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

Abstract

We give a presentation for the Floer cohomology ring $HF^*(Σ\times S^1)$, where $Σ$ 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 $Σ$. We study the spectrum of the action of the homology of $Σ$ on the Floer cohomology and prove a physical assumption made by Vafa et al.