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

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

2001-11-29

arxiv.org/abs/hep-th/0111274v2

Nucl.Phys. B628 (2002) 112-132

Physics

High Energy Physics

High Energy Physics - Theory

references added, 25 pages, no figures, Latex

Scientific paper

10.1016/S0550-3213(02)00042-1

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.

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