Mirror symmetry aspects for compact G_2 manifolds

TL;DR

This paper explores mirror symmetry for compact barely $G_2$ manifolds, proposing constructions and dualities with self-mirror Calabi-Yau families, and discusses implications for M-theory compactifications.

Contribution

It introduces a mirror symmetry framework for barely $G_2$ manifolds and identifies self-mirror Calabi-Yau structures related to Joyce manifolds.

Findings

Mirror of barely $G_2$ manifolds is another barely $G_2$ manifold built from mirror CY base.

Joyce manifolds of the first kind have a self-mirror CY with $h^{1,1}=h^{2,1}=19$.

No 5-brane instantons are present in certain M-theory compactifications.

Abstract

The present paper deals with mirror symmetry aspects of compact ``barely'' $G_2$ manifolds, that is, $G_2$ manifolds of the form (CY$\times S^1)/\mathbb{Z}_2$. We propose that the mirror of any barely $G_2$ manifold is another barely one and which is constructed as a fibration of the \emph{mirror} of the CY base. Also, we describe the Joyce manifolds of the first kind as ``barely'' and we show that the underlying CY of all the family is self-mirror with $h^{1,1}=h^{2,1}=19$. We thus propose that the mirror of a Joyce space of the first kind will be another Joyce space of the first kind.We also suggest that this self-mirror CY family is dual to K3$\times S^1$ in the heterotic/M-theory sense, and that arise as a particular case of the Borcea-Voisin construction. As a spin-off we conclude from this analysis that no 5-brane instantons are present in compactifications of eleven dimensional supergravity over Joyce manifolds of the first kind.