Supersymmetric Yang-Mills on S 3 in Plane-Wave Matrix Model at Finite Temperature

1. Supersymmetric Yang-Mills on S3in 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

2. 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

3. There are interesting construction of curved spaces by matrix models • Any d-dimensional manifold can be described in terms of dcovariant 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

4. 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

5. Table of contents • Introduction • Super-Yang-Mills on curved spaces in plane-wave matrix model • Super-Yang-Mills on S1×S3and plane-wave matrix model • Effective action of plane-wave matrix model • Summary K. Matsumoto

6. 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

7. S3configuration is constructed by 3 matrices : Spin representation of SU(2) K. Matsumoto

8. S3configuration is constructed by 3 matrices : Spin representation of SU(2) K. Matsumoto

9. S3configuration 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

10. Super-Yang-Mills on S1×S3and 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

11. 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

12. 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

13. We provide fuzzy spheres as S3configuration : 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

14. 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

15. 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

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

17. 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

18. 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 S3from 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

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

20. Relationship of coupling constants K. Matsumoto