Non-contractible periodic orbits, Gromov invariants, and Floer-theoretic torsions

Abstract

In a previous paper, the author introduced a Floer-theoretic torsion invariant I_F, which roughly takes the form of a product of a power series counting perturbed pseudo-holomorphic tori, and the Reidemeister torsion of the symplectic Floer complex. We pointed out the formal resemblance of I_F with a generating function of genus 1 Gromov invariant; furthermore, for heuristic reasons one also expects a relation with the 1-loop generating function in the A-model side of mirror symmetry, which counts genus 1 holomorphic curves. The present article makes this expected relation precise in the simplest cases, in two variants of the I_F defined in the earlier work: the lagrangian intersection version, I_F(L, L'), and an S^1-equivariant version, I_F^{S^1}. As a by-product, we obtain some existence results of noncontractible periodic orbits in symplectic dynamics. For example, the results of Gatien-Lalonde are extended to a much wider class of manifolds. The two versions I_F(L, L') and I_F^{S^1} are only minimally developed in this paper, leaving fuller accounts to future work. The lagrangian intersection version, I_F(L, L'), should be viewed as a simplest example of a rigorous definition of the higher-loop ``open Gromov-Witten invariants'' proposed by physicists.

arXiv comment73 pages, with 2 figures

References in corpus (3)

• Higher genus symplectic invariants and sigma model coupled with gravity
• Circle-valued Morse theory and Reidemeister torsion
• Non-contractible periodic trajectories of symplectic vector fields, Floer cohomology and symplectic torsion

Cited by in corpus (7)

• On Non-contractible Periodic Orbits of Hamiltonian Diffeomorphisms
• Noncontractible periodic orbits in cotangent bundles and Floer homology
• Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds
• Existence of noncontractible periodic orbits of Hamiltonian system separating two Lagrangian tori on $T^*\T^n$ with application to non convex Hamiltonian systems
• Heavy subsets and non-contractible trajectories
• Symplectic intersections and invariant measures
• Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles