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

2. Introduction

3. 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

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.  mass gap, many discrete vacua, SU(2|4) sym.(16 SUSY)  SYM on RxS 2 and RxS 3 /Z k 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 RxS 3 /Z k (4d) Consistent truncations of N=4 SYM on RxS 3. (PWMM) [Lin,Maldacena] [Maldacena,Sheikh-Jabbari,Raamsdonk] N=8 SYM on RxS 2 (3d) plane wave matrix model (1d) [Berenstein,Maldacena,Nastase][Kim,Klose,Plefka] “holonomy” “monopole” “fuzzy sphere”

6. Theories with SU(2|4) sym. N=4 SYM on RxS 3 /Z k (4d) Consistent truncations of N=4 SYM on RxS 3. (PWMM) [Lin,Maldacena] [Maldacena,Sheikh-Jabbari,Raamsdonk] N=8 SYM on RxS 2 (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 RxS 2 and RxS 3 /Z k from PWMM [Ishiki,SS,Takayama,Tsuchiya]  gravity dual corresponding to each vacuum of each theory is constructed (bubbling geometry in IIA SUGRA) [Lin,Maldacena]

7. 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, SS JHEP1302, 148 (2013)

8. Our Results 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] We also obtain the partition functions 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”. Asano, Ishiki, Okada, SS JHEP1302, 148 (2013)

9. 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 RxS 5

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

13. N=4 SYM on RxS 3 convention for S 3 right inv. 1-form: metric: Local Lorentz indices of S 3 Hereafter we focus on the spatial part (S 3 ) of the gauge fields. where

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

15. vacuum “Dirac monopole” In the second line we rewrite in terms of the gauge fields and the scalar field on S 2 as. plane wave matrix model monopole charge N=8 SYM on RxS 2

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

17. N=4 SYM on RxS 3 /Z k (4d) N=8 SYM on RxS 2 (3d) Plane wave matrix model (1d) commutative limit of fuzzy sphere Relations among theories with SU(2|4) symmetry T-duality in gauge theory [Taylor]

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

19. PWMM around the following fuzzy sphere vacuum N=8 SYM on RxS 2 from PWMM N=8 SYM on RxS 2 around the following monopole vacuum fixedwith

20. N=8 SYM on RxS 2 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

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

22. PWMM around a fuzzy sphere vacuum 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

23. PWMM around a fuzzy sphere vacuum

24. 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

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

26. Mode expansion N=8 SYM on RxS 2 PWMM Expand in terms of the monopole spherical harmonics Expand in terms of the fuzzy spherical harmonics

27. N=8 SYM on RxS 2 from PWMM PWMM around a fuzzy sphere vacuum N=8 SYM on RxS 2 around a monopole vacuum

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

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

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

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

32. This is achieved in the following way. Expand N=8 SYM on RxS 2 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 RxS 3 /Z k. Extension of Taylor’s T-duality to that on nontrivial fiber bundle [Ishiki,SS,Takayama,Tsuchiya] N=4 SYM on RxS 3 /Z k from N=8 SYM on RxS 2 from N=8 SYM on RxS 2

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

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

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

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

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

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

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 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 fixedwith

48. Partition function of N=8 SYM on RxS 2 trivial vacuum

49. Partition function of N=4 SYM on RxS 3 /Z k such that and impose orbifolding condition. 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

50. Partition function of N=4 SYM on RxS 3 /Z k When, N=4 SYM on RxS 3, 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]

51. Application of our result gauge/gravity duality for N=8 SYM gauge/gravity duality for N=8 SYM on RxS 2 around the trivial vacuum on RxS 2 around the trivial vacuum NS5-brane limit NS5-brane limit

52. 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.

53. 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 Little string theory on RxS 5 (# of NS5 = ) withandfixed In this limit, instantons are suppressed. So, we can check this conjecture by using our result.

54. 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.

55. NS5-brane limit is nicely fitted by with for various !

56. Summary

57. 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.

58. Future work take into account instantons N=8 SYM on RxS 2 ABJM on RxS 2 ? 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