Equivariant, string and leading order characteristic classes associated to fibrations

Infinite rank vector bundles often appear as pushdowns of finite rank bundles from the total space of a fibration to the base space. The infinite rank bundles have string and leading order characteristic classes related to the characteristic classes of the finite rank bundles. We rewrite the ^S1-index theorem as a statement about equivariant leading order classes on loop spaces, interpret certain Gromov-Witten invariants in terms of leading order and string classes, show that the generators of the cohomology of a loop group are Chern-Simons string classes, and relate Donaldson invariants to leading order currents.