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

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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

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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

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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 FF GG g¢g¢ FF GG 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

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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)

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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

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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 ; Nicolai, Samtleben ; de Wit, Samtleben, M.T ]

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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 ; Angelantonj, Ferrara, M.T ; de Wit, Samtleben, M.T …] Manifestly G-covariant formulation de Wit, Samtleben, M.T Emb. tensor from E 11 and tensor hiearachies [de Wit, Samtleben ; Riccioni, West ; de Wit, Nicolai, Samtleben, ]

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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

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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

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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

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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

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Dall’Agata, Inverso, M.T Borghese, Guarino, Roest,

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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, ] Inequivalent theories only for

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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 )

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