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A new large N reduction for Chern-Simons theory on S 3

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A new large N reduction for Chern-Simons theory on S3

Shinji Shimasaki (Kyoto U.)

In collaboration with G. Ishiki (KEK),

K. Ohta (Meiji GakuinU.) and A. Tsuchiya (Shizuoka U.)

(ref.) Ishiki-Ohta-SS-Tsuchiya, PLB 672 (2009) 289. arXiv:0811.3569[hep-th]

Ishiki-Ohta-SS-Tsuchiya, to appear

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]

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

Continuum limit

of fuzzy sphere

Dimensional

Reduction

N=1* matrix model

point

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

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

2. Relationship between reduced theories of

Chern-Simons theory on S3

Dimensional reduction

S1

S3

- 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

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

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

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

3. Chern-Simons theory on S3

form N=1* matrix model

Exact integration of N=1* matrix model

[Ishiki-Ohta-SS-Tsuchiya]

matrix

- Diagonalize and integrate and

- Use

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.

To 2d 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

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

Our matrix model - multi matrix model

Chern-Simons theory on S3 Chern-Simons matrix model

(cf)

Feynman rule for CSMM

Propagator:

Vertex:

(ex)

Feynman rule for our matrix model

Propagator:

Vertex:

(ex)

Free energy (connected diagrams)

Planar

Nonplanar

General connected planar diagrams of both theories are like

planar

Dashed lines ( ) should not form any loop

Correspondence between our matrix model and CSMM

- Let us look at the different part between two

For planar

our matrix model

complete agreement !!

CSMM

Correspondence between our matrix model and CSMM

For nonplanar

our matrix model

There is no correspondence

for nonplanar diagrams

CSMM

Wilson loop

Wilson loop in N=1* matrix model:

[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.

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