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

Symplectic Geometry