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 S 3. 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]

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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 s 3

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


A new large n reduction for chern simons theory on s 3

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]


A new large n reduction for chern simons theory on s 3

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


A new large n reduction for chern simons theory on s 3

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


A new large n reduction for chern simons theory on s 3

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


A new large n reduction for chern simons theory on s 3

2. Relationship between reduced theories of

Chern-Simons theory on S3


A new large n reduction for chern simons theory on s 3

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


A new large n reduction for chern simons theory on s 3

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


A new large n reduction for chern simons theory on s 3

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


A new large n reduction for chern simons theory on s 3

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


A new large n reduction for chern simons theory on s 3

3. Chern-Simons theory on S3

form N=1* matrix model


A new large n reduction for chern simons theory on s 3

Exact integration of N=1* matrix model

[Ishiki-Ohta-SS-Tsuchiya]

matrix

  • Diagonalize and integrate and

  • Use


A new large n reduction for chern simons theory on s 3

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.


A new large n reduction for chern simons theory on s 3

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


A new large n reduction for chern simons theory on s 3

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


A new large n reduction for chern simons theory on s 3

Our matrix model - multi matrix model

Chern-Simons theory on S3 Chern-Simons matrix model

(cf)


A new large n reduction for chern simons theory on s 3

Feynman rule for CSMM

Propagator:

Vertex:

(ex)


A new large n reduction for chern simons theory on s 3

Feynman rule for our matrix model

Propagator:

Vertex:

(ex)


A new large n reduction for chern simons theory on s 3

Free energy (connected diagrams)

Planar

Nonplanar


A new large n reduction for chern simons theory on s 3

General connected planar diagrams of both theories are like

planar

Dashed lines ( ) should not form any loop


A new large n reduction for chern simons theory on s 3

Correspondence between our matrix model and CSMM

  • Let us look at the different part between two

For planar

our matrix model

complete agreement !!

CSMM


A new large n reduction for chern simons theory on s 3

Correspondence between our matrix model and CSMM

For nonplanar

our matrix model

There is no correspondence

for nonplanar diagrams

CSMM


A new large n reduction for chern simons theory on s 3

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.


A new large n reduction for chern simons theory on s 3

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


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