M-theory compactifications on certain `toric' cones of G2 holonomy

We develop methods to study the singularities of certain G2 cones related to toric hyperkahler spaces and Einstein selfdual orbifolds. This allows us to determine the low energy gauge groups of chiral N = 1 compactifications of M-theory on a large family of such backgrounds, which includes the models recently studied by Acharya and Witten. All M-theory compactifications belonging to our family admit a T2 of isometries, and therefore T-dual IIA and IIB descriptions. We argue that reduction through such an isometry leads generically to systems of weakly and strongly coupled IIA 6-branes, T-dual to delocalized type-IIB 5-branes. We find a simple criterion for the existence of a `good' isometry which leads to IIA models containing only weakly-coupled D6-branes (up to possible unknown effects encoded by a CFT in 5 dimensions), and construct examples of such backgrounds.