Exact Results for perturbative partition functions of theories with SU(2|4) symmetry

Shinji Shimasaki

(Kyoto University) Based on the work in collaboration with Y. Asano (Kyoto U.), G. Ishiki (YITP) and T. Okada(YITP) JHEP1302, 148 (2013) (arXiv:1211.0364[hep-th]) and the work in progress

Introduction

Localization

Localization method is a powerful tool to exactly compute some physical quantities in quantum field theories.

i.e. Partition function, vev of Wilson loop in super Yang-Mills (SYM) theories in 4d, super Chern-Simons-matter theories in 3d, SYM in 5d, … M-theory(M2, M5-brane), AdS/CFT,…

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.

Consistent truncations of N=4 SYM on RxS 3 .

[Lin,Maldacena] N=4 SYM on RxS 3 /Z k (4d) “holonomy” N=8 SYM on RxS 2 (3d) “monopole” [Maldacena,Sheikh-Jabbari,Raamsdonk] plane wave matrix model (1d) (PWMM) “fuzzy sphere” [Berenstein,Maldacena,Nastase][Kim,Klose,Plefka]  mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)  gravity dual corresponding to each vacuum of each theory is constructed (bubbling geometry in IIA SUGRA) [Lin,Maldacena]  SYM on RxS 2 and RxS 3 /Z k from PWMM [Ishiki,SS,Takayama,Tsuchiya]

Theories with SU(2|4) sym.

Consistent truncations of N=4 SYM on RxS 3 .

[Lin,Maldacena] N=4 SYM on RxS 3 /Z k T-duality in gauge theory [Taylor] N=8 SYM on RxS 2 (4d) (3d) “holonomy” “monopole” commutative limit of fuzzy sphere plane wave matrix model (1d) (PWMM) [Maldacena,Sheikh-Jabbari,Raamsdonk] “fuzzy sphere” [Berenstein,Maldacena,Nastase][Kim,Klose,Plefka]  mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)  gravity dual corresponding to each vacuum of each theory is constructed (bubbling geometry in IIA SUGRA) [Lin,Maldacena]  SYM on RxS 2 and RxS 3 /Z k from PWMM [Ishiki,SS,Takayama,Tsuchiya]

•

Our Results

Asano, Ishiki, Okada, SS JHEP1302, 148 (2013) Using the localization method, we compute the partition function of PWMM up to instantons; • where : vacuum configuration characterized by In the ’t Hooft limit, our result becomes exact.

is written as a matrix integral. • We check that our result reproduces a one-loop result of PWMM.

•

Our Results

Asano, Ishiki, Okada, SS JHEP1302, 148 (2013) We also obtain the partition functions of N=8 SYM on RxS 2 that of PWMM and N=4 SYM on RxS 3 /Z k from by taking limits corresponding to “commutative limit of fuzzy sphere” and “T-duality in gauge theory”.

• We show that, in our computation, the partition function of N=4 SYM on RxS 3 ( N=4 SYM on RxS 3 /Z k 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]

Application of our result

Work in progress; Asano, Ishiki, Okada, SS • gauge/gravity correspondence for theories with SU(2|4) symmetry • Little string theory on RxS 5

Plan of this talk

1. Introduction 2. Theories with SU(2|4) symmetry 3. Localization in PWMM 4. Exact results of theories with SU(2|4) symmetry 5. Application of our result 6. Summary

Theories with SU(2|4) symmetry

N=4 SYM on RxS 3

: gauge field : scalar field (adjoint rep) + fermions • vacuum all fields=0 (Local Lorentz indices of RxS 3 )

N=4 SYM on RxS 3

Hereafter we focus on the spatial part (S 3 ) of the gauge fields.

Local Lorentz indices of S 3 where convention for S 3 right inv. 1-form: metric:

N=4 SYM on RxS 3 /Z k

Keep the modes with the periodicity in N=4 SYM on RxS 3 . • vacuum “holonomy” N=8 SYM on RxS 2 Angular momentum op. on S 2

N=8 SYM on RxS 2

In the second line we rewrite in terms of the gauge fields and the scalar field on S 2 as .

• vacuum “Dirac monopole” monopole charge plane wave matrix model

plane wave matrix model

• vacuum “fuzzy sphere” : spin rep. matrix

Relations among theories with SU(2|4) symmetry

N=4 SYM on RxS 3 /Z k T-duality in gauge theory [Taylor] N=8 SYM on RxS 2 (4d) (3d) commutative limit of fuzzy sphere Plane wave matrix model (1d)

N=8 SYM on RxS 2 from PWMM

N=4 SYM on RxS 3 /Z k (4d) N=8 SYM on RxS 2 (3d) commutative limit of fuzzy sphere Plane wave matrix model (1d)

N=8 SYM on RxS 2 from PWMM

 PWMM around the following fuzzy sphere vacuum with fixed  N=8 SYM on RxS 2 around the following monopole vacuum

N=8 SYM on RxS 2 around a monopole vacuum

• monopole vacuum • Expand the fields around a monopole vacuum • Decompose fields into blocks according to the block structure of the vacuum (s,t) block matrix

N=8 SYM on RxS 2 around a monopole vacuum

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

PWMM around a fuzzy sphere vacuum

• fuzzy sphere vacuum • Expand the fields around a fuzzy sphere vacuum • Decompose fields into blocks according to the block structure of the vacuum (s,t) block matrix

PWMM around a fuzzy sphere vacuum

 N=8 SYM on RxS 2 around a monopole vacuum : Angular momentum op. in the presence of a monopole with charge  PWMM around a fuzzy sphere vacuum

Spherical harmonics

 monopole spherical harmonics (basis of sections of a line bundle on S 2 ) [Wu,Yang]  fuzzy spherical harmonics (basis of rectangular matrix ) [Grosse,Klimcik,Presnajder; Baez,Balachandran,Ydri,Vaidya; Dasgupta,Sheikh-Jabbari,Raamsdonk;…] with fixed

Mode expansion

 N=8 SYM on RxS 2 Expand in terms of the monopole spherical harmonics  PWMM Expand in terms of the fuzzy spherical harmonics

N=8 SYM on RxS 2 from PWMM

 N=8 SYM on RxS 2 around a monopole vacuum  PWMM around a fuzzy sphere vacuum

N=8 SYM on RxS 2 from PWMM

 N=8 SYM on RxS 2 around a monopole vacuum  PWMM around a fuzzy sphere vacuum In the limit in which with PWMM coincides with N=8 SYM on RxS 2 .

fixed

N=4 SYM on RxS 3 /Z k from N=8 SYM on RxS 2

N=4 SYM on RxS 3 /Z k T-duality in gauge theory [Taylor] N=8 SYM on RxS 2 (4d) (3d) Plane wave matrix model (1d)

N=4 SYM on RxS 3 /Z k from N=8 SYM on RxS 2

 N=8 SYM on RxS 2 around the following monopole vacuum with Identification among blocks of fluctuations (orbifolding)  (an infinite copies of) N=4 SYM on RxS 3 /Z k the trivial vacuum around

N=4 SYM on RxS 3 /Z k from N=8 SYM on RxS 2

N=4 SYM on RxS 3 /Z k (S 3 /Z k : nontrivial S 1 bundle over S 2 ) KK expand along S 1 (locally) N=8 SYM on RxS 2 with infinite number of KK modes • These KK mode are sections of line bundle on S background in N=8 SYM on RxS 2 .

(monopole charge = KK momentum) 2 and regarded as fluctuations around a monopole • N=4 SYM on RxS N=8 SYM on RxS 3 2 /Z k can be obtained by expanding around an appropriate monopole background so that all the KK modes are reproduced.

N=4 SYM on RxS 3 /Z k from N=8 SYM on RxS 2

• This is achieved in the following way.

Extension of Taylor’s T-duality to that on nontrivial fiber bundle [Ishiki,SS,Takayama,Tsuchiya] Expand N=8 SYM on RxS 2 around the following monopole vacuum with • Make the identification among blocks of fluctuations (orbifolding) • Then, we obtain (an infinite copies of) N=4 U(N) SYM on RxS 3 /Z k .

Plan of this talk

1. Introduction 2. Theories with SU(2|4) symmetry 3. Localization in PWMM 4. Exact results of theories with SU(2|4) symmetry 5. Application of our result 6. Summary

Localization in PWMM

Localization

[Witten; Nekrasov; Pestun; Kapustin et.al.;…] Suppose that is a symmetry and there is a function such that Define is independent of

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 Indeed, such exist which satisfy [Berkovits] :Killing vector : invariant under the off-shell SUSY.

Saddle point

We choose Saddle point where is a constant matrix commuting with : const. matrix 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

Instanton

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

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

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)

[Yee,Yi;Lin;Bachas,Hoppe,Piolin] Here we neglect the instantons.

Plan of this talk

1. Introduction 2. Theories with SU(2|4) symmetry 3. Localization in PWMM 4. Exact results of theories with SU(2|4) symmetry 5. Application of our result 6. Summary

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

Partition function of PWMM

Partition function of PWMM with is given by where Eigenvalues of Contribution from the classical action

Partition function of PWMM

Trivial vacuum (cf.) partition function of 6d IIB matrix model [Kazakov-Kostov-Nekrasov] [Kitazawa-Mizoguchi-Saito]

Partition function of N=8 SYM on RxS 2

In order to obtain the partition function of N=8 SYM on RxS 2 from that of PWMM, we take the commutative limit of fuzzy sphere, in which with fixed

Partition function of N=8 SYM on RxS 2

trivial vacuum

Partition function of N=4 SYM on RxS 3 /Z k

In order to obtain the partition function of N=4 SYM on RxS 3 /Z k around the trivial background from that of N=8 SYM on RxS 2 , we take such that and impose orbifolding condition .

Partition function of N=4 SYM on RxS 3 /Z k

When , N=4 SYM on RxS the Vandermonde determinant.

3 , the measure factors completely cancel out except for Gaussian matrix model Consistent 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 RxS 2 around the trivial vacuum

•

NS5-brane limit

Gauge/gravity duality for N=8 SYM on RxS 2 around the trivial vacuum

Partition function of N=8 SYM on RxS 2 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 limit

Based on the gauge/gravity duality by Lin-Maldacena, Ling, Mohazab, Shieh, Anders and Raamsdonk proposed a double scaling limit of PWMM which gives little string theory (IIA NS5-brane theory) on RxS 5 .

Expand PWMM around and take the limit in which and with and fixed Little string theory on RxS 5 (# of NS5 = ) In this limit, instantons are suppressed.

So, we can check this conjecture by using our result.

NS5-brane limit

If this conjecture is true, the vev of an operator can be expanded as 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 RxS 2 and N=4 SYM on RxS 3 /Z k 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 RxS 5 .

Future work

 take into account instantons • N=8 SYM on RxS 2 ABJM on RxS 2 ?

• M-theory on 11d plane wave geometry • 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 • meaning of Q-closed operator in the gravity dual  can we say something about NS5-brane?