Para-Holomorphic Algebroids and Para-Complex Connections

TL;DR

This paper develops the theory of para-holomorphic algebroids on manifolds with para-complex structures, connecting them to para-Kähler geometry and Poisson-Lie groups through integrable structures and connections.

Contribution

It introduces para-holomorphic algebroids within Courant algebroid theory, extending geometric frameworks to include para-complex and para-Kähler structures.

Findings

Defined para-holomorphic algebroids on para-complex manifolds

Established connections between para-holomorphic algebroids and para-Kähler geometry

Provided examples linking the theory to Poisson-Lie groups

Abstract

The goal of this paper is to develop the theory of Courant algebroids with integrable para-Hermitian vector bundle structures by invoking the theory of Lie bialgebroids. We consider the case where the underlying manifold has an almost para-complex structure, and use this to define a notion of para-holomorphic algebroid. We investigate connections on para-holomorphic algebroids and determine an appropriate sense in which they can be para-complex. Finally, we show through a series of examples how the theory of exact para-holomorphic algebroids with a para-complex connection is a generalization of both para-K\"{a}hler geometry and the theory of Poisson-Lie groups.