# Quantum characteristic classes and the Hofer metric - Mathematics > Symplectic Geometry

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Abstract: Given a closed monotone symplectic manifold $M$, we define certaincharacteristic cohomology classes of the free loop space $L \text {Ham}M,\omega$ with values in $QH * M$, and their $S^1$ equivariant version. Theseclasses generalize the Seidel representation and satisfy versions of the axiomsfor Chern classes. In particular there is a Whitney sum formula, which givesrise 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 itsquantum product. As an application we prove an extension of a theorem of McDuffand Slimowitz on minimality in the Hofer metric of a semifree Hamiltoniancircle action, to higher dimensional geometry of the loop space $L \text{Ham}M, \omega$.

Autor: Yasha Savelyev

Fuente: https://arxiv.org/