  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.
