1 / 18

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

arch
Download Presentation

Finite N Index and Angular Momentum Bound from Gravity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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?

More Related