Equivariant, string and leading order characteristic classes associated to fibrations

We construct equivariant, string and leading order characteristic classes and Chern-Simons classes for certain infinite rank bundles associated to fibrations occurring in loop spaces, Gromov-Witten theory and gauge theory. Results include a restatement of the S^1 index theorem using equivariant classes on the tangent bundle to loop space; the expression of some GW invariants in terms of string and leading order classes for infinite rank bundles over moduli spaces of pseudoholomorphic curves for semipositive symplectic manifolds; the identification of the real cohomology of a loop group with certain string and leading order classes; the identification of Donaldson's nu-class for 4-manifolds with a leading order class for the fibration of irreducible connections A over the quotient A/G by the gauge group.