  Inspired by recent studies on string theory with non-geometric fluxes, we
develop a differential geometry calculus combining usual diffeomorphisms with
what we call beta-diffeomorphisms. This allows us to construct a manifestly
bi-invariant Einstein-Hilbert type action for the graviton, the dilaton and a
dynamical (quasi-)symplectic structure. The equations of motion of this
symplectic gravity theory, further generalizations and the relation to the
usual form of the string effective action are discussed. The Seiberg-Witten
limit, known for open strings to relate commutative with non-commutative
theories, makes an interesting appearance.
