We revisit AdS4 heterotic compactifications on nearly Kähler manifolds in the presence of H-flux and certain fermion condensates. Unlike previous studies, we do not assume the vanishing of the supersymmetry variations. Instead we determine the full equations of motion originating from the ten-dimensional action, and subsequently we provide explicit solutions to them on nearly Kähler manifolds at first order in α ′ . The Bianchi identity is also taken into account in order to guarantee the absence of all anomalies. In the presence of H-flux, which is identified with the torsion of the internal space, as well as of fermion condensates in the gaugino and dilatino sectors, new solutions are determined. These solutions provide a full classification of consistent backgrounds of heterotic supergravity under our assumptions. All the new solutions are non-supersymmetric, while previously known supersymmetric ones are recovered too. Our results indicate that fully consistent (supersymmetric or not) heterotic vacua on nearly Kähler manifolds are scarce, even on AdS4, and they can be completely classified.Calabi-Yau compactifications of string theory, despite their welcome features, suffer from the infamous moduli problem, the existence of flat directions in the four-dimensional potential corresponding to scalar fields which are not stabilized. The most plausible scenario which resulted from the attempts to resolve this problem was the introduction of fluxes, namely vacuum expectation values of the tensor fields of the theory along the compactification manifold. These fluxes typically lead to a deformation of the internal manifold away from the Calabi-Yau property and suggest the study of non-Kähler manifolds [1,2].Unlike type II string theories, where the plentitude of Ramond-Ramond fields offers considerable freedom in the introduction of internal fluxes, the heterotic string case is more restrictive. Indeed, in the heterotic string the only field which may acquire an expectation value is the three-form H of the common NS sector of string theory. Moreover, this field satisfies a more restrictive Bianchi identity than in the type II case. However, apart from the above field, it was suggested that, due to some strong dynamics in the hidden sector, fermion bilinears may also acquire some vacuum expectation value, thus forming a condensate [3,4]. From a Calabi-Yau perspective such condensates are related to supersymmetry breakdown. Supersymmetric AdS 4 heterotic compactifications on non-Kähler manifolds with fluxes and gaugino condensation were studied in [5,6]. Moreover, a study including dilatino condensation was performed in [7]. However, the above studies do not deal with the solution of the equations of motion of the theory but only with the Killing spinor equations and the Bianchi identity. Nevertheless, according to Ivanov [8], it is not straightforward that the solution of the latter imply that the field equations are satisfied. Therefore it is more natural to directly investigate solutions of the field equations of th...
