Manifolds of Cohomogeneity Two

First, a word from our sponsor (particle physics). The mathematical reader should not expect to understand the physics, but (paraphrasing A. N. Varchenko) just let the words wash over you like music. Acharya & Witten [1] and Atiyah & Witten [2] have demonstrated that singularities in G2 holonomy manifolds 1 can compactify M -theory yielding effective quantum field theories in 3+1 dimensional Minkowski space with realistic gauge groups, chirality and supersymmetry. The mathematician’s responsibility is to generate a zoo of such singularities, so that their physics can be probed, as possible candidates for a final physical theory. Enough physics. This article is not the first, but certainly the broadest in scope in this hunt for new singularities. Our aim is to classify all G2 holonomy manifolds of cohomogeneity two, i.e. with a symmetry group acting with orbits of codimension two. No symmetry group can act transitively, since G2 holonomy manifolds are Ricci flat, and homogeneous Ricci flat manifolds are flat (a result of Alekseevski; see Besse [3]). Therefore the minimal cohomogeneity of a symmetry group must be 1; these