Exact Results of theories with SU(2|4) symmetry and … · 2013. 5. 13. · super Yang-Mills (SYM) theories in 4d, super Chern-Simons-matter theories in 3d, ... The solutions to the

Exact Results of theories with SU(2|4) symmetry

and gauge/gravity correspondence

Shinji Shimasaki (Kyoto University)

JHEP1302, 148 (2013) (arXiv:1211.0364[hep-th])

Based on the work in collaboration with Y. Asano (Kyoto U.), G. Ishiki (YITP) and T. Okada(YITP)

and the work in progress

Introduction

Localization method is a powerful tool to exactly compute vev of some particular operators in quantum field theories.

Localization

super Yang-Mills (SYM) theories in 4d, super Chern-Simons-matter theories in 3d, SYM in 5d, …

M-theory(M2, M5-brane), AdS/CFT,…

i.e. Partition function, vev of Wilson loop in

In this talk, I’m going to talk about localization for SYM theories with SU(2|4) symmetry.

• gauge/gravity correspondence for theories with SU(2|4) symmetry • Little string theory on RxS5 (IIA NS5-brane)

Theories with SU(2|4) sym.

mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)

SU(2|4) theories from PWMM [Ishiki,SS,Takayama,Tsuchiya]

gravity duals corresponding to each vacuum of each theory are constructed (bubbling geometry in IIA SUGRA) [Lin,Maldacena]

N=4 SYM on RxS3/Zk (4d)

Consistent truncations of N=4 SYM on RxS3.

(BMN)

[Lin,Maldacena]

[Maldacena,Sheikh-Jabbari,Raamsdonk]

N=8 SYM on RxS2 (3d)

Plane wave matrix model (1d) [Berenstein,Maldacena,Nastase][Kim,Klose,Plefka]

“holonomy”

“monopole”

“fuzzy sphere”



Consistent truncations of N=4 SYM on RxS3.

(BMN)

[Lin,Maldacena]

[Maldacena,Sheikh-Jabbari,Raamsdonk]


Plane wave matrix model (1d)

“holonomy”

“monopole”

“fuzzy sphere”

T-duality in gauge theory [Taylor]

commutative limit of fuzzy sphere

[Berenstein,Maldacena,Nastase][Kim,Klose,Plefka]

mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)

SU(2|4) theories from PWMM [Ishiki,SS,Takayama,Tsuchiya]

gravity duals corresponding to each vacuum of each theory are constructed (bubbling geometry in IIA SUGRA) [Lin,Maldacena]

Our Results

• Using the localization method, we compute the partition function of PWMM up to instantons;

• We checked our result by comparing with perturbative computation.

where : vacuum configuration characterized by

In the ’t Hooft limit, our result becomes exact.

• is written as a matrix integral.

Asano, Ishiki, Okada, SS

Our Results

Partition func. = gaussian matrix model

• We checked our result in the k=1 case of N=4 SYM on RxS3/Zk, N=4 SYM on RxS

3, by comparing with the known result of N=4 SYM. [Pestun; Erickson,Semenoff,Zarembo; Drukker,Gross]

• We also obtain the partition function of N=8 SYM on RxS2 and N=4 SYM on RxS3/Zk from that of PWMM by taking limits corresponding to “commutative limit of fuzzy sphere” and “T-duality in gauge theory”.

Asano, Ishiki, Okada, SS

Applications of our results

• gauge/gravity correspondence for theories with SU(2|4) symmetry

Work in progress; Asano, Ishiki, Okada, SS

• Little string theory on RxS5

We will discuss the gauge/gravity correspondence for N=8 SYM on RxS2 around the trivial vacuum.

This theory is also one of theories with SU(2|4) symmetry.

Plan of this talk

1. Introduction

2. Theories with SU(2|4) symmetry

3. gauge/gravity correspondence for theories

with SU(2|4) symmetry

4. Localization in PWMM

5. Exact results for theories


6. Application of our result

7. Summary

Theories with SU(2|4) symmetry


N=4 SYM on RxS3

convention for S3

right inv. 1-from:

metric:

Local Lorentz indices of spatial directions

• vacuum “holonomy”

N=4 SYM on RxS3/Zk

Angular momentum op. on S2

Keep the modes with the periodicity in N=4 SYM on RxS3.


N=8 SYM on RxS2

• vacuum “monopole”

In the second line we rewrite in terms of the gauge fields and the scalar field as .

Plane wave matrix model

monopole charge


Plane wave matrix model (PWMM)

• vacuum “fuzzy sphere”

: spin rep. matrix


Plan of this talk

1. Introduction








7. Summary

gauge/gravity correspondence for theories


general solutions with 16 SUSY and RxSO(3)xSO(6) isometry

black region surrounded by white region

Lin-Lunin-Maldacena geometry

black region : shrinks

white region : shrinks

white region surrounded by black region

M2 flux

M5 flux

• 11d SUGRA

translational invariance along 10d IIA SUGRA

“bubbling geometry”

Lin-Maldacena geometry

• 10d IIA SUGRA

black region surrounded by white region

white region surrounded by black region

D2 flux

NS5 flux

general solutions with 16 SUSY and RxSO(3)xSO(6) isometry

black region : shrinks

white region : shrinks


• 10d IIA SUGRA

identify

vacua of SYM on RxS2

vacua of SYM on RxS3/Zk

vacua of PWMM


• 10d IIA SUGRA

identify

D2 flux

NS5 flux

D2 flux

NS5 flux

D2 flux

NS5 flux


• 10d IIA SUGRA

vacua of little string theory on RxS5



Plane wave matrix model (1d)

T-duality in gauge theory [Taylor]

commutative limit of fuzzy sphere

The relation among theories with SU(2|4) symmetry can be seen from the gravity duals.

D2 flux

NS5 flux

D2 flux

NS5 flux

• PWMM around • N=8 SYM on RxS2 around the trivial vacuum

N=8 SYM on RxS2 from PWMM

In field theory language, this limit corresponds to the commutative limit of fuzzy sphere.

PWMM around the following fuzzy sphere vacuum

N=8 SYM on RxS2 from PWMM

N=8 SYM on RxS2 around the following monopole vacuum

fixed with

• N=8 SYM on RxS2 around

In field theory language, this procedure corresponds to the Taylor’s T–duality.

with • N=4 SYM on RxS3/Zk around the trivial vauum

identify periodic

extract one period

N=4 SYM on RxS3/Zk from N=8 SYM on RxS2

N=8 SYM on RxS2 around the following monopole vacuum

Identification among blocks of fluctuations (orbifolding)

with

(an infinite copies of) N=4 SYM on RxS3/Zk around the trivial vacuum

N=4 SYM on RxS3/Zk from N=8 SYM on RxS2

Little string theory on RxS5 from PWMM ?

D2 flux

NS5 flux

D2 flux

NS5 flux

• PWMM around • Little string theory on RxS5 around the trivial vacuum

I will discuss this later..

[Ling, Mohazab, Shieh, Anders, Raamsdonk]

Plan of this talk

1. Introduction








7. Summary

Localization in PWMM

Localization

Suppose that is a symmetry

and there is a function such that

Define

is independent of

[Witten; Nekrasov; Pestun; Kapustin et.al.;…]

one-loop integral around the saddle points

We perform the localization in PWMM by constructing equivariant cohomology following Pestun.

Plane Wave Matrix Model

Off-shell SUSY in PWMM

SUSY algebra is closed if there exist spinors which satisfy

Indeed, such exist

• : invariant under the off-shell SUSY.

• :Killing vector

[Berkovits]

const. matrix where

Saddle point

We choose

Saddle point

In , and are vanishing.

Saddle points are characterized by reducible representations of SU(2), , and constant matrices

1-loop around a saddle point with integral of

The solutions to the saddle point equations we showed are the solutions when is finite.

In , some terms in the saddle point equations automatically vanish.

In this case, the saddle point equations for remaining terms are reduced to (anti-)self-dual equations.

(mass deformed Nahm equation)

In addition to these, one should also take into account the instanton configurations localizing at .

Here we neglect the instantons.

Instanton

[Yee,Yi;Lin;Bachas,Hoppe,Piolin]

: gauge transformation with parameter ;

: U(1) transformation (diagonal U(1) subgroup of SO(3)xSO(6)R)

Change of variables of fermion

Rewrite in the basis

SUSY transformation is rewritten as

In order to perform gauge-fixing simultaneously, we define and add ghosts

Field contents

SUSY transformation

SUSY+BRST

: linear diff. op. depend on

1-loop around saddle point

Relevant part of the 1-loop computation

Fluctuations are vanishing at infinity

Expand around the saddle point

1-loop around saddle point

: functional space of which vanishes at infinity

In the second equality, we used the fact that there is cancellation between and

Plan of this talk

1. Introduction








7. Summary

Exact results for theories with

SU(2|4) symmetry

Partition function of PWMM

Contribution from the classical action

Partition function of PWMM with is given by

where

Eigenvalues of

Partition function of PWMM

Trivial vacuum

(cf.) partition function of 6d IIB matrix model

[Moore-Nekrasov-Shatashvili][Kazakov-Kostov-Nekrasov]

[Kitazawa-Mizoguchi-Saito]

Partition function of N=8 SYM on RxS2

In order to obtain the partition function of N=8 SYM on RxS2 from that of PWMM, we take the limit in which

fixed with

Partition function of N=8 SYM on RxS2

trivial vacuum

Partition function of N=4 SYM on RxS3/Zk

such that

and impose orbifolding condition .

In order to obtain the partition function of N=4 SYM on RxS3/Zk around the trivial background from that of N=8 SYM on RxS2, we take

Partition function of N=4 SYM on RxS3/Zk

When , N=4 SYM on RxS3, the measure factors except for the Vandermonde determinant completely cancel out.

Gaussian matrix model

Consistent with the results for N=4 SYM [Pestun; Erickson,Semenoff,Zarembo; Drukker,Gross]

Application of our result

• gauge/gravity duality for N=8 SYM on RxS2 around trivial vacuum • NS5-brane limit

Gauge/gravity duality for N=8 SYM on RxS2 around trivial vacuum

Partition function of N=8 SYM on RxS2 around trivial vacuum

This can be solved in the large-N and the large ’t Hooft coupling limit;

The dependence of and is consistent with the gravity dual obtained by Lin and Maldacena.

NS5-brane limit

Based on the gauge/gravity duality by Lin-Maldacena, Ling, Mohazab, Shieh, Anders and Raamsdonk proposed a double scaling limit in which little string theory (IIA NS5 -brane theory) on RxS5 is obtained from PWMM.

Expand PWMM around and take the limit in which

and

Little string theory on RxS5

(# of NS5 = )

with and fixed

In this limit, instantons are suppressed. So, we can check this conjecture by using our result.

If this conjecture is true, the vev of an operator can be expanded as

NS5-brane limit

We checked this numerically in the case where

and for various .

NS5-brane limit

is nicely fitted by with for various !

Summary

Summary

• Using the localization method, we compute the partition function of PWMM up to instantons.

• We also obtain the partition function of N=8 SYM on RxS2 and N=4 SYM on RxS3/Zk from that of PWMM by taking limits corresponding to “commutative limit of fuzzy sphere” and “T-duality in gauge theory”.

• We may obtain some nontrivial evidence for the gauge/gravity duality for theories with SU(2|4) symmetry and the little string theory on RxS5.

where

Q-exact term

Documents

Exact Results of theories with SU(2|4) symmetry and … · 2013. 5. 13. · super Yang-Mills (SYM) theories in 4d, super Chern-Simons-matter theories in 3d, ... The solutions to the