34

Ž .where xsx x rx x and F x has an expan-12 34 13 24

sion in powers of x, without any logarithmic singu-larity. If there are logarithmic operators however, theOPE has to be modified and we have instead

12< <O x O x ; DqC ln xŽ . Ž . Ž .i 1 j 2 12D qD yDi j< <x12

q PPP 8Ž .

which together with the two-point functions for CŽ .and D leads to four point functions of the form 7 ,

x™0 2 D6Ž .but with F x x ln x. Indeed, logarithmicsingularities have been found in four point functions

w xcalculated in supergravity on AdS 12,13 , and it is5

possible that these could be an indication that thereis an LCFT on the boundary of AdS . However,5

these logarithms could also be accounted for as theperturbative expansion of anomalous dimensions in

w xCFT , with no need for logarithmic operators 14 .4

The clearest evidence for the existence of logarith-mic operators in AdSrCFT comes from calculationsof grey-body factors in AdS . Since grey-body fac-3

tors are related to two-point functions in CFT, loga-rithms here are a clear indication that we havelogarithmic operators on the boundary.

Ž .The grey-body factor or absorption cross sectionŽ .for a field in AdS which couples to a field O x in3

the CFT on the boundary is related to the two pointw xfunction in the CFT by 15,16

p2s s d x GG ty ie , x yGG tq ie , x 9Ž . Ž . Ž .Habs

v

Ž . ² Ž . Ž .:where GG t, x s O x,t O 0 is the thermalGreen’s function in imaginary time. This can bedetermined from the periodicity in imaginary time

w xand the singularities of the Green’s function 15 ,which if O is a primary field with weights h,h, aregiven by

CCO² :O t , x O 0 ; 10Ž . Ž . Ž .2 h 2 hx xq y

Ž .with x s t"x. GG t, x has the form"

2 hp T Rq

GG t , x sCCŽ . ž /sinh p T xŽ .q q

=

2 hp T Ry

11Ž .ž /sinh p T xŽ .y y

for a BTZ black hole with mass Msr 2 yr 2 ,q yangular momentum Js2 r r , left and right tem-q y

Ž .peratures T s r "r r2p , and Hawking temper-" q y

( )A. LewisrPhysics Letters B 480 2000 348–354350

w xature given by 2rT s1rT q1rT 17 . The ab-H q yw xsorption cross section is then 15,16

s h ,hŽ .abs

2 hy1 2 hy1p CC 2p T R 2p T R vŽ . Ž .q y

s sinh ž /v 2TG 2h G 2hŽ . Ž . H

=

2v v

G hq i G hq i 12Ž .ž / ž /4p T 4p Tq y

This expression can be obtained either using thew xeffective string method for supergravity 15 or using

w xthe AdsrCFT correspondence 16,18 .A large number of classical calculations of ab-

sorption cross sections have given results which areŽ . Žconsistent with 12 or a similar expression for

. w xfermions 15 , including calculations for severalw xfields for the BTZ black hole 19,16 . However, in

w xRef. 19 the cross section for gauge bosons withŽ .spin 2, which couple to fields with h,hs 2,0 or

Ž .0,2 on the boundary was found to have logarithmiccorrections which cannot be accounted for by Eq.Ž . w x12 . In Ref. 20 , the grey-body factor for singletonswas calculated, and while this does have a logarith-

Ž .mic correction to the cross section 12 , it was foundthat it still does not give the correct cross section forthe gauge bosons. The question we would like toaddress in this letter is, are there other kinds oflogarithmic operators in AdS rCFT , and can they3 2

correctly account for the greybody factor for thegauge bosons?

The greybody factor for the gauge bosons withspin ss2 in AdS , in the low temperature limit3

w xv4T , was found to be 19"

gb 2 2s sp v R 1qv Rsln 2v Rs 13Ž . Ž .abs

Ž .In the low temperature limit, Eq. 12 becomes, up toŽa normalization whch is proportional to CC Dshq

.h ,

2 Dy3s h ,h ;v 14Ž . Ž .abs

Ž .So that the second term in Eq. 13 is an indicationthat the gauge bosons cannot just couple to ordinaryprimary fields on the boundary. The greybody factor

Ž .for a singleton can also be calculated from Eq. 9 ,E² Ž . Ž .: ² Ž . Ž .:using the relation D t, x D 0 s C t, x D 0 ,ED

² Ž . Ž .:since C t, x D 0 is the same as the two point

function for an ordinary primary field. The greybodyfactor for a singleton is therefore given by s S sabs

Ž . w xEs h,h rED 20 , and soabs

2 hy1 2 hy1p CC 2p T R 2p T RŽ . Ž .q ySs sabsv G 2h G 2hŽ . Ž .

=v v

sinh G hq iž / ž /2T 4p TH q

=

2v 1 E CC

G hq i q ln 2p T RŽ .qž /4p T CC EDy

qln 2p T R yc h yc 2hŽ . Ž . Ž .y

v v1q c hq i qc hy i2 ½ ž / ž /4p T 4p Tq q

v vqc hq i qc hy i ,5ž / ž /4p T 4p Tq q

15Ž .

which in the low temperature limit reduces to

s S ;v 2 Dy3 2ln v R qcX 16Ž . Ž .Ž .abs

In an LCFT we always have the freedom to shift DŽ .by D™DqlC, which leaves Eq. 5 invariant, and

this can be used to choose any value for the constantX Ž . Ž .c . However, comparing Eqs. 16 and 13 , we can

Ž .see that the logarithmic term in 13 is multiplied byan extra factor of v R and is thus of a sub-leading

Ž .order compared to 16 . The gauge boson cannotw xtherefore be represented by a singleton in AdS 20 .3

However, we cannot immediately conclude, as wasw xsaid in Ref. 20 , that the gauge boson has nothing to

do with the AdSrLCFT correspondence, becausethere is potentially a much richer spectrum of loga-rithmic operators in a two dimensional LCFT thanhas been considered so far. The logarithmic opera-tors we have looked at so far arise when the dimen-

Ž .sions of two of the primary fields O in the OPE 6k

become degenerate, which leads to logarithms in thefour-point functions and the OPE has to be modifiedto include the logarithmic pair C and D, as in Eq.Ž .8 . In fact, logarithms will also arise in the four


point function if two of the primary fields havedimensions which are not equal, but differ by aninteger, so that it is a descendant of one primary fieldwhich becomes degenerate with the other primary

Ž .field. This is because the function F x in theŽ .four-point function 7 usually satisfies a Fuchsian

differential equation, such as a hypergeometric equa-tion, and when there are no degenerate dimensionsthe solutions have the form

`

D niF x ;x a x 17Ž . Ž .Ý nns0

but when two of the dimensions differ by an integer,say D sD qN, the second solution instead has the2 1

form

`

D n niF x ;x a x qb x log x 18Ž . Ž .Ž .Ý n nns0

in this case we again have a logarithmic pair with thehigher of the two dimensions which as before make

Ž .the Hamiltonian non-diagonalizable, as in Eq. 5 . Inaddition the C field satisfies in both cases the usualcondition for a primary field

< :L C s0, nG1 19Ž .n

However, in the earlier situation where two primaryfields became degenerate, D also satisfied this condi-tion, while in the case where we have two fieldswhose dimension differs by an integer N we haveinstead

N X< : < : < :L D sb C , L D s0, nG2 20Ž . Ž .1 n

where CX is another primary field, with conformalŽ .weights hyN,h , and b is some constant. C is

now not really a primary field, but rather a descen-X < : < X:dant of C : C ss C , where s is someyN yN

combination of Virasoro generators and, in general,the other generators of the chiral algebra of the CFT,

Ž .of dimension N. Eq. 19 then implies that C mustw xbe a null vector of the CFT, that is L ,s s0 forn yN

w x ŽnG1 21 which is why the two-point function² : .CC s0 . This type of logarithmic operator there-fore cannot exist with any dimension, but only withthose dimensions for which there are null vectors ofthe algebra. Because of this, we can only have thesegeneralized logarithmic operators in 2-dimensional

CFT, and we do not expect them in AdS forDq 1

D)2.The logarithmic pair C and D still have the same

Ž . Xcorrelation functions 4 , and C is just an ordinaryprimary field with the usual two point function

1X X² :C x , x C y0 ; 21Ž . Ž . Ž .q y 2ŽhyN . 2 hx xq y

so it seems that these new fields cannot give usanything new when we compute greybody factors.However, it is easy to see that we can reproduce the

Ž .greybody factor for the gauge bosons 13 if weassume that they correspond not to one of the fieldsC, D or CX in the LCFT, but to a linear combinationof all three. This might happen, for example, if thebosons can be thought of as arising from the fusionof two primary fields, since C, D and CX mustalways appear together in any OPE. Then if CX hasdimension Ds2, as is expected for the gauge bosonsw x X20 , and C ,C, D form a representation of the typediscussed above with Ns1, the greybody factorwill have exactly the right form, with the logarithmicterm being of sub-leading order. Of course, thiswould imply that the representation which includesthe primary field CX must have a null vector at level

X Ž . Ž . Ž1. This would be true if C has h,h s 0,2 orX XŽ .. < : Ž < :.2,0 , as then L C or L C is a null vector.y1 y1

X Ž . Ž .If C has h,h s 1,1 , there could still be a null;vector if, for example, the CFT on the boundary has

< X:a conserved current for which J C s0.y1

Now that we know there could be fields in anLCFT on the boundary that give the correct grey-body factor for the gauge bosons, the next questionwe address is, what sort of fields in the bulk cancouple to these fields on the boundary? To answerthis question, we start be reviewing how the confor-mal weights of fields on the boundary determine themass and spin of fields in the bulk when there are nologarithmic operators. We write the metric for AdS3

in the form

ds2 s l 2 ycosh2r dt 2 qsinh2r df 2 qdr 2 22Ž .Ž .

In these coordinates the Virasoro generators L , L ,0 "1w x w xwith commutators L , L s.L and L , L0 "1 "1 1 y1


Ž .s2 L , for spin s fields are ustqf, Õstyf0w x22,23 :

L s iE ,0 u

1yi uL s ie coth2 rE y Ey1 u Õž sinh2 r

i iq E y scoth r ,r /2 2

1iuL s ie coth2 rE y E1 u Õž sinh2 r

i iy E q scoth r 23Ž .r /2 2

and similarly for L , L with ulÕ and s™ys.0 "1

For a primary fields F , the conditions L FshF0

and L FshF , and L FsL Fs0 can then be0 1 1

solved to give sshyh and

yi ŽhuqhÕ.eF; 24Ž .

hqhcosh rŽ .

Ž .The second Casimir of sl 2, R is

12 2L s L L qL L yL 25Ž . Ž .1 y1 y1 1 02

2 Ž .and similarly for L , so that, using Eqs. 22 andŽ .23 , the sum of the two Casimirs is

2 2 2 m 2 2L qL syl E E qs coth r 26Ž .m

2 2Ž . Ž Ž .For a primary field, L qL Fs 2h hy1 qŽ .. Ž .2h hy1 F , so Eq. 26 can be written as

s2m 2yE E q Fsm F 27Ž .m 2 2ž /l sinh r

which is the equation of motion for a field with spins and mass m in AdS , with the mass3

2 2 2l m s2h hy1 q2 H hy1 ysŽ . Ž .

sD Dy2 28Ž . Ž .

So we can see that the conformal weights h and hon the boundary completely determine the mass and

Ž .spin of the fields in AdS and vice versa . We can3

repeat this analysis for a logarithmic pair C and D

on the boundary. C satisfies the same conditions asyi ŽhuqhÕ. hqhŽ .F above, so we find C;e r cosh r .

The conditions L DshDqC and L DshDqC0 0

then imply that

Ds uqÕq f r C 29Ž . Ž .

Ž .where the function f r will depend on what type oflogarithmic operator we have. In the simplest case,which we expect to give us singletons, we haveL DsL Ds0, which has the solution1 1

Ds uqÕy2 iln cosh r qd C 30Ž . Ž .

where d is an arbitrary constant, which we can set toany value using the freedom to shift D by an amountproportional to C. Evaluating the second Casimirsgives the equations of motion for the fields in AdS3

which will couple to C and D:

s2m 2yE E q Csm C ,m 2 2ž /l sinh r

s2m 2yE E q Dsm Dq4 Dy1 C 31Ž . Ž .m 2 2ž /l sinh r

2 Ž .with m again given by Eq. 28 . When ss0, theseare just the expected equations of motion for single-

Ž . Žton dipole-pair 2 apart from a different normaliza-.tion of C , with the expected relation between the

singleton mass m and the dimension of the logarith-mic operator D. Thus we can see that the mass andspin in AdS are still completely determined by the3

data of the LCFT on the boundary when we havesingletons and logarithmic operators. This also give

w xus another way of seeing, as was found in Ref. 5that there can be no logarithmic operators with Ds1,corresponding to m2 sy1.

Now we can use this map between AdS and3

CFT to see what kind of operators will couple to2

the other kinds of logarithmic operators. In this casewe have three fields C, D and CX, but since C and CX

are both primary fields they will both have the sameŽ .form as before, but with weights h,h for C and

XŽ . Ž .hyN,h for C . D is then given by Eq. 29 withŽ . Ž .f r a solution of the Nq1 ’th order differential


Ž .Nq1equation L Ds0. The second casimirs then1

give the equations of motion in AdS as3

s2m 2yE E q Csm C ,m 2 2ž /l sinh r

s2m 2yE E q Dsm Dq4 Dy1 CqCŽ .m 2 2ž /l sinh r

32Ž .

Which is the same as the equations of motion for thesingleton except that we have the new field is CsL L D. C will therefore be a descendant of CX, ory1 1

in AdS it will correspond to some derivative of the3

field which couples to the primary field CX, and theŽ .action for the singleton 3 should be modified by

adding a term which couples the singleton to the newfield. We will therefore have an interacting theoryinstead of a free singleton, with an action of the form

1mn 2 2'Ss d3 x g g E AE Bym ABy BŽH m n 2

ql AC 33Ž ..where C is a derivative of a field with spin N, for aspinless singleton. In addition, it is important that Cis also a descendant in this case, and so the field Ain the above action is also a derivative of the fieldwhich couples to CX and is not a fundamental fielditself. This is especially significant for the case whenNs1, since then CX has no descendant at level 1except C itself, and so the action in AdS in this3

case is the same as for the ordinary singleton, exceptthat B is now a derivative of a field BX with spin 1.Of course, we also need to add to the action thekinetic and mass terms for the field BX to treat thisfield properly.

Although we have seen that new kinds of loga-rithmic operators can exist in AdS rLCFT , they3 2

cannot exist for just any values of m2 and s - wehave to have a null vector in the CFT on theboundary. This means that to determine if such fieldsreally exist we need to know more about the struc-ture of the CFT, or to calculate four-point functions,from which the OPE could be deduced. However, itseems to be clear that at least one example of thistype of operator is needed to give the correct grey-

body factor for the spin-2 gauge bosons. It is aninteresting question why the same interactions can-not be introduced for singletons which do not havespecial values for the mass, which would lead to acontradiction in the CFT, but is not obviously forbid-den from the three-dimensional point of view. Possi-bly related is the question of why these type of fieldscan exist in AdS but not in AdS for D)2 –3 Dq1

since the full Virasoro algebra applies only to CFTin 2 dimensions, there are no null vectors in D)2and so these type of logarithmic do not exist, al-though there can be singletons and the ordinarylogarithmic pair of C and D in any dimension.

Acknowledgements

This work was supported by Enterprise Irelandgrant no. SCr98r739.

References

w x Ž .1 J.M. Maldacena, Adv. Theor. Math. Phys. 2 1998 231,hep-thr9711200.

w x2 S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Phys. Lett. BŽ .428 1998 105, hep-thr9802109.

w x Ž .3 E. Witten, Adv. Theor. Math. Phys. 2 1998 253, hep-thr9802150.

w x4 A.M. Ghezelbash, M. Khorrami, A. Aghamohammadi, Int. J.Ž .Mod. Phys. A 14 1999 2581, hep-thr9807034.

w x Ž .5 I.I. Kogan, Phys. Lett. B 458 1999 66, hep-thr9903162.w x Ž .6 M. Flato, C. Fronsdal, Commun. Math. Phys. 108 1987

Ž .469; Phys. Lett. B 189 1987 145.w x Ž .7 V. Gurarie, Nucl. phys. B 410 1993 535.w x8 J.S. Caux, I.I. Kogan, A.M. Tsvelik, Nucl. Phys. B 466

Ž .1996 444, hep-thr9511134; M.A.I. Flohr, Int. J. Mod.Ž .Phys. A 11 1996 4147, hep-thr9509166; Int. J. Mod. Phys.

Ž .A 12 1997 1943, hep-thr9605151; M.R. Rahimi Tabar, A.Ž .Aghamohammadi, M. Khorrami, Nucl. Phys. B 497 1997

555, hep-thr9610168; M.R. Gaberdiel, H.G. Kausch, Nucl.Ž .Phys. B 538 1999 631, hep-thr9807091; I.I. Kogan, A.

Ž .Lewis, O.A. Soloviev, Int. J. Mod. Phys. A 13 1998 1345,hep-thr9703028; I.I. Kogan, A. Lewis, Nucl. Phys. B 509Ž .1998 687, hep-thr9705240.

w x Ž .9 I.I. Kogan, N.E. Mavromatos, Phys. Lett. B 375 1996 111,hep-thr9512210.

w x10 I.I. Kogan, N.E. Mavromatos, J.F. Wheater, Phys. Lett. BŽ .387 1996 483, hep-thr9606102; V. Periwal, O. Tafjord,

Ž .Phys. Rev. D 54 1996 3690, hep-thr9603156; W. Fischler,Ž .S. Paban, M. Rozali, Phys. Lett. B 381 1996 62, hep-

thr9604014.


w x11 J. Ellis, N.E. Mavromatos, E. Winstanley, hep-thr9909068.w x12 E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis, L.

Rastelli, hep-thr9903196.w x Ž .13 Sanjay, Mod. Phys. Lett. A 14 1999 1413, hep-thr9906099.w x14 M. Bianchi, S. Kovacs, G. Rossiand, Y.S. Stanev, JHEP

Ž .9908 1999 020, hep-thr9906188.w x Ž .15 S.S. Gubser, Phys. Rev. D 56 1997 7854, hep-thr9706100.w x Ž .16 E. Teo, Phys. Lett. B 436 1998 269, hep-thr9805014.w x17 M. Banados, M. Henneaux, C. Teitelboim, J. Zanelli, Phys.

Ž .Rev. D 48 1993 1506, gr-qcr9302012.

w x18 H.J.W. Muller-Kirsten, N. Ohta, J. Zhou, Phys. Lett. B 445¨Ž .1999 287, hep-thr9809193.

w x19 J.Y. Kim, H.W. Lee, Y.S. Myung, hep-thr9812016.w x Ž .20 Y.S. Myung, H.W. Lee, JHEP 9910 1999 009, hep-

thr9904056.w x21 F. Rohsiepe, hep-thr9611160.w x Ž .22 J. Maldacena, A. Strominger, JHEP 9812 1998 005, hep-

thr9804085.w x Ž .23 J. de Boer, Nucl. Phys. B 548 1999 139, hep-thr9806104.

logarithmic operators in ads3/cft2

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