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Mirror map for Fermat polynomials with a nonabelian group of symmetries

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Abstract

We study Landau–Ginzburg orbifolds \((f,G)\) with \(f=x_1^n+\cdots+x_N^n\) and \(G=S\ltimes G^d\), where \(S\subseteq S_N\) and \(G^d\) is either the maximal group of scalar symmetries of \(f\) or the intersection of the maximal diagonal symmetries of \(f\) with \(SL_N(\mathbb{C})\). We construct a mirror map between the corresponding phase spaces and prove that it is an isomorphism restricted to a certain subspace of the phase space when \(n=N\) is a prime number. When \(S\) satisfies the parity condition of Ebeling–Gusein-Zade, this subspace coincides with the full space. We also show that two phase spaces are isomorphic for \(n=N=5\).

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Acknowledgments

The authors are grateful to the anonymous referee for many important remarks.

Funding

The authors acknowledge partial support by the RSF (grant no. 19-71-00086) and by the International Laboratory of Cluster Geometry, HSE University (RF Government grant, agreement no. 075-15-2021-608 from 08.06.2021). In particular, the proof of Theorem 1 was obtained under the support of the RSF (grant no. 19-71-00086) and the proof of Theorem 2 was obtained under the support of the International Laboratory of Cluster Geometry, HSE University (RF Government grant).

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Authors and Affiliations

  1. Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia

    A. A. Basalaev & A. A. Ionov

  2. Skolkovo Institute of Science and Technology, Moscow, Russia

    A. A. Basalaev

  3. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    A. A. Ionov

Authors
  1. A. A. Basalaev
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  2. A. A. Ionov
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Correspondence to A. A. Basalaev.

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The authors declare no conflicts of interest.

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 209, pp. 205–223 https://doi.org/10.4213/tmf10104.

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Basalaev, A.A., Ionov, A.A. Mirror map for Fermat polynomials with a nonabelian group of symmetries. Theor Math Phys 209, 1491–1506 (2021). https://doi.org/10.1134/S0040577921110015

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  • DOI: https://doi.org/10.1134/S0040577921110015

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