Fundamentals of Lie categories and Yang-Mills theory for multiplicative Ehresmann connections

Abstract

The first and shorter part of this thesis deals with the structural assumption of invertibility in a Lie groupoid. When this assumption is dropped, we obtain the notion of a Lie category: a small category, endowed with a compatible differentiable structure. We introduce various examples of Lie categories, examine their differences and similarities with Lie groupoids, and research the notions emerging naturally from the lack of invertibility of arrows. The aim of the second and principal part of this thesis is to provide a far-reaching generalization of Yang-Mills theory, extending it from the classical setting of principal bundles to general Lie groupoids and algebroids. The notion of a principal bundle connection is now replaced with that of a more general multiplicative Ehresmann connection. In obtaining this generalization, we make various advances to the theory of such connections, as well as invariant linear connections on representations. We develop the obstruction classes for their existence, generalize the (horizontal) exterior covariant derivative to the representation-valued Bott-Shulman-Stasheff and Weil complexes, and inspect their relationship with the van Est map. We research the class of multiplicative connections with cohomologically trivial curvature, which are central to obtaining the desired generalization. Applying the variational principle to this framework rests upon our developed formulae for affine deformations of multiplicative connections. Ultimately, we develop the extension of Yang-Mills theory to a non-integrable and non-transitive setting: the classical Yang-Mills equation is upgraded to a gauge-invariant pair of equations, which now describe the dynamics of gauge fields in both the longitudinal and transversal directions with respect to the (singular) orbit foliation. As an example, we obtain a Yang-Mills theory for $S^1$-bundle gerbes.