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

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

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**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**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) “fuzzy sphere” [Berenstein,Maldacena,Nastase][Kim,Klose,Plefka] (PWMM) • 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. [Lin,Maldacena] N=4 SYM on RxS3/Zk (4d) “holonomy” T-duality in gauge theory [Taylor] N=8 SYM on RxS2 (3d) “monopole” commutative limit of fuzzy sphere [Maldacena,Sheikh-Jabbari,Raamsdonk] plane wave matrix model (1d) “fuzzy sphere” [Berenstein,Maldacena,Nastase][Kim,Klose,Plefka] (PWMM) • 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 RxS2and N=4 SYM on RxS3/Zkfrom • 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**N=4 SYM on RxS3**: gauge field (Local Lorentz indices of RxS3) : scalar field (adjoint rep) + fermions all fields=0 • vacuum**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 in terms of the gauge fields and the scalar field on S2 as . • vacuum monopole charge “Dirac monopole” 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 fixed PWMM coincides with N=8 SYM on RxS2.**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**[Witten; Nekrasov; Pestun; Kapustin et.al.;…] Suppose that is a symmetry and there is a function such that Define is independent of**We perform the localization in PWMM**following Pestun,**Off-shell SUSY in PWMM**SUSY algebra is closed if there exist spinors which satisfy [Berkovits] Indeed, such exist • : invariant under the off-shell SUSY. • :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**Partition function of PWMM**Partition function of PWMM with is given by Eigenvalues of Contribution from the classical action where**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]