Compactification of M(atrix) theory on noncommutative toroidal orbifolds

It was shown by A. Connes, M. Douglas and A. Schwarz that noncommutative tori arise naturally in consideration of toroidal compactifications of M(atrix) theory. A similar analysis of toroidal Z2 orbifolds leads to the algebra Bθ that can be defined as a crossed product of noncommutative torus and the group Z2 . Our paper is devoted to the study of projective modules over Bθ ( Z2-equivariant projective modules over a noncommutative torus). We analyze the Morita equivalence (duality) for Bθ algebras working out the two-dimensional case in detail.