Aspects of Type I String Theory in 4d with N=2 Supersymmetry

In chapter one we review the construction, the spectrum and the basic properties of four dimensional N = 2 theories arising from the type I string compactified on the six-manifold K3 x T2. We then briefly review what D-branes and orientfold are, in order to construct general type I compactifications involving these objects. In chapter two, as already anticipated, a survey on higher derivative F-terms of the form Fg W^2g is given, pointing out their importance in string dualities. Chapter 3 is devoted to type I-heterotic duality, where this is analyzed for d <= 10 and in particular for d = 4. A dual pair of four dimensional models is presented and then it is performed the one-loop computation of the Fg's in type I, to test the aforementioned duality. The structure of threshold corrections in type I K3 X T^2 compactifications and their relation with the N = 2 supersymmetric index is presented in chapter four, where it is then shown the purely dependence of this index on the BPS states of the theory. After that, we show the one-loop moduli dependence of the Fg couplings for orientifold models, pointing out their BPS dependence. Some comments and remarks are finally given in the conclusions, while technical details about the one-loop computations are reported in the appendix.