A nonperturbative definition of N=4 Super Yang-Mills by the plane wave matrix model

1. A nonperturbative definition of N=4 Super Yang-Mills bythe plane wave matrix model Shinji Shimasaki (Osaka U.) In collaboration with T. Ishii (Osaka U.), G. Ishiki (Osaka U.) and A. Tsuchiya (Shizuoka U.) (ref.) Ishii-Ishiki-SS-Tsuchiya, arXiv:0807.2352[hep-th] Ishiki-SS-Takayama-Tsuchiya, JHEP 11(2006)089 [hep-th/0610038]

2. Motivation and Introduction ☆ AdS/CFT correspondence IIB string on AdS5xS5 N=4 Super Yang-Mills strong coupling classical gravity • In order to verify the correspondence, we need understand the N=4 SYM in strong coupling regime, in particular, its non-BPS sectors. • A nonperturbative definition of N=4 SYM would enable us to study its strong coupling regime. Matrix regularization of N=4 SYM

3. What we would like to talk about • N=4 SYM on RxS3 can be described by the theory around • a certain vacuum of the plane wave matrix model with • periodicity condition imposed. Ishiki-SS-Takayama-Tsuchiya, JHEP 11(2006)089 [hep-th/0610038] • Our proposal: Matrix regularization of N=4 SYM on RxS3 by the plane wave matrix model Our method has the following features: (cf.) lattice theory given by Kaplan-Katz-Unsal • PWMM is massive no flat direction • gauge symmetry as a matrix model • SU(2|4) sym. ⊂ SU(2,2|4) sym. 16 supercharges 32 supercharges • We perform a perturbative analysis (1-loop) We provide some evidences that our regularization indeed works

4. Plan of this talk • Motivation and Introduction • N=4 SYM on RxS3 • from the plane wave matrix model • 3. Perturbative analysis • 4. Summary and Outlook

5. N=4 SYM on RxS3 from the plane wave matrix model

6. Action [Lin-Maldacena] [Kim-Klose-Plefka] SYM on RxS3 SU(2,2|4) (32 SUSY) SYM on RxS2 SU(2|4) (16 SUSY) plane wave matrix model SU(2|4) (16 SUSY)

7. [Ishiki-SS-Takayama-Tsuchiya] N=4 SYM on R×S3 SU(2,2|4) (32 SUSY) (2) Large N reduction Dimensional Reduction SYM on R×S2 (1)+(2) SU(2|4) (16 SUSY) Ishiki’s talk (1) Continuum limit of fuzzy sphere Dimensional Reduction plane wave matrix model (cf.) [Lin-Maldacena] SU(2|4) (16 SUSY) IIA SUGRA sol. with SU(2|4) sym.

8. plane wave matrix model SU(2) generator vacuum fuzzy sphere In order to obtain the SYM on RxS3, we consider the theory around the following vacuum configuration. (1) (Commutative limit of fuzzy sphere) (2) (large N reduction)

9. continuum limit of fuzzy sphere (1) We obtain SYM on RxS2 around the monopole background SYM on RxS2 Monopole background (vacuum) Monopole charge We can verify this by using harmonic expansion Ishiki’s talk (PWMM) (SYM on RxS2) Fuzzy spherical harmonics Monopole spherical harmonics

10. [Eguchi-Kawai][Parisi][Gross-Kitazawa] [Bhanot-Heller-Neuberger][Gonzalez-Arroyo - Okawa]… (2) Large N reduction A gauge theory in the planar limit is equivalent to the matrix model obtained by dimensionally reducing it to zero dimension if U(1)D sym. is unbroken. (Review) quantum mechanics : NxN hermitian matrix Reduction procedure UV cutoff IR cutoff

11. Free energy ( direction = R) planar nonplanar Suppressed compared to the planar diagrams ☆How about compact (S1) case? No suppression ??

12. Free energy ( direction = S1) (new) planar KK momentum nonplanar Suppressed compared to the planar diagrams !!

13. We apply this large N reduction to the construction of N=4 SYM on RxS3 from SYM on RxS2 SYM on RxS2 Monopole background (vacuum) Monopole charge play a role of nontrivial U(1) bundle Extension of the large N reduction to a non-trivial S1 fibration Planar N=4 SYM on RxS3

14. There may be • UV/IR mixing The loop effect may cause the deviation between SYM on RxS2 and PWMM • perturbative and nonperturbative instability of the vacuum Our theory is massive and has 16 supersymmetries and we take the planar limit There is no UV/IR mixing and no instability of the vacuum.

15. ☆ Our proposal [Ishii-Ishiki-SS-Tsuchiya, arXiv:0807.2352[hep-th]] We obtain the matrix regularization of planar N=4 SYM on RxS3 by the theory around the vacuum of the plane wave matrix model with to be finite. Nonperturbative definition of N=4 SYM on RxS3 massive, gauge symmetry, SU(2|4) symmetry(16 SUSYs)

16. Perturbative analysis We perform a perturbative calculation at the 1-loop order. We adopt the Feynman-type gauge Tadpole SYM on RxS2 decoupling of overall U(1) and SYM on RxS3 Restoration of SO(4)

17. Fermion self-energy no dependent divergences SYM on RxS2 2+1 dim. theory is super-renormalizable and agree with the calculation in the continuum theory (Feynman gauge) SYM on RxS3 logarithmic divergence in

18. Summary • We propose a nonperturbative definition of planar N=4 SYM on RxS3 by the plane wave matrix model . • Our regularization keeps the gauge sym. and the SU(2|4) sym. • The planar limit and 16 SUSY protect us from the instanton effect and the UV/IR mixing. • By performing the 1-loop analysis and comparing the results with those in continuum N=4 SYM, we provide some evidences that our regularization for N=4 SYM indeed works. Outlook • strong evidence for the restoration of the SU(2,2|4) symmetry • numerical simulation [Hanada-Nishimura-Takeuchi] [Anagnostopoulos-Hanada-Nishimura-Takeuchi] [Catterall-Wiseman] • Wilson loop [Ishii-Ishiki-Ohta-SS-Tsuchiya] [Erickson-Semenoff-Zarembo][Drukker-Gross]