Orbifold zeta functions for dual invertible polynomials

TL;DR

This paper investigates the relationship between orbifold zeta functions of dual pairs of invertible polynomials and their symmetry groups, revealing a reciprocal relationship depending on the number of variables.

Contribution

It introduces a study of reduced orbifold zeta functions for dual pairs of invertible polynomials and demonstrates their reciprocal or identical nature based on variable count.

Findings

Orbifold zeta functions of dual pairs either coincide or are inverses.

The relationship depends on whether the number of variables is even or odd.

Provides insight into mirror symmetry in Landau-Ginzburg models.

Abstract

An invertible polynomial in $n$ variables is a quasihomogeneous polynomial consisting of $n$ monomials so that the weights of the variables and the quasi-degree are well defined. In the framework of the construction of mirror symmetric orbifold Landau--Ginzburg models, P.~Berg\-lund, T.~H\"ubsch 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 $(\widetilde{f}, \widetilde{G})$. Here we study the reduced orbifold zeta functions of dual pairs $(f,G)$ and $(\widetilde{f}, \widetilde{G})$ and show that they either coincide or are inverse to each other depending on the number $n$ of variables.