Moscow Mathematical Journal
Volume 21, Issue 4, October–December 2021 pp. 695–736.
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 $G$). 2020 Math. Subj. Class. Primary: 53D18; Secondary: 53C15.
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:
Keywords: Courant algebroids, generalized Kähler structures, generalized complex structures, generalized hypercomplex structures,
generalized hyper-Kähler structures, generating Dirac operators.
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