Quantum characteristic classes and the Hofer metric

Details Quantum characteristic classes and the Hofer metric Quantum characteristic classes and the Hofer metric

2007-09-27

arxiv.org/abs/0709.4510v4

Geometry & Topology 12 (2008) 2277-2326

Mathematics

Symplectic Geometry

published version, fixed missing incorrectly compiled citations

Scientific paper

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

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