The integration problem for principal connections

TL;DR

This paper introduces the Integration Problem for principal connections, exploring the relationship between continuous and discrete connections, and providing conditions for unique solutions based on curvature and group properties.

Contribution

It formalizes the Integration Problem for principal connections, analyzes solutions for flat and abelian cases, and establishes conditions for uniqueness and existence of discrete connections.

Findings

Unique solution for flat principal connections among flat discrete connections.

Existence of solutions under mild conditions on the structure group.

Unique solution when the structure group is abelian with compatible curvatures.

Abstract

In this paper we introduce the Integration Problem for principal connections. Just as a principal connection on a principal bundle $\phi:Q\rightarrow M$ may be used to split $TQ$ into horizontal and vertical subbundles, a discrete connection may be used to split $Q\times Q$ into horizontal and vertical submanifolds. All discrete connections induce a connection on the same principal bundle via a process known as the Lie or derivative functor. The Integration Problem consists of describing, for a principal connection $\mathcal{A}$, the set of all discrete connections whose associated connection is $\mathcal{A}$. Our first result is that for \emph{flat} principal connections, the Integration Problem has a unique solution among the \emph{flat} discrete connections. More broadly, under a fairly mild condition on the structure group $G$ of the principal bundle $\phi$, we prove that the existence part of the Integration Problem has a solution that needs not be unique. Last, we see that, when $G$ is abelian, given compatible continuous and discrete curvatures the Integration Problem has a unique solution constrained by those curvatures.