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 for regular Courant algebroids. As an application we provide a criterion for the integrability of generalized almost Hermitian structures and generalized almost hyper-Hermitian structures defined on a regular Courant algebroid E, in terms of canonically defined differential operators on spinor bundles associated to E.