LargeNgauge theories from orientifolds

We consider four-dimensional $\mathcal{N}=1\mathrm{}$ supersymmetric gauge theories obtained via orientifolds of type IIB on Abelian ${\mathbf{C}}^3/\Gamma$ orbifolds. We construct all such theories that have a 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 a world-sheet description is inadequate due to nonperturbative (from the orientifold viewpoint) states arising from D-branes wrapping (collapsed) two-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 nontrivial check for correctness of the corresponding conclusions. We also find nontrivial agreement with various field theory expectations, and point out their origin in string language. The orientifold gauge theories that do possess a 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 $\mathcal{N}=4\mathrm{}$ oriented theory.