On the kernel of holonomy

A connection on a principal $G$-bundle may be identified with a smooth group morphism $\Cal H:\Cal G\Cal L^{\infty}(M)\rightarrow G$, called a holonomy, where $\Cal G\Cal L^{\infty}(M)$ is a group of equivalence classes of loops on the base $M$. The present article focuses on the kernel of this morphism, which consists of the classes of loops along which parallel transport is trivial. Use is made of a formula expressing the gauge potential as a suitable derivative of the holonomy, allowing a different proof of a theorem of Lewandowski's, which states that the kernel of the holonomy contains all the information about the corresponding connection. Some remarks are made about non-smooth holonomies in the context of quantum Yang-Mills theories.