We consider string perturbative expansion in the presence of D-branes imbedded in orbifolded space-time. In the regime where the string coupling is weak and α ′ → 0, the string perturbative expansion coincides with 't Hooft's large N expansion. We specifically concentrate on theories with d = 4 and N = 0, 1, 2, 4, and use world-sheet orbifold techniques to prove vanishing theorems for the field theory β-functions to all orders in perturbation theory in the large N limit. This is in accord with recent predictions.

We consider six dimensional Nϭ1 space-time supersymmetric type IIB orientifolds with a nonzero untwisted NS-NS sector B field. The B field is quantized due to the requirement that the type IIB spectrum be left-right symmetric. The presence of the B field results in rank reduction of both 99 and 55 open string sector gauge groups. We point out that in some of the models with a nonzero B field there are extra tensor multiplets in the Z 2 twisted closed string sector, and we explain their origin in a simple example. Also, the 59 open string sector states come with a multiplicity that depends on the B field. These two facts are in accord with anomaly cancellation requirements. We point out relations between various orientifolds with and without the B field, and also discuss the F-theory duals of these models. ͓S0556-2821͑98͒05116-9͔

We consider four dimensional N = 1 supersymmetric gauge theories obtained via orientifolds of Type IIB on Abelian C 3 /Γ orbifolds. We construct all such theories that have well defined world-sheet expansion. The number of such orientifolds is rather limited. We explain this fact in the context of recent developments in four dimensional Type IIB orientifolds. In particular, we elaborate these issues in some examples of theories where world-sheet description is inadequate due to non-perturbative (from the orientifold viewpoint) states arising from D-branes wrapping (collapsed) 2-cycles in the orbifold. We find complete agreement with the corresponding statements recently discussed in the context of Type I compactifications on toroidal orbifolds. This provides a non-trivial check for correctness of the corresponding conclusions. We also find non-trivial agreement with various field theory expectations, and point out their origin in string language. The orientifold gauge theories that do possess well defined world-sheet description have the property that in the large N limit computation of any M -point correlation function in these theories reduces to the corresponding computation in the parent N = 4 oriented theory.