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 hyperKähler ALE 4-manifolds fibered over compact flat 3-manifolds $T^3/K$ and certain moduli spaces of flat $G_{ADE}^{\mathbb{C}}$ connections on $T^3/K$.