# On Mirror Maps for Manifolds of Exceptional Holonomy

## TL;DR

This paper explores mirror symmetry for manifolds with exceptional holonomy groups $G_2$ and Spin(7), constructing mirror pairs via generalized connected sums and verifying these through conformal field theory analyses.

## Contribution

It introduces a new geometric construction of mirror manifolds for Spin(7) spaces and provides CFT checks for Joyce orbifold models, including the novel phenomenon of frozen singularities.

## Key findings

- Mirror pairs constructed for Spin(7) and $G_2$ manifolds.
- Verification of mirror symmetry via CFT analysis of Joyce orbifolds.
- Identification of frozen singularities in mirror models.

## 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.