Consider $T^*S^1$ as symplectic manifold, with hamiltonian function $H(x,y) = y^2$ (y is the fiber direction, I know this is morse bott but it can be perturbed). consider the set of maps $u: \Sigma \rightarrow T^*S^1$ satisfying floer's equation: $(du+X_H\otimes dt)^{0,1} =0$ (with approrpiate modifications for when $\Sigma$ is a punctured riemann surface with higher genus). Is this set of maps understood? asked differently, is there an invariant (floer homology theory, gromov witten etc etc) defined by counting these maps that has been computed? (I know symplectic homology of $T^*S^1$ has been computed and is shown to be isomorphic to string topology of $S^1$ (with corresponding product/coproduct structures, but that's in genus 0), but what about these potentially "higher genus" invariants?
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