Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory

Abstract

In this paper we first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entovi-Polterovich theory of spectral symplectic quasi-states and quasimorphisms by incorporating \emph{bulk deformations}, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher [Us4] in a slightly less general context. Then we explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasimorphisms and new Lagrangian intersection results on toric manifolds. The most novel part of this paper is to use open-closed Gromov-Witten theory (operator $\frak q$ in [FOOO1] and its variant involving closed orbits of periodic Hamiltonian system) to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasimorphism to the Lagrangian Floer theory (with bulk deformation). We use this open-closed Gromov-Witten theory to produce new examples. Especially using the calculation of Lagrangian Floer homology with bulk deformation in [FOOO3,FOOO4], we produce examples of compact toric manifolds $(M,\omega)$ which admits uncountably many independent quasimorphisms $\widetilde{\operatorname{Ham}}(M,\omega) \to \mathbb R$. We also obtain a new intersection result of Lagrangian submanifolds on $S^2 \times S^2$ discovered in [FOOO6]. Many of these applications were announced in [FOOO3,FOOO4,FOOO6].