The caloron correspondence and higher string classes for loop groups

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.