Where are the mirror manifolds?

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 P 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 model 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 Z 2 orbifold representation of the D invariant, implying a hidden symmetry in tensor products of minimal models.