Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2001-11-29
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.
Bilal Adel
Derendinger Jean-Pierre
Sfetsos Konstadinos
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