AdS strings with torsion: Noncomplex heterotic compactifications

Combining the effects of fluxes and gaugino condensation in heterotic supergravity, we use a ten-dimensional approach to find a new class of four-dimensional supersymmetric AdS₄ compactifications on almost-Hermitian manifolds of SU(3) structure. Computation of the torsion allows a classification of the internal geometry, which for a particular combination of fluxes and condensate, is nearly Kähler. We argue that all moduli are fixed, and we show that the Kähler potential and superpotential proposed in the literature yield the correct AdS₄ radius. In the nearly Kähler case, we are able to solve the H Bianchi identity using a nonstandard embedding. Finally, we point out subtleties in deriving the effective superpotential and understanding the heterotic supergravity in the presence of a gaugino condensate.