M theory is considered in its low-energy limit on a $G_2$ manifold with nonvanishing flux. Using the Killing spinor equations for linear flux, an explicit set of first-order bosonic equations for supersymmetric solutions is found. These solutions describe a warped product of a domain wall in four-dimensional space-time and a deformed $G_2$ manifold. It is shown how these domain walls arise from the perspective of the associated four-dimensional $N=1$ effective supergravity theories. We also discuss the inclusion of membrane and M5-brane sources.

• Received 2 December 2004

DOI:https://doi.org/10.1103/PhysRevD.71.046006