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

## TL;DR

This paper characterizes the integrability of various generalized structures on Courant algebroids using torsion-free generalized connections and introduces a new approach for Dirac generating operators, providing criteria for integrability in terms of differential operators on spinor bundles.

## Contribution

It offers a novel, self-contained framework linking torsion-free generalized connections with integrability conditions for generalized structures on Courant algebroids, including new criteria involving spinor bundles.

## Key findings

- Characterization of integrability via torsion-free generalized connections.
- Development of a new approach for Dirac generating operators on Courant algebroids.
- Criteria for integrability of generalized almost Hermitian and hyper-Hermitian structures.

## Abstract

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 with scalar product of neutral signature, in terms of canonically defined differential operators on spinor bundles associated to E_{\pm} (the subbundles of E determined by the generalized metric G).