1/4 BPS States and Non-Perturbative Couplings in N=4 String Theories

We compute certain 2K+4-point, one-loop couplings in the type IIA string compactified on K3 x T^2, which are related to a topological index on this manifold. Their special feature is that they are sensitive to only short and intermediate BPS multiplets. The couplings derive from underlying prepotentials G[K](T,U), which can be nicely summed up into a fundamental generating function. In the dual heterotic string on T^6, the amplitudes describe non-perturbative gravitational corrections to K-loop amplitudes due to bound states of fivebrane instantons with heterotic world-sheet instantons. We argue, as a consequence, that our results also give information about instanton configurations in six dimensional Sp(2k) gauge theories on T^6.