Counting higher genus curves in a Calabi-Yau manifold

We explicitly evaluate the low energy coupling F-g in a d = 4, N = 2 compac tification of the heterotic string. The holomorphic piece of this expressio n 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. Th e heterotic model has a dual Calabi-Yau compactification of the type II str ing. We compare the answer with the general form expected from curve-counti ng formulae and find good agreement. As a corollary of this comparison we p redict some numbers of higher genus curves in a specific Calabi-Yau, and ex tract some intersection numbers on the moduli space of genus-g Riemann surf aces. (C) 1999 Elsevier Science B.V.