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

TL;DR

This paper revisits six-dimensional orientifold vacua with non-zero Kalb-Ramond backgrounds, clarifying gauge group choices, constraining solutions via tadpole conditions, and verifying unitarity constraints for consistent string models.

Contribution

It extends previous constructions by analyzing gauge group independence, tadpole constraints, and unitarity in specific orbifold backgrounds with reduced rank.

Findings

Only the N=4 orbifold with a rank-two Kalb-Ramond background admits integer solutions.

Most cases involve fractional branes, which are inconsistent.

Unitarity constraints confirm the consistency of certain Z2 and Z4 vacua.

Abstract

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.