Supergravity with Doubled Spacetime Structure

Double Field Theory (DFT) is a low-energy effective theory of a manifestly $O(D,D)$ invariant formulation of the closed string theory when toroidally compactified dimensions are present. The theory is based on a doubled spacetime structure and, in order to preserve the gauge symmetry provided by the invariance under generalized diffeomorphisms, a constraint has to be imposed on fields and gauge parameters. In this paper, we propose a DFT-inspired Supergravity by using a suitable {\em star product} that implements such constraint and the corresponding algebraic structure is explored. We get a consistent DFT in which also an orthogonality condition of momenta is necessary for having a closed gauge algebra. In constructing this theory, we start from the simplest case of doubling one spatial dimension where the action is uniquely determined, without ambiguities, by the gauge symmetry. Then, the extension to the generic $O(D, D)$ case is presented. The result is consistent with the closed string field theory.