IX Simp´ osio Latino Americano de F´ ısica de Altas Energias SILAFAE 2012 N = 8 gauged supergravity and Extended field theory Jos´ e Alejandro Rosabal Rodriguez (in collaboration with G. Aldazabal, M. Gra˜ na, D. Marqu´ es) Instituto Balseiro - Universidad Nacional de Cuyo - Centro At´ omico Bariloche - CONICET, Bariloche, Argentina Abstract We present a unified description of the bosonic sector of N = 8 gauged supergravity (GSG) in four dimensions. The theory is based on an extended manifold of 1 + 3 + 56 dimensions. The bosonic degrees of freedom are unified in a generalized metric. The diffeomorphism and gauge symmetries are promoted to generalized diffeomorphism Diff (4) × E 7(7) × R + . Motivation Different attempts to promote T-duality, a distinctive symmetry of string theory, to a symmetry of a field theory, can be grouped into the name of Double Field theory (DFT) [1]. Interestingly enough, the Scherk-Schwarz (SS) reduction of (bosonic) DFT leads to an action that can be identified with the bosonic sector of N = 4 gauged supergravities [2]. The aim of the present work is to extend the above ideas to incorporate the full stringy U-duality symmetry group in four dimensions. We name it Extended Field Theory (EFT). Generalized diffeomorphisms on E 7(7) × R + We start by defining the generalized diffeomorphism in the E 7(7) × R + group compatible with supersymmetry. Generalized Lie derivative over an E 7(7) frame (L E ¯ A E ¯ B ) M = E P ¯ A ∂ P E M ¯ B - 12P M Q P R ∂ Q E R ¯ A E P ¯ B + ω 2 ∂ P E P ¯ A E M ¯ B (L E ¯ A E ¯ B ) M = F ¯ C ¯ A ¯ B E M ¯ C ; F ¯ C ¯ A ¯ B = X ¯ C ¯ A ¯ B + T ¯ C ¯ A ¯ B P : 56 × 56 → 133 = Adj (E 7(7) ) ; ω conformal weight of R + Linear constraints T ¯ C ¯ A ¯ B = -ϑ ¯ A δ ¯ C ¯ B + 8ϑ ¯ K P ¯ K ¯ C ¯ A ¯ B ; P ¯ C ¯ M ¯ B ¯ N X ¯ N ¯ A ¯ M = X ¯ C ¯ A ¯ B X ¯ B ¯ A ¯ B = X ¯ A ¯ A ¯ B = 0 ; X ¯ A[ ¯ B ¯ C ] = 0 X ( ¯ A ¯ B ¯ C ) = 0 ; X ¯ C ¯ A ¯ B ∈ 912 ; ϑ ¯ A ∈ 56 Closure condition (quadratic constraints) Generalized bracket [[E ¯ A , E ¯ B ]] = 1 2 (L E ¯ A E ¯ B - L E ¯ B E ¯ A ) We demand [L E ¯ A , L E ¯ B ]E ¯ C - L [[E ¯ A ,E ¯ B ]] E ¯ C = [ ¯ A ¯ B ] = 0 On the other hand gaugings expressed in SU (8) indices should transform as scalars δ ξ F ¯ N ¯ R ¯ M = ξ ¯ P D ¯ P F ¯ N ¯ R ¯ M - ξ ¯ Q ¯ N ¯ P ( ¯ Q ¯ M ) ⇒ ¯ N ¯ P ( ¯ Q ¯ M ) = 0 δ ξ ϑ ¯ M = ξ ¯ P D ¯ P ϑ ¯ M - ξ ¯ Q ˆ ¯ Q ¯ M ⇒ ˆ ¯ Q ¯ M = 0 Remarkably, in the particular case when the gaugins are constant we find exactly the conditions found in [3] [F ¯ M , F ¯ N ]= -F ¯ R ¯ M ¯ N F ¯ R ; F ¯ P ( ¯ M ¯ N ) F ¯ Q ¯ P ¯ R = F ¯ P ( ¯ M ¯ N ) ϑ ¯ P = 0 Space-time extension We now want to couple the gaugings with a four-dimensional metric g μν and vectors A M μ and perform the SS reduction. Generalized diffeomorphism E M ¯ A (x , Y )= E B ¯ A (x )U M B (Y ) L ξ V M = ξ P ∂ P V M - A M P N Q ∂ P ξ Q V N The mixed components of A M P N Q can only take the form A μ M ν N = a 1 δ ν μ δ N M ; A M μ N ν = a 2 δ ν μ δ N M A M μ ν N = a 3 δ ν μ δ N M ; A μ M N ν = a 4 δ ν μ δ N M Master flux (L E ¯ A E ¯ B ) M = F ¯ C ¯ A ¯ B E M ¯ C Bein E M ¯ A (x )= e μ ¯ α e ν ¯ α A M ν 0 Φ M ¯ A U M B (Y )= δ μ β 0 0 U M B Gauge symmetry and four dimensional diffeomorphisms ξ M =(λ μ , Λ M ) ˆ δ e μ ¯ α = L λ e μ ¯ α + 1 2 ϑ N Λ N e μ ¯ α ˆ δ A M μ = L λ A M μ - F M NP Λ N A P μ - ∂ μ Λ M - 1 2 ϑ N Λ N A M μ ˆ δ Φ M ¯ A = L λ Φ M ¯ A - F M NP Λ N Φ P ¯ A F MN P = F MN P (gaugings) F μν M = 2∂ [μ A ν ] M + F NP M A μ N A ν P F [μM ] P = (Φ -1 ) M ¯ N ∂ μ Φ ¯ N P + F [RM ] P A μ R F (μM ) P = F (RM ) P A μ R F μν γ = 2 Γ [μν ] γ = -2(e -1 ) [ν ¯ β ∂ μ] e ¯ β γ Conclusions We have shown that the idea of DFT can be extended to a more general context to include the whole U-duality group. We treated the diffeomorphisms and gauge transformations in a unified way. We provide a higher dimensional origin for the gaugings of maximal supergravity in terms of an extended geometry over a generalized manifold. The generalized Lie derivative provides gaugings that are compatible with supersymmetry, therefore satisfying automatically the linear constraint. We also compute the closure conditions of the algebra and extract the quadratic constraints, which generalize those found in maximal supergravity to the case of non-constant fluxes. Based on these results, we give a first step towards an action written in terms of the generalized fluxes which we hope can be interpreted as generalized Ricci scalar. References [1] C. Hull and B. Zwiebach, “Double Field Theory”, JHEP 0909 (2009) 099 [arXiv:0904.4664 [hep-th]]. [2] G. Aldazabal, W. Baron, D. Marques and C. Nunez, “The effective action of Double Field Theory”, JHEP 1111 (2011) 052 [arXiv:1109.0290 [hep-th]]. [3] Arnaud Le Diffon, Henning Samtleben, Mario Trigiante, “N = 8 Supergravity with Local Scaling Symmetry”, JHEP 04 (2011) 079 [arXiv:1103.2785 [hep-th]]. [email protected] http://fisica.cab.cnea.gov.ar/particulas