On heterotic vacua with fermionic expectation values

We study heterotic backgrounds with non-trivial H-flux and non-vanishing expectation values of fermionic bilinears, often referred to as gaugino condensates. The gaugini appear in the low energy action via the gauge-invariant three-form bilinear Sigma(MNP) = tr (chi)over bar>F-MNP chi. For Calabi-Yau compactifications to four dimensions, the gaugino condensate corresponds to an internal three-form Sigma(mnp) that must be a singlet of the holonomy group. This condition does not hold anymore when an internal H-flux is turned on and O(alpha') effects are included. In this paper we study flux compactifications to three and fourdimensions on G-structure manifolds. We derive the generic conditions for supersymmetric solutions. We use integrability conditions and Lichnerowicz type arguments to derive a set of constraints whose solution, together with supersymmetry, is sufficient for finding backgrounds with gaugino condensate.