A nonperturbative definition of N4 Super YangMills by

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 ☆ Ad. S/CFT correspondence N=4 Super Yang-Mills strong coupling IIB string on Ad. S 5 x. S 5 classical gravity Ø In order to verify the correspondence, we need understand the N=4 SYM in strong coupling region, in particular, its non-BPS sectors. Ø A nonperturbative definition of N=4 SYM would enable us to study its strong coupling region.

What we would like to discuss n N=4 SYM on Rx. S 3 can be described by theory around a certain vacuum of the plane wave matrix model with Ishiki-SS-Takayama-Tsuchiya, periodicity condition imposed. JHEP 11(2006)089 [hep-th/0610038] Our proposal: Matrix regularization of N=4 SYM on Rx. S 3 by the plane wave matrix model Our method has the following features: • PWMM is massive no flat direction • gauge symmetry as a matrix model (cf. ) lattice theory by Kaplan-Katz-Unsal • SU(2|4) sym. ⊂ SU(2, 2|4) sym. 16 supercharges 32 supercharges n We perform a perturbative analysis (1 -loop) The Ward identity is satisfied. The beta function vanishes in a continuum limit.

Plan of this talk 1. Motivation and Introduction ü　 2. N=4 SYM on Rx. S 3 and the plane wave matrix model 3. Check of the validity (perturbative calculation) 4. Summary and Perspective

N=4 SYM on Rx. S 3 and the plane wave matrix model [Kim-Klose-Plefka] [Ishiki-SS-Takayama-Tsuchiya] N=4 SYM on R×S 3 SU(2, 2|4) (32 SUSY) Dimensional Reduction (1)+(2) (cf. ) [Lin-Maldacena] IIA SUGRA sol. with SU(2|4) sym. (1) Matrix T-duality SYM on R×S 2 SU(2|4) (16 SUSY) Dimensional Reduction plane wave matrix model SU(2|4) (16 SUSY) (2) Continuum limit of fuzzy sphere

plane wave matrix model vacuum: SU(2) generator fuzzy sphere In order to obtain the SYM on Rx. S 3, we consider theory around the following vacuum configuration. periodicity

We expand matrices by using the fuzzy spherical harmonics • Fuzzy spherical harmonics (basis of a rectangular matrix) [Grosse-Klimcik-Presnajder, Baez-Balachandran-Ydri-Vaidya, Dasgupta-Sheikh. Jabbari-Raamsdonk, ] Clebsch-Gordan coeff. The modes become those of SYM on Rx. S 2 and SYM on Rx. S 3.

continuum limit of fuzzy sphere We obtain SYM on Rx. S 2 around the monopole background SYM on Rx. S 2 Monopole background where are identified with the modes of SYM on Rx. S 2 around the monopole background.

Moreover & periodicity condition matrix T-duality SYM on Rx. S 3 momentum of S 1 Completely consistent in classical level How about quantum level ? [Taylor] orbifolding

☆ some comments n 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 n Possible UV/IR mixing is avoided due to the planar limit and 16 SUSY n Instanton effect is suppressed by taking We obtain the regularized theory of planar N=4 SYM on Rx. S 3 by theory around the vacuum of the plane wave matrix model with to be finite. Nonperturbative definition of N=4 SYM on Rx. S 3 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 l We propose a nonperturbative definition of planar N=4 SYM on Rx. S 3 by the plane wave matrix model. l Our regularization keeps the gauge sym. and the SU(2|4) sym. l The planar limit and 16 SUSY protect us from the instanton effect and the UV/IR mixing. l We verified that the Ward identity is actually satisfied and the beta function vanishes in the continuum limit at 1 -loop level. Outlook l Is the superconformal symmetry SU(2, 2|4) indeed restored ? due to 16 SUSY ? l Wilson loop [Ishii-Ishiki-Ohta-SS-Tsuchiya][Drukker-Gross] l finite temperature [Aharony et. al. ]