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

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.