Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

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 $\left(M,\omega \right)$ 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 $\left(M,\omega \right)$ to admit a Calabi quasimorphism and a symplectic quasistate. 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.