A note on the holonomy of connections in twisted bundles

TL;DR

This paper explores the holonomy of connections in twisted principal bundles, proposing a functorial approach that generalizes previous work and enhances understanding of twisted K-theory applications in string theory.

Contribution

It introduces a functorial definition of holonomy in twisted principal bundles, extending Kapustin's work and providing a broader framework for twisted K-theory.

Findings

Holonomy can be best defined as a functor rather than a map.

The approach generalizes Kapustin's results.

Provides a new perspective on connections in twisted bundles.

Abstract

Recently twisted K-theory has received much attention due to its applications in string theory and the announced result by Freed, Hopkins and Telemann relating the twisted equivariant K-theory of a compact Lie group to its Verlinde algebra. Rather than considering gerbes as separate objects, in twisted K-theory one considers a gerbe as being part of the data for a twisted vector bundle. There is also a notion of a connection in a twisted vector bundle and Kapustin has studied some aspects of the holonomy of such connections. In this note I study the holonomy of connections in twisted principal bundles and show that it can best be defined as a functor rather than a map. Even for the case which Kapustin studied the results in this paper give a more general picture.