N=2 Type I-Heterotic Duality and Higher Derivative F-Terms

We test the conjectured Type I-Heterotic Duality in four dimensions by analyzing a given class of higher derivative F-terms of the form $F_gW^{2g}$, with W the N=2 gravitational superfield. We study a particular dual pair of theories, the O(2,2) heterotic model and a type I model based on the K3 $Z_2$ orbifold theory constructed by Gimon and Polchinski, further compactified on a torus. The $F_g$ couplings appear at 1-loop on both theories; because of the weak-weak nature of this duality in four dimensions, it is meaningful to compare the heterotic $F_g$'s with the corresponding type I couplings perturbatively. We compute the $F_g$'s in type I, showing that they receive contributions only from N=2 BPS states and that in the appropriate limit they coincide with the heterotic couplings, in agreement with the given duality.