Deformed Hamiltonian Floer theory, capacity estimates, and Calabi quasimorphisms

TL;DR

This paper introduces deformations of Hamiltonian Floer theory that connect to quantum homology, leading to new capacity bounds and criteria for Calabi quasimorphisms in symplectic manifolds.

Contribution

It develops a family of deformations in Floer theory that relate to quantum homology, providing new bounds and criteria for symplectic invariants.

Findings

Bounds the Hofer-Zehnder capacity using Floer-theoretic invariants.

Provides algebraic criteria for the existence of Calabi quasimorphisms.

Shows the criteria hold for manifolds with semisimple quantum homology and blowups.

Abstract

We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold (M,\omega) which upon passing to homology yields ring isomorphisms with the big quantum homology of M. By studying the properties of the resulting deformed version of the Oh-Schwarz spectral invariants, we obtain a Floer-theoretic interpretation of a result of Lu which bounds the Hofer-Zehnder capacity of M when M has a nonzero Gromov-Witten invariant with two point constraints, and we produce a new algebraic criterion for (M,\omega) to admit a Calabi quasimorphism and a symplectic quasi-state. This latter criterion is found to hold whenever M has generically semisimple quantum homology in the sense considered by Dubrovin and Manin (this includes all compact toric M), and also whenever M is a point blowup of an arbitrary closed symplectic manifold.