$N=2\to 0$ super no-scale models and moduli quantum stability

We consider a class of heterotic $N=2\to 0$ super no-scale $Z_2$-orbifold models. An appropriate stringy Scherk–Schwarz supersymmetry breaking induces tree level masses to all massless bosons of the twisted hypermultiplets and therefore stabilizes all twisted moduli. At high supersymmetry breaking scale, the tachyons that occur in the $N=4\to 0$ parent theories are projected out, and no Hagedorn-like instability takes place in the $N=2\to 0$ models (for small enough marginal deformations). At low supersymmetry breaking scale, the stability of the untwisted moduli is studied at the quantum level by taking into account both untwisted and twisted contributions to the 1-loop effective potential. The latter depends on the specific branch of the gauge theory along which the background can be deformed. We derive its expression in terms of all classical marginal deformations in the pure Coulomb phase, and in some mixed Coulomb/Higgs phases. In this class of models, the super no-scale condition requires having at the massless level equal numbers of untwisted bosonic and twisted fermionic degrees of freedom. Finally, we show that $N=1\to 0$ super no-scale models are obtained by implementing a second $Z_2$ orbifold twist on $N=2\to 0$ super no-scale $Z_2$-orbifold models.