Discrete connections on principal bundles: abelian group case

Abstract

In this note we consider a few interesting properties of discrete connections on principal bundles when the structure group of the bundle is an abelian Lie group. In particular, we show that the discrete connection form and its curvature can be interpreted as singular $1$ and $2$ cochains respectively, with the curvature being the coboundary of the connection form. Using this formalism we prove a discrete analogue of a formula for the holonomy around a loop given by Marsden, Montgomery and Ratiu for (continuous) connections in a similar setting.