On Mirror Symmetry for Manifolds of Exceptional Holonomy

Abstract

We consider Type II string theories on ${\mathbf{T}}^{\mathbf{n}}/{{\mathbf{Z}}_2}^m$ Joyce orbifolds. This class contains orbifolds which can be desingularised to give manifolds of $G_2$ $(\mathbf{n}$$=$$7)$ and $Spin(7)$ holonomy $(\mathbf{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 ${\mathbf{T}}^3$ (${\mathbf{T}}^4$) 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.

20 Pages, Latex

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Cited by in corpus (32)

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• G_2-manifolds and associative submanifolds via semi-Fano 3-folds
• A Note on Compactifications on Spin(7)-Holonomy Manifolds
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• Fluxes and Branes in Type II Vacua and M-theory Geometry with G(2) and Spin(7) Holonomy
• Duality and Fibrations on G_2 Manifolds
• Rolling among G_2 vacua
• Mirror Symmetry for $G_2$-Manifolds: Twisted Connected Sums and Dual Tops
• Four curious supergravities
• Generalised discrete torsion and mirror symmetry for G_2 manifolds
• Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps
• Generalized mirror symmetry and trace anomalies
• Open G2 Strings
• Towards Generalized Mirror Symmetry for Twisted Connected Sum $G_2$ Manifolds
• The Topological G2 String
• Superconformal algebras for twisted connected sums and $G_2$ mirror symmetry
• Marginal deformations of heterotic $G_2$ sigma models
• Heterotic strings on G_2 orbifolds
• On the Global Structure of Some Natural Fibrations of Joyce Manifolds
• On Mirror Maps for Manifolds of Exceptional Holonomy
• $\mathcal{G}$-structure symmetries and anomalies in $(1,0)$ non-linear $\sigma$-models
• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes
• Superconformal algebras for generalized Spin(7) and G${}_2$ connected sums
• Coassociative cones that are ruled by 2-planes
• Mirror symmetry aspects for compact G_2 manifolds
• A Note on Mirror Symmetry for Manifolds with Spin(7) Holonomy
• New G2 holonomy metrics, D6 branes with inherent U(1)xU(1) isometry and gamma-deformations
• Towards A Topological G_2 String
• G${}_2$-Manifolds and M-Theory Compactifications
• Cayley fibrations in the Bryant-Salamon $Spin(7)$ manifold
• Discrete Torsion in Singular G_2-Manifolds and Real LG
• On K3-Thurston 7-manifolds and their deformation space: A case study with remarks on general K3T and M-theory compactification