1. Finite N Index and Angular Momentum Bound from Gravity “KEK Theory Workshop 2007” Yu Nakayama, 13 th. Mar. 2007. (University of Tokyo) Based on hep-th/0701208

2. 0. Introduction Classification of (S)CFT  2 dimension CFT (BPZ…) Central charge Character  2 Dimension SCFT Witten index Elliptic genus  Witten index  Central charge (a-theorem, a-maximization)  Character?  Index for 4-dimensional SCFT  Geometrical classification via AdS-CFT? Similar classification exists for 4-dimensional SCFT?

3. Witten index for supersymmetric field theory Witten Index on R 4 (or T 3 ×R) captures vacuum structure of the supersymmetric (field) theories  Bose-Fermi cancellation Only vacuum (H=0) states contribute Does not depend on  Many applications Study on vacuum structure Implication for SUSY breaking Derivation of index theorem (geometry)

4. The index for 4d SCFT Consider SCFT on S 3 × R. The index (Romelsberger, Kinney et al) can be defined by a similar manner.  Properties Only short multiplets (Δ=0) states contribute Does not depend on β No dep on continuous deformation of SCFT The index is unique (KMMR) Captures a lot more information of SCFT!

5. AdS-CFT @ Finite N Index can be studied in the strongly coupled regime  AdS/CFT duality Large N limit  SUGRA approximation  Excellent agreement  N=4 SYM (KMMR)  Orbifolds and conifold (Nakayama) Finite N case?  1/N ~ g s  Quantized string coupling?  What is the fundamental degrees of freedom? Index does not depend on the coupling constant

6. Finite N Index and Angular Momentum Bound Finite N Index and Angular Momentum Bound from Gravity Yu Nakayama

7. Index for N=4 SYM (g YM = 0) Only states with will contribute.

8. Contribution to Index Chiral LH multiplets and LH semi-long multiplets contribute to the Index Chiral LH multiplet LH semi-long multiplet

9. Computation of index from matrix model (AMMPR) Strategy to determine S eff  Count Δ=0 single letter states  Integrate over U  Or direct path integral Path integral on S 3 ×R reduces to a matrix integral over the holonomy (Polyakov loop)

10. Large N Limit vs Finite N Introduce eigenvalue density  evaluate saddle point Saddle point is trivial  leading contribution is just Gaussian fluctuation Finite N  seems difficult.  Even for SU(2), we have to evaluate Explicit integration is possible in the large N limit

11. Maximal Angular Momentum Limit We take  Only states with will contribute. Why do we call maximal angular momentum limit?  The limit prevents us from taking too large j 1 with fixed j 2.  Not protected by any BPS algebra!! We propose a new limit, where the matrix integral is feasible

12. Index in maximal angular momentum limit For SU(2), we have Similarly, they are trivial for SU(N). Agrees with gravity (large N limit). No finite N corrections Index is trivial nontrivially! No finite N corrections!

13. Partition function For SU(2) For SU(3) For SU(∞) Partition function does have finite N corrections in the maximal angular momentum limit Does not agree with gravity computation Partition function is nontrivial with finite N corrections

14. Maximal Angular Momentum Limit from Gravity Finite N Index and Angular Momentum Bound from Gravity Yu Nakayama

15. Physical meaning of angular momentum bound? No consistent interacting theory with (finitely many) massless particles spin > 2.  Gives the maximal angular momentum bound for dual CFTs. Highest weight state should satisfy j 1 ≦ 1, j 2 ≦ 1.  Only decoupled free DOF contributes to the index in this limit.  Any CFTs with dual gravity description (e.g. any Sasaki-Einstein) should satisfy this bound.  Again there is no general proof from field theoy. Nontrivial bound! SUGRA admits only massless particle spin up to 2!

16. Contribution from BH? Asymptotically AdS (extremal = BPS) Black holes have charge They do not satisfy maximal angular momentum bound.  consistent with our results They are not exhaustive? In high energy regime, black holes may contribute to the index

17. Summary and Outlook Finite N Index and Angular Momentum Bound from Gravity Yu Nakayama

18. Summary and Outlook Counting states (index) for finite N gauge theory is of great significance.  Basic building blocks for nonperturbative string theory  Nature of quantum gravity Difficult problem in general. Maximal Angular Momentum Limit was proposed. No finite N corrections for index in this limit. Finite N corrections for full index?