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:0811.3569[hep-th] Ishiki-Ohta-SS-Tsuchiya, to appear

• Nonperturbative definition (regularization) of large N gauge theory (Large N reduction)

Matrix model

• Nonperturbative definition of superstring theory

[Eguchi-Kawai][Parisi][Gross-Kitazawa][Bhanot-Heller-Neuberger][Gonzalez-Arroyo – Okawa]…

[Banks-Fischler-Shenker-Susskind][Ishibashi-Kawai-Kitazawa-Tsuchiya][Dijkgraaf-Verlinde-Verlinde]

Introduction

YM on RDMatrix Model (0-dim)

☆ Can we describe curved spaces and topological invariants by matrices ?

• Description of curved spaces by matrices [Hanada-Kawai-Kimura]

• gauge theory on S1, T2, flux on T2, S2(fuzzy sphere), monopoles on S2,…

[Madore][Grosse-Madore][Grosse-Klimcik-Presnajder][Carow-Watamura – Watamura][Ishiki-SS-Takayama-Tsuchiya]…

• gauge/gravity correspondence [Lin-Lunin-Maldacena][Lin-Maldacena]

planar

In this talk, we give a new large N reduction

large N reduction for Chern-Simons theory on S3

DimensionalReduction

DimensionalReduction

large N reductionto make S1

Continuum limitof fuzzy sphere

S2

S3 = S1 on S2

point

Chern-Simons theory on S3

BF theory + mass term on S2 = YM on S2

N=1* matrix model

Reduced theories of Chern-Simons theory on S3

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 expansionwith respect to ‘t Hooft coupling

Results

In this talk, we give a new large N reduction

large N reduction for Chern-Simons theory on S3

1. Introduction

2. 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

Plan of this talk

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

Chern-Simons theory on S3

Dimensional reduction

right-invariant Killing vector on S3

: right-invariant 1-form on S3

S3 S1

S2

• Fourier expansion along the S1 fiber :

: angular momentum op. on S2

KK momenta along the S1 fiber monopole charge on S2

•

•

angular momentum op. in the presence of magnetic charge

BF theory + mass term on S2 = YM on S2

Dimensional reduction

N=1* matrix model

(cf)mass deformed superpotentialof N=4 SYM

Integrating out

• Expand around a classical solution

BF + mass term on S2 around a monopole background

N=1* matrix model

Classical relationship

Continuum limit of fuzzy sphere

fuzzy sphere

take in all monopole charge

BF + mass term on S2 around a monopole background

Planar Chern-Simons theory on S3

Classical relationship

large N reduction for nontrivial S1 fiber

reproduce all KK momentaalong the S1 fiber

=

3. Chern-Simons theory on S3

form N=1* matrix model

matrix

Diagonalize and integrate and

Use

Exact integration of N=1* matrix model[Ishiki-Ohta-SS-Tsuchiya]

The integral is decomposed into sectors which are characterizedby -dimensional representation of SU(2). (partition of )

specifies irreducible representationsand its multiplicity:

Each sector seems to be the contribution around each classical solutionof N=1* matrix model.

: irreducible rep.: multiplicity

Extract -block sector

To 2d YM on S2

partition function of SU(K) YM on S2

Equal size block configuration is dominant

and take

Set and take

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)

Planar

Nonplanar

Free energy (connected diagrams)

General connected planar diagrams of both theories are like

planar

Dashed lines ( ) should not form any loop

Correspondence between our matrix model and CSMM

our matrix model

CSMM

For planar

complete agreement !!

• Let us look at the different part between two

our matrix model

CSMM There is no correspondence for nonplanar diagrams

For nonplanar

Correspondence between our matrix model and CSMM

Wilson loop

our matrix model (great circle on S3)

CSMM (Unknot, fundamental rep.)

Wilson loop in N=1* matrix model:

For great circle on S3,

We can also see the planar correspondence for these two.

[Ishii-Ishiki-Ohta-SS-Tsuchiya]

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

Wilson loops (various contour, deformation)

We can also show that N=1* MM includes sectors corresponding to various nontrivial vacua of CS on S3/Zk.

Localization