Discrete connections on principal bundles: abelian group case

TL;DR

This paper explores properties of discrete connections on principal bundles with abelian groups, interpreting connection forms and curvature as cochains, and establishes a discrete holonomy formula analogous to the continuous case.

Contribution

It introduces a cochain-based formalism for discrete connections and proves a discrete holonomy formula, extending continuous connection concepts to the discrete setting.

Findings

Discrete connection form as a singular 1-cochain

Curvature as a singular 2-cochain and coboundary of the connection form

Discrete holonomy formula analogous to the continuous 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.