A nonperturbative definition of N=4 Super Yang-Mills bythe plane wave matrix modelShinji Shimasaki (Osaka U.)

In collaboration withT. 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]

Motivation and Introduction A nonperturbative definition of N=4 SYM would enable us to study its strong coupling regime.N=4 Super Yang-MillsIIB string on AdS5xS5classical 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

Our proposal: Matrix regularization of N=4 SYM on RxS3 by the plane wave matrix model 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] PWMM is massive no flat directionWhat 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 superchargesOur method has the following features:

Motivation and Introduction

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

3. Perturbative analysis

4. Summary and OutlookPlan of this talk

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

SYM on RxS3SYM on RxS2plane wave matrix modelActionSU(2,2|4) (32 SUSY)SU(2|4) (16 SUSY)SU(2|4) (16 SUSY)[Lin-Maldacena][Kim-Klose-Plefka]

SYM on RS2plane wave matrix modelN=4 SYM on RS3(1)+(2)SU(2,2|4) (32 SUSY)SU(2|4) (16 SUSY)SU(2|4) (16 SUSY)Dimensional ReductionDimensional 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]Ishikis talk

plane wave matrix modelvacuumfuzzy sphereSU(2) generatorIn order to obtain the SYM on RxS3,we consider the theory around the following vacuum configuration.(2)(1)(large N reduction)(Commutative limit of fuzzy sphere)

We obtain SYM on RxS2 around the monopole backgroundcontinuum limit of fuzzy sphereSYM on RxS2Monopole background (vacuum)(1)We can verify this by using harmonic expansionFuzzy spherical harmonicsMonopole spherical harmonics(PWMM)(SYM on RxS2)Ishikis talkMonopole charge

(2) Large N reduction: NxN hermitian matrixIR cutoffReduction procedureUV cutoff(Review)A gauge theory in the planar limit is equivalent to the matrix model obtainedby 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

Free energy ( direction = R)Suppressed compared to the planar diagramsHow about compact (S1) case?planarnonplanarNo suppression ??

Free energy ( direction = S1)planarnonplanarSuppressed compared to the planar diagrams !!(new)KK momentum

SYM on RxS2Monopole background (vacuum)We apply this large N reduction to the constructionof N=4 SYM on RxS3 from SYM on RxS2Planar N=4 SYM on RxS3nontrivial U(1) bundleplay a role ofExtension of the large N reduction to a non-trivial S1 fibrationMonopole charge

perturbative and nonperturbative instability of the vacuum UV/IR mixingThe loop effect may cause the deviation between SYM on RxS2 and PWMMOur theory is massive and has 16 supersymmetriesand we take the planar limitThere is no UV/IR mixing and no instability of the vacuum.There may be

We obtain the matrix regularization of planar N=4 SYM on RxS3by the theory around the vacuum of the plane wave matrix modelwith to be finite.Nonperturbative definition of N=4 SYM on RxS3 Our proposalmassive, gauge symmetry, SU(2|4) symmetry(16 SUSYs)[Ishii-Ishiki-SS-Tsuchiya, arXiv:0807.2352[hep-th]]

Tadpoledecoupling of overall U(1)Restoration of SO(4)andWe perform a perturbative calculation at the 1-loop order.We adopt the Feynman-type gaugePerturbative analysisSYM on RxS2SYM on RxS3

no dependent divergencesFermion self-energyand2+1 dim. theory is super-renormalizablelogarithmic divergence inagree with the calculation in the continuumtheory (Feynman gauge)SYM on RxS2SYM on RxS3

strong evidence for the restoration of the SU(2,2|4) symmetryOutlook 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 RxS3 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]