Non-invertible quasihomogeneous singularities and their Landau-Ginzburg orbifolds

TL;DR

This paper classifies non-invertible quasihomogeneous singularities, explores their possible polynomial forms, and establishes orbifold equivalences with explicit Frobenius algebra isomorphisms.

Contribution

It introduces a method to determine all possible additive polynomials for non-invertible singularities and constructs explicit Landau-Ginzburg orbifold equivalences.

Findings

Classification of additive polynomials for non-invertible singularities

Construction of explicit orbifold equivalences

Isomorphism between Frobenius algebras in the orbifold setting

Abstract

According to the classification of quasihomogeneus singularities, any polynomial $f$ defining such singularity has a decomposition $f = f_\kappa + f_{add}$. The polynomial $f_\kappa$ is of the certain form while $f_{add}$ is only restricted by the condition that the singularity of $f$ should be isolated. The polynomial $f_{add}$ is zero if and only if $f$ is invertible, and in the non-invertible case $f_{add}$ is arbitrary complicated. In this paper we investigate all possible polynomials $f_{add}$ for a given non-invertible $f$. For a given $f_\kappa$ we introduce the specific small collection of monomials that build up $f_{add}$ such that the polynomial $f = f_\kappa + f_{add}$ defines an isolated quasihomogeneus singularity. If $(f,\mathbb{Z}/2\mathbb{Z})$ is Landau-Ginzburg orbifold with such non-invertible polynomial $f$, we provide the quasihomogeneus polynomial $\bar{f}$ such that the orbifold equivalence $(f,\mathbb{Z}/2\mathbb{Z}) \sim (\bar{f}, \{id\})$ holds. We also give the explicit isomorphism between the corresponding Frobenius algebras.