Orbifold Euler characteristics for dual invertible polynomials

TL;DR

This paper investigates the orbifold Euler characteristics of Milnor fibers associated with dual invertible polynomials in mirror symmetry, revealing a sign-coincidence property under group actions.

Contribution

It establishes a relationship between the orbifold Euler characteristics of dual pairs of invertible polynomials, advancing understanding of mirror symmetry in Landau-Ginzburg models.

Findings

Orbifold Euler characteristics of Milnor fibers coincide up to a sign.

Dual pairs of invertible polynomials exhibit related topological invariants.

Supports mirror symmetry conjectures through topological invariants.

Abstract

To construct mirror symmetric Landau-Ginzburg models, P.Berglund, 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 Euler characteristics of the Milnor fibres of $f$ and $\widetilde f$ with the actions of the groups $G$ and $\widetilde G$ respectively and show that they coincide up to a sign.