Spin(7)-manifolds in compactifications to four dimensions

TL;DR

This paper explores the geometric structures of eight-dimensional manifolds with $Spin(7)$-structure in the context of M-theory compactifications to four dimensions, linking $G_2$- and $Spin(7)$-structures and their applications.

Contribution

It introduces a framework using tensors $rak{S}$ to describe $Spin(7)$-structures and their relation to $G_2$-structures, elliptic fibrations, and F-theory applications.

Findings

Established a mathematical relation between $G_2$ and $Spin(7)$ structures.

Described how $rak{S}$ encodes elliptic fibrations relevant for F-theory.

Provided conditions under which $rak{S}$ induces $Spin(7)$-structures from $G_2$-structures.

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.