Quantum characteristic classes and the Hofer metric

TL;DR

This paper introduces new characteristic classes for the free loop space of Hamiltonian diffeomorphisms on symplectic manifolds, extending known representations and applying them to analyze minimality in the Hofer metric.

Contribution

It defines generalized characteristic classes that extend the Seidel representation and establishes a ring homomorphism linking loop space homology to quantum homology, with applications to Hofer metric minimality.

Findings

Defined characteristic classes generalizing Seidel representation.

Established a ring homomorphism from loop space homology to quantum homology.

Extended results on minimality in the Hofer metric for Hamiltonian circle actions.

Abstract

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 $L \text {Ham}(M, \omega)$.