LOOP GROUPS, STRING CLASSES AND EQUIVARIANT COHOMOLOGY

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 a result for characteristic classes for based loop groups for the free loop group. These classes have a natural interpretation in equivariant cohomology and we give equivariant differential form representatives for the universal case in all odd dimensions.

CIRCLE ACTIONS, CENTRAL EXTENSIONS AND STRING STRUCTURES

The caloron correspondence can be understood as an equivalence of categories between G-bundles over circle bundles and LG ⋊ρ S1-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 ⋊ρ S1-bundle.