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We develop methods to study the singularities of certain $G_2$ cones relatedto toric hyperkahler spaces and Einstein selfdual orbifolds. This allows us todetermine the low energy gauge groups of chiral N=1 compactifications ofM-theory on a large family of such backgrounds, which includes the modelsrecently studied by Acharya and Witten. All M-theory compactificationsbelonging to our family admit a $T^2$ of isometries, and therefore T-dual IIAand IIB descriptions. We argue that reduction through such an isometry leadsgenerically to systems of weakly and strongly coupled IIA 6-branes, T-dual todelocalized type IIB 5-branes. We find a simple criterion for the existence ofa `good' isometry which leads to IIA models containing only weakly-coupledD6-branes, and construct examples of such backgrounds. Some of the methods wedevelop may also apply to different situations, such as the study of certainsingularities in the hypermultiplet moduli space of N=2 supergravity in fourdimensions.Comment: 83 pages, numerous figures v2,3: corrected typos, some references  adde

M-theory compactifications on certain `toric' cones ofG2holonomy

M-theory compactifications on certain `toric' cones ofG₂holonomy