(Weak) G_2 Holonomy from Self-duality, Flux and Supersymmetry

TL;DR

This work shows that compactifications preserving four-dimensional supersymmetry can be understood via reduced holonomy, with G2 holonomy (Ricci-flat) arising from a self-dual spin connection in seven dimensions and weak G2 holonomy arising when flux yields an Einstein space. By exploiting octonionic invariants, the authors prove equivalences between self-duality (and its generalized, inhomogeneous form) of the spin connection and the existence of Killing or conformal Killing spinors, linking to $G_2$ and weak $G_2$ geometries, respectively. They further connect these seven-dimensional structures to eight-dimensional ${\rm spin}(7)$ holonomy via cohomogeneity-one constructions and provide explicit first-order equations for broad metric ansatze. The results illuminate how flux in eleven-dimensional supergravity constrains internal geometry and demonstrate a unifying, efficient framework for generating and analyzing $G_2$ and weak $G_2$ manifolds in the context of M-theory compactifications.

Abstract

The aim of this paper is two-fold. First, we provide a simple and pedagogical discussion of how compactifications of M-theory or supergravity preserving some four-dimensional supersymmetry naturally lead to reduced holonomy or its generalization, reduced weak holonomy. We relate the existence of a (conformal) Killing spinor to the existence of certain closed and co-closed p-forms, and to the metric being Ricci flat or Einstein. Then, for seven-dimensional manifolds, we show that octonionic self-duality conditions on the spin connection are equivalent to G_2 holonomy and certain generalized self-duality conditions to weak G_2 holonomy. The latter lift to self-duality conditions for cohomogeneity-one spin(7) metrics. To illustrate the power of this approach, we present several examples where the self-duality condition largely simplifies the derivation of a G_2 or weak G_2 metric.