From quantum difference equation to Dubrovin connection of affine type A quiver varieties
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913576982151168 |
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| author | Zhu, Tianqing |
| author_facet | Zhu, Tianqing |
| contents | This is the continuation of the article \cite{Z23}. In this article we will give a detailed analysis of the quantum difference equation of the equivariant $K$-theory of the affine type $A$ quiver varieties. We will give a good representation of the quantum difference operator $\mathbf{M}_{\mathcal{L}}(z)$ such that the monodromy operator $\mathbf{B}_{\mathbf{m}}(z)$in the formula can be written in the $U_{q}(\mathfrak{sl}_2)$-form or in the $U_{q}(\hat{\mathfrak{gl}}_1)$-form. We also give the detailed analysis of the connection matrix for the quantum difference equation in the nodal limit $p\rightarrow0$. Using these two results, we prove that the degeneration limit of the quantum difference equation is the Dubrovin connection for the quantum cohomology of the affine type $A$ quiver varieties, and the monodromy representation for the Dubrovin connection is generated by the monodromy operators $\mathbf{B}_{\mathbf{m}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02473 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From quantum difference equation to Dubrovin connection of affine type A quiver varieties Zhu, Tianqing Representation Theory Mathematical Physics Quantum Algebra This is the continuation of the article \cite{Z23}. In this article we will give a detailed analysis of the quantum difference equation of the equivariant $K$-theory of the affine type $A$ quiver varieties. We will give a good representation of the quantum difference operator $\mathbf{M}_{\mathcal{L}}(z)$ such that the monodromy operator $\mathbf{B}_{\mathbf{m}}(z)$in the formula can be written in the $U_{q}(\mathfrak{sl}_2)$-form or in the $U_{q}(\hat{\mathfrak{gl}}_1)$-form. We also give the detailed analysis of the connection matrix for the quantum difference equation in the nodal limit $p\rightarrow0$. Using these two results, we prove that the degeneration limit of the quantum difference equation is the Dubrovin connection for the quantum cohomology of the affine type $A$ quiver varieties, and the monodromy representation for the Dubrovin connection is generated by the monodromy operators $\mathbf{B}_{\mathbf{m}}$. |
| title | From quantum difference equation to Dubrovin connection of affine type A quiver varieties |
| topic | Representation Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2405.02473 |