Coulomb and Higgs Phases of $G_2$-manifolds

Ricci flat manifolds of special holonomy are a rich framework as models of the extra dimensions in string/$M$-theory. At special points in vacuum moduli space, special kinds of singularities occur and demand a physical interpretation. In this paper we show that the topologically distinct $G_2$-holonomy manifolds arising from desingularisations of codimension four orbifold singularities due to Joyce and Karigiannis correspond physically to Coulomb and Higgs phases of four dimensional gauge theories. The results suggest generalisations of the Joyce-Karigiannis construction to arbitrary ADE-singularities and higher order twists which we explore in detail in explicitly solvable local models. These models allow us to derive an isomorphism between moduli spaces of Ricci flat metrics on these non-compact $G_2$-manifolds and flat ADE-connections on compact flat 3-manifolds which we establish explicitly for $\operatorname{SU}(n)$.Comment: 22 page