Deformed graded Poisson structures, Generalized Geometry and Supergravity

In recent years, a close connection between supergravity, string effective actions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded geometry, introducing an approach based on deformations of graded Poisson structures and derive the corresponding gravity actions. We consider in particular natural deformations of the $2$-graded symplectic manifold $T^{*}[2]T[1]M$ that are based on a metric $g$, a closed Neveu-Schwarz $3$-form $H$ (locally expressed in terms of a Kalb-Ramond 2-form $B$) and a scalar dilaton $\phi$. The derived bracket formalism relates this structure to the generalized differential geometry of a Courant algebroid, which has the appropriate stringy symmetries, and yields a connection with non-trivial curvature and torsion on the generalized "doubled" tangent bundle $E \cong TM \oplus T^{*}M$. Projecting onto $TM$ with the help of a natural non-isotropic splitting of $E$, we obtain a connection and curvature invariants that reproduce the NS-NS sector of supergravity in 10~dimensions. Further results include a fully generalized Dorfman bracket, a generalized Lie bracket and new formulas for torsion and curvature tensors associated to generalized tangent bundles. A byproduct is a unique Koszul-type formula for the torsionful connection naturally associated to a non-symmetric metric, which resolves ambiguity problems and inconsistencies of traditional approaches to non-symmetric gravity theories.