Ricci Flat Metrics, Flat Connections and $G_{2}$-Manifolds

Abstract

Inspired by considerations in $M$-theory, we prove the equivalence between the moduli spaces of (suitably complexified) torsion free $G_{2}$-structures on 7-manifolds which are families of hyperK\"ahler ALE 4-manifolds fibered over compact flat 3-manifolds $T^{3}/K$ and certain moduli spaces of flat $G^{\mathbb{C}}_{ADE}$ connections on $T^{3}/K$.