On Mirror Maps for Manifolds of Exceptional Holonomy

Abstract

We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups $G_2$ and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum $G_2$ manifolds, mirrors of such Spin(7) manifolds can be found by applying mirror symmetry to the pair of non-compact manifolds they are glued from. To provide non-trivial checks for such geometric mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds in several new instances for both the Spin(7) and the $G_2$ case. For all of these models we find possible assignments of discrete torsion phases, work out the action of mirror symmetry, and confirm the consistency with the geometrical construction. A novel feature appearing in the examples we analyse is the possibility of frozen singularities.

arXiv comment45 pages

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Cited by in corpus (6)

• On TCS $G_2$ manifolds and 4D Emergent Strings
• Evidence for an Algebra of ${\mathbf{G}}_2$ Instantons
• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes
• Odd dimensional analogue of the Euler characteristic
• Superconformal algebras for generalized Spin(7) and G${}_2$ connected sums
• $\mathcal{S}\mathcal{W}$-algebras and strings with torsion