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Based on the structure of a Lie algebroid for non-geometric fluxes in stringtheory, a differential-geometry calculus is developed which combines usualdiffeomorphisms with so-called \beta-diffeomorphisms emanating from gaugesymmetries of the Kalb-Ramond field. This allows to construct a bi-invariantaction of Einstein-Hilbert type comprising a metric, a (quasi-)symplecticstructure \beta and a dilaton. As a salient feature, this symplectic gravityaction and the resulting equations of motion take a form which is similar tothe standard action and field equations. Furthermore, the two actions turn outto be related via a field redefinition reminiscent of the Seiberg-Witten limit.Remarkably, this redefinition admits a direct generalization to higher-order\alpha'-corrections and to the additional fields and couplings appearing in theeffective action of the superstring. Simple solutions to the equations ofmotion of the symplectic gravity action, including Calabi-Yau geometries, arediscussed.

Non-geometric strings, symplectic gravity and differential geometry of Lie algebroids