Counting higher genus curves in a Calabi-Yau manifold

We explicitly evaluate the low energy coupling $F_g$ in a $d=4,\mathcal{N}=2$ compactification of the heterotic string. The holomorphic piece of this expression provides the information not encoded in the holomorphic anomaly equations, and we find that it is given by an elementary polylogarithm with index $3-2g$, thus generalizing in a natural way the known results for $g=0,1$. The heterotic model has a dual Calabi-Yau compactification of the type II string. We compare the answer with the general form expected from curve-counting formulae and find good agreement. As a corollary of this comparison we predict some numbers of higher genus curves in a specific Calabi-Yau, and extract some intersection numbers on the moduli space of genus $g$ Riemann surfaces.