On 2-Holonomy

TL;DR

This paper constructs a 2-holonomy cycle in higher Hochschild homology for non-abelian gerbes on a torus, extending the concept of holonomy from principal bundles using the Baez-Schreiber connection.

Contribution

It introduces a method to represent 2-holonomy as a cycle in higher Hochschild homology, leveraging abelian structures in crossed modules of principal 2-bundles.

Findings

Constructs a 2-holonomy cycle in higher Hochschild homology.

Uses the Baez-Schreiber connection 1-form.

Shows the possibility of simplifying crossed modules to abelian cases.

Abstract

We construct a cycle in higher Hochschild homology associated to the 2-dimensional torus which represents 2-holonomy of a non-abelian gerbe in the same way the ordinary holonomy of a principal G-bundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1-form of Baez-Schreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module mu: g -> h of the principal 2-bundle, the Lie algebra h is abelian, up to equivalence of crossed modules.