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.

arXiv comment50 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

References in corpus (26)

• String Theory Dynamics In Various Dimensions
• Evidence for F-Theory
• Compactifications of F-Theory on Calabi--Yau Threefolds -- II
• Flux Compactification
• Flux compactifications in string theory: a comprehensive review
• Compactifications of F-Theory on Calabi--Yau Threefolds -- I
• M-Theory on Eight-Manifolds
• Moduli Stabilization from Fluxes in a Simple IIB Orientifold
• Generalised Geometry for M-Theory
• Superstrings with Intrinsic Torsion
• M-theory, exceptional generalised geometry and superpotentials
• String Theory on Calabi-Yau Manifolds
• Les Houches Lectures on Constructing String Vacua
• The geometry of three-forms in six and seven dimensions
• Physics of String Flux Compactifications
• Lectures on Generalized Complex Geometry for Physicists
• Supersymmetric M-theory compactifications with fluxes on seven-manifolds and G-structures
• Exceptional Lie groups
• Generalised $G_2$-manifolds
• M-theory Compactifications on Manifolds with G2 Structure
• Special metric structures and closed forms
• M-theory on eight-manifolds revisited: N=1 supersymmetry and generalized Spin(7) structures
• Non-Supersymmetric F-Theory Compactifications on Spin(7) Manifolds
• F-Theory on Spin(7) Manifolds: Weak-Coupling Limit
• Generalized N=1 and N=2 structures in M-theory and type II orientifolds
• Holonomy Groups in Riemannian Geometry

Cited by in corpus (10)

• Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds
• Foliated eight-manifolds for M-theory compactification
• 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