TY - JOUR

T1 - Orbifold E-functions of dual invertible polynomials

AU - Ebeling, Wolfgang

AU - Gusein-Zade, Sabir M.

AU - Takahashi, Atsushi

N1 - Funding information: We would like to thank the referee for carefully reading our paper and useful comments. This work is partially supported by the DFG-programme SPP1388 “Representation Theory” and by a DFG-Mercator fellowship. The second named author is also supported by RFBR-16-01-00409 and NSh-9789.2016.1. The third named author is also supported by Grant-in Aid for Scientific Research grant numbers 20360043 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

PY - 2016/8/1

Y1 - 2016/8/1

N2 - An invertible polynomial is a weighted homogeneous 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 (f~,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 and a simple change 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.

AB - An invertible polynomial is a weighted homogeneous 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 (f~,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 and a simple change 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.

KW - E-function

KW - Mirror symmetry

KW - Mixed Hodge structure

KW - Monodromy

KW - Orbifold

KW - Singularity

UR - http://www.scopus.com/inward/record.url?scp=84962920229&partnerID=8YFLogxK

UR - https://arxiv.org/abs/1509.04101

U2 - 10.1016/j.geomphys.2016.03.026

DO - 10.1016/j.geomphys.2016.03.026

M3 - Article

AN - SCOPUS:84962920229

VL - 106

SP - 184

EP - 191

JO - Journal of geometry and physics

JF - Journal of geometry and physics

SN - 0393-0440

ER -