Heterotic Calabi-Yau Compactifications with Flux

Compactifications of the heterotic string with NS flux normally require non Calabi-Yau internal spaces which are complex but no longer K\"ahler. We point out that this conclusion rests on the assumption of a maximally symmetric four-dimensional space-time and can be avoided if this assumption is relaxed. Specifically, it is shown that an internal Calabi-Yau manifold is consistent with the presence of NS flux provided four-dimensional space-time is taken to be a domain wall. These Calabi-Yau domain wall solutions can still be associated with a covariant four-dimensional N=1 supergravity. In this four-dimensional context, the domain wall arises as the "simplest" solution to the effective supergravity due to the presence of a flux potential with a runaway direction. Our main message is that NS flux is a legitimate ingredient for moduli stabilization in heterotic Calabi-Yau models. Ultimately, the success of such models depends on the ability to stabilize the runaway direction and thereby "lift" the domain wall to a maximally supersymmetric vacuum.