$b$-Hurwitz numbers from Whittaker vectors for $\mathcal{W}$-algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917408765116416 |
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| author | Chidambaram, Nitin K. Dołęga, Maciej Osuga, Kento |
| author_facet | Chidambaram, Nitin K. Dołęga, Maciej Osuga, Kento |
| contents | We show that $b$-Hurwitz numbers with a rational weight are obtained by taking an explicit limit of a Whittaker vector for the $\mathcal{W}$-algebra of type $A$. Our result is a vast generalization of several previous results that treated the monotone case, and the cases of quadratic and cubic polynomial weights. It also provides an interpretation of the associated Whittaker vector in terms of generalized branched coverings that might be of independent interest. Our result is new even in the special case $b=0$ that corresponds to classical hypergeometric Hurwitz numbers, and implies that they are governed by the topological recursion of Eynard-Orantin. This gives an independent proof of the recent result of Bychkov-Dunin-Barkowski-Kazarian-Shadrin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_12814 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $b$-Hurwitz numbers from Whittaker vectors for $\mathcal{W}$-algebras Chidambaram, Nitin K. Dołęga, Maciej Osuga, Kento Algebraic Geometry Mathematical Physics Combinatorics Representation Theory 14N10, 05E05, 05E16, 81R10 We show that $b$-Hurwitz numbers with a rational weight are obtained by taking an explicit limit of a Whittaker vector for the $\mathcal{W}$-algebra of type $A$. Our result is a vast generalization of several previous results that treated the monotone case, and the cases of quadratic and cubic polynomial weights. It also provides an interpretation of the associated Whittaker vector in terms of generalized branched coverings that might be of independent interest. Our result is new even in the special case $b=0$ that corresponds to classical hypergeometric Hurwitz numbers, and implies that they are governed by the topological recursion of Eynard-Orantin. This gives an independent proof of the recent result of Bychkov-Dunin-Barkowski-Kazarian-Shadrin. |
| title | $b$-Hurwitz numbers from Whittaker vectors for $\mathcal{W}$-algebras |
| topic | Algebraic Geometry Mathematical Physics Combinatorics Representation Theory 14N10, 05E05, 05E16, 81R10 |
| url | https://arxiv.org/abs/2401.12814 |