Shinji Shimasaki (Osaka U.) 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.) (ref.) Ishiki-SS-Takayama-Tsuchiya, JHEP 11(2006)089 [hep-th/0610038] Ishii-Ishiki-SS-Tsuchiya, in preparation

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

What we would like to discuss 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 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) The Ward identity is satisfied. The beta function vanishes in a continuum limit.

Plan of this talk Motivation and Introduction N=4 SYM on RxS3 and the plane wave matrix model 3. Check of the validity (perturbative calculation) 4. Summary and Perspective

N=4 SYM on RxS3 and the plane wave matrix model [Kim-Klose-Plefka] [Ishiki-SS-Takayama-Tsuchiya] (cf.) [Lin-Maldacena] N=4 SYM on R×S3 IIA SUGRA sol. with SU(2|4) sym. SU(2,2|4) (32 SUSY) (1) Matrix T-duality Dimensional Reduction SYM on R×S2 (1)+(2) SU(2|4) (16 SUSY) Dimensional Reduction (2) Continuum limit of fuzzy sphere plane wave matrix model SU(2|4) (16 SUSY)

plane wave matrix model vacuum: SU(2) generator fuzzy sphere In order to obtain the SYM on RxS3, we consider the theory around the following vacuum configuration. periodicity

We expand matrices by using the fuzzy spherical harmonics (basis of a rectangular matrix) [Grosse-Klimcik-Presnajder, Baez-Balachandran-Ydri-Vaidya, Dasgupta-SheikhJabbari-Raamsdonk,] Clebsch-Gordan coeff. The modes become those of SYM on RxS2 and SYM on RxS3.

continuum limit of fuzzy sphere We obtain SYM on RxS2 around the monopole background SYM on RxS2 where Monopole background are identified with the modes of SYM on RxS2 around the monopole background.

& periodicity condition matrix T-duality Moreover & periodicity condition matrix T-duality [Taylor] momentum of S1 SYM on RxS3 orbifolding Completely consistent in classical level How about quantum level ?

☆ some comments We want to set to be finite. But the periodicity condition is incompatible with finite . If we restrict ourselves to the planar limit, we do not need the periodicity condition. Reduced model Possible UV/IR mixing is avoided due to the planar limit and 16 SUSY Instanton effect is suppressed by taking We obtain the regularized theory 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 We perform the perturbative analysis in our regularized theory.

Check of the validity (1-loop calculation) Ward Identity Beta function no quadratic divergence due to the gauge inv. regularization

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. We verified that the Ward identity is actually satisfied and the beta function vanishes in the continuum limit at 1-loop level. Outlook Is the superconformal symmetry SU(2,2|4) indeed restored ? due to 16 SUSY ? Wilson loop [Ishii-Ishiki-Ohta-SS-Tsuchiya][Drukker-Gross] finite temperature [Aharony et.al.]