Group bundles and group connections

TL;DR

This paper studies smooth families of Lie groups called group bundles and their compatible connections, characterizing their structure, properties, and moduli space via representations of the fundamental group.

Contribution

It provides a detailed characterization of group connections, shows their relation to the Ambrose-Singer theorem, and constructs the moduli space as a representation space.

Findings

Group connections form an affine space modeled over cocycles in the Lie algebra bundle.

Group connections satisfy the Ambrose-Singer theorem.

The moduli space of group connections is described as a space of fundamental group representations.

Abstract

We consider smooth families of Lie groups (group bundles) and connections that are compatible with the group operation. We characterize the space of group connections on a group bundle as an affine space modeled over the vector space of $1$-forms with values cocycles in the Lie algebra bundle of the aforementioned group bundle. We show that group connections satisfy the Ambrose-Singer theorem and that group bundles can be seen as a particular case of associated bundles realizing group connections as associated connections. We give a construction of the Moduli space of group connections with fixed base and fiber, as an space of representations of the fundamental group of the base.