Reidemeister torsion in symplectic Floer theory and counting pseudo-holomorphic tori

The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consider one such instance—the Reidemeister torsion τF of the Floer complex of (possibly non-hamiltonian) symplectomorphisms. τF turns out not to be invariant under hamiltonian isotopies, but we introduce a “correction term”—the Floertheoretic zeta function ζF , such that the product IF := τF ζF is invariant under hamiltonian isotopies. When the symplectic manifoldM is monotone, IF is in fact invariant under general symplectic isotopy; this enables us to compute IF in the special case specified in Corollary 1.8—it reduces to the classical Milnor torsion of M in this case. The zeta function ζF is defined by counting perturbed pseudoholomorphic tori in a way very similar to the genus 1 Gromov invariant. In fact, the analogy of the formula IF = τF ζF to the definition of Quillen metric via the holomorphic analytic torsion suggests that the genus 1 mirror symmetry described in [BCOV] might have an explanation as a correspondence between the torsion on the complex side (the holomorphic analytic torsion) and the torsion on the symplectic side (the genus 1 Gromov invariant as a “symplectic torsion”). Because the torsion invariant we consider is a non-homotopic invariant, the continuation method used in typical invariance proofs of Floer theory does not apply; instead, we work out the detailed bifurcation analysis. This is the first time such bifurcation analysis is carried out in the Floer theory literature; the same analysis may be adapted to other versions of Floer theory (some of them will appear in upcoming papers), and should be generalizable to define other more refined invariants of Floer theory. In this paper we only draw some immediate applications to the existence of symplectic fixed points. As examples of results obtainable via the new invariant IF , which are on the other hand inaccessible to Floer homologies, we state some applications to the existence of noncontractible periodic orbits of symplectic vector fields, and leave the proofs to upcoming papers.