Summary: We construct new string vacua featuring Brane Supersymmetry Breaking, based on \(T^4/\mathbb{Z}_N\) orientifolds with \(\mathrm{O}5_+\) planes and \(\overline{\mathrm{D}5}\) branes. Differently from the original construction, in these vacua the cancellation of twisted R-R charges of D-branes and orientifold planes is achieved in a non-trivial way, and results in rigid configurations with few open-string moduli, which highly restrict the possible deformations of the model. The breaking of supersymmetry and, consequently, the non-vanishing untwisted and twisted NS-NS tadpoles, generates a rich potential both for the dilaton and the blown-up moduli.
We also uncover the stringy origin of the \(J\) form entering the gauge kinetic functions in the low-energy effective action, and display its relation to the NS-NS tadpoles for the scalars in the tensor multiplets. As a result, the \(J\) form can be consistently identified also when supersymmetry is broken, thus solving an embarrassing puzzle related to its very existence. We also discuss the unitarity constraints for one-dimensional defects in these vacua, where the Kač-Moody algebra for D9 and \(\overline{\mathrm{D}5}\) can be realised both in the left-moving and right-moving sectors.