A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base
In this paper, we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group GLᵣ(KP¹ₓ) with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at ∞∈ℙ₁. The restriction of the family is that the matrix elements in the defining representati...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2019 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2019
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210191 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base / R. Frassek, V. Pestun // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 64 назв. — англ. |
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Frassek, R. Pestun, V. 2025-12-03T14:33:27Z 2019 A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base / R. Frassek, V. Pestun // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 64 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16T25; 53D30; 81R12 arXiv: 1808.00799 https://nasplib.isofts.kiev.ua/handle/123456789/210191 https://doi.org/10.3842/SIGMA.2019.031 In this paper, we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group GLᵣ(KP¹ₓ) with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at ∞∈ℙ₁. The restriction of the family is that the matrix elements in the defining representation are linear functions of x. We study how the symplectic leaves in this family are obtained by the fusion of certain fundamental symplectic leaves. These symplectic leaves arise as minimal examples of (i) moduli spaces of multiplicative Higgs bundles on ℙ¹ with prescribed singularities, (ii) moduli spaces of U(r) monopoles on ℝ²×S¹ with Dirac singularities, (iii) Coulomb branches of the moduli space of vacua of 4d N = 2 supersymmetric Aᵣ₋₁ quiver gauge theories compactified on a circle. While degree 1 symplectic leaves regular at ∞∈ℙ¹ (Coulomb branches of the superconformal quiver gauge theories) are isomorphic to co-adjoint orbits in glᵣ and their Darboux parametrization and quantization are well known, the case irregular at infinity (asymptotically free quiver gauge theories) is novel. We also explicitly quantize the algebra of functions on these moduli spaces by presenting the corresponding solutions to the quantum Yang-Baxter equation valued in the Heisenberg algebra (free field realization). We would like to thank Chris Elliott and Alexei Sevastyanov for multiple helpful discussions. We further thank Oleksandr Tsymbaliuk and the very helpful anonymous referees for comments on the manuscript. R.F. is supported by the IHES´ visitor program. The research of V.P. on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (QUASIFT grant agreement 677368). V.P. also acknowledges grant RFBR 16-02-01021. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base |
| spellingShingle |
A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base Frassek, R. Pestun, V. |
| title_short |
A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base |
| title_full |
A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base |
| title_fullStr |
A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base |
| title_full_unstemmed |
A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base |
| title_sort |
family of glᵣ multiplicative higgs bundles on rational base |
| author |
Frassek, R. Pestun, V. |
| author_facet |
Frassek, R. Pestun, V. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
In this paper, we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group GLᵣ(KP¹ₓ) with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at ∞∈ℙ₁. The restriction of the family is that the matrix elements in the defining representation are linear functions of x. We study how the symplectic leaves in this family are obtained by the fusion of certain fundamental symplectic leaves. These symplectic leaves arise as minimal examples of (i) moduli spaces of multiplicative Higgs bundles on ℙ¹ with prescribed singularities, (ii) moduli spaces of U(r) monopoles on ℝ²×S¹ with Dirac singularities, (iii) Coulomb branches of the moduli space of vacua of 4d N = 2 supersymmetric Aᵣ₋₁ quiver gauge theories compactified on a circle. While degree 1 symplectic leaves regular at ∞∈ℙ¹ (Coulomb branches of the superconformal quiver gauge theories) are isomorphic to co-adjoint orbits in glᵣ and their Darboux parametrization and quantization are well known, the case irregular at infinity (asymptotically free quiver gauge theories) is novel. We also explicitly quantize the algebra of functions on these moduli spaces by presenting the corresponding solutions to the quantum Yang-Baxter equation valued in the Heisenberg algebra (free field realization).
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210191 |
| citation_txt |
A Family of GLᵣ Multiplicative Higgs Bundles on Rational Base / R. Frassek, V. Pestun // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 64 назв. — англ. |
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2025-12-07T21:24:41Z |
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2025-12-07T21:24:41Z |
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