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. The theory is based on a doubling of the $D$-dimensional 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 consider a DFT-inspired supergravity by using a suitable commutative star product of fields and gauge parameters with the aim of studying the corresponding algebraic structure. This star product is not associative, implying the nonclosure of the gauge transformations. Despite this, we get a consistent theory in which an orthogonality condition of momenta associated with double fields is also necessary to have a closed gauge algebra. We first start from the simplest case of one compact spatial dimension. Then, the extension to the case of $d$ spatial dimensions is taken into consideration.