# Moscow Mathematical Journal

Volume 21, Issue 4, October–December 2021 pp. 695–736.

Generalized Connections, Spinors, and Integrability of Generalized Structures on Courant Algebroids

**Authors**:
Vicente Cortés (1) and Liana David (2)

**Author institution:**(1) Department of Mathematics and Center for Mathematical Physics, University of Hamburg, Bundesstrasse 55, D-20146, Hamburg, Germany

(2) Institute of Mathematics Simion Stoilow of the Romanian Academy, Calea Grivitei no. 21, Sector 1, 010702, Bucharest, Romania

**Summary: **

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 $G$).

2020 Math. Subj. Class. Primary: 53D18; Secondary: 53C15.

**Keywords:**Courant algebroids, generalized Kähler structures, generalized complex structures, generalized hypercomplex structures, generalized hyper-Kähler structures, generating Dirac operators.

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