Almost contact structures on manifolds with a $G_2$ structure

TL;DR

This paper reviews the construction of almost contact structures on manifolds with $G_2$ structures, exploring their properties, torsion, and connections to supersymmetry and submanifolds in string and M-theory.

Contribution

It introduces the study of ACM(3) structures on $G_2$ manifolds, computes their torsion, and explores their topological and geometric properties, linking to supersymmetric theories.

Findings

Computed torsion of associated $SU(3)$ structures.

Identified the infinite-dimensional space of ACM(3) structures.

Discovered links between ACM(3) structures and special submanifolds.

Abstract

We review the construction of almost contact metric (three-) structures, abbreviated ACM(3)S, on manifolds with a $G_2$ structure. These are of interest for certain supersymmetric configurations in string and M-theory. We compute the torsion of the $SU(3)$ structure associated to an ACMS and apply these computations to heterotic $G_2$ systems and supersymmetry enhancement. We initiate the study of the space of ACM3Ss, which is an infinite dimensional space with a local product structure and interesting topological features. Tantalising links between ACM3Ss and associative and coassociative submanifolds are observed.