Loop groups and characteristic classes

Suppose $G$ is a compact Lie group, $LG$ its (free) loop group and $\Omega G$ its based loop group. Let $P \to M$ be a principal bundle with structure group one of these loop groups. In general, differential form representatives of characteristic classes for principal bundles can be easily obtained using the Chern-Weil homomorphism, however for infinite-dimensional bundles such as $P$ this runs into analytical problems and classes are more difficult to construct. In this talk I will explain some new results on characteristic classes for loop group bundles which demonstrate how to construct certain classes---which we call string classes---for such bundles. These are obtained by making heavy use of a certain $G$-bundle associated to any loop group bundle (which allows us to avoid the problems of dealing with infinite-dimensional bundles). We shall see that the free loop group case naturally involves equivariant cohomology. This is joint work with Michael Murray.

Attachment	Size
Vozzo-Carey60.pdf	109.49 KB