Combinatorics of Non-Abelian Gerbes with Connection and Curvature

TL;DR

This paper develops a functorial combinatorial framework for non-Abelian gerbes with connection and curvature, linking them to gauge theories and providing a geometric perspective on strongly coupled gauge models.

Contribution

It introduces a functorial, combinatorial definition of non-Abelian gerbes with connective structure and curvature, extending the theory to simplicial complexes.

Findings

Defined a fibered category with a functorial connection over edge-paths.

Computed curvature on infinitesimal 4-simplices, recovering cocycle identities.

Linked gerbes to $BF$-theories, suggesting applications in gauge theories.

Abstract

We give a functorial definition of $G$-gerbes over a simplicial complex when the local symmetry group $G$ is non-Abelian. These combinatorial gerbes are naturally endowed with a connective structure and a curving. This allows us to define a fibered category equipped with a functorial connection over the space of edge-paths. By computing the curvature of the latter on the faces of an infinitesimal 4-simplex, we recover the cocycle identities satisfied by the curvature of this gerbe. The link with $BF$-theories suggests that gerbes provide a framework adapted to the geometric formulation of strongly coupled gauge theories.