Let $P \to B$ be a principal $G$-bundle. Then any connection on $P$ defines a continious representation of the loop group $Hol \colon \Omega B \to G$.
The connection is flat iff this representation factors through the fundamental group. And what can one say about its kernel if the connection is not flat? Or at least about its intersection with the connected component of unity?
Is it possible to describe it implicitly knowing the curvature?