1 / 4 BPS States andNon – Perturbative Couplings in N = 4 String Theories

We compute certain (2K + 4)–point, one–loop couplings in the type IIA string compactified on K3 × T 2 , which are related to the topologi-cal index Tr(−1) J L +J R J L J R 2K+1 q L 0 q L 0 on this manifold. Their special feature is that they are sensitive to only short and intermediate BPS mul-tiplets. The couplings derive from underlying prepotentials of the form G [2K+4] (T, U) ∼ (∂ V) 2K ln χ 10 (T, U, V), where χ 10 (T, U, V) is the helicity partition function of 1/4 BPS states. 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 .