Smoothness of the holonomy map

In this talk, smoothness properties of the holonomy map of a smooth principal $G$-bundle with connection will be discussed. For a natural infinite-dimensional manifold structure on the space $LM$ of piecewise smooth loops, the holonomy turns out to be smooth as a map $\mathrm{Hol}: LM \to G$. We will give an explicit formula for the derivative of the holonomy map at a loop γ in terms of the curvature and the horizontal lift of γ. Conversely, every smooth map $H: LM \to G$ satisfying certain natural morphism properties gives rise to a smooth principal $G$-bundle with a connection having $H$ as its holonomy. This reconstruction theorem and Maurer-Cartan theory in the infinite-dimensional setting allows to redrive well-known facts about the topological classification of principal bundles and Chern-Weil theory. If time permits, an application of these results to the moduli space of Yang-Mills connections over a Riemannian surface is outlined.