TY - JOUR

T1 - On the orbifold Euler characteristics of dual invertible polynomials with non-abelian symmetry groups

AU - Ebeling, Wolfgang

AU - Gusein-Zade, Sabir M.

N1 - Funding information: Received November 5, 2018. 2010 Mathematics Subject Classification: 14J33, 57R18, 32S55, 19A22. ?Partially supported by DFG. The work of the second author (Sections 1, 3, 4) was supported by the grant 16-11-10018 of the Russian Science Foundation.

PY - 2020

Y1 - 2020

N2 - In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P. Berglund, T. Hübsch and M. Henningson considered a pair (f,G) consisting of an invertible polynomial f and a finite abelian group G of its diagonal symmetries and associated to this pair a dual pair (f˜,G˜). A. Takahashi suggested a generalization of this construction to pairs (f,G) where G is a non-abelian group generated by some diagonal symmetries and some permutations of variables. In a previous paper, the authors showed that some mirror symmetry phenomena appear only under a special condition on the action of the group G: a parity condition. Here we consider the orbifold Euler characteristic of the Milnor fibre of a pair (f,G). We show that, for an abelian group G, the mirror symmetry of the orbifold Euler characteristics can be derived from the corresponding result about the equivariant Euler characteristics. For non-abelian symmetry groups we show that the orbifold Euler characteristics of certain extremal orbit spaces of the group G and the dual group G˜ coincide. From this we derive that the orbifold Euler characteristics of the Milnor fibres of dual periodic loop polynomials coincide up to sign.

AB - In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P. Berglund, T. Hübsch and M. Henningson considered a pair (f,G) consisting of an invertible polynomial f and a finite abelian group G of its diagonal symmetries and associated to this pair a dual pair (f˜,G˜). A. Takahashi suggested a generalization of this construction to pairs (f,G) where G is a non-abelian group generated by some diagonal symmetries and some permutations of variables. In a previous paper, the authors showed that some mirror symmetry phenomena appear only under a special condition on the action of the group G: a parity condition. Here we consider the orbifold Euler characteristic of the Milnor fibre of a pair (f,G). We show that, for an abelian group G, the mirror symmetry of the orbifold Euler characteristics can be derived from the corresponding result about the equivariant Euler characteristics. For non-abelian symmetry groups we show that the orbifold Euler characteristics of certain extremal orbit spaces of the group G and the dual group G˜ coincide. From this we derive that the orbifold Euler characteristics of the Milnor fibres of dual periodic loop polynomials coincide up to sign.

KW - Berglund–Hübsch–Henningson duality

KW - Equivariant Euler char-acteristic

KW - Group action

KW - Invertible polynomial

KW - Mirror symme-try

KW - Saito duality

UR - http://www.scopus.com/inward/record.url?scp=85096897722&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1811.05781

DO - 10.48550/arXiv.1811.05781

M3 - Article

VL - 16

SP - 1099

EP - 1113

JO - Pure and Applied Mathematics Quarterly

JF - Pure and Applied Mathematics Quarterly

SN - 1558-8599

IS - 4

ER -