Poisson-generalized Geometry and R-flux

We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of [Formula: see text]-diffeomorphisms and [Formula: see text]-transformations. It is a starting point of an alternative version of the generalized geometry based on the cotangent bundle, such as Dirac structures and generalized Riemannian structures. In particular, [Formula: see text]-fluxes are formulated as a twisting of this Courant algebroid by a local [Formula: see text]-transformations, in the same way as [Formula: see text]-fluxes are the twist of the generalized tangent bundle. It is a three-vector classified by Poisson three-cohomology and it appears in a twisted bracket and in an exact sequence.