Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation
We show that Shakirov's non-stationary difference equation, when it is truncated, implies the quantum Knizhnik-Zamolodchikov (-KZ) equation for ᵥ(⁽¹⁾₁) with generic spins. Namely, we can tune mass parameters so that the Hamiltonian acts on the space of finite Laurent polynomials. Then the repre...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2024 |
| Автори: | , , , , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2024
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212343 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation. Hidetoshi Awata, Koji Hasegawa, Hiroaki Kanno, Ryo Ohkawa, Shamil Shakirov, Jun'ichi Shiraishi and Yasuhiko Yamada. SIGMA 20 (2024), 077, 55 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862663218645696512 |
|---|---|
| author | Awata, Hidetoshi Hasegawa, Koji Kanno, Hiroaki Ohkawa, Ryo Shakirov, Shamil Shiraishi, Jun'ichi Yamada, Yasuhiko |
| author_facet | Awata, Hidetoshi Hasegawa, Koji Kanno, Hiroaki Ohkawa, Ryo Shakirov, Shamil Shiraishi, Jun'ichi Yamada, Yasuhiko |
| citation_txt | Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation. Hidetoshi Awata, Koji Hasegawa, Hiroaki Kanno, Ryo Ohkawa, Shamil Shakirov, Jun'ichi Shiraishi and Yasuhiko Yamada. SIGMA 20 (2024), 077, 55 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We show that Shakirov's non-stationary difference equation, when it is truncated, implies the quantum Knizhnik-Zamolodchikov (-KZ) equation for ᵥ(⁽¹⁾₁) with generic spins. Namely, we can tune mass parameters so that the Hamiltonian acts on the space of finite Laurent polynomials. Then the representation matrix of the Hamiltonian agrees with the -matrix, or the quantum 6 symbols. On the other hand, we prove that the K-theoretic Nekrasov partition function from the affine Laumon space is identified with the well-studied Jackson integral solution to the -KZ equation. Combining these results, we establish that the affine Laumon partition function gives a solution to Shakirov's equation, which was a conjecture in our previous paper. We also work out the base-fiber duality and four-dimensional limit in relation to the -KZ equation.
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| first_indexed | 2026-03-16T06:16:33Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212343 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T06:16:33Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Awata, Hidetoshi Hasegawa, Koji Kanno, Hiroaki Ohkawa, Ryo Shakirov, Shamil Shiraishi, Jun'ichi Yamada, Yasuhiko 2026-02-05T09:54:08Z 2024 Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation. Hidetoshi Awata, Koji Hasegawa, Hiroaki Kanno, Ryo Ohkawa, Shamil Shakirov, Jun'ichi Shiraishi and Yasuhiko Yamada. SIGMA 20 (2024), 077, 55 pages 1815-0659 2020 Mathematics Subject Classification: 14H70; 81R12; 81T40; 81T60 arXiv:2309.15364 https://nasplib.isofts.kiev.ua/handle/123456789/212343 https://doi.org/10.3842/SIGMA.2024.077 We show that Shakirov's non-stationary difference equation, when it is truncated, implies the quantum Knizhnik-Zamolodchikov (-KZ) equation for ᵥ(⁽¹⁾₁) with generic spins. Namely, we can tune mass parameters so that the Hamiltonian acts on the space of finite Laurent polynomials. Then the representation matrix of the Hamiltonian agrees with the -matrix, or the quantum 6 symbols. On the other hand, we prove that the K-theoretic Nekrasov partition function from the affine Laumon space is identified with the well-studied Jackson integral solution to the -KZ equation. Combining these results, we establish that the affine Laumon partition function gives a solution to Shakirov's equation, which was a conjecture in our previous paper. We also work out the base-fiber duality and four-dimensional limit in relation to the -KZ equation. We would like to thank S. Arthamonov, M. Bershtein, P. Gavrylenko, M. Ito, M. Noumi, M. Schlosser, and G. Shibukawa for useful discussions. Our work is supported in part by Grants-in-Aid for Scientific Research (Kakenhi): 18K03274 (H.K.), 23K03087 (H.K.), 21K03180 (R.O.), 19K03512 (J.S.), 19K03530 (J.S.), and 22H01116 (Y.Y.). The work of R.O. was partly supported by Osaka Central Advanced Mathematical Institute: MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849, and the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation Article published earlier |
| spellingShingle | Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation Awata, Hidetoshi Hasegawa, Koji Kanno, Hiroaki Ohkawa, Ryo Shakirov, Shamil Shiraishi, Jun'ichi Yamada, Yasuhiko |
| title | Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation |
| title_full | Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation |
| title_fullStr | Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation |
| title_full_unstemmed | Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation |
| title_short | Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation |
| title_sort | non-stationary difference equation and affine laumon space ii: quantum knizhnik-zamolodchikov equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212343 |
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