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

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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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hal-00286459 , version 1 (09-06-2008)

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A. Bilal, J. -P. Derendinger, K. Sfetsos. (Weak) G_2 Holonomy from Self-duality, Flux and Supersymmetry. 2002. ⟨hal-00286459⟩
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