On Mirror Symmetry for Manifolds of Exceptional Holonomy

TL;DR

This paper explores mirror symmetry and T-duality transformations in string theories compactified on orbifolds leading to manifolds with exceptional holonomy, proposing new dualities and fibrations for G2 and Spin(7) manifolds.

Contribution

It introduces generalized T-duality transformations for G2 and Spin(7) holonomy manifolds and suggests their fibrations, extending mirror symmetry concepts beyond Calabi-Yau spaces.

Findings

Two types of T-duality for G2 holonomy spaces are identified.

A T-duality for Spin(7) holonomy spaces is established.

Evidence suggests mirror manifolds admit supersymmetric torus fibrations.

Abstract

We consider Type II string theories on ${\bf T^n}/{{\bf Z_2}^m}$ Joyce orbifolds. This class contains orbifolds which can be desingularised to give manifolds of $G_2$ $({\bf n}$$=$$7)$ and $Spin(7)$ holonomy $({\bf n}$$=$$8)$. In the $G_2$ holonomy case we present two types of $T$-duality transformation which are clearly generalisations of the $T$-duality/mirror transformation in Calabi-Yau spaces. The first maps Type IIA theory on one such space from this class to Type IIB theory on another such space. The second maps Type IIA (IIB) to Type IIA (IIB). In the case of $Spin(7)$ holonomy we present a $T$-duality transformation which maps Type IIA (IIB) theory on one such space to Type IIA (IIB) on another such space. As orbifold conformal field theories these $T$-dual target spaces are related via the inclusion/exclusion of discrete torsion and the $T$-duality is proven to genus $g$ in string perturbation theory. We then apply a Strominger, Yau, Zaslow type argument which suggests that manifolds of $G_2$ holonomy which have a ``mirror'' of the first (second) type admit supersymmetric ${\bf T^3}$ (${\bf T^4}$) fibrations and that manifolds of $Spin(7)$ holonomy for which a mirror exists admit fibration by supersymmetric $4$-tori. Further evidence for this suggestion is given by examining the moduli space structure of wrapped D-branes.