Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers
We derive an explicit formula for the connected (, )-point functions associated with an arbitrary diagonal tau-function (⁺, ⁻) of the 2d Toda lattice hierarchy using fermionic computations and the boson-fermion correspondence. Then, for fixed ⁻, we compute the KP-affine coordinates of (⁺, ⁻). As app...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212046 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers. Zhiyuan Wang and Chenglang Yang. SIGMA 19 (2023), 085, 33 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862664749517373440 |
|---|---|
| author | Wang, Zhiyuan Yang, Chenglang |
| author_facet | Wang, Zhiyuan Yang, Chenglang |
| citation_txt | Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers. Zhiyuan Wang and Chenglang Yang. SIGMA 19 (2023), 085, 33 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We derive an explicit formula for the connected (, )-point functions associated with an arbitrary diagonal tau-function (⁺, ⁻) of the 2d Toda lattice hierarchy using fermionic computations and the boson-fermion correspondence. Then, for fixed ⁻, we compute the KP-affine coordinates of (⁺, ⁻). As applications, we present a unified approach to compute various types of connected double Hurwitz numbers, including the ordinary double Hurwitz numbers, the double Hurwitz numbers with completed -cycles, and the mixed double Hurwitz numbers. We also apply this method to the computation of the stationary Gromov-Witten invariants of ℙ¹ relative to two points.
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| first_indexed | 2026-03-16T07:50:14Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212046 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T07:50:14Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wang, Zhiyuan Yang, Chenglang 2026-01-23T10:12:08Z 2023 Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers. Zhiyuan Wang and Chenglang Yang. SIGMA 19 (2023), 085, 33 pages 1815-0659 2020 Mathematics Subject Classification: 37K10; 14N10; 14N35 arXiv:2210.08712 https://nasplib.isofts.kiev.ua/handle/123456789/212046 https://doi.org/10.3842/SIGMA.2023.085 We derive an explicit formula for the connected (, )-point functions associated with an arbitrary diagonal tau-function (⁺, ⁻) of the 2d Toda lattice hierarchy using fermionic computations and the boson-fermion correspondence. Then, for fixed ⁻, we compute the KP-affine coordinates of (⁺, ⁻). As applications, we present a unified approach to compute various types of connected double Hurwitz numbers, including the ordinary double Hurwitz numbers, the double Hurwitz numbers with completed -cycles, and the mixed double Hurwitz numbers. We also apply this method to the computation of the stationary Gromov-Witten invariants of ℙ¹ relative to two points. We thank the anonymous referees for helpful suggestions. We also thank Professor Jian Zhou for the online course on Hurwitz numbers in TMCSC, and thank Professor Huijun Fan, Professor Xiaobo Liu, and Professor Xiangyu Zhou for their encouragement. The second author is supported by the National Natural Science Foundation of China (No. 12288201). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers Article published earlier |
| spellingShingle | Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers Wang, Zhiyuan Yang, Chenglang |
| title | Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers |
| title_full | Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers |
| title_fullStr | Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers |
| title_full_unstemmed | Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers |
| title_short | Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected (, )-Point Functions, and Double Hurwitz Numbers |
| title_sort | diagonal tau-functions of 2d toda lattice hierarchy, connected (, )-point functions, and double hurwitz numbers |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212046 |
| work_keys_str_mv | AT wangzhiyuan diagonaltaufunctionsof2dtodalatticehierarchyconnectedpointfunctionsanddoublehurwitznumbers AT yangchenglang diagonaltaufunctionsof2dtodalatticehierarchyconnectedpointfunctionsanddoublehurwitznumbers |