Type IIB orientifolds, F-theory, Type I strings on orbifolds and Type I-heterotic duality

We consider six- and four-dimensional N = 1 supersymmetric orientifolds of Type IIB compactified on orbifolds. We give the conditions under which the perturbative world-sheet orientifold approach is adequate, and list the four-dimensional N = 1 orientifolds (which are rather constrained) that satisfy these conditions. We argue that in most cases orientifolds contain non-perturbative sectors that are missing in the world-sheet approach. These non-perturbative sectors can be thought of as arising from D-branes wrapping various collapsed 2-cycles in the orbifold, Using these observations, we explain certain "puzzles" in the literature on four-dimensional orientifolds, In particular, in some four-dimensional orientifolds the "naive" tadpole cancellation conditions have no solution. However, these tadpole cancellation conditions are derived using the world-sheet approach which we argue to be inadequate in these cases due to appearance of additional non-perturbative sectors. The main tools in our analyses are the map between F-theory and orientifold vacua and Type I-heterotic duality. Utilizing the consistency conditions we have found in this paper, we discuss consistent four-dimensional chiral N = 1 Type I vacua which are non-perturbative from the heterotic viewpoint. (C) 1998 Elsevier Science B.V.