Generalized Calabi-Yau Manifolds and the Mirror of a Rigid Manifold

TL;DR

This paper explores the mirror symmetry of a Z orbifold by representing it as a generalized Calabi-Yau manifold of dimension five or seven, maintaining superconformal properties suitable for string compactification.

Contribution

It introduces a class of generalized Calabi-Yau manifolds as mirrors of the Z orbifold, extending mirror symmetry to higher-dimensional manifolds relevant for string theory.

Findings

Computed the structure of the complex structure moduli space of the mirror.

Reproduced known Yukawa couplings and metric results for the orbifold.

Confirmed instanton corrections match expected physical properties.

Abstract

We describe the mirror of the Z orbifold as a representation of a class of generalized Calabi-Yau manifolds that can be realized as manifolds of dimension five and seven. Despite their dimension these correspond to superconformal theories with $c=9$ and so are perfectly good for compactifying the heterotic string to the four dimensions of space-time. As a check of mirror symmetry we compute the structure of the space of complex structures of the mirror and check that this reproduces the known results for the Yukawa couplings and metric appropriate to the Kahler class parameters on the Z orbifold together with their instanton corrections.