On the geometry of double field theory

Double field theory was developed by theoretical physicists as a way to encompass T-duality. In this paper, we express the basic notions of the theory in differential-geometric invariant terms in the framework of para-Kähler manifolds. We define and study the metric algebroids that possess a bracket which is analogous to the Courant bracket of generalized geometry. We show that a double field gives rise to two canonical connections, whose scalar curvatures can be integrated to obtain actions. Finally, in analogy with Dirac structures, we define and study para-Dirac structures on double manifolds.