The integration problem for principal connections

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.