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

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  1. 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

  2. Introduction

  3. 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,…

  4. 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)

  5. 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]

  6. 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]

  7. 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.

  8. 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]

  9. 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

  10. 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

  11. Theories with SU(2|4) symmetry

  12. N=4 SYM on RxS3 : gauge field (Local Lorentz indices of RxS3) : scalar field (adjoint rep) + fermions all fields=0 • vacuum

  13. 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:

  14. 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

  15. 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

  16. plane wave matrix model • vacuum “fuzzy sphere” : spin rep. matrix

  17. 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)

  18. 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)

  19. 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

  20. 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

  21. N=8 SYM on RxS2 around a monopole vacuum : Angular momentum op. in the presence of a monopole with charge

  22. 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

  23. PWMM around a fuzzy sphere vacuum

  24. 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

  25. 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

  26. Mode expansion • N=8 SYM on RxS2 Expand in terms of the monopole spherical harmonics • PWMM Expand in terms of the fuzzy spherical harmonics

  27. N=8 SYM on RxS2 from PWMM • N=8 SYM on RxS2 around a monopole vacuum • PWMM around a fuzzy sphere vacuum

  28. 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.

  29. 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)

  30. 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

  31. 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.

  32. 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.

  33. 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

  34. Localization in PWMM

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

  36. one-loop integral around the saddle points

  37. We perform the localization in PWMM following Pestun,

  38. Plane Wave Matrix Model

  39. 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

  40. Saddle point We choose Saddle point where is a constant matrix commuting with : const. matrix In , and are vanishing.

  41. Saddle points are characterized by reducible representations of SU(2), , and constant matrices 1-loop around a saddle point with integral of

  42. 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.

  43. 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

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

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

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

  47. 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

  48. Partition function of N=8 SYM on RxS2 trivial vacuum

  49. 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 .

  50. 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]