Orbifold E-functions of dual invertible polynomials

TL;DR

This paper investigates the orbifold E-functions associated with dual pairs of invertible polynomials and demonstrates their equality up to a sign, revealing a symmetry in their monodromy and Hodge structures.

Contribution

It establishes that orbifold E-functions of Berglund-Henningson dual pairs of invertible polynomials are equal up to a sign, linking monomials and symmetry group elements.

Findings

Orbifold E-functions of dual pairs coincide up to a sign.

A relation between monomials and symmetry group elements is established.

The result supports mirror symmetry in Landau-Ginzburg models.

Abstract

An invertible polynomial is a quasihomogeneous polynomial with the number of monomials coinciding with the number of variables and such that the weights of the variables and the quasi-degree are well defined. In the framework of the search for mirror symmetric orbifold Landau-Ginzburg models, P.~Berglund and M.~Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and an abelian group $G$ of its symmetries together with a dual pair $(\widetilde{f}, \widetilde{G})$. We consider the so-called orbifold E-function of such a pair $(f,G)$ which is a generating function for the exponents of the monodromy action on an orbifold version of the mixed Hodge structure on the Milnor fibre of $f$. We prove that the orbifold E-functions of Berglund-Henningson dual pairs coincide up to a sign depending on the number of variables. The proof is based on a relation between monomials (say, elements of a monomial basis of the Milnor algebra of an invertible polynomial) and elements of the whole symmetry group of the dual polynomial.