Open Hurwitz numbers and the mKP hierarchy
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909832672444416 |
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| author | Buryak, Alexandr Tessler, Ran J. Troshkin, Mikhail |
| author_facet | Buryak, Alexandr Tessler, Ran J. Troshkin, Mikhail |
| contents | We give a natural definition of open Hurwitz numbers, where the weight of each ramified covering includes an integer parameter $N$ taken to the power that is equal to the number of boundary components of a Riemann surface with boundary mapping to $\mathbb{CP}^1$. We prove that the resulting sequence of partition functions, depending on $N\in\mathbb{Z}$, is a tau-sequence of the mKP hierarchy, or in other words it is a sequence of tau-functions of the KP hierarchy where each tau-function is obtained from the previous one by a Bäcklund-Darboux transformation. Our result is motivated by a previous observation of Alexandrov and the first two authors that the refined intersection numbers on the moduli spaces of Riemann surfaces with boundary give a tau-sequence of the mKP hierarchy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08038 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Open Hurwitz numbers and the mKP hierarchy Buryak, Alexandr Tessler, Ran J. Troshkin, Mikhail Mathematical Physics Algebraic Geometry We give a natural definition of open Hurwitz numbers, where the weight of each ramified covering includes an integer parameter $N$ taken to the power that is equal to the number of boundary components of a Riemann surface with boundary mapping to $\mathbb{CP}^1$. We prove that the resulting sequence of partition functions, depending on $N\in\mathbb{Z}$, is a tau-sequence of the mKP hierarchy, or in other words it is a sequence of tau-functions of the KP hierarchy where each tau-function is obtained from the previous one by a Bäcklund-Darboux transformation. Our result is motivated by a previous observation of Alexandrov and the first two authors that the refined intersection numbers on the moduli spaces of Riemann surfaces with boundary give a tau-sequence of the mKP hierarchy. |
| title | Open Hurwitz numbers and the mKP hierarchy |
| topic | Mathematical Physics Algebraic Geometry |
| url | https://arxiv.org/abs/2510.08038 |