1. A new large N reduction for Chern-Simons theory on S3
Shinji Shimasaki (Kyoto U.)
In collaboration with G. Ishiki (KEK),
K. Ohta (Meiji Gakuin U.) and A. Tsuchiya (Shizuoka U.)
(ref.) Ishiki-Ohta-SS-Tsuchiya, PLB 672 (2009) 289. arXiv: [hep-th]
Ishiki-Ohta-SS-Tsuchiya, to appear

2. Introduction Matrix model
Nonperturbative definition (regularization) of large N gauge theory
(Large N reduction)
[Eguchi-Kawai][Parisi][Gross-Kitazawa]
[Bhanot-Heller-Neuberger][Gonzalez-Arroyo – Okawa]…
YM on RD
Matrix Model (0-dim)
planar
Nonperturbative definition of superstring theory
[Banks-Fischler-Shenker-Susskind][Ishibashi-Kawai-Kitazawa-Tsuchiya]
[Dijkgraaf-Verlinde-Verlinde]
☆ Can we describe curved spaces and topological invariants
by matrices ?
[Madore][Grosse-Madore]
[Grosse-Klimcik-Presnajder]
[Carow-Watamura – Watamura]
[Ishiki-SS-Takayama-Tsuchiya]…
gauge theory on
S1, T2, flux on T2, S2(fuzzy sphere),
monopoles on S2,…
gauge/gravity correspondence
[Lin-Lunin-Maldacena][Lin-Maldacena]
Description of curved spaces by matrices
[Hanada-Kawai-Kimura]

3. large N reduction for Chern-Simons theory on S3
In this talk, we give a new large N reduction
large N reduction for Chern-Simons theory on S3
Reduced theories of Chern-Simons theory on S3
Chern-Simons theory on S3
Dimensional
Reduction
large N reduction
to make S1
S3 = S1 on S2
BF theory + mass term on S2
= YM on S2 S2
Dimensional
Reduction
Continuum limit
of fuzzy sphere
N=1* matrix model
point

4. large N reduction for Chern-Simons theory on S3
In this talk, we give a new large N reduction
large N reduction for Chern-Simons theory on S3
Results
A particular sector of N=1* matrix model reproduce
the planar limit of Chern-Simons theory on S3.
Planar free energy and Wilson loop (unknot) of CS on S3
is reproduced from our matrix model
This is the first explicitly shown large N reduction on S3.
Interesting application to topological field theory
Alternative regularization of CS on S3
All order correspondence for perturbative expansion
with respect to ‘t Hooft coupling

5. Plan of this talk Introduction
Relationships between reduced theories of
Chern-Simons theory on S3
3. Chern-Simons theory on S3
from N=1* matrix model
4. Summary and Outlook

6. 2. Relationship between reduced theories of
Chern-Simons theory on S3

7. Dimensional reduction
Chern-Simons theory on S3 S2
: right-invariant 1-form on S3
right-invariant Killing vector on S3
: angular momentum op. on S2
Fourier expansion along the S1 fiber :
angular momentum op.
in the presence of magnetic charge
KK momenta along the S1 fiber monopole charge on S2

8. Dimensional reduction
BF theory + mass term on S2 = YM on S2
Integrating out
N=1* matrix model
(cf)
mass deformed superpotential
of N=4 SYM

9. Classical relationship
N=1* matrix model
Expand around a classical solution
fuzzy sphere
Continuum limit of fuzzy sphere
BF + mass term on S2 around a monopole background

10. = Classical relationship
BF + mass term on S2 around a monopole background
large N reduction for nontrivial S1 fiber
take in all monopole charge =
reproduce all KK momenta
along the S1 fiber
Planar Chern-Simons theory on S3

11. 3. Chern-Simons theory on S3
form N=1* matrix model

12. Exact integration of N=1* matrix model
[Ishiki-Ohta-SS-Tsuchiya]
matrix
Diagonalize and integrate and
Use

13. The integral is decomposed into sectors which are characterized
by -dimensional representation of SU(2). (partition of )
specifies irreducible representations
and its multiplicity:
: irreducible rep.
: multiplicity
Each sector seems to be the contribution around each classical solution
of N=1* matrix model.

14. To 2d YM on S2 partition function of SU(K) YM on S2
Extract block sector
and take
Equal size block configuration is dominant
Set
and take
partition function of SU(K) YM on S2

15. To Chern-Simons on S3 Extract the following sector
We expect that in the limits
the planar limit of the partition function of CS on S3 is reproduced. In

16. Our matrix model - multi matrix model
Chern-Simons theory on S Chern-Simons matrix model
(cf)

17. Feynman rule for CSMM
Propagator:
Vertex:
(ex)

18. Feynman rule for our matrix model
Propagator:
Vertex:
(ex)

19. Free energy (connected diagrams)
Planar
Nonplanar

20. General connected planar diagrams of both theories are like
Dashed lines ( ) should not form any loop

21. Correspondence between our matrix model and CSMM
Let us look at the different part between two
For planar
our matrix model
complete agreement !!
CSMM

22. Correspondence between our matrix model and CSMM
For nonplanar
our matrix model
There is no correspondence
for nonplanar diagrams
CSMM

23. Wilson loop Wilson loop in N=1* matrix model: For great circle on S3,
[Ishii-Ishiki-Ohta-SS-Tsuchiya]
For great circle on S3,
our matrix model (great circle on S3)
CSMM (Unknot, fundamental rep.)
We can also see the planar correspondence for these two.

24. 4. Summary and Outlook
We give a new type of the large N reduction extended
to curved space, S3, and its application to CS theory.
In the planar limit, a particular sector of N=1* matrix model
reproduce the planar Chern-Simons theory on S3.
Free energy and Wilson loop are reprodeced
We can also show that N=1* MM includes sectors
corresponding to various nontrivial vacua of CS on S3/Zk.
Wilson loops (various contour, deformation)
Localization