Spectral Invariants With Bulk, Quasi-Morphisms 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 Entov-Polterovich theory of spectral symplectic quasi-states and quasi-morphisms by incorporating 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 (2011) 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, quasi-morphisms and new Lagrangian intersection results on toric and non-toric manifolds The most novel part of this paper is to use open-closed Gromov-Witten-Floer theory (operator q in Fukaya, et al. (2009) and its variant involving closed orbits of periodic Hamiltonian system) to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasi-morphism to the Lagrangian Floer theory (with bulk deformation). We use this open-closed Gromov-Witten-Floer theory to produce new examples. Especially using the calculation of Lagrangian Floer cohomology with bulk deformation in Fukaya, et al. (2010, 2011, 2016), we produce examples of compact symplectic manifolds (M, omega) which admits uncountably many independent quasi-morphisms (Ham) over bar (M, omega) -> R. We also obtain a new intersection result for the Lagrangian submanifold in S-2 x S-2 discovered in Fukaya, et al. (2012). Many of these applications were announced in Fukaya, et al. (2010, 2011, 2012).