The caloron correspondence and higher string classes for loop groups

TL;DR

This paper reviews the caloron correspondence linking G-bundles on M×S^1 and loop group bundles on M, and introduces higher string classes for loop groups via transgression of G-bundle characteristic classes.

Contribution

It extends the concept of string classes to higher dimensions and provides a framework for defining characteristic classes for loop group bundles.

Findings

Defined higher string classes for loop group bundles.

Generalized Killingback's string class to higher cohomology.

Connected caloron correspondence with characteristic class transgression.

Abstract

We review the caloron correspondence between $G$-bundles on $M \times S^1$ and $\Omega G$-bundles on $M$, where $\Omega G$ is the space of smooth loops in the compact Lie group $G$. We use the caloron correspondence to define characteristic classes for $\Omega G$-bundles, called string classes, by transgression of characteristic classes of $G$-bundles. These generalise the string class of Killingback to higher dimensional cohomology.