Integrability of N=6 Super Chern-Simons Theories

Dongsu Bak

University of Seoul

with S. J. Rey and D. Kang

(KIAS, 9/24/2008)

A new concrete example of AdS/CFT correspondence

N=6 SCS AdS4 X CP3

N=6 Superconformal Chern-Simons Theories ABJM

N=4 SYM AdS5 X S5

The N=6 SCS theory is dual to the type IIA string theory

on AdS4 X CP3 geometry.

N=6 SCS theory IIA Strings on AdS4 X CP31 to 1

We are interested in the planar limit where

with

Strong weak coupling duality

The perturbative expansion is a small lambda expansion. On the other hand the tension of the string

plays the role of in the Nambu-Goto string action.

Hence, for the strings,

expansion corresponds to the

expansion of the usual quantum mechanics.

In this sense the duality here is again a strong-weak coupling duality.

1/N expansion

1/N expansion of the CS theories

the higher genus worldsheet string interactions.

planar non-planar

We like to test

Large N planar limit of N=6 SCS theory

IIA String on AdS4 X CP3 geometry

dual to

Even this version of duality is very nontrivial and the keytechnique would be the integrability.

In this talk we like to compute the operator dimensionsof single trace operators and show integrability of the field theory. Minahan+ Zarembo

Field contents

Covariant derivatives

Action

Potentials

It is a mixed chain of 4 and 4-bar of SU(4).

Single trace gauge invariant operators

Three site interactions + mixed chains (de Vega +Wynarovich)

Yang Baxter equations

Solutions

where

1 2 3

The other set of Yang Baxter equations

with

Introduce T-matrices

and

T-matrices satisfy

and

Then

and

Use the definition

and

We get

Using the parity symmetry, one is led to

with

Perturbative computation of the spin chain Hamiltonian

Anomalous dimension matrix

For this, we compute

We find

Scalar three site interactions

This leads to

where we used

Two site interactions

The gauge two site interactions

Yukawa two site interactions

One gets

One site interaction: wave function renormalization

1/12

Chern-Simons interactions

Paramagnetic gauge interactions 2/3

1/3

Fermion pair contributions

Two loop vacuum polarization

4/3 1

- 8/3

Total Hamiltonian

Wave function renormalization

Therefore,

Wrapping interactions at two loops

15+1

Bethe Ansatz Diagonalization

Eigenvalue of T-matrix

BAE

Momentum and energy

l m r

Nl =Nr= Nm =1 case

Two decoupled SU(2) XXX spin chains!

Nm=0 case

Introduce

No bound states!

Complete two loop Hamiltonian?

Future problems

Higher loops?

Comparisons to the string spectrum?

Osp(6|4)

Gromov Vieira, Ahn NepomachieGrignani Harmark Orselli SemenoffBH Lee Panigrahi C Park …

Geometry

Sigma model action

Magnon spectrum