Landau-Ginzburg String Vacua

TL;DR

This paper investigates a broad class of $(2,2)$ string vacua realized as Landau--Ginzburg theories with isolated singularities, connecting them to Calabi--Yau geometries and analyzing the extent of mirror symmetry within this framework. It employs spectral data from the chiral ring, ${\mathbb Z}_d$ orbifolds, and Bertini-type results to construct and classify LG vacua without requiring a full polynomial classification, including configurations in products of weighted projective spaces. A finite census under the constraint $c=9$ is provided, revealing thousands of models and nearly 3,000 distinct spectra, while examining how orbifolding enhances mirror symmetry within this landscape. The work clarifies how LG constructions relate to CY manifolds beyond naive LG potentials and highlights both the reach and limitations of this approach for mapping the heterotic string vacuum space.

Abstract

We investigate a class of (2,2) supersymmetric string vacua which may be represented as Landau--Ginzburg theories with a quasihomogeneous potential which has an isolated singularity at the origin. There are at least three thousand distinct models in this class. All vacua of this type lead to Euler numbers which lie in the range $-960 \leq χ\leq 960$. The Euler characteristics do not pair up completely hence the space of Landau--Ginzburg ground states is not mirror symmetric even though it exhibits a high degree of symmetry. We discuss in some detail the relation between Landau--Ginzburg models and Calabi--Yau manifolds and describe a subtlety regarding Landau--Ginzburg potentials with an arbitrary number of fields. We also show that the use of topological identities makes it possible to relate Landau-Ginzburg theories to types of Calabi-Yau manifolds for which the usual Landau-Ginzburg framework does not apply.