(Weak) \(G_2\) holonomy from self-duality, flux and supersymmetry. (English) Zbl 0992.83073
Summary: 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.
MSC:
| 83E30 | String and superstring theories in gravitational theory |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 53C29 | Issues of holonomy in differential geometry |
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