Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence
We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, c...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2024 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
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Інститут математики НАН України
2024
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212645 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence. Nafiz Ishtiaque, Seyed Faroogh Moosavian and Yehao Zhou. SIGMA 20 (2024), 099, 95 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862704439430742016 |
|---|---|
| author | Ishtiaque, Nafiz Moosavian, Seyed Faroogh Zhou, Yehao |
| author_facet | Ishtiaque, Nafiz Moosavian, Seyed Faroogh Zhou, Yehao |
| citation_txt | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence. Nafiz Ishtiaque, Seyed Faroogh Moosavian and Yehao Zhou. SIGMA 20 (2024), 099, 95 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d = 2 quiver gauge theories. The Bethe/gauge correspondence relates such a gauge theory to an isotropic/elliptic superspin chain, and the stable envelopes compute the -matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic (1|1) spin chain with fundamental representations using the corresponding 3d = 2 SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on × for an interval and an elliptic curve compute the elliptic stable envelopes, and in turn the geometric elliptic -matrix, of the anisotropic (1|1) spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the -matrix. The reduction to 2d gives the K-theoretic stable envelopes, and the trigonometric -matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational -matrix. The latter recovers Rimányi-Rozansky's results that appeared recently in the mathematical literature.
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| first_indexed | 2026-03-18T19:16:00Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212645 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T19:16:00Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ishtiaque, Nafiz Moosavian, Seyed Faroogh Zhou, Yehao 2026-02-09T09:31:59Z 2024 Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence. Nafiz Ishtiaque, Seyed Faroogh Moosavian and Yehao Zhou. SIGMA 20 (2024), 099, 95 pages 1815-0659 2020 Mathematics Subject Classification: 81R12; 81T60; 55N34 arXiv:2308.12333 https://nasplib.isofts.kiev.ua/handle/123456789/212645 https://doi.org/10.3842/SIGMA.2024.099 We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d = 2 quiver gauge theories. The Bethe/gauge correspondence relates such a gauge theory to an isotropic/elliptic superspin chain, and the stable envelopes compute the -matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic (1|1) spin chain with fundamental representations using the corresponding 3d = 2 SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on × for an interval and an elliptic curve compute the elliptic stable envelopes, and in turn the geometric elliptic -matrix, of the anisotropic (1|1) spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the -matrix. The reduction to 2d gives the K-theoretic stable envelopes, and the trigonometric -matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational -matrix. The latter recovers Rimányi-Rozansky's results that appeared recently in the mathematical literature. We thank Kevin Costello, Yuan Miao, Hiraku Nakajima, Tadashi Okazaki, Surya Raghavendran, Junya Yagi, Masahito Yamazaki, and Zijun Zhou for useful discussions. We thank Davide Gaiotto and Andrei Okounkov for reading the draft and sending us valuable feedback. Special thanks go to Mykola Dedushenko for explaining some parts of his work with Nikita Nekrasov and for providing extensive commentary on an earlier version of this work. We thank the Banff International Research Station for generously hosting us during the last stage of this project. N.I. is supported by the Huawei Young Talents Program Fellowship at IHES. The work of S.F.M. is funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) funding number SAPIN-2022-00028, and also in part by the Alfred P. Sloan Foundation, grant FG2020-13768. S.F.M would like to thank Davide Gaiotto for his support during the visit to the Perimeter Institute. Kavli IPMU is supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence Article published earlier |
| spellingShingle | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence Ishtiaque, Nafiz Moosavian, Seyed Faroogh Zhou, Yehao |
| title | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence |
| title_full | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence |
| title_fullStr | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence |
| title_full_unstemmed | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence |
| title_short | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence |
| title_sort | elliptic stable envelopes for certain non-symplectic varieties and dynamical -matrices for superspin chains from the bethe/gauge correspondence |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212645 |
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