Universal string classes and equivariant cohomology

TL;DR

This paper develops a classifying theory for loop group bundles, calculates their string class as an equivariant cohomology class, and introduces higher characteristic classes with explicit differential form representatives.

Contribution

It introduces a new classifying framework for LG-bundles, computes the universal string class as an equivariant cohomology class, and defines higher characteristic classes using the caloron correspondence.

Findings

String class is an equivariant cohomology class.

Explicit equivariant differential form representatives are provided.

Higher characteristic classes are defined for LG-bundles.

Abstract

We give a classifying theory for $LG$-bundles, where $LG$ is the loop group of a compact Lie group $G$, and present a calculation for the string class of the universal $LG$-bundle. We show that this class is in fact an equivariant cohomology class and give an equivariant differential form representing it. We then use the caloron correspondence to define (higher) characteristic classes for $LG$-bundles and to prove for the free loop group an analogue of the result for characteristic classes for based loop groups in Murray-Vozzo (J. Geom. Phys., 60(9), 2010). These classes have a natural interpretation in equivariant cohomology and we give equivariant differential form representatives for the universal case in all odd dimensions.