Non-contractible periodic orbits, Gromov invariants, and Floer-theoretic torsions
Abstract
Description
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.
73 pages, with 2 figures
73 pages, with 2 figures