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.
Highlights
We show that S can be naturally used to describe regular as well as a singular elliptic fibrations on M8, which may be relevant for F-theory applications, and prove several mathematical results concerning the relation between topological G2-structures in seven dimensions and topological Spin(7)-structures in eight dimensions
Since M8 contains in a natural way a family of seven-dimensional manifolds with G2 structure, which are the leaves of the foliation, we find a natural correspondence between the G2-structure manifolds and the corresponding Spin (7)-structure manifolds, which may have a physical meaning in terms of dualities in String/M/F-theory
In this paper we have studied N = 1 M-theory compactifications down to four dimensions in terms of an eight-dimensional manifold M8 endowed with an intermediate structure S, whose presence is motivated by the exceptionally generalized geometric description of such compactifications
Summary
We are interested in bosonic solutions to eleven-dimensional Supergravity, and we will give from the onset a zero vacuum expectation value to the Majorana gravitino Ψ. Where M denotes the eleven-dimensional space-time differentiable, orientable and spinnable manifold, dV is the canonical volume form induced by the metric g, and G is the closed four-form flux associated to C, i.e. locally we can write G = dC. Given the product structure (2.3) of the space-time manifold M, the tangent bundle splits as follows. Let us respectively denote by S1+,3 and S7R the corresponding spin bundles over M1,3 and M7. Given the decomposition (2.7), M7 is equipped with a globally defined no-where vanishing complex spinor η, that is a globally defined section of the complexified spin bundle S7R⊗C → M7. Its real components η1 and η2 can a priori vanish at points or become parallel, as long as they do not simultaneously vanish
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