integrability of n=6 super chern-simons theories dongsu bak university of seoul with s. j. rey and...
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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