$T$-duality for transitive Courant algebroids

We develop a theory of $T$-duality for transitive Courant algebroids. We show that $T$-duality between transitive Courant algebroids $E \to M$ and $\tilde{E} \to \tilde{M}$ induces a map between the spaces of sections of the corresponding canonical weighted spinor bundles $\mathbb{S}_E$ and $\mathbb{S}_\tilde{E}$ intertwining the canonical Dirac generating operators. The map is shown to induce an isomorphism between the spaces of invariant spinors, compatible with an isomorphism between the spaces of invariant sections of the Courant algebroids. The notion of invariance is defined after lifting the vertical parallelisms of the underlying torus bundles $M \to B$ and $\tilde{M} \to B$ to the Courant algebroids and their spinor bundles. We prove a general existence result for $T$-duals under assumptions generalizing the cohomological integrality conditions for $T$-duality in the exact case. Specializing our construction, we find that the $T$-dual of an exact or a heterotic Courant algebroid is again exact or heterotic, respectively.