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A Combinatorial Description of the Dormant Miura Transformation

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Abstract

The aim of the present paper is to describe Miura \({\mathfrak {s}}{\mathfrak {l}}_2\)-opers and Miura transformations in terms of graph-theoretic objects. We construct a bijective correspondence between dormant generic Miura \({\mathfrak {s}}{\mathfrak {l}}_2\)-opers on a totally degenerate curve in positive characteristic and certain branch numberings on a 3-regular graph. This correspondence allows us to completely identify dormant generic Miura \({\mathfrak {s}}{\mathfrak {l}}_2\)-opers on totally degenerate curves. Also, we investigate how this result can be related to the combinatorial description of dormant \({\mathfrak {s}}{\mathfrak {l}}_2\)-opers given by S. Mochizuki, F. Liu, and B. Osserman.

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Acknowledgements

We would like to thank the referee for reading carefully the paper and for valuable comments and suggestions. The author was partially supported by the Grant-in-Aid for Scientific Research (KAKENHI No. 18K13385, 21K13770).

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  1. Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, 565-0871, Japan

    Yasuhiro Wakabayashi

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  1. Yasuhiro Wakabayashi
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Correspondence to Yasuhiro Wakabayashi.

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Communicated by Malihe Yousofzadeh.

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Wakabayashi, Y. A Combinatorial Description of the Dormant Miura Transformation. Bull. Iran. Math. Soc. 49, 81 (2023). https://doi.org/10.1007/s41980-023-00822-3

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  • DOI: https://doi.org/10.1007/s41980-023-00822-3

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