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arXiv · hep-th/0111274

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

A. Bilal·J. -P. Derendinger·K. Sfetsos
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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.

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A. Bilal, J. -P. Derendinger, K. Sfetsos. 2002-01-29. (Weak) G_2 Holonomy from Self-duality, Flux and Supersymmetry. https://doi.org/10.1016/s0550-3213(02)00042-1

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