The Mirror Map for Invertible LG Models

TL;DR

This paper explicitly constructs the mirror map for invertible Landau-Ginzburg models, demonstrating the Berglund--Hübsch construction's validity and extending mirror symmetry to complex orbifolds with non-abelian twists and discrete torsion.

Contribution

It provides an explicit mirror map for invertible LG models, confirming the Berglund--Hübsch construction and extending mirror symmetry to general orbifolds.

Findings

The mirror map matches monomials to twisted states in orbifolds.

OP selection rules correspond to twist selection rules.

The construction applies to non-abelian orbifolds and includes discrete torsion.

Abstract

Calculating the (a,c) ring of the maximal phase orbifold for `invertible' Landau--Ginzburg models, we show that the Berglund--H"ubsch construction works for all potentials of the relevant type. The map that sends a monomial in the original model to a twisted state in the orbifold representation of the mirror is constructed explicitly. Via this map, the OP selection rules of the chiral ring exactly correspond to the twist selection rules for the orbifold. This shows that we indeed arrive at the correct point in moduli space, and that the mirror map can be extended to arbitrary orbifolds, including non-abelian twists and discrete torsion, by modding out the appropriate quantum symmetries.