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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,ω) 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,ω) 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.
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