Towards a T-dual Emergent Gravity

Abstract

Emergent gravity provides a geometric realization of noncommutative U(1) gauge theory, in which gauge-field deformations of a symplectic structure are absorbed by diffeomorphisms through the Darboux theorem, giving rise to an effective Riemannian metric. Independently, topological T-duality relates principal torus bundles with different geometric and flux data whose associated sigma models are physically equivalent. We unify these two constructions within the framework of generalized geometry by formulating emergent gravity in terms of generalized metrics on exact Courant algebroids. In this description, the Seiberg-Witten map is naturally interpreted as a composition of B-transformations and theta-transformations acting on a flat background. Using the Gualtieri-Cavalcanti isomorphism of Courant algebroids, we construct a notion of T-dual emergent gravity for principal torus bundles and derive the corresponding dual generalized metrics. For flat backgrounds with vanishing H-flux, we show that the T-dual generalized metric again admits an emergent gravity interpretation, encoded by a commutative diagram in which T-duality exchanges the order of the transformations generating the emergent metric. For general two-torus fibrations, we obtain explicit expressions for the dual generalized metric and demonstrate that the dual background generically carries nontrivial H-flux, obstructing a conventional symplectic formulation of emergent gravity and motivating an extension to non-exact Courant algebroids. These results establish a precise mathematical correspondence between emergent gravity and topological T-duality, and identify generalized geometry as the natural framework for a T-duality-covariant formulation of emergent gravity in both geometric and flux backgrounds.