RING STRUCTURE OF THE FLOER COHOMOLOGY OF Σ×S1

We give a presentation for the Floer cohomology ring HF∗(Σ×S1), where Σ is a Riemann surface of genus g⩾1, which coincides with the conjectural presentation for the quantum cohomology ring of the moduli space of odd degree rank two stable vector bundles on Σ with fixed determinant. We study the spectrum of the action of H∗(Σ) on HF∗(Σ×S1) and prove a physical assumption made in [1].