1. Maximal super Yang-Mills theories on curved background with off-shell supercharges 総合研究大学院大学 藤塚 理史 共同研究者： 吉田 豊 氏 (KEK), 本多 正純 氏 ( 総研大 /KEK) based on M. F, M. Honda and Y. Yoshida, arxiv:1209.4320[hep-th] 2012. 10.24. String Advanced Lectures (SAL) at KEK

2. Our motivation Gauge/Gravity correspondence SUGRA solution for N Dp-branes [Maldacena ‘97] The well known example: dual !? Different views of low energy effective theory on D-branes (or M-branes) (p+1)-dim. U(N) SYM ・ N D3-branes at near horizon In non-conformal field theory, the correspondence is also expected. ex.) [Itzhaki-Maldacena-Sonnenschein-Yankielowicz ‘98]

3. Our motivation Gauge/Gravity correspondence SUGRA solution for N Dp-branes [Maldacena ‘97] dual !? Different views of low energy effective theory on D-branes (or M-branes) (p+1)-dim. U(N) SYM at near horizon This duality is a strong/weak duality: strong coupling region weak coupling region “Localization” method

4. Localization In these days it has been performed various exact calculations using “ Localization ” method in SUSY gauge theories. ・ a Supercharge Q such that ・ V such that Deformation of the expectation value: Then we note

5. Localization In these days it has been performed various exact calculations using “ Localization ” method in SUSY gauge theories. ・ a Supercharge Q such that Then we note Off-shell

6. Localization In these days it has been performed various exact calculations using “ Localization ” method in SUSY gauge theories. ・ a Supercharge Q such that Off-shell If we consider it on flat space, “infrared effect” and “flat directions” divergence ! compact space mass terms

7. Localization If we consider it on flat space, “infrared effect” and “flat directions” divergence ! compact space mass terms Ex) 4D N=4 SYM on [Pestun ‘07]

8. etc… [Pestun ‘07] [Kapustin-Willet-Yaakov ‘09] [Hosomichi –Seong-Terashima ‘12] [Hama-Hosomichi-Lee ‘11], [Imamura-Yokoyama ‘11] [Hama-Hosomichi ‘12] [Imamura ‘12] [Gang ‘09] [Hama-Hosomichi-Lee ‘10], [Jafferis ‘10] [Imamura-Yokoyama ‘12] Many off-shell SUSY theories on curved space have been studied in these days: Round sphere Squashed sphere Others

9. 4D N=1 [Festuccia-Seiberg ‘11] [Dumitrescu-Festuccia-Seiberg ‘12] The more the number of SUSY and dimension grows, the more difficult we construct the off-shell SUSY theories on curved space generally. It has been constructed partially by Berkovits. [Berkovits ‘93] However its general formalism has not been known. Off-shell SUSY ex.) off-shell maximal SYM on flat space Rigid SUSY on curved space ex.)

10. We can construct off-shell maximal SYM on curved space on which a Killing spinor exists. Our research purpose Maximal SYM It’s important for gauge/gravity duality. Off-shell formulation on curved space Localization Main result Off-shell maximal SYM on flat sp. [Berkovits ‘93]

11. Contents 1. Off-shell maximal SYM on flat sp. 2. Off-shell maximal SYM on curved sp. 3. Some examples 4. Summary and discussions

12. 1. Off-shell maximal SYM on flat sp.

13. SUSY tr. Notation where Berkovits method [Berkovits ‘93] on-shell SYM on flat space: where is a 16 components Majorana-Weyl spinor, and. maximal SYM dred Charge conjugation matrix: Note where is a constant bosonic spinor.

14. Notation where Berkovits method [Berkovits ‘93] on-shell SYM on flat space: where is a 16 components Majorana-Weyl spinor, and. Charge conjugation matrix: In off-shell, 7-(bosonic) auxiliary fields

15. where is a (bosonic) pure imaginary auxiliary field. where depends on, and is (bosonic) spinor satisfying Off-shell maximal SYM on flat space SUSY tr. For any nonzero, there exist which satisfy above constraint.

16. solution The number of off-shell supercharges [Berkovits ‘93] [Evans ‘94] Given any, we can construct which solves the constraints. linear in conventional SUSY d.o.f of the number of off-shell supercharges We can construct the solution which has 9 off-shell supercharges at least. more than 9 ?? 16-components

17. The number of off-shell supercharges We impose the restriction to as (1). 8 off-shell supercharges By using the, Reduce “d.o.f of “ to 1/2 8 off-shell supercharges Next in the case of 9 off-shell charges solution… We have to introduce concrete notation. 16-components the solution of 8 the solution of 9

18. eigenspinors of In this representation, the solution with 8 off-shell charges: where is the anti-symmetric matrix satisfying Notation

19. eigenspinors of In this representation, the solution with 8 off-shell charges: (2). 9 off-shell supercharges Note in 8 off-shell charges, We can construct a solution in which is nonzero:

20. (2). 9 off-shell supercharges Note in 8 off-shell charges, We can construct a solution in which is nonzero: Introduce a matrix: Then, 9 off-shell supercharges

21. 2. Off-shell maximal SYM on curved sp.

22. On curved space SUST tr. Same as the flat one Constant spinor doesn’t exist on the curved space in general. Note

23. On curved space SUST tr. Same as the flat one Constant spinor doesn’t exist on the curved space in general. Note Then,

24. On curved space Then, The condition for invariance is

25. Parallel spinors The condition for the invariance is Existence of the above spinors can be characterized by the holonomy group. [Hitchin ‘74] [Wang ‘89] For example these don’t include spheres.

26. is the odd product of internal gamma matrices. Killing spinors extension where is a constant that depends on a space, and The above eq. implies

27. Examples of spaces We take, where is satisfied by. [Hijazi ‘86]

28. These have been also classified:

29. Next we consider whether SUSY theories can be constructed on curved space on which Killing spinor exists.

30. On curved space Then, The condition for invariance is So the action is not invariant. Deformation of the action and SUSY tr.

31. Class 1 (d=4) We modify the action and transformation in the following way, SUSY tr. Using the Killing spinor eq.

32. Note that this is the equivalent to the well known theory on conformally flat space. [Pestun ‘07] etc. ex.) Thus, the action is invariant under the transformation

33. SUSY algebra We consider the square of the SUSY tr. of the each field,

34. Class 2 ( ) We modify the action and transformation in the following way, SUSY tr.

35. There is a unique nontrivial solution, Using the Killing spinor eq.

36. SUSY algebra We consider the square of the SUSY tr. of the each field, The dilatation vanishes automatically in this class because of anti-symmetry of.

37. Thus, the action is invariant under the transformation

38. 3. Some examples

39. BMN matrix model [Berenstein-Maldacena-Nastase ‘02] If we integrate out, this is the on-shell BMN matrix model. We take d=1 and in class 2, then SUSY tr.

40. BMN matrix model [Berenstein-Maldacena-Nastase ‘02] β-function and Wilson loop of ・ Non-perturbative formulation of [Ishiki-Shimasaki-Tsuchiya ‘11] conformal map [Ishii-Ishiki-Shimasaki-Tsuchiya ‘08] ・ Gravity dual corresponding to theory around each vacuum [Lin-Maldacena ‘06] flat direction Large-N equivalence

41. 6D N=(1,1) SYM on We take d=6 and in class 2. SUSY tr.

42. 3D N=8 SYM on There are 2 ways of constructing this theory: (1). Applying to the class 2 directly i.e. we take d=3 and in class 2. (2). Dimensional reduction of the class 1 on to These theories are different! main difference R-symmetry: reduction from 4D. (1) (2)

43. 4. Summary and discussions

44. ・ We have constructed off-shell maximal SYM on curved space on which a Killing spinor exists. ・ This class of the space contains and so on. ・ We have also constructed the different maximal SYM with same number of supercharges on same space. Ex.) d=3, N=8 SYM on Summary

45. Future work ・ Gauge/Gravity duality ex.) ・ Localization of BMN matrix model Non-perturbative verification of the large-N equivalence ・ Extending to more larger class of curved space ex.) spaces which include a connection and so on.

46. Supplements

47. The rewriting of Killing spinor eq. Killing spinor eq. We can decompose as Then we can rewrite the Killing spinor eq. where D is the Dirac op.

48. Here we take and we define Then,

49. Therefore the existence of the Killing spinors can be characterized by the holonomy group similarly: Killing spinors Killing spinor eq. Here we introduce “cone” over Then the Killing spinor eq. can be rewritten as where is covariant derivative on the cone.

50. -action: subgroup of isometry. Since Killing spinors on are constants along, so the former is also the Killing spinors on. Therefore the number of off-shell supercharges is 4 at least. Also there are orbifolds in which exist Killing spinors :

51. Also the solution of the Killing spinor eq. is where is any constant spinor. The solution of the Killing spinor eq. is The solution for Killing spinor eq. We give the metric:

52. Also the solution of the Killing spinor eq. is where is any constant spinor. The solution of the Killing spinor eq. is We give the metric:

53. Furthermore we can construct another class of maximal SYM on with off-shell SUSY : d=3, N=8 SYM on dred

54. The action of the class 2 is Remarks ・ In the case of d=2 and R<0 ( ), the mass terms of scalar fields become negative. ・ In the case of d≠4, the action becomes non-hermitian because of the 3-pt. term of scalar fields and fermionic mass-term. Integrability condition Comments