AdS strings with torsion: Noncomplex heterotic compactifications

Abstract

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 ${\mathrm{AdS}}_{4}$ 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 K\"ahler. We argue that all moduli are fixed, and we show that the K\"ahler potential and superpotential proposed in the literature yield the correct ${\mathrm{AdS}}_{4}$ radius. In the nearly K\"ahler 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.