Higher Gauge Theory: 2-Connections on 2-Bundles

TL;DR

This paper introduces the concept of 2-connections on principal 2-bundles, generalizing classical connections to higher categories, and explores their local data, holonomies, and relation to nonabelian gerbes with applications in geometry and physics.

Contribution

It formalizes 2-connection theory on 2-bundles, linking it to nonabelian gerbes and establishing conditions for 2-holonomies and parallel transport in higher gauge theory.

Findings

Defined 2-connection on principal 2-bundles.

Connected 2-holonomies to nonabelian gerbes.

Ensured existence of 2-holonomies under vanishing fake curvature.

Abstract

Connections and curvings on gerbes are beginning to play a vital role in differential geometry and mathematical physics -- first abelian gerbes, and more recently nonabelian gerbes. These concepts can be elegantly understood using the concept of `2-bundle' recently introduced by Bartels. A 2-bundle is a generalization of a bundle in which the fibers are categories rather than sets. Here we introduce the concept of a `2-connection' on a principal 2-bundle. We describe principal 2-bundles with connection in terms of local data, and show that under certain conditions this reduces to the cocycle data for nonabelian gerbes with connection and curving subject to a certain constraint -- namely, the vanishing of the `fake curvature', as defined by Breen and Messing. This constraint also turns out to guarantee the existence of `2-holonomies': that is, parallel transport over both curves and surfaces, fitting together to define a 2-functor from the `path 2-groupoid' of the base space to the structure 2-group. We give a general theory of 2-holonomies and show how they are related to ordinary parallel transport on the path space of the base manifold.