Double Yangian, Factorization, and qKZ-equation for Cotangent Lie Algebras
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| Format: | Preprint |
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2025
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| _version_ | 1866915685635981312 |
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| author | Abedin, Raschid Niu, Wenjun |
| author_facet | Abedin, Raschid Niu, Wenjun |
| contents | In this paper, we construct the dual $Y^*_\hbar(\mathfrak d)$ and double $DY_\hbar (\mathfrak d)$ of the Yangian $Y_\hbar (\mathfrak d)$ associated with a cotangent Lie algebra $\mathfrak d=T^*\mathfrak g$. We define a coherent factorization algebra version of the dual Yangian $Y_\hbar^*(\mathfrak d)^{\mathrm{co-op}}$ with opposite coproduct. Furthermore, we define a quantum vertex algebra structure on the quantum vacuum module $V_{\hbar,k}(\mathfrak d)$ of central extensions $\widehat{DY}_{\hbar,\ell} (\mathfrak d)$ of this double Yangian and show that its conformal blocks satisfy quantum KZ equations. We discuss examples of $\mathfrak d$ that arise from 3d $N=4$ gauge theories via the work of Costello-Gaiotto. These examples include Takiff Lie algebras $T^*\mathfrak g$, whose affine VOA is a large subalgebra of the chiral differential operator algebra of $G$, as well as the smallest type-A Lie superalgebra $\mathfrak{gl} (1|1)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_16996 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Double Yangian, Factorization, and qKZ-equation for Cotangent Lie Algebras Abedin, Raschid Niu, Wenjun Quantum Algebra Mathematical Physics Rings and Algebras 17B37, 17B69, 81R50 In this paper, we construct the dual $Y^*_\hbar(\mathfrak d)$ and double $DY_\hbar (\mathfrak d)$ of the Yangian $Y_\hbar (\mathfrak d)$ associated with a cotangent Lie algebra $\mathfrak d=T^*\mathfrak g$. We define a coherent factorization algebra version of the dual Yangian $Y_\hbar^*(\mathfrak d)^{\mathrm{co-op}}$ with opposite coproduct. Furthermore, we define a quantum vertex algebra structure on the quantum vacuum module $V_{\hbar,k}(\mathfrak d)$ of central extensions $\widehat{DY}_{\hbar,\ell} (\mathfrak d)$ of this double Yangian and show that its conformal blocks satisfy quantum KZ equations. We discuss examples of $\mathfrak d$ that arise from 3d $N=4$ gauge theories via the work of Costello-Gaiotto. These examples include Takiff Lie algebras $T^*\mathfrak g$, whose affine VOA is a large subalgebra of the chiral differential operator algebra of $G$, as well as the smallest type-A Lie superalgebra $\mathfrak{gl} (1|1)$. |
| title | Double Yangian, Factorization, and qKZ-equation for Cotangent Lie Algebras |
| topic | Quantum Algebra Mathematical Physics Rings and Algebras 17B37, 17B69, 81R50 |
| url | https://arxiv.org/abs/2512.16996 |