Conifold transitions and five-brane condensation in M-theory on Spin(7) manifolds

We conjecture a topology-changing transition in M-theory on a non-compact asymptotically conical Spin(7) manifold, where a 5-sphere collapses and a CP^2 bolt grows. We argue that the transition may be understood as the condensation of M5-branes wrapping S^5. Upon reduction to ten dimensions, it has a physical interpretation as a transition of D6-branes lying on calibrated submanifolds of flat space. In yet another guise, it may be seen as a geometric transition between two phases of type IIA string theory on a G2 holonomy manifold with either wrapped D6-branes, or background Ramond-Ramond flux. This is the first non-trivial example of a topology-changing transition with only 1/16 supersymmetry.