M-theory moduli spaces and torsion-free structures

TL;DR

This paper explores a geometric framework for M-theory compactifications using torsion-free structures on higher-dimensional manifolds, establishing a link between $Spin(7)$ and $G_2$-structures to better understand moduli spaces.

Contribution

It introduces a novel approach to geometrize M-theory fluxes by relating compactification spaces to higher-dimensional manifolds with torsion-free structures, including a bijection between $Spin(7)$ and $G_2$-structures.

Findings

Constructed a bijection between $Spin(7)$-structures on 8D $S^1$-bundles and $G_2$-structures on the base.

Characterized $G_2$-torsion classes via torsion-free $Spin(7)$-structures.

Proposed a method to study M-theory moduli spaces through higher-dimensional torsion-free structures.

Abstract

Motivated by the description of $\mathcal{N}=1$ M-theory compactifications to four-dimensions given by Exceptional Generalized Geometry, we propose a way to geometrize the M-theory fluxes by appropriately relating the compactification space to a higher-dimensional manifold equipped with a torsion-free structure. As a non-trivial example of this proposal, we construct a bijection from the set of $Spin(7)$-structures on an eight-dimensional $S^{1}$-bundle to the set of $G_{2}$-structures on the base space, fully characterizing the $G_{2}$-torsion clases when the total space is equipped with a torsion-free $Spin(7)$-structure. Finally, we elaborate on how the higher-dimensional manifold and its moduli space of torsion-free structures can be used to obtain information about the moduli space of M-theory compactifications.