New comments on six-dimensional orientifold vacua with reduced rank and unitarity constraints

We revisit and extend the construction of six-dimensional orientifolds built upon the $T^4/\mathbb{Z}_N$ orbifolds with a non-vanishing Kalb-Ramond background, both in the presence of $\mathcal{N}=(1,0)$ supersymmetry and Brane Supersymmetry Breaking, thus amending some statements present in the literature. In the $N=2$ case, we show how the gauge groups on unoriented D9 and D5 (anti-)branes do not need to be correlated, but can be independently chosen complex or real. For $N>2$ we find that the Diophantine tadpole conditions severely constrain the vacua. Indeed, only the $N=4$ orbifold with a rank-two Kalb-Ramond background may admit integer solutions for the Chan-Paton multiplicities, if the $\mathbb{Z}_4$ fixed points support $\text{O}5_-$ planes, both with and without supersymmetry. All other cases would involve a fractional number of branes, which is clearly unacceptable. We check the consistency of the new $\mathbb{Z}_2$ and $\mathbb{Z}_4$ vacua by verifying the unitarity constraints for string defects coupled to Ramond-Ramond two-forms entering the Green-Schwarz-Sagnotti mechanism.