Quantum difference equation for the affine type $A$ quiver varieties I: General Construction
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929589319630848 |
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| author | Zhu, Tianqing |
| author_facet | Zhu, Tianqing |
| contents | In this article we use the philosophy in [OS22] to construct the quantum difference equation of affine type $A$ quiver varieties in terms of the quantum toroidal algebra $U_{q,t}(\hat{\hat{\mathfrak{sl}}}_{r})$. In the construction, and we define the set of wall for each quiver varieties by the action of the universal $R$-matrix, which is shown to be almost equivalent to that of the $K$-theoretic stable envelope on each interval in $H^2(X,\mathbb{Q})$. We also give the examples of the instanton moduli space $M(n,r)$ and the equivariant Hilbert scheme $\text{Hilb}_{n}([\mathbb{C}^2/\mathbb{Z}_{r}])$ to show the explicit form of the quantum difference operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00550 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantum difference equation for the affine type $A$ quiver varieties I: General Construction Zhu, Tianqing Representation Theory High Energy Physics - Theory Mathematical Physics Quantum Algebra In this article we use the philosophy in [OS22] to construct the quantum difference equation of affine type $A$ quiver varieties in terms of the quantum toroidal algebra $U_{q,t}(\hat{\hat{\mathfrak{sl}}}_{r})$. In the construction, and we define the set of wall for each quiver varieties by the action of the universal $R$-matrix, which is shown to be almost equivalent to that of the $K$-theoretic stable envelope on each interval in $H^2(X,\mathbb{Q})$. We also give the examples of the instanton moduli space $M(n,r)$ and the equivariant Hilbert scheme $\text{Hilb}_{n}([\mathbb{C}^2/\mathbb{Z}_{r}])$ to show the explicit form of the quantum difference operator. |
| title | Quantum difference equation for the affine type $A$ quiver varieties I: General Construction |
| topic | Representation Theory High Energy Physics - Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2308.00550 |