Supersymmetric M3-branes and G₂ Manifolds

We obtain a generalisation of the original complete Ricci-flat metric of G_2 holonomy on R^4\times S^3 to a family with a non-trivial parameter \lambda. For generic \lambda the solution is singular, but it is regular when \lambda={-1,0,+1}. The case \lambda=0 corresponds to the original G_2 metric, and \lambda ={-1,1} are related to this by an S_3 automorphism of the SU(2)^3 isometry group that acts on the S^3\times S^3 principal orbits. We then construct explicit supersymmetric M3-brane solutions in D=11 supergravity, where the transverse space is a deformation of this class of G_2 metrics. These are solutions of a system of first-order differential equations coming from a superpotential. We also find M3-branes in the deformed backgrounds of new G_2-holonomy metrics that include one found by A. Brandhuber, J. Gomis, S. Gubser and S. Gukov, and show that they also are supersymmetric.