Tropical twisted Hurwitz numbers for elliptic curves
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909363847823360 |
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| author | Hahn, Marvin Anas Markwig, Hannah |
| author_facet | Hahn, Marvin Anas Markwig, Hannah |
| contents | Hurwitz numbers enumerate branched morphisms between Riemann surfaces. For a fixed elliptic target, Hurwitz numbers are intimately related to mirror symmetry following work of Dijkgraaf. In recent work of Chapuy and Dolega a new variant of Hurwitz numbers with fixed genus $0$ target was introduced that includes maps between between non-orientiable surfaces. These numbers are called $b$-Hurwitz numbers and are polynomials in a parameter $b$ which measures the non-orientability of the involved maps. An interpretation in terms of factorisations of $b$-Hurwitz numbers for $b=1$, so-called twisted Hurwitz numbers, was found in work of Burman and Fesler. In previous work, the authors derived a tropical geometry interpretation of these numbers. In this paper, we introduce a natural generalisation of twisted Hurwitz numbers with elliptic targets within the framework of symmetric groups. We derive a tropical interpretation of these invariants, relate them to Feynman integrals and derive an expression as a matrix element of an operator in the bosonic Fock space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00333 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tropical twisted Hurwitz numbers for elliptic curves Hahn, Marvin Anas Markwig, Hannah Combinatorics Algebraic Geometry 14T15, 14N10, 57M12, 05C30 Hurwitz numbers enumerate branched morphisms between Riemann surfaces. For a fixed elliptic target, Hurwitz numbers are intimately related to mirror symmetry following work of Dijkgraaf. In recent work of Chapuy and Dolega a new variant of Hurwitz numbers with fixed genus $0$ target was introduced that includes maps between between non-orientiable surfaces. These numbers are called $b$-Hurwitz numbers and are polynomials in a parameter $b$ which measures the non-orientability of the involved maps. An interpretation in terms of factorisations of $b$-Hurwitz numbers for $b=1$, so-called twisted Hurwitz numbers, was found in work of Burman and Fesler. In previous work, the authors derived a tropical geometry interpretation of these numbers. In this paper, we introduce a natural generalisation of twisted Hurwitz numbers with elliptic targets within the framework of symmetric groups. We derive a tropical interpretation of these invariants, relate them to Feynman integrals and derive an expression as a matrix element of an operator in the bosonic Fock space. |
| title | Tropical twisted Hurwitz numbers for elliptic curves |
| topic | Combinatorics Algebraic Geometry 14T15, 14N10, 57M12, 05C30 |
| url | https://arxiv.org/abs/2403.00333 |