Z_N x Z_M orientifolds with and without discrete torsion

TL;DR

The paper analyzes compact four-dimensional ${\mathbf Z}_N\times{\mathbf Z}_M$ Type IIB orientifolds, classifying phase data from discrete torsion and vector structure to enumerate consistent models. It develops a constructive framework to derive closed- and open-string spectra and tadpole cancellation conditions for each model, including supersymmetric cases with specific vector-structure and discrete-torsion choices and nonsupersymmetric variants built with anti-D5-branes. A comprehensive set of explicit spectra is presented across six orbifold groups, revealing how discrete torsion reduces gauge rank and how brane configurations must be arranged at fixed points to satisfy tadpoles. The work provides a practical methodological toolkit for orientifold model building, enabling systematic exploration of semi-realistic spectra with controlled SUSY breaking and gauge groups.

Abstract

We discuss compact four-dimensional Z_N x Z_M type IIB orientifolds. We take a systematic approach to classify the possible models and construct them explicitely. The supersymmetric orientifolds of this type have already been constructed some time ago. We find that there exist several consistent orientifolds for each of the discrete groups Z_2 x Z_2, Z_2 x Z_4, Z_4 x Z_4, Z_2 x Z_6, Z_2 x Z_6' and Z_6 x Z_6 if anti-D5-branes are introduced. Supersymmetry is broken by the open strings ending on antibranes. The rank of the gauge group is reduced by a factor two if the underlying orbifold space has discrete torsion.