Generalized connections, spinors, and integrability of generalized structures on Courant algebroids

We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined on Courant algebroids. We develop a new, self-contained, approach for the theory of Dirac generating operators on regular Courant algebroids with scalar product of neutral signature. As an application we provide a criterion for the integrability of generalized almost Hermitian structures $(G, \mathcal J)$ and generalized almost hyper-Hermitian structures $(G, \mathcal J_{1}, \mathcal J_{2}, \mathcal J_{3})$ defined on a regular Courant algebroid $E$ in terms of canonically defined differential operators on spinor bundles associated to $E_{\pm}$ (the subbundles of $E$ determined by the generalized metric $G$).