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, J ) and generalized almost hyper-Hermitian structures (G, J₁, J₂, J₃) defined on a regular Courant algebroid E in terms of canonically defined differential operators on spinor bundles associated to E_± (the subbundles of E determined by the generalized metric G).