Fixed point Floer cohomology and closed-string mirror symmetry for nodal curves

TL;DR

This paper connects fixed point Floer cohomology with genus-zero Gromov-Witten theory for singular hypersurfaces and proves mirror symmetry for nodal curves of genus ≥ 2 using Floer homology calculations.

Contribution

It introduces a new approach to describe Gromov-Witten invariants via fixed point Floer cohomology for singular spaces and provides a direct proof of mirror symmetry for certain nodal curves.

Findings

Gromov-Witten theory can be expressed as a limit of fixed point Floer cohomology.

Established mirror symmetry for nodal curves of genus ≥ 2.

Computed product structures on Floer homology of Dehn twists.

Abstract

We show that for singular hypersurfaces, a version of their genus-zero Gromov-Witten theory may be described in terms of a direct limit of fixed point Floer cohomology groups, a construction which is more amenable to computation and easier to define than the technical foundations of the enumerative geometry of more general singular symplectic spaces. As an illustration, we give a direct proof of closed-string mirror symmetry for nodal curves of genus greater than or equal to 2, using calculations of (co)product structures on fixed point Floer homology of Dehn twists due to Yao-Zhao.