1. 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 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 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

4. 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
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5. 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
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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
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7. S3 configuration is constructed by 3 matrices
: Spin representation
of SU(2)
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8. S3 configuration is constructed by 3 matrices
: Spin representation
of SU(2)
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9. 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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10. 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
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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
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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
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13. 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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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
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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
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16. 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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17. 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
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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 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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19. Appendix Two-loop effective action
Feynman diagrams of two-loop corrections
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20. Relationship of coupling constants
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