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arXiv · 1509.04101

Orbifold E-functions of dual invertible polynomials

Wolfgang Ebeling·Sabir M. Gusein-Zade·Atsushi Takahashi
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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.

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Wolfgang Ebeling, Sabir M. Gusein-Zade, Atsushi Takahashi. 2015-09-14. Orbifold E-functions of dual invertible polynomials. https://doi.org/10.1016/j.geomphys.2016.03.026

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