Supersymmetric YangMills on S3 in PlaneWave Matrix Model at Finite Temperature

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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 _at_
YITP
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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-Ts
  uchiya (1997)
• BFSS matrix model Banks-Fischler-Shenker-S
  usskind (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

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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 contents

• Introduction
• Super-Yang-Mills on curved spaces in plane-wave
  matrix model
• Super-Yang-Mills on S1S3 and plane-wave matrix
  model
• Effective action of plane-wave matrix model
• Summary

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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

N4 super-Yang-Mills on RS3
Dimensional reduction
Large N
N4 super Yang-Mills on RS2
Dimensional reduction
Large N
Plane-wave matrix model
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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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Super-Yang-Mills on S1S3 and plane-wave matrix
model

• We derive the super-Yang-Mills theory on S1S3
• 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
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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 S1S3

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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

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

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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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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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Appendix

• Two-loop effective action

Feynman diagrams of two-loop corrections
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• Relationship of coupling constants