Maximal super Yang-Mills theories on curved background with off-shell supercharges (KEK), ( /KEK) based on M. F, M. Honda and Y. Yoshida, arxiv: [hep-th] String Advanced Lectures (SAL) at KEK 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] 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 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 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 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 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] 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 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.) We can construct off-shell maximal SYM on curved space on which a Killing spinor exists. Our research purpose Maximal SYM Its important for gauge/gravity duality. Off-shell formulation on curved space Localization Main result Off-shell maximal SYM on flat sp. [Berkovits 93] Contents 1. Off-shell maximal SYM on flat sp. 2. Off-shell maximal SYM on curved sp. 3. Some examples 4. Summary and discussions 1. Off-shell maximal SYM on flat sp. 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. 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 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. 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 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 eigenspinors of In this representation, the solution with 8 off-shell charges: where is the anti-symmetric matrix satisfying Notation 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: (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 2. Off-shell maximal SYM on curved sp. On curved space SUST tr. Same as the flat one Constant spinor doesnt exist on the curved space in general. Note On curved space SUST tr. Same as the flat one Constant spinor doesnt exist on the curved space in general. Note Then, On curved space Then, The condition for invariance is 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 dont include spheres. 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 Examples of spaces We take, where is satisfied by. [Hijazi 86] These have been also classified: Next we consider whether SUSY theories can be constructed on curved space on which Killing spinor exists. On curved space Then, The condition for invariance is So the action is not invariant. Deformation of the action and SUSY tr. Class 1 (d=4) We modify the action and transformation in the following way, SUSY tr. Using the Killing spinor eq. 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 SUSY algebra We consider the square of the SUSY tr. of the each field, Class 2 ( ) We modify the action and transformation in the following way, SUSY tr. There is a unique nontrivial solution, Using the Killing spinor eq. 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. Thus, the action is invariant under the transformation 3. Some examples 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. BMN matrix model [Berenstein-Maldacena-Nastase 02] -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 6D N=(1,1) SYM on We take d=6 and in class 2. SUSY tr. 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) 4. Summary and discussions 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 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. Supplements 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. Here we take and we define Then, 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. -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 : 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: 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: Furthermore we can construct another class of maximal SYM on with off-shell SUSY : d=3, N=8 SYM on dred The action of the class 2 is Remarks In the case of d=2 and R