Orientifolds with discrete torsion

We show how discrete torsion can be implemented in D=4, N=1 type IIBorientifolds. Some consistency conditions are found from the closed string andopen string spectrum and from tadpole cancellation. Only real values of thediscrete torsion parameter are allowed, i.e. epsilon=+-1. Orientifold modelsare related to real projective representations. In a similar way as complexprojective representations are classified by H^2(Gamma,C^*)=H^2(Gamma,U(1)),real projective representations are characterized byH^2(Gamma,R^*)=H^2(Gamma,Z_2). Four different types of orientifoldconstructions are possible. We classify these models and give the spectrum andthe tadpole cancellation conditions explicitly.Comment: Latex, 48 pages, v2: several misprints and sign errors corrected, clarifications and references adde