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

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.

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Cited by in corpus (4)

• Adjusted Parallel Transport for Higher Gauge Theories
• Higher Gauge Theory
• Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid
• Parallel 2-transport and 2-group torsors