Based on collaboration with Y. Kitazawa (KEK, SOKENDAI) Supersymmetric Yang-Mills on S3 in Plane-Wave Matrix Model at Finite Temperature K. Matsumoto (KEK) Based on collaboration with Y. Kitazawa (KEK, SOKENDAI) YITP workshop on “Development of Quantum Field Theory and String Theory” 28 Jul ~ 1 Aug 2008 @ YITP

Introduction We want to understand the phenomena including the gravity at quantum level completely Matrix models are strong candidates for the non-perturbative formulation of the superstring theory or M-theory IKKT matrix model [Ishibashi-Kawai-Kitazawa-Tsuchiya (1997)] BFSS matrix model 　 [Banks-Fischler-Shenker-Susskind (1997)] However, matrix models were originally constructed on flat spaces We have the problem that it is unclear how curved spaces are described in matrix models K. Matsumoto

There are interesting construction of curved spaces by matrix models Any d-dimensional manifold can be described in terms of d covariant derivatives acting on an infinite-dimensional space [Hanada-Kawai-Kimura (2005)] The curved space can be realized by a generalized compactification procedure in the S1 direction [Ishiki-Shimasaki-Takayama-Tsuchiya (2006)] ISTT showed that the relationships between super-Yang-Mills theories on curved spaces and matrix model K. Matsumoto

We have investigated the relationship between Relationship between a large N gauge theories on flat spaces and matrix models Large N reduced model [Eguchi-Kawai (1982)] Quenched reduced model [Bhanot-Heller-Neuberger (1982), Das-Wadia (1982), Gross-Kitazawa (1982), Parisi (1982)] Twisted reduced model [Gonzalez-Arroyo-Okawa (1983)] We have investigated the relationship between the super-Yang-Mills on S3 and the plane-wave matrix model at finite temperature K. Matsumoto

Table of contents Introduction Super-Yang-Mills on curved spaces in plane-wave matrix model Super-Yang-Mills on S1×S3 and plane-wave matrix model Effective action of plane-wave matrix model Summary K. Matsumoto

Super-Yang-Mills on curved spaces in plane-wave matrix model [Ishiki-Shimasaki-Takayama-Tsuchiya (2006)] Relationships between super-Yang-Mills theories on curved spaces and the plane-wave matrix model in the large N limit N=4 super-Yang-Mills on R×S3 Dimensional reduction Large N N=4 super Yang-Mills on R×S2 Dimensional reduction Large N Plane-wave matrix model K. Matsumoto

S3 configuration is constructed by 3 matrices : Spin representation of SU(2) K. Matsumoto

S3 configuration is constructed by 3 matrices : Spin representation of SU(2) K. Matsumoto

S3 configuration is constructed by 3 matrices : Spin representation of SU(2) In order to make the connection between the super-Yang-Mills on S3 and the plane-wave matrix model K. Matsumoto

Super-Yang-Mills on S1×S3 and plane-wave matrix model We derive the super-Yang-Mills theory on S1×S3 from the plane-wave matrix model by taking a large N limit : Temperature : Radius of S3 The action of the plane-wave matrix model : Bosonic : Fermionic N × N Hermitian matrices K. Matsumoto

Let us consider a large N limit For example: where the metric tensor on S3 is obtained by the Killing vectors We can obtain the action of super-Yang-Mills theory on S1×S3 K. Matsumoto

Effective action of plane-wave matrix model We calculate the effective action of the plane-wave matrix model at finite temperature up to two-loop Background field method Backgrounds Quantum fluctuations K. Matsumoto

We provide fuzzy spheres as S3 configuration : Spin representation of SU(2) Cutoff for matrices size of : Cutoff for the number of fuzzy spheres: We set the magnitude relation for two cutoff scales K. Matsumoto

For example, we consider the leading terms of the one-loop effective action In analogy with the large N reduced model on flat spaces K. Matsumoto

For example, we consider the leading terms of the one-loop effective action We divide the sums over because the effective action for the plane- wave matrix model is consistent with it for the large N reduced model of the super-Yang-Mills on S3 K. Matsumoto

We consider the following cutoff scale region We approximate sums over by integrals over We take the following high temperature limit K. Matsumoto

the super-Yang-Mills on S3 We summarize the effective action of the plane-wave matrix model at finite temperature up to the two-loop level One-loop Two-loop One-loop where we divided the effective action by the volume of S3 The two-loop effective action which we obtained is consistent with times the free energy density of the super-Yang-Mills on S3 K. Matsumoto

Summary We have derived the action of the super-Yang-Mills on S3 from it of the plane-wave matrix model by taking the large N limit We have derived the free energy of the super-Yang-Mills on S3 from the effective action of the plane-wave matrix model up to the two-loop level Our results serve as a non-trivial check that the plane-wave matrix model is consistent with the large N reduced model of the super-Yang-Mills on S3 K. Matsumoto

Appendix Two-loop effective action Feynman diagrams of two-loop corrections K. Matsumoto

Relationship of coupling constants K. Matsumoto