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 G-bundle $\varphi :Q\to M$ 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 $A$, the set of all discrete connections whose associated connection is $A$. Our first result is that for flat principal connections, the Integration Problem has a unique solution among the flat discrete connections. More broadly, we prove that the existence part of the Integration Problem always has a solution that needs not be unique. We also see that, when G is abelian, given compatible continuous and discrete curvatures the Integration Problem has a unique solution constrained by those curvatures. Last, extending the notion of discrete connection, we introduce a notion of discrete secondary Ehresmann connection associated to suitable morphisms of local Lie groupoids; then we state the Integration Problem in this context, proving that in the flat case, there is an essentially unique solution.