Circle actions, central extensions and string structures

TL;DR

This paper explores the caloron correspondence and lifting bundle gerbes to derive an explicit differential form formula for the string class of certain loop group bundles, advancing understanding of string structures.

Contribution

It provides a new explicit differential form formula for the string class of loop group bundles using the caloron correspondence and lifting bundle gerbes.

Findings

Derived an explicit differential form formula for the string class.

Connected caloron correspondence with bundle gerbes.

Enhanced understanding of string structures in bundle theory.

Abstract

The caloron correspondence can be understood as an equivalence of categories between $G$-bundles over circle bundles and $LG \rtimes_\rho S^1$-bundles where $LG$ is the group of smooth loops in $G$. We use it, and lifting bundle gerbes, to derive an explicit differential form based formula for the (real) string class of an $LG \rtimes_\rho S^1$-bundle.