A gravitational action with stringy Q and R fluxes via deformed differential graded Poisson algebras

We study a deformation of a 2-graded Poisson algebra where the functions of the phase space variables are complemented by linear functions of parity odd velocities. The deformation is carried by a 2-form B-field and a bivector Π, that we consider as gauge fields of the geometric and non-geometric fluxes H, f, Q and R arising in the context of string theory compactification. The technique used to deform the Poisson brackets is widely known for the point particle interacting with a U(1) gauge field, but not in the case of non-abelian or higher spin fields. The construction is closely related to Generalized Geometry: with an element of the algebra that squares to zero, the graded symplectic picture is equivalent to an exact Courant algebroid over the generalized tangent bundle E ≅ TM ⊕ T∗M, and to its higher gauge theory. A particular idempotent graded canonical transformation is equivalent to the generalized metric. Focusing on the generalized differential geometry side we construct an action functional with the Ricci tensor of a connection on covectors, encoding the dynamics of a gravitational theory for a contravariant metric tensor and Q and R fluxes. We also extract a connection on vector fields and determine a non-symmetric metric gravity theory involving a metric and H-flux.