1. Shinji Shimasaki (Osaka U.)
A nonperturbative definition of N=4 Super Yang-Mills by the plane wave matrix model
Shinji Shimasaki (Osaka U.)
In collaboration with
T. Ishii (Osaka U.), G. Ishiki (Osaka U.) and A. Tsuchiya (Shizuoka U.)
(ref.) Ishiki-SS-Takayama-Tsuchiya, JHEP 11(2006)089 [hep-th/ ]
Ishii-Ishiki-SS-Tsuchiya, in preparation

2. Motivation and Introduction
☆ AdS/CFT correspondence
N=4 Super Yang-Mills
IIB string on AdS5xS5
strong coupling
classical gravity
In order to verify the correspondence, we need
understand the N=4 SYM in strong coupling region,
in particular, its non-BPS sectors.
A nonperturbative definition of N=4 SYM would enable us to
study its strong coupling region.

3. What we would like to discuss
N=4 SYM on RxS3 can be described by the theory around
a certain vacuum of the plane wave matrix model with
periodicity condition imposed.
Ishiki-SS-Takayama-Tsuchiya,
JHEP 11(2006)089 [hep-th/ ]
Our proposal: Matrix regularization of N=4 SYM on RxS3
by the plane wave matrix model
Our method has the following features:
(cf.) lattice theory
by Kaplan-Katz-Unsal
PWMM is massive no flat direction
gauge symmetry as a matrix model
SU(2|4) sym. ⊂ SU(2,2|4) sym.
16 supercharges
32 supercharges
We perform a perturbative analysis (1-loop)
The Ward identity is satisfied.
The beta function vanishes in a continuum limit.

4. Plan of this talk Motivation and Introduction N=4 SYM on RxS3
and the plane wave matrix model
3. Check of the validity (perturbative calculation)
4. Summary and Perspective

5. N=4 SYM on RxS3 and the plane wave matrix model
[Kim-Klose-Plefka]
[Ishiki-SS-Takayama-Tsuchiya]
(cf.) [Lin-Maldacena]
N=4 SYM on R×S3
IIA SUGRA sol.
with SU(2|4) sym.
SU(2,2|4) (32 SUSY)
(1) Matrix
T-duality
Dimensional Reduction
SYM on R×S2
(1)+(2)
SU(2|4) (16 SUSY)
Dimensional Reduction
(2) Continuum limit
of fuzzy sphere
plane wave matrix model
SU(2|4) (16 SUSY)

6. plane wave matrix model
vacuum:
SU(2) generator
fuzzy sphere
In order to obtain the SYM on RxS3, we consider
the theory around the following vacuum configuration.
periodicity

7. We expand matrices by using the fuzzy spherical harmonics
(basis of a rectangular matrix)
[Grosse-Klimcik-Presnajder, Baez-Balachandran-Ydri-Vaidya, Dasgupta-SheikhJabbari-Raamsdonk,]
Clebsch-Gordan coeff.
The modes become those of SYM on RxS2
and SYM on RxS3.

8. continuum limit of fuzzy sphere
We obtain SYM on RxS2 around the monopole background
SYM on RxS2
where
Monopole background
are identified with the modes of SYM on RxS2
around the monopole background.

9. & periodicity condition matrix T-duality
Moreover
& periodicity condition
matrix T-duality
[Taylor]
momentum of S1
SYM on RxS3
orbifolding
Completely consistent in classical level
How about quantum level ?

10. ☆ some comments We want to set to be finite.
But the periodicity condition is incompatible with finite .
If we restrict ourselves to the planar limit, we do not
need the periodicity condition.
Reduced model
Possible UV/IR mixing is avoided due to the planar limit
and 16 SUSY
Instanton effect is suppressed by taking
We obtain the regularized theory of planar N=4 SYM on RxS3
by the theory around the vacuum of the plane wave matrix
model with to be finite.
Nonperturbative definition of N=4 SYM on RxS3
We perform the perturbative analysis in our regularized theory.

11. Check of the validity (1-loop calculation)
Ward Identity
Beta function
no quadratic divergence
due to the gauge inv. regularization

12. Summary
We propose a nonperturbative definition of planar N=4 SYM on RxS3
by the plane wave matrix model .
Our regularization keeps the gauge sym. and the SU(2|4) sym.
The planar limit and 16 SUSY protect us from the instanton effect
and the UV/IR mixing.
We verified that the Ward identity is actually satisfied
and the beta function vanishes in the continuum limit at 1-loop level.
Outlook
Is the superconformal symmetry SU(2,2|4) indeed restored ?
due to 16 SUSY ?
Wilson loop
[Ishii-Ishiki-Ohta-SS-Tsuchiya][Drukker-Gross]
finite temperature
[Aharony et.al.]