Thin homotopy and the holonomy approach to gauge theories

We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of this approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy equivalence relations -- such as the so-called thin homotopy -- and the resulting interpretation of gauge fields as group homomorphisms to a Lie group $G$ satisfying a suitable smoothness condition, encoding the holonomy of a gauge orbit of smooth connections on a principal $G$-bundle. We also prove several structural results on thin homotopy, and in particular we clarify the difference between thin equivalence and retrace equivalence for piecewise-smooth based loops on a smooth manifold, which are often used interchangeably in the physics literature. We conclude by listing a set of questions on topological and functional analytic aspects of groups of based loops, which we consider to be fundamental to establish a rigorous differential geometric foundation of the holonomy formulation of gauge theory.