Heterotic Strings on Mirror Half-Flat Manifolds

In this talk we report on progress made in the study of $E_8 \times E_8$ heterotic string theory on mirror half-flat manifolds. We are motivated to study this system because mirror half-flat manifolds offer a way to fix some of the moduli of heterotic string theory on Calabi-Yau manifolds. We argue that the analogue of standard embedding in the half-flat case is to embed the natural torsionful connection into the gauge connection. The surviving subgroup is still $E_6 \times E_8$ as in Calabi-Yau compactification. We show this by thinking of the heterotic string on a half-flat manifold as a ``reduction" of $R^{1,2}\times Z_7$, where $Z_7$ is non-compact $G_2$ holonomy cylinder foliated by compact mirror half-flat leaves. We then report progress on working out the effective action of heterotic string theory on these manifolds.