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

TL;DR

This paper explores how self-duality conditions relate to G_2 and weak G_2 holonomy in seven-dimensional manifolds, simplifying the derivation of special metrics relevant for M-theory compactifications.

Contribution

It establishes equivalences between octonionic self-duality and G_2 holonomy, providing a pedagogical framework and new examples for constructing special holonomy metrics.

Findings

Self-duality conditions correspond to G_2 and weak G_2 holonomy.

Simplified derivation of G_2 and weak G_2 metrics using self-duality.

Examples demonstrating the approach's effectiveness.

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.