Spin(7)-manifolds in compactifications to four dimensions

Abstract

We describe off-shell $\mathcal{N}=1$ M-theory compactifications down to four dimensions in terms of eight-dimensional manifolds equipped with a topological $Spin(7)$-structure. Motivated by the exceptionally generalized geometry formulation of M-theory compactifications, we consider an eight-dimensional manifold ${\mathcal{M}}_8$ equipped with a particular set of tensors $\mathfrak{S}$ that allow to naturally embed in ${\mathcal{M}}_8$ a family of $G_2$-structure seven-dimensional manifolds as the leaves of a codimension-one foliation. Under a different set of assumptions, $\mathfrak{S}$ allows to make ${\mathcal{M}}_8$ into a principal $S^1$ bundle, which is equipped with a topological $Spin(7)$-structure if the base is equipped with a topological $G_2$-structure. We also show that $\mathfrak{S}$ can be naturally used to describe regular as well as a singular elliptic fibrations on ${\mathcal{M}}_8$, which may be relevant for F-theory applications, and prove several mathematical results concerning the relation between topological $G_2$-structures in seven dimensions and topological $Spin(7)$-structures in eight dimensions.

50 pages. We have included Proposition 6.4 about elliptic fibrations in relation to a pair of vector fields. We have also included Remark 5.13, thanks to an internal communication by Dominic Joyce. Discussion about the relation of singular foliations and D7-branes included

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• Generalised Geometry for M-Theory
• Les Houches Lectures on Constructing String Vacua
• Physics of String Flux Compactifications
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• Exceptional Lie groups
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Cited by in corpus (10)

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• Generalised Structures for $\mathcal{N}=1$ AdS Backgrounds
• M-theory on non-Kähler manifolds
• Spinors of real type as polyforms and the generalized Killing equation
• A class of non-geometric M-theory compactification backgrounds
• Five-dimensional null & time-like supersymmetric geometries
• Singular foliations for M-theory compactification
• Heterotic solitons on four-manifolds
• M-theory moduli spaces and torsion-free structures