Toward a T-Dual Emergent Gravity

Abstract

Emergent gravity provides a geometric interpretation of noncommutative $U\left(1\right)$ gauge theory, in which deformations of a symplectic structure induced by a gauge field are absorbed by diffeomorphisms via the Darboux theorem, giving rise to an effective Riemannian metric. Independently, topological T-duality identifies pairs of principal torus bundles equipped with distinct geometric and flux data whose associated sigma models are physically equivalent. In this work, we place both constructions within the framework of generalized geometry, thereby enabling a uniform treatment of emergent metrics and T-duality transformations. We describe emergent gravity in terms of generalized metrics on exact Courant algebroids and interpret the Seiberg–Witten map as a composition of $B$ and $\theta$ transformations acting on a flat background. Using the Gualtieri–Cavalcanti formulation of T-duality as an isomorphism of Courant algebroids, we construct a natural notion of a T-dual emergent gravity for principal torus bundles. For flat spacetimes with trivial H-flux, we show that the T-dual generalized metric again admits an emergent gravity interpretation. At the level of generalized geometry, this result is encoded in a commutative diagram in which T-duality interchanges the order of the transformations responsible for the emergence of the metric. For general $T^2$-fibrations, we derive explicit formulas for the T-dual generalized metric and show that the dual background generically carries a nontrivial H-flux, which obstructs a direct interpretation in terms of conventional symplectic emergent gravity and necessitates an extension of the framework to nonexact Courant algebroids. Our results provide a precise mathematical link between emergent gravity and T-duality and indicate that generalized geometry furnishes the natural setting for a T-duality–covariant formulation of emergent gravity, including backgrounds with nonvanishing flux.