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

2. Motivation and Introduction  A nonperturbative definition of N=4 SYM would enable us to study its strong coupling regime. N=4 Super Yang-Mills IIB string on AdS 5 xS 5 classical gravitystrong coupling ☆ AdS/CFT correspondence  In order to verify the correspondence, we need understand the N=4 SYM in strong coupling regime, in particular, its non-BPS sectors. Matrix regularization of N=4 SYM

3. Our proposal: Matrix regularization of N=4 SYM on RxS 3 by the plane wave matrix model N=4 SYM on RxS 3 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] PWMM is massive no flat direction What we would like to talk about We perform a perturbative analysis (1-loop) We provide some evidences that our regularization indeed works (cf.) lattice theory given by Kaplan-Katz-Unsal gauge symmetry as a matrix model SU(2|4) sym. ⊂ SU(2,2|4) sym. 16 supercharges32 supercharges Our method has the following features:

4. 1. Motivation and Introduction 2. N=4 SYM on RxS 3 from the plane wave matrix model 3. Perturbative analysis 4. Summary and Outlook Plan of this talk

5. N=4 SYM on RxS 3 from the plane wave matrix model

6. SYM on RxS 3 SYM on RxS 2 plane wave matrix model Action SU(2,2|4) (32 SUSY) SU(2|4) (16 SUSY) [Lin-Maldacena] [Kim-Klose-Plefka]

7. SYM on R×S 2 plane wave matrix model N=4 SYM on R×S 3 (1)+(2) SU(2,2|4) (32 SUSY) SU(2|4) (16 SUSY) Dimensional Reduction (2) Large N reduction (1) Continuum limit of fuzzy sphere (cf.) [Lin-Maldacena] IIA SUGRA sol. with SU(2|4) sym. [Ishiki-SS-Takayama-Tsuchiya] Ishiki’s talk

8. plane wave matrix model vacuum fuzzy sphere SU(2) generator In order to obtain the SYM on RxS 3, we consider the theory around the following vacuum configuration. (2) (1) (large N reduction) (Commutative limit of fuzzy sphere)

9. We obtain SYM on RxS 2 around the monopole background continuum limit of fuzzy sphere SYM on RxS 2 Monopole background (vacuum) (1) We can verify this by using harmonic expansion Fuzzy spherical harmonics Monopole spherical harmonics (PWMM) (SYM on RxS 2 ) Ishiki’s talk Monopole charge

10. (2) Large N reduction : NxN hermitian matrix IR cutoff Reduction procedure UV cutoff (Review) 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. [Eguchi-Kawai][Parisi][Gross-Kitazawa] [Bhanot-Heller-Neuberger][Gonzalez-Arroyo - Okawa]… quantum mechanics

11. Free energy ( direction = R) Suppressed compared to the planar diagrams ☆ How about compact (S 1 ) case? planar nonplanar No suppression ??

12. Free energy ( direction = S 1 ) planar nonplanar Suppressed compared to the planar diagrams !! (new) KK momentum

13. SYM on RxS 2 Monopole background (vacuum) We apply this large N reduction to the construction of N=4 SYM on RxS 3 from SYM on RxS 2 Planar N=4 SYM on RxS 3 nontrivial U(1) bundle play a role of Extension of the large N reduction to a non-trivial S 1 fibration Monopole charge

14. perturbative and nonperturbative instability of the vacuum UV/IR mixing The loop effect may cause the deviation between SYM on RxS 2 and PWMM 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. There may be

15. We obtain the matrix regularization of planar N=4 SYM on RxS 3 by the theory around the vacuum of the plane wave matrix model with to be finite. Nonperturbative definition of N=4 SYM on RxS 3 ☆ Our proposal massive, gauge symmetry, SU(2|4) symmetry(16 SUSYs) [Ishii-Ishiki-SS-Tsuchiya, arXiv:0807.2352[hep-th]]

16. Tadpole decoupling of overall U(1) Restoration of SO(4) and We perform a perturbative calculation at the 1-loop order. We adopt the Feynman-type gauge Perturbative analysis SYM on RxS 2 SYM on RxS 3

17. no dependent divergences Fermion self-energy and 2+1 dim. theory is super-renormalizable logarithmic divergence in agree with the calculation in the continuum theory (Feynman gauge) SYM on RxS 2 SYM on RxS 3

18. strong evidence for the restoration of the SU(2,2|4) symmetry Outlook Wilson loop [Ishii-Ishiki-Ohta-SS-Tsuchiya] [Erickson-Semenoff-Zarembo][Drukker-Gross] 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. Summary We propose a nonperturbative definition of planar N=4 SYM on RxS 3 by the plane wave matrix model. The planar limit and 16 SUSY protect us from the instanton effect and the UV/IR mixing. Our regularization keeps the gauge sym. and the SU(2|4) sym. numerical simulation [Hanada-Nishimura-Takeuchi] [Anagnostopoulos-Hanada-Nishimura-Takeuchi] [Catterall-Wiseman]