Elliptic Quantum Toroidal Algebra $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ and Elliptic Stable Envelopes for the $A^{(1)}_{N-1}$ Quiver Varieties
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912671890145280 |
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| author | Konno, Hitoshi Smirnov, Andrey |
| author_facet | Konno, Hitoshi Smirnov, Andrey |
| contents | We propose a new construction of vertex operators of the elliptic quantum toroidal algebra $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ by combining representations of the algebra and formulas of the elliptic stable envelopes for the $A^{(1)}_{N-1}$ quiver variety ${\cal M}(v,w)$. Compositions of the vertex operators turn out consistent to the shuffle product formula of the elliptic stable envelopes. Their highest to highest expectation values provide K-theoretic vertex functions for ${\cal M}(v,w)$. We also derive exchange relation of the vertex operators and construct a $L$-operator satisfying the $RLL=LLR^*$ relation with $R$ and $R^*$ being elliptic dynamical $R$-matrices defined as transition matrices of the elliptic stable envelopes. Assuming a universal form of $L$ and defining a comultiplication $Δ$ in terms of it, we show that our vertex operators are intertwining operators of the $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$-modules w.r.t $Δ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22992 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elliptic Quantum Toroidal Algebra $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ and Elliptic Stable Envelopes for the $A^{(1)}_{N-1}$ Quiver Varieties Konno, Hitoshi Smirnov, Andrey Representation Theory High Energy Physics - Theory Quantum Algebra 17B37, 14N10, 81R10 We propose a new construction of vertex operators of the elliptic quantum toroidal algebra $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ by combining representations of the algebra and formulas of the elliptic stable envelopes for the $A^{(1)}_{N-1}$ quiver variety ${\cal M}(v,w)$. Compositions of the vertex operators turn out consistent to the shuffle product formula of the elliptic stable envelopes. Their highest to highest expectation values provide K-theoretic vertex functions for ${\cal M}(v,w)$. We also derive exchange relation of the vertex operators and construct a $L$-operator satisfying the $RLL=LLR^*$ relation with $R$ and $R^*$ being elliptic dynamical $R$-matrices defined as transition matrices of the elliptic stable envelopes. Assuming a universal form of $L$ and defining a comultiplication $Δ$ in terms of it, we show that our vertex operators are intertwining operators of the $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$-modules w.r.t $Δ$. |
| title | Elliptic Quantum Toroidal Algebra $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ and Elliptic Stable Envelopes for the $A^{(1)}_{N-1}$ Quiver Varieties |
| topic | Representation Theory High Energy Physics - Theory Quantum Algebra 17B37, 14N10, 81R10 |
| url | https://arxiv.org/abs/2510.22992 |