Rigid D6-branes on T6/(Z2 x Z2M x ΩR) with discrete torsion

We give a complete classification of T-6/(Z(2) x Z(2M) x Omega R) orientifolds on factorisable tori and rigid D6-branes on them. The analysis includes the supersymmetry, RR tadpole cancellation and K-theory conditions and complete massless open and closed string spectrum (i.e. non-chiral as well as chiral) for fractional or rigid D6-branes for all inequivalent compactification lattices, without and with discrete torsion. We give examples for each orbifold background, which show that on Z(2) x Z(6) and Z(2) x Z'(6) there exist completely rigid D6-branes despite the self-intersections of orbifold image cycles. This opens up a new avenue for improved Standard Model building. On the other hand, we show that Standard and GUT model building on the Z(2) x Z(4) background is ruled out by simple arguments.