BPS spectra and non-perturbative gravitational couplings in N=2,4 supersymmetric string theories

We study the BPS spectrum in D=4, N=4 heterotic string compactifications, with some emphasis on intermediate N=4 BPS states. These intermediate states, which can become short in N=2 compactifications, are crucial for establishing an S-T exchange symmetry in N=2 compactifications. We discuss the implications of a possible S-T exchange symmetry for the N=2 BPS spectrum. Then we present the exact result for the 1-loop corrections to gravitational couplings in one of the heterotic N=2 models recently discussed by Harvey and Moore. We conjecture this model to have an S-T exchange symmetry. This exchange symmetry can then be used to evaluate non-perturbative corrections to gravitational couplings in some of the non-perturbative regions (chambers) in this particular model and also in other heterotic models.