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Lattices for Landau–Ginzburg orbifolds

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Abstract

We consider a pair consisting of an invertible polynomial and a finite abelian group of its symmetries. Berglund, Hübsch, and Henningson proposed a duality between such pairs giving rise to mirror symmetry. We define an orbifoldized signature for such a pair using the orbifoldized elliptic genus. In the case of three variables and based on the homological mirror symmetry picture, we introduce two integral lattices, a transcendental and an algebraic one. We show that these lattices have the same rank and that the signature of the transcendental one is the orbifoldized signature. Finally, we give some evidence that these lattices are interchanged under the duality of pairs.

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Acknowledgements

This work has been partially supported by DFG. The second named author is also supported by JSPS KAKENHI Grant Number 16H06337. The authors would like to thank the referee for useful comments.

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Authors and Affiliations

  1. Institut für Algebraische Geometrie, Leibniz Universität Hannover, Postfach 6009, 30060, Hannover, Germany

    Wolfgang Ebeling

  2. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Atsushi Takahashi

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  1. Wolfgang Ebeling
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  2. Atsushi Takahashi
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Correspondence to Wolfgang Ebeling.

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Ebeling, W., Takahashi, A. Lattices for Landau–Ginzburg orbifolds. Math. Z. 296, 639–659 (2020). https://doi.org/10.1007/s00209-019-02441-3

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