Where are the Mirror Manifolds?

TL;DR

This paper investigates the mirror symmetry of Landau-Ginzburg models related to Calabi-Yau manifolds, revealing transformations that connect models with unknown mirrors to those with known constructions, and uncovering hidden symmetries.

Contribution

It introduces non-linear transformations that relate models without known mirrors to those with established mirror constructions, expanding understanding of mirror symmetry in complex models.

Findings

Identifies transformations linking models with unknown mirrors to known cases

Reveals hidden symmetries in tensor products of minimal models

Suggests new pathways for constructing mirror pairs in complex Landau-Ginzburg models

Abstract

The recent classification of Landau--Ginzburg potentials and their abelian symmetries focuses attention on a number of models with large positive Euler number for which no mirror partner is known. All of these models are related to Calabi--Yau manifolds in weighted $\IP_4$, with a characteristic structure of the defining polynomials. A closer look at these potentials suggests a series of non-linear transformations, which relate the models to configurations for which a construction of the mirror is known, though only at certain points in moduli space. A special case of these transformations generalizes the $\ZZ_2$ orbifold representation of the $D$ invariant, implying a hidden symmetry in tensor products of minimal models.