Geometry of orientifolds with NS-NSB flux

We discuss geometry underlying orientifolds with nontrivial NS-NS B flux. I f D-branes wrap a torus with B flux the rank of the gauge group is reduced due to noncommuting Wilson lines whose presence is implied by the B flux. I n the case of D-branes transverse to a torus with B flux the rank reduction is due to a smaller number of D-branes required by tadpole cancellation co nditions in the presence of B flux as some of the orientifold planes now ha ve the opposite orientifold projection. We point out that T duality in the presence of B Aux is more subtle than in the case with trivial B Bur, and i t is precisely consistent with the qualitative difference between the afore mentioned two setups. In the case where both types of branes are present, t he states in the mixed (e.g. 59) open string sectors come with a nontrivial multiplicity, which we relate to a discrete gauge symmetry due to nonzero B Aux, and construct vertex operators for the mixed sector states. Using th ese results we revisit K3 orientifolds with B flux (where K3 is a T-4/Z(M) orbifold) and point out various subtleties arising in some of these models. For instance, in the Zz case the conformal field theory orbifold does not appear to be the consistent background for the corresponding orientifolds w ith B flux. This is related to the fact that nonzero B flux requires the pr esence of both O5(-)- as well as O5(+)-planes at various Z(2) orbifold fixe d points, which appears to be inconsistent with the presence of the twisted B flux in the conformal field theory orbifold. We also consider four-dimen sional N = 2 and N = 1 supersymmetric orientifolds. We construct consistent four-dimensional models with B flux which do not suffer from difficulties encountered in the K3 cases.