Connections on Lie groupoids and Chern-Weil theory

TL;DR

This paper develops a Chern-Weil theory for principal bundles over Lie groupoids with integrable connections, defining a new de Rham cohomology and characteristic classes in this generalized setting.

Contribution

It introduces a de Rham cohomology for Lie groupoids with integrable distributions and extends Chern-Weil theory to principal bundles over these structures and differentiable stacks.

Findings

Defined a de Rham cohomology for Lie groupoids with integrable distributions.

Extended Chern-Weil theory to principal G-bundles over Lie groupoids and stacks.

Described characteristic classes in this new framework.

Abstract

Let $\mathbb{X}=[X_1\rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $\mathcal{H} \subset T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution $\mathcal{H}$is integrable, we define a version of de Rham cohomology for the pair $(\mathbb{X}, \mathcal{H})$, and we study connections on principal $G$-bundles over $(\mathbb{X}, \mathcal{H})$ in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern-Weil theory and describe characteristic classes of principal G-bundles over a pair $(\mathbb{X}, \mathcal{H})$.