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Finite N Index and Angular Momentum Bound from Gravity

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. Classification of (S)CFT 2 dimension CFT (BPZ…) Central charge Character 2 Dimension SCFT

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Finite N Index and Angular Momentum Bound from Gravity

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  1. Finite N Index and Angular Momentum Bound from Gravity “KEK Theory Workshop 2007” Yu Nakayama, 13th. 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 R4 (or T3 ×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 S3 × 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 does not depend on the coupling constant • 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 ~ gs • Quantized string coupling? • What is the fundamental degrees of freedom?

  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 (gYM = 0) • Only states with will contribute.

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

  9. Computation of index from matrix model (AMMPR) Path integral on S3 ×R reduces to a matrix integral over the holonomy (Polyakov loop) • Strategy to determine Seff • Count Δ=0 single letter states • Integrate over U • Or direct path integral

  10. Large N Limit vs Finite N Explicit integration is possible in the large N limit • 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

  11. Maximal Angular Momentum Limit We propose a new limit, where the matrix integral is feasible • We take • Only states with will contribute. • Why do we call maximal angular momentum limit? • The limit prevents us from taking too large j1 with fixed j2. • Not protected by any BPS algebra!!

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

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

  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? SUGRA admits only massless particle spin up to 2! • 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 j1 ≦ 1, j2 ≦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!

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

  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?

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