Generalised geometry for supersymmetric flux backgrounds

Abstract

We present a geometric description of flux backgrounds in supergravity that preserve eight supercharges using the language of (exceptional) generalised geometry. These “exceptional Calabi–Yau” geometries generalise complex, symplectic and hyper-Kahler geometries, where integrability is equivalent to supersymmetry for the background. The integrability conditions take the form of vanishing moment maps for the “generalised diffeomorphism group”, and the moduli spaces of structures appear as hyper-Kahler and symplectic quotients. Our formalism applies to generic D=4,5,6 backgrounds preserving eight supercharges in both type II and eleven-dimensional supergravity. We include a number of examples of flux backgrounds that can be reformulated as exceptional Calabi–Yau geometries.We extend this analysis and show that generic AdS flux backgrounds in D=4,5 are also described by exceptional generalised geometry, giving what one might call “exceptional Sasaki–Einstein” geometry. These backgrounds always admit a “generalised Reeb vector” that generates a Killing symmetry of the background, corresponding to the R-symmetry of the dual field theory. We also discuss the relation between generalised structures and quantities in the dual field theory.We then consider deformations of these generalised structures. For AdS5 backgrounds in type IIB, a first-order deformation amounts to turning on three-form fluxes that preserve supersymmetry. We find the general form of these fluxes for any Sasaki–Einstein space and show that higher-order deformations are obstructed by the moment map for the symmetry group of the undeformed background. In the dual field theory, this corresponds to finding those marginal deformations that are exactly marginal. We give a number of examples and match to known expressions in the literature. We also apply our formalism to AdS5 backgrounds in M-theory, where the first-order deformation amounts to turning on a four-form flux that preserves supersymmetry.