1. Gauged Supergravities in Different Frames Dr. Mario Trigiante (Politecnico di Torino) F.Cordaro, P.Frè, L.Gualtieri, P.Termonia, M.T. 9804056 Wit, Samtleben, M.T. 0311224 Wit, Samtleben, M.T. 0311224 ; Dall’Agata, Inverso, M.T. 1209.0760

2. Plan of the Talk Overview and Motivations: Gauged Supergravity and string/M-theory compactifications. Embedding tensor formulation of D=4 gauged SUGRAs and duality Conclusions Relevance of symplectic frames: New N=8 SUGRAs with SO(8) local symmetry

3. Introduction D=4 Supergravity from Superstring/M-theory: Superstring M-theory M 1,3 x M Ricci flat Flux = 0 D=4 ungauged Supergravity Global symmetries DualitiesMinimal coupl. mass. def. V(  ) D=4 gauged Supergravity M 1,3 x M Flux  0 Embedding tensor Mass deformations: spontaneous SUSY breaking Scalar potential: moduli stabilization in Minkoswki, dS or AdS vacua

4. Ungauged (extended) Supergravities Electric-magnetic duality symmetry of Maxwell equations now must also involve the scalar fields (Gaillard-Zumino) G = Isom( M scal ) Non-linear action on  Linear action FF GG g¢g¢ FF GG Sp(2 n v, R)  E/M duality promotes G to global sym. of f.eqs. E B. ids. g = 2 G AB CD Smaller symmetry of the action: Scalar fields (described by a non-lin. Sigma-model) are non- minimally coupled to the vector ones

5. The Issue of Symplectic Frames Different symplectic frames (SF) may yield inequivalent actions with different global symmetry groups G e but same physics In the SUGRA description of string/M-theory compactifications, SF fixed by the resulting scalar-vector couplings Symplectic FrameCoupling of scalar fields to vectors is fixed up to a symplectic transfomation on F and G (Symplectic Frame)

6. Parity as an anti-Symplectic Duality Split total scalars so that: isometry is an invariance of the theory is realized on the vector fields and their magnetic duals by an anti-symplectic duality transformation Distinction between the scalar/pseudo-scalar fields depends on the choice of the symplectic frame

7. Gauging Gauging consists in promoting a group G ½ G e ½ G from global to local symmetry of the action. Different SF ) different choices for G. Local invariance w.r.t. G Description of gauging which is independent of the SF: E symplectic 2n v x 2n v matrix All information about the gauging encoded in a G-tensor: the embedding tensor [Cordaro, Frè, Gualtieri, Termonia, M.T. 9804056; Nicolai, Samtleben 0010076; de Wit, Samtleben, M.T. 0311224 ]

8. Restore SUSY of the action: Mass terms: Scalar potential: Fermion shifts: Closure: Locality Linear : String/M-theory origin: [D’Auria, Gargiulo, Ferrara, M.T., Vaulà 0303049; Angelantonj, Ferrara, M.T. 0306185; de Wit, Samtleben, M.T. 0311224…] Manifestly G-covariant formulation de Wit, Samtleben, M.T. 0507289 Emb. tensor from E 11 and tensor hiearachies [de Wit, Samtleben 0501243; Riccioni, West 0705.0752; de Wit, Nicolai, Samtleben, 0801.1294]

9. N=8, D=4 SUGRA Scalar fields in non-linear  -model with target space (1) g    A  (28) A AB    ABC   ABCD gravitational A,B 2 8 of SU(8) R M scal = = 32 supercharges

10. Linear constraints Quad. constraints Looking for SO(8): First gauging: [de Wit, Nicolai ’82] Original dWN gauging Hull’s CSO(p,q,r)-gaugings Same groups gauged by the magnetic vectors Gaugings defined by

11. Take generic Quadratic constraints Gauge connection: Choice corresponds to an SO(8)-gauging in a different SF in which A’ IJ are electric Features of E: it centralizes so(8) in Sp(56) and is NOT in E 7(7) for generic angle: but not in SU(28) for generic 

12.  analogue of de Roo-Wagemann’s angle in N=4, N=2: parametrizes inequivalent theories. Vacua of original dWN theory  =0  studied by Warner and recently by Fischbacher (found several critical points, not complete yet) Studied vacua with a G 2 residual symmetry: suffices to restrict to G 2 singlets Scalar potential: whereand de Wit, Samtleben, M.T. 0705.2101

13. Dall’Agata, Inverso, M.T. 1209.0760 Borghese, Guarino, Roest, 1209.3003

14. Discrete symmetries of V eff : (Parity) (SO(8) Triality) originate from non trivial symmetries of the whole theory  does not affect action terms up to second order in the fluctuations about the N=8 vacuum (mass spectrum). Possible relation to compactifiation of D=11 SUGRA on with torsion (  ) (ABJ) [Aharony, Bergman, Jafferis, 0807.4924] Inequivalent theories only for

15. Conclusions Showed in a given example how initial choice of SF determines, after gauging, physical properties of the model Study vacua of the new family of SO(8)-gauged maximal SUGRAS RG flow from new N=0 G 2 vacuum to N=8 SO(8) one (both stable AdS 4 )