Connections on non-abelian Gerbes and their Holonomy

TL;DR

This paper develops an axiomatic framework for the parallel transport and surface holonomy of non-abelian gerbes, extending known concepts from abelian cases and revealing new features like mapping class group actions.

Contribution

It introduces a systematic axiomatic approach to non-abelian gerbe connections and holonomy, unifying various existing models and extending surface holonomy to non-abelian cases.

Findings

Framework reproduces known gerbe concepts

Extends surface holonomy to non-abelian gerbes

Reveals new features like mapping class group actions

Abstract

We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the role of a band, or structure 2-group. Upon choosing certain examples of Lie 2-groups, our axiomatic framework reproduces in a systematical way several known concepts of gerbes with connection: non-abelian differential cocycles, Breen-Messing gerbes, abelian and non-abelian bundle gerbes. These relationships convey a well-defined notion of surface holonomy from our axiomatic framework to each of these concrete models. Till now, holonomy was only known for abelian gerbes; our approach reproduces that known concept and extends it to non-abelian gerbes. Several new features of surface holonomy are exposed under its extension to non-abelian gerbes; for example, it carries an action of the mapping class group of the surface.