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

A bstract We revisit and extend the construction of six-dimensional orientifolds built upon the T 4 / ℤ N orbifolds with a non-vanishing Kalb-Ramond background, both in the presence of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = (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 &gt; 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 ℤ 4 fixed points support O5 − 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 ℤ 2 and ℤ 4 vacua by verifying the unitarity constraints for string defects coupled to Ramond-Ramond two-forms entering the Green-Schwarz-Sagnotti mechanism.