The A-model partition function is a topological invariant of any hyperbolic surface. So, what is this invariant?
I'm not sure if there is a way to normalize these since the usual perscription $Z(S^2) = 1$ doesn't work here, but I figure there's a way to do it anyway.
EDIT: It seems like the invariant should just be the "volume" of the moduli space of complex structures on the surface. In, eg. "Enumerative Geometry of Calabi-Yau 4-folds" http://arxiv.org/abs/math/0702189 this is written
$N^X_{g,\beta} = \int_{[\mathcal{M}_g(X,\beta)]^{vir}} 1$
This depends only on the genus $g>1$, the target space $X$ (and at most its Kahler structure), and the degree of the map $\beta \in H_2(X,\mathbb{Z})$.
Algebraicly, one uses stability of curves of genus at least two (few algebraic automorphisms) to define the virtual fundamental class on this space. I'm trying to understand how this stability condition is related to the existence of a hyperbolic metric on the surface. I would like a more geometric formula using a hyperbolic structure on the surface for the quantity above.
