Type IIB Orientifolds with NS-NS Antisymmetric Tensor Backgrounds

We consider six dimensional N=1 space-time supersymmetric Type IIB orientifolds with non-zero 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 non-zero 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.