Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety
Let X be a holomorphic symplectic variety with a torus T action and a finite fixed point set of cardinality k. We assume that an elliptic stable envelope exists for X. Let AI, J=Stab(J)|I be the k×k matrix of restrictions of the elliptic stable envelopes of X to the fixed points. The entries of this...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2019 |
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| Мова: | English |
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Інститут математики НАН України
2019
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210295 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety / R. Rimányi, A. Smirnov, A. Varchenko, Z. Zhou // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 42 назв. — англ. |
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Rimányi, R. Smirnov, A. Varchenko, A. Zhou, Z. 2025-12-05T09:23:41Z 2019 Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety / R. Rimányi, A. Smirnov, A. Varchenko, Z. Zhou // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 42 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 55N34; 32C35; 55R40 arXiv: 1906.00134 https://nasplib.isofts.kiev.ua/handle/123456789/210295 https://doi.org/10.3842/SIGMA.2019.093 Let X be a holomorphic symplectic variety with a torus T action and a finite fixed point set of cardinality k. We assume that an elliptic stable envelope exists for X. Let AI, J=Stab(J)|I be the k×k matrix of restrictions of the elliptic stable envelopes of X to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the Kähler parameters and equivariant parameters of X. We say that two such varieties X and X′ are related by the 3d mirror symmetry if the fixed point sets of X and X′ have the same cardinality and can be identified so that the restriction matrix of X becomes equal to the restriction matrix of X′ after transposition and interchanging the equivariant and Kähler parameters of X, respectively, with the Kähler and equivariant parameters of X′. The first examples of pairs of 3d symmetric varieties were constructed in [Rimányi R., Smirnov A., Varchenko A., Zhou Z., arXiv:1902.03677], where the cotangent bundle T*Gr(k,n) to a Grassmannian is proved to be a 3d mirror to a Nakajima quiver variety of Aₙ₋₁-type. In this paper, we prove that the cotangent bundle of the full flag variety is 3d mirror self-symmetric. That statement in particular leads to nontrivial theta-function identities. The authors would like to thank M. Aganagic and A. Okounkov for their insights on 3d mirror symmetries and elliptic stable envelopes that motivate this work. We thank I. Cherednik for his interest in this work and useful comments. R.R. is supported by the Simons Foundation grant 523882. A.S. is supported by RFBR grant 18-01-00926 and by the AMS travel grant. A.V. is supported in part by NSF grant DMS-1665239. Z.Z. is supported by FRG grant 1564500. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety |
| spellingShingle |
Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety Rimányi, R. Smirnov, A. Varchenko, A. Zhou, Z. |
| title_short |
Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety |
| title_full |
Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety |
| title_fullStr |
Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety |
| title_full_unstemmed |
Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety |
| title_sort |
three-dimensional mirror self-symmetry of the cotangent bundle of the full flag variety |
| author |
Rimányi, R. Smirnov, A. Varchenko, A. Zhou, Z. |
| author_facet |
Rimányi, R. Smirnov, A. Varchenko, A. Zhou, Z. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Let X be a holomorphic symplectic variety with a torus T action and a finite fixed point set of cardinality k. We assume that an elliptic stable envelope exists for X. Let AI, J=Stab(J)|I be the k×k matrix of restrictions of the elliptic stable envelopes of X to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the Kähler parameters and equivariant parameters of X. We say that two such varieties X and X′ are related by the 3d mirror symmetry if the fixed point sets of X and X′ have the same cardinality and can be identified so that the restriction matrix of X becomes equal to the restriction matrix of X′ after transposition and interchanging the equivariant and Kähler parameters of X, respectively, with the Kähler and equivariant parameters of X′. The first examples of pairs of 3d symmetric varieties were constructed in [Rimányi R., Smirnov A., Varchenko A., Zhou Z., arXiv:1902.03677], where the cotangent bundle T*Gr(k,n) to a Grassmannian is proved to be a 3d mirror to a Nakajima quiver variety of Aₙ₋₁-type. In this paper, we prove that the cotangent bundle of the full flag variety is 3d mirror self-symmetric. That statement in particular leads to nontrivial theta-function identities.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210295 |
| citation_txt |
Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety / R. Rimányi, A. Smirnov, A. Varchenko, Z. Zhou // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 42 назв. — англ. |
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2025-12-07T21:25:03Z |
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2025-12-07T21:25:03Z |
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