Sangmin LeeSeoul National University

29 Nov. 2008, PAQFT08, Singapore

Kazuo Hosomichi, Ki-Myeong Lee, SL, Sungjay Lee, Jaemo Park, Piljin Yi [JHEP0807:091, JHEP0809:002, JHEP0811:058]

New Superconformal Chern-Simons Theories

and M2-branes

Holographic duality in various dimensions

(3+1)-dim :

(2+1)-dim :

(1+1)-dim : techniques special to 2d CFTs available.

gravity

well-knowne.g. O(N) vector

??

M-theory??

gravity

Super-Yang-Mills IIB string theory

QFT

Super-Chern-Simons Theories

Super-YM-CS cannot go beyond N = 3. [Kao-K.Lee 92]

CS without YM can go beyond N = 3 ! - useful for N = 8 M2-brane theory ? [Schwarz 04]

Bagger-Lambert theory [Bagger-Lambert 06-07][Gustavsson 07]

- Super-conformal N = 8 CS-matter based on a Lie-3-algebra.

Gaiotto-Witten construction [Gaiotto-Witten 0804.2907]

- Large class of new N = 4 super-conformal CS theories. - List not complete / Relation to M2-branes unclear

N = 4 and twisted multiplets

N = 4 super-algebra includes SO(4) = SU(2)L x SU(2)R R-symmetry

(hyper)

(vector)

SU(2)L SU(2)Rq! 2 1"! 1 2Aµ 1 1#!! 2 2s!$ 1 3

SU(2)L SU(2)R1 2 q!2 1 "!1 1 Aµ2 2 #!!3 1 s!$

(ordinary) (twisted)

Gaiotto-Witten : most general N = 4 with ordinary hypers only.

Our contribution : 1) Added twisted hypers to GW : complete classification2) Explained relation to M2-brane world-volume theories

Gaiotto-Witten construction [Gaiotto-Witten 0804.2907]

Key steps :

SU(2)diag ! SU(2)L " SU(2)R global symmetry.

Begin with N = 1 SUSY theory with

Adjust the N = 1 super-potential to restore .SU(2)L ! SU(2)R

The SO(4) R-symmetry together with N = 1 SUSY generate a full N = 4 structure.

Rewrite the action and SUSY transformation rules in a manifestly N = 4 covariant way.

Field content

(Am)µ,!mGauge :

Gauge group G ! Sp(2n)

(tm)AB satisfy [tm, tn] = f mn

ptp, Tr(tmtn) = kmn tmAB ! !AC(tm)C

B = tmBA

qA! ,"

A! ,F

A!Matter : Sp(2n) with !AB

q!A = (qA! )† = "!#$ABqB

#reality condition :

kmntm(ABtn

C)D = 0 (“fundamental identity”)

Conditions for N = 4 SUSY

Classification in terms of Lie super-algebra [Gaiotto-Witten]

[Mm, Mn] = f mnp Mp, [Mm, QA] = QB(tm)B

A, {QA, QB} = tmAB Mm.

Fundamental identity implies existence of a Lie super-algebra :

[{QA, QB}, QC] + (cyclic) = 0 !" kmntm(ABtn

C)D = 0

Lie super-algebras classifies Gaiotto-Witten theories completely!

G1

G2

q

q

G1 G2q

Typical examples : U(N|M), OSp(N|M)

Adding twisted hypers

G3

G4

G1

G2

G2G1 G3 G4

q

q

q

q

q

q

q qq

Hypers and twisted hypers should independently form super-algebra structure.

Generically, they form quivers with the two types of hypers alternating between gauge groups.

The quiver can either have open ends or form a closed loop.

Short loops exhibit enhanced (N > 4) SUSY.

G1

G2

q q

q qG1 G2

q

q

Summary

!qA" = +i# "

" $A" , !qA

" = !i#""$A

" , !Amµ = 2%i#""&µ('m"" ! 'm

""),

!$A" = +

!DqA

" +2%

3(tm)A

BqB((µm)(

"

"#"

" ! 2%(tm)ABqB

((µm)("#

(

(,

!$A" = !

!DqA

" +2%

3(tm)A

BqB((µm)(

"

"# "

" + 2%(tm)ABqB

((µm)(

"#(

( .

L =!µ"#

4$

!kmn Am

m%" An# +

13

fmnp Amµ An

" Ap#

"

+12

&AB

#!'()DqA

( DqB) + i'()*A

( D/ *B)

$+

12

&AB

#!'()DqA

( DqB)

+ i'()*A( D/ *B

)

$

!i$kmn'()'+, -m(+ -n

),! i$kmn'()'+, -m

(+ -n),

+ 4$ikmn'(+'), -m()

-n,+

+i$kmn

#'(+'),µm

()*A

+ tnAB*B

,+ '(+'),µm

()*A+ tn

AB*B,

$

!$2

6fmnp(µm)(

)(µn))+(µp)+

( !$2

6fmnp(µm)(

)(µn))+(µp)+

(

+$2(µmn)++(µm)(

)(µn))( + $2(µmn)+

+(µm)()(µn))

(,

Embedding in String/M-theory [Gaiotto-Witten] [Hosomichi-Lee3-Park]

NS5 NS5D5

D3(O3)

0 1 2 3 4 5 6 7 8 9D3(O3) ! ! ! !NS5 ! ! ! ! ! !D5 ! ! ! ! ! !

IIB brane configuration

M : Replace (C2/Zn)2 by Taub-NUT(Zn)2

“T-dual” !

N ! 4 Chern-Simons Theories = M2-branes on Orbifolds

[N = 4] [Hosomichi-Lee3-Park][Benna-Klebanov-Klose-Smedback][Terashima-Yagi]

Circular quiver with (2m) gauge group factorsand alternating hyper/twisted-hypers

The theory describes N M2-branes on

Linear quivers with U(N|M) or OSp(N|M)

C4/(Zm !Zmk) or C4/(Zm !Dmk)

Ai

Bi

Di

Ci

Relation to “brane-crystal” model [Kim-SL-Lee-Park]

[N = 5] [Hosomichi-Lee3-Park]

R-symmetry enhancement from SU(2) x SU(2) to USp(4) = SO(5)

Hypers and twisted hypers together form an N = 5 multiplet.

The theory describes N M2-branes on Dk orbifolds.

A derivation of the field theory from M-IIB T-duality and brane construction is given.

Extended OSp(N|M)

[N = 6] [Aharony-Bergman-Jafferis-Maldacena]

R-symmetry enhancement from SU(2) x SU(2) to SU(4).Hypers and twisted hypers together form an N = 6 multiplet.

When M = N, at CS level k, the theory is claimed to describe N M2-branes on C4/Zk orbifold.

A derivation of the field theory from M-IIB T-duality and brane construction is given.

ABJM = Extended U(N|M)

[N = 8] Bagger-Lambert

Based on Lie-3-algebra [Bagger-Lambert 06-07][Gustavsson 07]!

f abcd, hab"

Positive allows nothing but “SO(4)”hab f abcd = !abcd [Papadopoulos][Gauntlett-Gutowski]

SO(4) Bagger-Lambert = Extended PSU(2|2)

From N = 8 to N = 4 :

SO(8) ! SO(4)1 " SO(4)2

# (SU(2)L " SU(2)A)" (SU(2)B " SU(2)R)

X ! q : (2, 2, 1, 1) " q : (1, 1, 2, 2),! ! ! : (2, 1, 2, 1) " ! : (1, 2, 1, 2)," ! # : (2, 1, 1, 2) " # : (1, 2, 2, 1).

The SO(4) theory is an ordinary CS-matter theory [Raamsdonk]

Instantons in 3-dim Yang-Mills

D3 D3

D1

Monopole in 4-dim

D2 D2

D0

Instanton in 3-dim

Idea : compare world-volume theory and bulk theory A non-perturbative test of the ABJM proposal!

Instantons in ABJM theory [Hosomichi-Lee3-Park-Yi]

Classical instanton action D0 : DBI + RR

Strings stretched between D2’sHiggs mechanism

Zero-instanton effective action Super-graviton exchange

D0 exchangeOne-instanton effective action

ABJM model 11/10-dim bulk

subtleties due to imaginary value of Chern-Simons action, etc.

Discussions

Superconformal index [Bhattacharya-Minwalla][Choi-SL-Song]

Beta-deformation [Jeong-Jin-Kim-SL, work in progress]

Vortex solitons, non-relativistic AdS/CFT

Topological twisting of CS-matter theories : relation to pure-CS and Rozansky-Witten theories?

Solution to the N3/2 problem?