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Periods of the multiple Berglund–Hübsch–Krawitz mirrors

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Abstract

We consider the multiple Calabi–Yau mirror phenomenon which appears in Berglund–Hübsch–Krawitz (BHK) mirror symmetry. We show that for any pair of Calabi–Yau orbifolds that are BHK mirrors of a loop–chain-type pair of Calabi–Yau threefolds in the same weighted projective space the periods of the holomorphic nonvanishing form coincide.

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Notes

  1. With a slight abuse of notation, we use here h for the Hodge number of the orbifold, which we used before for the original manifold \(X_M\).

  2. See also [19] for related discussion.

  3. Below \(A_i\) are defined according to Eqs. (2.3) and (2.4).

  4. This was calculated using Mathematica, the algorithm is based on the definition of the polynomials \(W ^0_M\) of chain and loop types given above. First, we find all the cases for these two types separately, then the intersection of the two sets is found.

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Acknowledgements

The work of A. Belavin has been supported by the Russian Science Foundation under the grant 18-12-00439. The authors thank the reviewers for their careful reading and very helpful comments and remarks.

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Authors and Affiliations

  1. Landau Institute for Theoretical Physics, Chernogolovka, Russia, 142432

    Alexander Belavin

  2. Physics Department, Ariel University, 40700, Ariel, Israel

    Vladimir Belavin

  3. Moscow Institute of Physics and Technology, Dolgoprudny, Russia, 141700

    Alexander Belavin

  4. Kharkevich Institute for Information Transmission Problems, Moscow, Russia, 127994

    Alexander Belavin & Gleb Koshevoy

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  1. Alexander Belavin
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  2. Vladimir Belavin
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Correspondence to Alexander Belavin.

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In memory of Boris Dubrovin.

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Appendix

Appendix

Table 1 Here, \(\{k_i\}\) are weights arising simultaneously in loop and chain types, with the corresponding exponents \(\{A_i\}^C\) and \(\{A_i\}^L\) in (2.4)
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Belavin, A., Belavin, V. & Koshevoy, G. Periods of the multiple Berglund–Hübsch–Krawitz mirrors. Lett Math Phys 111, 93 (2021). https://doi.org/10.1007/s11005-021-01439-5

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