Multiple M2 branes and Nambu bracket

Yutaka Matsuo (U. Tokyo)

Talk at YITPJuly 29th, 2008

Developments of Multiple M2Bagger-Lambert (arXiv:0711.0955) ~100 citations (April 2008 ~)

M2-M5 Nahm eq.

BLG

Lie 3 algebra

N=4

Mass deformation ABJM

AdS/Integrability

LorentzianBLG

M5 & NP

Basu-Harvey

Gauntlett-GutowskiPapadopoulosHo-Hou-MFigueroa-O’Farrill et.al.

Ho-MHo-Imamura-M-ShibaChu-Ho-M-Shiba

Gomis-Milanesi-RussoVerlinde et. al. Ho-Imamura-M

Hosomichi –Lee3-Park

Minahan-ZaremboGaiotto-Giombi-Yin

Bagger-Lambert

Bagger-LambertGustavsson

Gaiotto-Witten

Aharony-Bergman-Jafferis-Maldacena

Goal?

I.

II.

III.IV.

V.

VI.

I. Before BLG

Nambu bracket

Y. Nambu (1973) Generalization of Poisson bracket

ff1; f2; f3g=X

i jk

² i jk@i f1@jf2@kf3

Dynamical system with two Hamiltonians

H;K : two HamiltoniansdOdt

= fO;H;Kg

Application to the motion of tops

[f1; f2; f3]=X

¾

(- 1)¾f¾(1)f¾(2)f¾(3)?Quantization

Long lasting difficult questions!

Nambu bracket and membrane

S =Zd3¾

p- detGi j +C¹ º ¸@0X¹ @1Xº@2X¸

Gi j =@X¹

@¾i¢@X¹

@¾j

detG = fX¹ ;Xº ;X¸gfX¹ ;Xº ;X¸gf²;²;²g: Nambu bracket

Membrane action can be written in terms of Nambu bracket

Multiplicities of M-branesN D-branes : Gauge symmetry U(N)DOF=N2=Number of open stringsExpressed by Matrices

M-theory (Klebanov-Tseytlin ‘96)

N M2-branes O(N3=2)N M5-branes O(N3)

•AdS/CFT absorption cross section•Brane thermodynamics•Anomaly cancellation

Review : D. Berman arXiv:0710.1707

Basu-Harvey hep-th/0412310

D1-D3 system

x1;2;3

x9

D3: 0 1 2 3D1: 0 9

D3

N D1=BPS monopole

From D3: BPS Monopole solution

X9 =N¼®0

p(X1)2 +(X2)2 +(X3)2

(multiple) D1 viewpoint

Nahm equation:@Xi

@X9§

i2² i jk[Xj;Xk]= 0 Xi : N £ N matrices

Solution Xi =§1

2X9®i ; [®i ;®j]= 2i² i jk®k

®i : N dimrepresentation of SU(2)

R=

s(2¼®0)2

N

X

i

Tr(Xi)2 »¼®0NX9

Match!

M2-M5

x1

x2

x3;4;5;6

s

M2 0 1 2M5 0 1 3 4 5 6

Self-dual string on M5

s »NR2

; \ ridge" solution

R2 = (X3)2 +(X4)2 +(X5)2 +(X6)2

Eq. that corresponds to Nahm’s equation

dXi

ds=

¸M 311

4!8¼² i jkl [G;Xj;Xk;Xl ]

Xi : fuzzy S3; G2 = 1

How to derive Basu-Harvey equation? Motivation for Bagger-Lambert-Gustavsson model

II. BLG model and early development

J. Bagger, N. Lambert, arXiv:0611108, 0711.0955, 0712.3738A. Gustavsson: 0709.1260

0803.3218 Mukhi, Papageorgakis0803.3803 Raamsdonk0804.1114 Lambert, Tong0804.1256 Distler, Mukhi, Papageorgakis, Raamsdonk0804.1784 Gran, Nilsson, Petersson

Lie 3-algebraT (a =1» dimA =D) basis\ Lie3-algebra": [T ;T ;T ]= f T

\Metric": hT ;T i =h

\Fundamental identity (FI)"

[T ;T ;[T ;T ;T ]]= [[T ;T ;T ];T ;T ]+[T ;[T ;T ;T ];T ]+[T ;T ;[T ;T ;T ]]

Invarianceof metrich[T ;T ;T ];T i +hT ;[T ;T ;T ]i = 0

BL action

L = -12hD¹ XI ;D¹ XI i +

i2h¹ª ;¡¹ D¹ ª i

+i4h¹ª ;¡I J [XI ;XJ ;ª ]i - V(X) +LCS

V(X) =112

h[XI ;XJ ;XK];[XI ;XJ ;XK]i

LCS =12² ¹ º ¸(fabcdA ¹ ab@ºA¸cd +

23fcda

gfefgbA ¹ abAº cdA¸ef)

XI =XIaT

a (I = 1» 8); ª = ª aTa

(D¹ XI )a = @¹ XIa - fcdb

aA ¹ cdXIb

Gauge symmetry

±¤ XIa = fbcd

a¤ bcXId

±¤ ª a = fbcda¤ bcª d

±¤ A ¹ ab = @¤ ab - fcdeaA ¹ cd¤ eb +fcde

bA ¹ cd¤ ea

Supersymmetry (N=8 maximal SUSY)

±XI = i¹² ¡ Iª

±ª =D¹ XI ¡ ¹ ¡ I ² -16[XI ;XJ ;XK]¡I J K²

±(~A ¹ )ba = i¹² ¡¹ ¡IXIaª dfcdb

a; (~A ¹ )ba = fcdbaA ¹ cd

Properties of BLG model

• Gauge symmetry based on Lie 3-algebra• Maximal SUSY (N=8)• No arbitrary parameter except for structure

constant• Gauge field described by Chern-Simons

Lagrangian (No propagating d.o.f)• Lagrangian is defined only through structure

constant (adjoint representation)• BPS equation takes the form of Basu-Harvey

Study of A4 modelFirst example: A4 -- SO(4) inv. algebra (BLG)

fabcd = ² abcd ; hab =±ab cf. Kawamura

Initial confusion: how many M2 branes ?Number of moduli: elements that satisfies [Ta;Tb;Tc]=0

One may take such generators as T1, T2 (moduli=2) If we add 1 for center of M2 branes, the number of M2 may be 3. This initial confusion was resolved by introducing Higgs-like mechanism　 (Mukhi and Papageorgakis) : number of M2 is 2 For level k theory, the moduli is conjectured to be (R8)2/D2k

III. Study of Lie 3 algebra

0803.3242 Bandres, Lipstein, Schwarz0804.2110 Ho, Hou, Matsuo0804.2662 Papadopoulos0804.3078 Gauntlett, Gutowski0805.4363 Mendeiros, Figueroa-O’Farril, Mendez-Escobar0806.3242 Mendeiros, Figueroa-O’Farrill, Mendez-Escobar

NO GO theoremOther algebra? many studies

• FI• h > 0• finite D•anti-symmetric

Only possible Lie 3-algebra is A4 and its direct sum !(NO GO THEOREM)

HHM(1) ConjecturePapadopoulos 0804.2662Gauntlett & Gutowski 0804.3078

Escape from NO-GO theoremWith milder condition, there exists other Lie 3-algebras

• negative/null norm generators

• infinite D

Lorentzian BLG model

M5

• non anti-symmetric ABJM model (N=6 SUSY)Bagger-Lambert

Ho-M, Ho-Imamura-M-Shiba

Lorentzian Lie 3-algebraLie algebra + 1 extra generator

¯Ti (i = 1;¢¢¢;dimg)T0; T- 1 cf. Awata, Li, Minic, Yoneya ‘99

Lie 3-algebra [T0;Ti ;Tj]= f i jkTk

[Ti ;Tj;Tk]= f i jkT- 1

[T- 1;Ta;Tb]= 0

T0 appear only on LHS

T- 1 center

Invariant metrichT- 1;T0i = - 1; hTi ;Tj i = hi j Killing form

Lorentzian signature

Further study: Mendeiros, Figueroa-O’Farril, Mendez-Escobar

Nambu-Poisson bracketff1; f2; f3g= P¹ 1;¹ 2;¹ 3(x)@¹ 1f1@¹ 2f2@¹ 3f3

FIff1; f2; ff3; f4; f5gg= fff1; f2; f3g;f4; f5g

+ff3; ff1; f2; f4g;f5g+ff3; f4; ff1; f2; f5gg

anti-symmetric tensor

It implies a very strong constraint on P !!

Decomposability

Pi := P¹i @¹ (i = 1;2;3)

P =P¹ 1¹ 2¹ 3@¹ 1 ^@¹ 2 ^@¹ 3 =P1 ^P2 ^P3

Nambu-Poisson bracket exists only in 3 dimensionsInfinite dim. Lie 3-algebra

Non-antisymmetric 3-algebraTo keep N=6 SUSY, we do not need antisymmetry 3-algebra

[Ta;Tb;Tc]= fabcdTd ; fabc

ehed = fabcd

fabcd = - fbacd = - fabdc = (fcdab)¤

Such 3-algebra can be realized by rectangle matrices

A = f(Ta)rsg; r = 1;¢¢¢;n;s =1;¢¢¢;m

where

[A;B;C]=ACyB - BCyA; hA;Bi =Tr(AyB)

When n=m, BLG model based on this 3-algebra is ABJM model!

with fundamental identity for f and invariance of metric h

IV. Lorentzian BLGHIM(3) arXiv:0805.1202J.Gomis, G.Milousi, J.G.Russo 0805.1012S.Benvenuti, D.Rodriguez-Gomez, E.Tonin, H.Verlinde arXiv: 0805.1087

Generic feature of Lorentzian systemT0 : Generators never produced by 3-commutator (X)

T- 1 : Center(s) of 3-algebra (Y)T : Other generators (Z)

@2X =0; @2Y = f1(X;Z); @2Z = f2(X;Z)

Equation of motion

Symmetry transformation (SUSY, gauge transf.)

±X = 0; ±Y = g1(X;Z); ±Z =g2(X;Z)

• The value of Y fields do not affect other fields: Y are irrelevant with dynamics

• X are free fields : Their VEV does not break symmetry of the system (like coupling const?)

More explicitly

XI = XI0T

0 +XI- 1T

- 1 +X; XI = XIiT

i

A ¹ = T- 1 Ð A ¹ (- 1) - A ¹ (- 1) Ð T- 1

+T0 Ð A ¹ - A ¹ Ð T0 +A ¹ i jTi Ð Tj

Aº = A ¹ 0iTi ; A 0¹ :=A ¹ i jf i j

kTk

The bosonic part of the lagrangian becomes

L = -12(D¹ XI - A 0

¹ XI0)

2 +14(X0)2[XI ; XJ ]2 -

12(XI

0[XI ; XJ ])2

+12² ¹ º ¸ F¹ ºA 0

¸ +(fermion) +Lgh

where

D¹ XI = @¹ XI - [A ¹ ; XI ]F¹ º = @¹ Aº - @º A ¹ - [A ¹ ; Aº ]

Lgh = - @¹ XI0A

0¹ X

I +@¹ XI0@¹ XI

- 1 +(fermion)

Treatment of ghost lagrangian

Variation of X- 1

@2XI0 = 0; ¡ ¹ @¹ ª 0 =0

X0 and ª 0 are free¯eld!

We can treat them as classical fields!

One can setwithout losing consistency of e.o.m.

XI0 =const.(= v±I

10); ª 0 =0

SUSY and Gauge symmetry can be kept

After thisLgh=0!No ghost in the theory

D2 actionOne can integrate A’ m in the Lagrangian

cf. Mukhi, Papageorgakis 0803.3218

Leff = -12(D¹ XA )2 +

v2

4[XA ;XB]2 +

i4¹ª ¡ ¹ D¹ ª

-1

4v2F2¹ º (A;B =3» 9)

Higgs-like mechanism : X10 is converted into d.o.f. of gauge fields

Removal of ghosts was carried out by introducing extra gauge symmetry by Bandres-Lipstein-Schwarz (0806.0054) and Gomis-Rodriguez-Gomez-Raamsdonk-Verlinde (0806.0738).

M2 or D2 ?•By dualizing diagonal part of U(N) gauge fields, one obtains 8th extra dimensions which describe the motion of M2 in 11th dimension.

•Moduli space: (R8) N/SN which coincides with moduli of M2

•The M2 limit corresponds to v ∞ : It implies that the coupling is infinitely large : So it may not be a practical description to study multiple M2

•Recent study of the structure of the vacua of the mass deformed theory (Gomis et. al. 0807.1074) or membrane scattering (Verlinde, 0807.2121) also imply that Lorentzian BLG model does not give a good description of multiple M2 branes.

•Therefore it would be more practical to regard Lorentzian BLG model as the description of D2 branes in the context of BLG

V. ABJM

0805.3662 Hosomichi, Lee, Lee, Lee, Park0806.1519 Benna, Klebanov, Klose, Smedback0806.3391 Nishioka, Takayanagi0806.3498 Honma, Iso, Sumitomo, Zhang0806.3727 Imamura, Kimura0806.3951 Minahan, Zarembo0807.1074 Gomis, Rodriguez-Gomez, Raamsdonk, Verlinde

0806.1218 Aharony, Bergman, Jafferis, Maldacena

Definition of ABJM modelLagrangian (Component form from Benna-Klebanov-Klose-Smedback, Bagger-Lambert )

• U(N)x U(N) gauge symmetry: ZA complex NxN matrices• Chern-Simons term: level k + level (-k)• N=6 SUSY (SU(4) R-symmetry: Rotation of ZA)• No freely adjustable parameters (except for N and k)• 3-algebra NOT NEEDED

Structure of Moduli

Moduli space :(C4=Zk)N

SN=N M2branes on orbifold:

C4

Zk

• Orbifold projection breaks N=8 to N=6 for k>2• For k=1,2, we expect to have N=8 SUSY• For N=2, it reduces to A4 BLG model

Origin of orbifold projection: quantization of CS term

by gauge transformation

Brane constructionIIB string NS5+D3+D5 system

NS5 NS5

D3 : 0 1 2 _ _ _6 _ _ _NS5: 0 1 2 3 4 5 _ _ _ _D5 : 0 1 2 3 4 _ _ _ _ 9

k D5N D3

U(N)xU(N) gauge sym.N=2 SUSY

Mass deformation

NS5

D5

(1,k) 5 brane

Generation of level k & (-k) CS term

(1,k) 5 brane

N=3 SUSY

IIA theory

T-duality

D3 D2NS5 KK monopole(1,k)5 KK monopole

M-theory

D2 M2KK momentum C4/Zk background

Gravity dual

• M theory AdS/CFT: AdS4 x S7

• S7 : S1 fibration over CP3

• level k theory: S1 S1 /Zk

• ‘t Hooft coupling : λ= N/k• M theory description is valid for k5 << N• IIA description : λ =(Rstr)4<< 1, N<< k5

• IIA theory : AdS4 x CP3

• Spectrum of SUGRA field coincides with chiral primary in field theory !

Integrable modelColor singlet operator:

Tr(Z¹ZZ¹Z¢¢¢Z¹Z)

IIB AdS4 x S5 IIA AdS4 x CP3 : integrability at two loops

Spin chain Hamiltonian

alternating spin system

Osp(2,2|6) invariant Bethe ansats system

VI. Nambu bracket, M5 & Entropy law

0804.3629 Ho-M0805.2898 Ho-Imamura-M-Shiba0806.0335 Park-Sochichiu0806.4777 Bandos-Townsend0807.0812 Chu-Ho-M-Shiba

:original membrane worldvolume :3 dim mfd where NP structure is defined

x¹

y _¹

M2 (3d) M5 (6d)

M M £ N

M

N

XIaT

a = XIaÂ

a(y) ! XI (x;y)

ª aTa = ª aÂa(y) ! ª (x;y)A ¹ abÂa(y)Âb(y0) ! A ¹ (x;y;y0)

Gauge field bilocal in N?　 No! They appear only in the combination

A ¹ abfabcd = A ¹ abhfÂa;Âb;Âcg;Âdi

=Z

N² _¹ _º _

@@y _¹

@@y0_º

A ¹ (x;y;y0)¯¯¯¯y 0=y

@Âc

@y _Âd

So what we need is,

@@y0_º

A ¹ (x;y;y0)¯¯¯¯y 0=y

=b¹ _º (x;y)

local field in 6d

Division of transverse space

I = 1» 8

_¹ = _1» _3 : identify with N

i = 1» 5 : transversedirection of M5

We put,

X_¹ (x;y) = y _¹ +b_¹ (x;y) = y _¹ +12² _¹ _º _

b_º _(x;y)

Fields on M5Xi ;ª ;b¹ _º ;b_¹ _º

Self-dual two form after field redefinition

Gauge symmetry on M5

±¤ © = ¤ ab(x)fabcd©cÂd(y)

= ¤ abfÂa;Âb;©g

= ² _¹ _º _¤ ab(x)@_¹ Âa@_ºÂb@_©

= ±¤ y_@_©

±¤ y_

= ²_ _¹ _º@_¹ ¤ _º (x;y)

¤ _¹ (x;y) = @0_¹ (¤ ab(x)Âa(y)Âb(y0))jy 0=y

@_¹ ±¤ y _¹ = 0 $ ±¤ : Volumepreservingdi®eo

BLG gauge symmetry reduces to volume preserving diffeo of N

M5 ActionQuadratic part (HM2, complete formula HIMS4)

Lquad = -12[(@¹ Xi)2 +(@_¹ Xi)2]+

i2h¹ª ;(¡ ¹ @¹ +¡ _¹ @_¹ )ª i

-14F2¹ _º _ -

112

F_¹ _º _2 -

12² ¹ º ¸² _¹ _º _

@¹ bº _¹ @_ºb¸ _

Lorentz covariant for Xi and Ψ but non-covariant for b fields

However, the e.o.m becomes covariant even for b fields

self-dual two form fields on M5

Derivation of N3/2 lawChu-Ho-M-Shiba

Cut-off version of Nambu-Poisson bracket

ff;g;hgN = ¼N (ff;g;hg)

Truncate the Hilbert space to polynomials with degree ≤ N

This 3-bracket still satisfies fundamental identity can define N=8 SUSY equation of motion

Number of moduli: mutually commuting elements ff;g;hgN = 0

f, g, h: function of two variables, x1 and x2 #M » N2

Degree of freedom: #D » N3

#D » (#M )3=2!

N3/2 law is a universal feature of (quantum) 3-algebra system?

Quantum Nambu bracket

Cubic matrices (Awata et. al., Kawamura)

Matrix realization (Curtright-Zachos)Fuzzy S3 (Raamsdonk)

Generalization of Moyal product Zariski Quantization (Dito et. al.)

Open Membrane and noncommutative vortex string (Sasakura, Matsuo-Shibusa, Pioline)

Intertwine M2 M5 by taking large N limit?

VII. Conclusion

• BLG model is a real breakthrough to understand M-theory

• It is the first realistic model where 3-algebra becomes the gauge symmetry

• (multiple) D2 and (single) M5 can be described within BLG framework

• There appeared a promising proposal (ABMJ) which would describe multiple M2 for large k

Future directions• N3/2 law was derived from cut-off version of Nambu-

Poisson bracket but cut-off symmetry is not good enough

• ABJM model is good for k>>1 (string regime) but strong interaction makes it difficult to derive N3/2 law for k=O(1)

• BLG model based on genuine 3-algebra (quantum Nambu bracket) would be necessary to describe M-theory regime of M2 brane

• Description of multiple M5 brane is totally missing yet!

Why 3-algebra is needed for M2? Gauge invariant operators:

Two algebra

Spin chain: string like object

Three algebra

Decomposition of S2!

Relation with3d lattice gravityHo-M. (‘07)