On the global $2$-holonomy for a $2$-connection on a $2$-bundle

TL;DR

This paper develops an explicit algorithm to compute the global 2-holonomy of a 2-connection on a 2-bundle, integrating local data via transition 2-arrows and cocycles to produce a well-defined global invariant.

Contribution

It introduces a concrete method for calculating the global 2-holonomy by gluing local holonomies using transition 2-arrows and cocycles, advancing the understanding of 2-connection geometry.

Findings

Explicit algorithm for global 2-holonomy calculation

Use of transition 2-arrows for gluing local holonomies

Well-defined global 2-holonomy constructed from local data

Abstract

A crossed module constitutes a strict $2$-groupoid $\mathcal{G}$ and a $\mathcal{G}$-valued cocycle on a manifold defines a $2$-bundle. A $2$-connection on this $2$-bundle is given by a Lie algebra $\mathfrak g$ valued $1$-form $A $ and a Lie algebra $\mathfrak h$ valued $2$-form $B $ over each coordinate chart together with $2$-gauge transformations between them, which satisfy the compatibility condition. Locally, the path-ordered integral of $A $ gives us the local $1$-holonomy, and the surface-ordered integral of $(A ,B )$ gives us the local $2$-holonomy. The transformation of local $2$-holonomies from one coordinate chart to another is provided by the transition $2$-arrow, which is constructed from a $2$-gauge transformation. We can use the transition $2$-arrows and the $2$-arrows provided by the $\mathcal{G}$-valued cocycle to glue such local $2$-holonomies together to get a global one, which is well defined. Namely we give an explicit algorithm for calculating the global $2$-holonomy.