Quantum characteristic classes and the Hofer metric

Given a closed monotone symplectic manifold M , we define certain characteristic cohomology classes of the free loop space LHam(M, ω) with values in QH∗(M) , and their S1 equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern classes. In particular there is a Whitney sum formula, which gives rise to a graded ring homomorphism from the ring H∗(ΩHam(M, ω),Q) , with its Pontryagin product to QH2n+∗(M) with its quantum product. As an application we prove an extension to higher dimensional geometry of the loop space LHam(M, ω) of a theorem of McDuff and Slimowitz on minimality in the Hofer metric of a semifree Hamiltonian circle action.