Quantum characteristic classes and the Hofer metric

Abstract

Given a closed monotone symplectic manifold $M$, we define certain characteristic cohomology classes of the free loop space $LHam\left(M,\omega \right)$ with values in $Q{H}_{\ast }\left(M\right)$, and their $S^1$ 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 $H_{\ast }\left(\Omega Ham\left(M,\omega \right),\mathbb{Q}\right)$, with its Pontryagin product to $Q{H}_{2n+\ast }\left(M\right)$ with its quantum product. As an application we prove an extension to higher dimensional geometry of the loop space $LHam\left(M,\omega \right)$ of a theorem of McDuff and Slimowitz on minimality in the Hofer metric of a semifree Hamiltonian circle action.