The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module

TL;DR

This paper introduces a new mathematical structure called the thin fundamental Gray 3-groupoid for smooth manifolds and develops a framework for 3-dimensional holonomies using 2-crossed modules, with applications to Wilson 3-sphere observables.

Contribution

It defines the thin fundamental Gray 3-groupoid of a smooth manifold and constructs 3-dimensional holonomies via differential geometric data using 2-crossed modules.

Findings

Defined the thin fundamental Gray 3-groupoid $S_3(M)$ for smooth manifolds.

Constructed 3-dimensional holonomies as smooth strict Gray 3-groupoid maps.

Introduced Wilson 3-sphere observables based on this framework.

Abstract

We define the thin fundamental Gray 3-groupoid $S_3(M)$ of a smooth manifold $M$ and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps $S_3(M) \to C(H)$, where $H$ is a 2-crossed module of Lie groups and $C(H)$ is the Gray 3-groupoid naturally constructed from $H$. As an application, we define Wilson 3-sphere observables.