Lattices for Landau-Ginzburg orbifolds

TL;DR

This paper explores the lattice structures associated with Landau-Ginzburg orbifolds, establishing their properties and dualities within the context of mirror symmetry and elliptic genera.

Contribution

It introduces integral lattices for Landau-Ginzburg orbifolds and demonstrates their properties and duality relations based on the orbifoldized elliptic genus.

Findings

Lattices have the same rank and the transcendental lattice's signature matches the orbifoldized signature.

Evidence suggests these lattices are exchanged under duality of pairs.

The work connects lattice theory with mirror symmetry in Landau-Ginzburg models.

Abstract

We consider a pair consisting of an invertible polynomial and a finite abelian group of its symmetries. Berglund, H\"ubsch, and Henningson proposed a duality between such pairs giving rise to mirror symmetry. We define an orbifoldized signature for such a pair using the orbifoldized elliptic genus. In the case of three variables and based on the homological mirror symmetry picture, we introduce two integral lattices, a transcendental and an algebraic one. We show that these lattices have the same rank and that the signature of the transcendental one is the orbifoldized signature. Finally, we give some evidence that these lattices are interchanged under the duality of pairs.