Report

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 RxS3. [Lin,Maldacena] N=4 SYM on RxS3/Zk (4d) “holonomy” N=8 SYM on RxS2 (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 RxS2 and RxS3/Zk from PWMM [Ishiki,SS,Takayama,Tsuchiya] Theories with SU(2|4) sym. Consistent truncations of N=4 SYM on RxS3. N=4 SYM on RxS3/Zk (4d) T-duality in gauge theory [Lin,Maldacena] “holonomy” [Taylor] N=8 SYM on RxS2 (3d) commutative limit of fuzzy sphere “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 RxS2 and RxS3/Zk 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 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 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] 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 RxS5 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 RxS3 : gauge field : scalar field (adjoint rep) + fermions • vacuum all fields=0 (Local Lorentz indices of RxS3) N=4 SYM on RxS3 Hereafter we focus on the spatial part (S3) of the gauge fields. Local Lorentz indices of S3 where convention for S3 right inv. 1-form: metric: N=4 SYM on RxS3/Zk Keep the modes with the periodicity in N=4 SYM on RxS3. • vacuum “holonomy” N=8 SYM on RxS2 Angular momentum op. on S2 N=8 SYM on RxS2 In the second line we rewrite and the scalar field on S2 as • vacuum in terms of the gauge fields . “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 RxS3/Zk (4d) T-duality in gauge theory [Taylor] N=8 SYM on RxS2 (3d) commutative limit of fuzzy sphere Plane wave matrix model (1d) N=8 SYM on RxS2 from PWMM N=4 SYM on RxS3/Zk (4d) N=8 SYM on RxS2 (3d) commutative limit of fuzzy sphere Plane wave matrix model (1d) N=8 SYM on RxS2 from PWMM PWMM around the following fuzzy sphere vacuum with fixed N=8 SYM on RxS2 around the following monopole vacuum N=8 SYM on RxS2 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 RxS2 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 RxS2 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 S2) [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 RxS2 Expand in terms of the monopole spherical harmonics PWMM Expand in terms of the fuzzy spherical harmonics N=8 SYM on RxS2 from PWMM N=8 SYM on RxS2 around a monopole vacuum PWMM around a fuzzy sphere vacuum N=8 SYM on RxS2 from PWMM N=8 SYM on RxS2 around a monopole vacuum PWMM around a fuzzy sphere vacuum In the limit in which with PWMM coincides with N=8 SYM on RxS2. fixed N=4 SYM on RxS3/Zk from N=8 SYM on RxS2 N=4 SYM on RxS3/Zk (4d) T-duality in gauge theory [Taylor] N=8 SYM on RxS2 (3d) 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 with Identification among blocks of fluctuations (orbifolding) (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 N=4 SYM on RxS3/Zk (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 can be obtained by expanding N=8 SYM on RxS2 around an appropriate monopole background so that all the KK modes are reproduced. N=4 SYM on RxS3/Zk from N=8 SYM on RxS2 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 RxS2 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 RxS3/Zk. 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 Suppose that [Witten; Nekrasov; Pestun; Kapustin et.al.;…] is a symmetry and there is a function Define is independent of such that 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 [Berkovits] : invariant under the off-shell SUSY. • • exist which satisfy :Killing vector 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 Eigenvalues of where with is given by 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 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 with fixed Partition function of N=8 SYM on RxS2 trivial vacuum Partition function of N=4 SYM on RxS3/Zk 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 such that and impose orbifolding condition . 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 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 RxS2 around the trivial vacuum • NS5-brane limit Gauge/gravity duality for N=8 SYM on RxS2 around the trivial vacuum Partition 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 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 RxS5. Expand PWMM around and take the limit in which and with and fixed Little string theory on RxS5 (# 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 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? • 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?