On the positivity of black hole degeneracies in string theory

Certain helicity trace indices of charged states in N=4 and N=8 superstring theory have been computed exactly using their explicit weakly coupled microscopic description. These indices are expected to count the exact quantum degeneracies of black holes carrying the same charges. In order for this interpretation to be consistent, these indices should be positive integers. We prove this positivity property for a class of four/five dimensional black holes in type II string theory compactified on T^6/T^5 and on K3 \times T^2/S^1. The proof relies on the mock modular properties of the corresponding generating functions.