A bi-invariant Einstein-Hilbert action for the non-geometric string

Inspired by recent studies on string theory with non-geometric fluxes, we develop a differential geometry calculus combining usual diffeomorphisms with what we call β-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.