Finite N Index and Angular Momentum Bound from

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

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

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

The index for 4 d SCFT n 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 n Does not depend on β n No dep on continuous deformation of SCFT n The index is unique (KMMR) n Captures a lot more information of SCFT! n

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

Finite N Index and Angular Momentum Bound from Gravity Yu Nakayama

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

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

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

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

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

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

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

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

Physical meaning of angular momentum bound? SUGRA admits only massless particle spin up to 2! n No consistent interacting theory with (finitely many) massless particles spin > 2. Gives the maximal angular momentum bound for dual CFTs. n 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!

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

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

Summary and Outlook n 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. n Maximal Angular Momentum Limit was proposed. n No finite N corrections for index in this limit. n Finite N corrections for full index? n