The caloron correspondence and higher string classes for loop groups

We review the caloron correspondence between GG-bundles on M×S1M×S1 and ΩG-bundles on MM, where ΩG is the space of smooth loops in the compact Lie group GG. We use the caloron correspondence to define characteristic classes for ΩG-bundles, called string classes, by transgression of characteristic classes of GG-bundles. These generalise the string class of Killingback to higher-dimensional cohomology.