On two-Dimensional Holonomy

TL;DR

This paper introduces a framework for two-dimensional holonomy on smooth manifolds using categorical groups and connections, enabling the definition of Wilson spheres and advancing higher gauge theory.

Contribution

It constructs categorical holonomies via categorical connections and defines Wilson spheres, extending the concept of holonomy to two dimensions in a smooth manifold setting.

Findings

Defined the thin fundamental categorical group of a manifold.

Constructed smooth categorical holonomies using categorical connections.

Introduced Wilson spheres in the context of higher gauge theory.

Abstract

We define the thin fundamental categorical group ${\mathcal P}_2(M,*)$ of a based smooth manifold $(M,*)$ as the categorical group whose objects are rank-1 homotopy classes of based loops on $M$, and whose morphisms are rank-2 homotopy classes of homotopies between based loops on $M$. Here two maps are rank-$n$ homotopic, when the rank of the differential of the homotopy between them equals $n$. Let $\C(\Gc)$ be a Lie categorical group coming from a Lie crossed module ${\Gc= (\d\colon E \to G,\tr)}$. We construct categorical holonomies, defined to be smooth morphisms ${\mathcal P}_2(M,*) \to \C(\Gc)$, by using a notion of categorical connections, being a pair $(\w,m)$, where $\w$ is a connection 1-form on $P$, a principal $G$ bundle over $M$, and $m$ is a 2-form on $P$ with values in the Lie algebra of $E$, with the pair $(\w,m)$ satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.