1. Integrability of N=6 Super Chern-Simons Theories Dongsu Bak University of Seoul with S. J. Rey and D. Kang (KIAS, 9/24/2008) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A AA A

2. A new concrete example of AdS/CFT correspondence N=6 SCSAdS4 X CP3 N=6 Superconformal Chern-Simons Theories ABJM N=4 SYMAdS5 X S5

3. The N=6 SCS theory is dual to the type IIA string theory on AdS 4 X CP 3 geometry. N=6 SCS theory IIA Strings on AdS 4 X CP 3 1 to 1 We are interested in the planar limit where with

4. Strong weak coupling duality The perturbative expansion is a small lambda expansion. On the other hand the tension of the string plays the role ofin 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.

5. 1/N expansion 1/N expansion of the CS theories the higher genus worldsheet string interactions. planarnon-planar

6. We like to test Large N planar limit of N=6 SCS theory IIA String on AdS 4 X CP 3 geometry dual to Even this version of duality is very nontrivial and the key technique would be the integrability. In this talk we like to compute the operator dimensions of single trace operators and show integrability of the field theory. Minahan+ Zarembo

7. Field contents Covariant derivatives

8. Action Potentials

9. 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 )

10. Yang Baxter equations Solutions where 1 2 3

11. The other set of Yang Baxter equations with Introduce T-matrices and

12. T-matrices satisfy and

13. Then and

14. Use the definition and We get

15. Using the parity symmetry, one is led to with

16. Perturbative computation of the spin chain Hamiltonian Anomalous dimension matrix For this, we compute

17. We find Scalar three site interactions

18. This leads to where we used

19. Two site interactions The gauge two site interactions Yukawa two site interactions

20. One gets One site interaction: wave function renormalization 1/12

21. Chern-Simons interactions Paramagnetic gauge interactions2/3 1/3

22. Fermion pair contributions Two loop vacuum polarization 4/3 1 - 8/3

23. Total Hamiltonian Wave function renormalization Therefore,

24. Wrapping interactions at two loops  15+1 

25. Bethe Ansatz Diagonalization Eigenvalue of T-matrix

26. BAE Momentum and energy lmr

27. Nl =Nr= Nm =1 case Two decoupled SU(2) XXX spin chains! N m =0 case

28. Introduce No bound states!

29. Complete two loop Hamiltonian? Future problems Higher loops? Comparisons to the string spectrum? Osp(6|4) Gromov Vieira, Ahn Nepomachie Grignani Harmark Orselli Semenoff BH Lee Panigrahi C Park …

30. Geometry

31. Sigma model action Magnon spectrum