Exact and Asymptotic Degeneracies of Small Black Holes

19pp. harvmac b-modeWe examine the recently proposed relations between black hole entropy and the topological string in the context of type II/heterotic string dual models. We consider the degeneracies of perturbative heterotic BPS states. These correspond to `small' black holes which have zero horizon area in the classical limit, but nonzero area due to quantum corrections. In several examples with N=4 and N=2 supersymmetry, we show that the macroscopic degeneracy of small black holes agrees to all orders with the microscopic degeneracy deduced from an appropriate index, but misses non-perturbative corrections which are computable on the heterotic side. Using these examples we refine the previous proposals and comment on their domain of validity