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 one-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.