Integrability of generalized structures on odd exact Courant algebroids using generalized connections

TL;DR

This paper investigates the integrability conditions of generalized complex and pseudo-Kähler structures on odd exact Courant algebroids using the framework of adapted generalized connections.

Contribution

It characterizes integrability of these structures via the existence of specific adapted generalized connections and describes their affine spaces.

Findings

Integrability is characterized by the existence of adapted generalized connections.

Describes the affine space of such connections for integrable structures.

Provides a geometric framework for understanding generalized structures on Courant algebroids.

Abstract

Odd exact Courant algebroids constitute a simple class of transitive Courant algebroids. Their underlying vector bundle is of odd rank and differs from a generalized tangent bundle by the addition of a line bundle. In this article we study natural analogues of almost complex and almost pseudo-Hermitian structures on such Courant algebroids, which are called B_n-generalized almost complex/pseudo-Hermitian structures. The corresponding integrable structures are known as B_n-generalized complex structures and B_n-generalized pseudo-K\"{a}hler structures, respectively. We characterize the integrability of B_n-generalized almost complex/pseudo-Hermitian structures on odd exact Courant algebroids in terms of existence of adapted generalized connections. We describe the affine spaces of adapted generalized connections for such integrable generalized structures.