Mirror Symmetry and the Moduli Space for Generic Hypersurfaces in Toric Varieties

TL;DR

This paper explores the moduli space of Calabi-Yau hypersurfaces in toric varieties using mirror symmetry, extending previous work from Fermat-type to arbitrary quasi-homogeneous singularities, and investigates transitions between different theories.

Contribution

It generalizes the analysis of moduli dependence in superstring compactifications to include arbitrary singularities and studies the structure of moduli space transitions via mirror symmetry.

Findings

Derived moduli dependence of models on complexified Kähler parameters.

Mapped non-algebraic deformations to algebraic ones, broadening moduli space analysis.

Connected moduli space transitions to black hole condensation phenomena.

Abstract

The moduli dependence of $(2,2)$ superstring compactifications based on Calabi--Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with $c=9$ whose potential is a sum of $A$-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at $c=9$. We use mirror symmetry to derive the dependence of the models on the complexified K\"ahler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic (``twisted'') deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent work of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactified on Calabi-Yau manifolds.