exact results for perturbative partition functions of theories with su(2|4) symmetry

Exact Results for perturbative partition functions of theories

with SU(2|4) symmetryShinji 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 some physical quantities 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 ((IIA) NS5-brane)

Theories with SU(2|4) sym.

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

SYM on RxS2 and RxS3/Zk from PWMM [Ishiki,SS,Takayama,Tsuchiya]

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

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

Consistent truncations of N=4 SYM on RxS3.

(PWMM)

[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”

Theories with SU(2|4) sym.


Consistent truncations of N=4 SYM on RxS3.

(PWMM)

[Lin,Maldacena]

[Maldacena,Sheikh-Jabbari,Raamsdonk] N=8 SYM on RxS2 (3d)

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)

SYM on RxS2 and RxS3/Zk from PWMM [Ishiki,SS,Takayama,Tsuchiya]

gravity dual corresponding to each vacuum of each theory is 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 check that our result reproduces a one-loop result of PWMM.

where : vacuum configuration characterized by

In the ’t Hooft limit, our result becomes exact.• is written as a matrix integral.

Asano, Ishiki, Okada, SSJHEP1302, 148 (2013)

Our Results

• We show that, in our computation, the partition function of N=4 SYM on RxS3(N=4 SYM on RxS3/Zk with k=1) is given by the gaussian matrix model. This is consistent with the known result of N=4 SYM. [Pestun; Erickson,Semenoff,Zarembo; Drukker,Gross]

• We also obtain the partition functions 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, SSJHEP1302, 148 (2013)

Application of our result

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

Work in progress; Asano, Ishiki, Okada, SS

• Little string theory on RxS5

Plan of this talk1. Introduction2. Theories with SU(2|4) symmetry3. Localization in PWMM4. Exact results of theories with SU(2|4) symmetry5. Application of our result6. Summary

Theories with SU(2|4) symmetry

N=4 SYM on RxS3

(Local Lorentz indices of RxS3)

• vacuum all fields=0

: gauge field: scalar field (adjoint rep)

+ fermions

N=4 SYM on RxS3

convention for S3

right inv. 1-form:

metric:

Local Lorentz indices of S3

Hereafter we focus on the spatial part (S3) of the gauge fields.

where

• vacuum“holonomy”

Angular momentum op. on S2

Keep the modes with the periodicityin N=4 SYM on RxS3.

N=4 SYM on RxS3/Zk

N=8 SYM on RxS2

• vacuum “Dirac monopole”

In the second line we rewrite in terms of the gauge fieldsand the scalar field on S2 as .

plane wave matrix model

monopole charge

N=8 SYM on RxS2

• vacuum “fuzzy sphere”

: spin rep. matrix

plane wave matrix model


N=8 SYM on RxS2 (3d)

Plane wave matrix model (1d)commutative limit of fuzzy sphere

Relations among theorieswith SU(2|4) symmetry




Plane wave matrix model (1d)commutative limit of fuzzy sphere

N=8 SYM on RxS2 from PWMM

PWMM around the following fuzzy sphere vacuum


N=8 SYM on RxS2 around the following monopole vacuum

fixedwith

N=8 SYM on RxS2 around a monopole vacuum

matrix

• Decompose fields into blocks according to the block structure of the vacuum

• monopole vacuum

(s,t) block

• Expand the fields around a monopole vacuum

: Angular momentum op. in the presence of a monopole with charge


PWMM around a fuzzy spherevacuum• fuzzy sphere vacuum

• Decompose fields into blocks according to the block structure of the vacuum

matrix

(s,t) block

• Expand the fields around a fuzzy sphere vacuum

PWMM around a fuzzy spherevacuum

PWMM around a fuzzy sphere vacuum


: Angular momentum op. in the presence of a monopole with charge

Spherical harmonics monopole spherical harmonics

fuzzy spherical harmonics

(basis of sections of a line bundle on S2)

(basis of rectangular matrix )

with fixed

[Grosse,Klimcik,Presnajder; Baez,Balachandran,Ydri,Vaidya; Dasgupta,Sheikh-Jabbari,Raamsdonk;…]

[Wu,Yang]

Mode expansion N=8 SYM on RxS2

PWMM

Expand in terms of the monopole spherical harmonics

Expand in terms of the fuzzy spherical harmonics




fixed

In the limit in which

with

PWMM coincides with N=8 SYM on RxS2.



Plane wave matrix model (1d)


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



(S3/Zk : nontrivial S1 bundle over S2)

KK expand along S1 (locally)

N=8 SYM on RxS2 with infinite number of KK modes• These KK mode are sections of line bundle on S2

and regarded as fluctuations around a monopole background in N=8 SYM on RxS2. (monopole charge = KK momentum)

N=4 SYM on RxS3/Zk

• N=4 SYM on RxS3/Zk can be obtained by expanding N=8 SYM on RxS2 around an appropriate monopole background so that all the KK modes are reproduced.

This is achieved in the following way.

• Expand N=8 SYM on RxS2 around the following monopole vacuum

• Make the identification among blocks of fluctuations (orbifolding)

with

• Then, we obtain (an infinite copies of) N=4 U(N) SYM on RxS3/Zk.

Extension of Taylor’s T-duality to that on nontrivial fiber bundle [Ishiki,SS,Takayama,Tsuchiya]


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 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.

is a constant matrix commuting with :

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 equationsautomatically vanish.

In this case, the saddle point equations for remainingterms 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]

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

Partition function of PWMM

Contribution from the classical action

Partition function of PWMM with is given by

whereEigenvalues of

Partition function of PWMMTrivial vacuum

(cf.) partition function of 6d IIB matrix model[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 commutative limit of fuzzy sphere, in which

fixedwith

Partition function of N=8 SYM on RxS2

trivial vacuum

Partition function of N=4 SYM on RxS3/Zk

such thatand 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 completely cancel out except for the Vandermonde determinant.

Gaussian matrix modelConsistent with the result of N=4 SYM

[Pestun; Erickson,Semenoff,Zarembo; Drukker,Gross]

Application of our result

• gauge/gravity duality for N=8 SYM on RxS2 around the trivial vacuum

• NS5-brane limit

Gauge/gravity duality for N=8 SYM on RxS2 around the trivial vacuumPartition function of N=8 SYM on RxS2 around the trivial vacuum

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

The and dependences are consistent with the gravity dual obtained by Lin and Maldacena.

NS5-brane limitBased on the gauge/gravity duality by Lin-Maldacena,Ling, Mohazab, Shieh, Anders and Raamsdonk proposed a double scaling limit of PWMM which giveslittle string theory (IIA NS5-brane theory) on RxS5.

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.

Future work take into account instantons

• N=8 SYM on RxS2 ABJM on RxS2?

• What is the meaning of the full partition function in the gravity(string) dual? geometry change?

baby universe? (cf) Dijkgraaf-Gopakumar-Ooguri-Vafa

precise check of the gauge/gravity duality

can we say something about NS5-brane?• meaning of Q-closed operator in the gravity dual

• M-theory on 11d plane wave geometry

exact results for perturbative partition functions of theories with su(2|4) symmetry

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