Ring structure of the Floer Cohomology of $\Sigma \times S^1$

TL;DR

This paper presents a detailed description of the Floer cohomology ring of a surface cross a circle, linking it to the quantum cohomology of moduli spaces and confirming a physical conjecture.

Contribution

It provides an explicit presentation of the Floer cohomology ring for surface-cross-circle manifolds and verifies a physical assumption about its spectrum.

Findings

Floer cohomology ring presentation matches conjectural quantum cohomology

Spectrum of the homology action is characterized and matches physical predictions

Confirmed a physical assumption by Vafa et al.

Abstract

We give a presentation for the Floer cohomology ring $HF^*(\Sigma \times S^1)$, where $\Sigma$ is a Riemann surface of genus bigger than one, which coincides with the conjectural presentation for the quantum cohomology ring of the moduli space of flat SO(3)-connections of odd degree over $\Sigma$. We study the spectrum of the action of the homology of $\Sigma$ on the Floer cohomology and prove a physical assumption made by Vafa et al.