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 $L\text{Ham}(M,\omega )$ with values in $Q{H}_{\ast }(M)$, 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 $H_{\ast }(L\text{Ham}(M,\omega ),\mathbb{Q})$, with its Pontryagin product to $Q{H}_{2n+\ast }(M)$ 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 $L\text{Ham}(M,\omega )$.

published version, fixed missing incorrectly compiled citations

References in corpus (1)

• Homotopy properties of Hamiltonian group actions

Cited by in corpus (8)

• Bott periodicity and stable quantum classes
• Applications of the theory of Floer to symmetric spaces
• Non-contractible Hamiltonian loops in the kernel of Seidel's representation
• K-theoretic invariants of Hamiltonian fibrations
• Quantum characteristic classes, moment correspondences and the Hamiltonian groups of coadjoint orbits
• Global Fukaya category II: singular connections, quantum obstruction theory, and other applications
• Gromov K-area and jumping curves in CP^n
• Floer theory and topology of $Diff(S^2)$