Orientifolds with discrete torsion

TL;DR

This work extends discrete torsion to four-dimensional ${\cal N}=1$ type IIB orientifolds, showing that consistency across closed and open string sectors and tadpole cancellation restricts the torsion parameter to real values $\epsilon=\pm1$. By treating orientifolds via real projective representations, it classifies four distinct vector-structure cases for ${\bf Z}_N\times{\bf Z}_M$ and provides explicit spectra and tadpole conditions, including the need for D7-branes in many setups. The analysis covers non-compact orbifolds with D3-branes at ${\mathbb C}^3/\Gamma$, detailing the closed/open string spectra, deformation/resolution of singularities, and how discrete torsion shifts the superpotential while preserving consistency. In the orientifold context, the work demonstrates that only real torsion is compatible with the projection, outlines the corresponding tadpole constraints, and explains how deformations are constrained (e.g., conifold deformations can be frozen). Overall, the paper builds a coherent framework linking discrete torsion, real projective representations, spectrum, tadpoles, and geometric resolutions in non-compact orientifold models, with implications for compactifications and holographic duals.

Abstract

We show how discrete torsion can be implemented in D=4, N=1 type IIB orientifolds. Some consistency conditions are found from the closed string and open string spectrum and from tadpole cancellation. Only real values of the discrete torsion parameter are allowed, i.e. epsilon=+-1. Orientifold models are related to real projective representations. In a similar way as complex projective representations are classified by H^2(Gamma,C^*)=H^2(Gamma,U(1)), real projective representations are characterized by H^2(Gamma,R^*)=H^2(Gamma,Z_2). Four different types of orientifold constructions are possible. We classify these models and give the spectrum and the tadpole cancellation conditions explicitly.