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総合研究大学院大学 藤塚 理史

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). b ased on M. F, M. Honda and Y. Yoshida, arxiv: 1209.4320[hep-th]. 2012. 10.24. String Advanced Lectures (SAL) at KEK. Our motivation.

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総合研究大学院大学 藤塚 理史

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  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 [Maldacena ‘97] Different views of low energy effective theory on D-branes (or M-branes) SUGRA solution for N Dp-branes (p+1)-dim. U(N) SYM dual !? at near horizon The well known example: ・ ND3-branes In non-conformal field theory, the correspondence is also expected. [Itzhaki-Maldacena-Sonnenschein-Yankielowicz ‘98] ex.)

  3. Our motivation Gauge/Gravity correspondence [Maldacena ‘97] Different views of low energy effective theory on D-branes (or M-branes) SUGRA solution for N Dp-branes (p+1)-dim. U(N) SYM dual !? 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. ・ aSupercharge 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. ・ aSupercharge Q such that Off-shell Then we note

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  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 (2) (1) R-symmetry: reduction from 4D.

  43. 4. Summary and discussions

  44. Summary ・ 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

  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. 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. Therefore the existence of the Killing spinors can be characterized by the holonomy group similarly:

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

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