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We construct equivariant, string and leading order characteristic classes andChern-Simons classes for certain infinite rank bundles associated to fibrationsoccurring in loop spaces, Gromov-Witten theory and gauge theory. Resultsinclude a restatement of the S^1 index theorem using equivariant classes on thetangent bundle to loop space; the expression of some GW invariants in terms ofstring and leading order classes for infinite rank bundles over moduli spacesof pseudoholomorphic curves for semipositive symplectic manifolds; theidentification of the real cohomology of a loop group with certain string andleading order classes; the identification of Donaldson's nu-class for4-manifolds with a leading order class for the fibration of irreducibleconnections A over the quotient A/G by the gauge group.

Equivariant, string and leading order characteristic classes associated to fibrations