K. Matsumoto 1

Supersymmetric Yang-Mills on S3 in

Plane-Wave Matrix Model at Finite Temperature

Supersymmetric Yang-Mills on S3 in

Plane-Wave Matrix Model at Finite Temperature

K. Matsumoto (KEK)

Based on collaboration withY. Kitazawa (KEK, SOKENDAI)

YITP workshopon

“Development of Quantum Field Theory and String Theory”28 Jul ~ 1 Aug 2008 @ YITP

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1. Introduction1. 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 (199

7)]

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

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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

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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

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Table of contentsTable of contents

1. Introduction2. Super-Yang-Mills on curved spaces in

plane-wave matrix model3. Super-Yang-Mills on S1×S3 and plane-

wave matrix model4. Effective action of plane-wave matrix

model5. Summary

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2. Super-Yang-Mills on curved spaces in plane-wave matrix model

2. Super-Yang-Mills on curved spaces in plane-wave matrix model

[Ishiki-Shimasaki-Takayama-Tsuchiya (2006)]

N=4 super-Yang-Mills on R×S3

Dimensional reduction

Dimensional reduction

Large N

N=4 super Yang-Mills on R×S2

Large N

Plane-wave matrix model

Relationships between super-Yang-Mills theories on curved spaces and the plane-wave matrix model in the large N limit

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S3 configuration is constructed by 3 matrices

: Spin representation

of SU(2)

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S3 configuration is constructed by 3 matrices

: Spin representation

of SU(2)

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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

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3. Super-Yang-Mills on S1×S3 and plane-wave matrix model

3. 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

The action of the plane-wave matrix model

: Bosonic : Fermionic

N × N Hermitian matrices

: Temperature: Radius of S3

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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

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4. Effective action of plane-wave matrix model

4. 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

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We provide fuzzy spheres as S3 configuration

Cutoff for matrices size of :

Cutoff for the number of fuzzy spheres:

We set the magnitude relation for two cutoff scales

: Spin representation

of SU(2)

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For example, we consider the leading terms of the one-loop effective action

In analogy with the large N reduced model on flat spaces

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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

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We consider the following cutoff scale region

We approximate sums over by integrals over

We take the following high temperature limit

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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

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5. Summary5. 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

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AppendixAppendix

Feynman diagrams of two-loop corrections

Two-loop effective action

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Relationship of coupling constants