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| Format: | Preprint |
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2020
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| Online-Zugang: | https://arxiv.org/abs/2002.07344 |
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| _version_ | 1866912999499890688 |
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| author | Frenkel, Edward Koroteev, Peter Sage, Daniel S. Zeitlin, Anton M. |
| author_facet | Frenkel, Edward Koroteev, Peter Sage, Daniel S. Zeitlin, Anton M. |
| contents | We introduce the notions of $(G,q)$-opers and Miura $(G,q)$-opers, where $G$ is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of $(G,q)$-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. This may be viewed as a $q$DE/IM correspondence between the spectra of a quantum integrable model (IM) and classical geometric objects ($q$-differential equations). If $\mathfrak{g}$ is simply-laced, the Bethe Ansatz equations we obtain coincide with the equations that appear in the quantum integrable model of XXZ-type associated to the quantum affine algebra $U_q \widehat{\mathfrak{g}}$. However, if $\mathfrak{g}$ is non-simply laced, then these equations correspond to a different integrable model, associated to $U_q {}^L\widehat{\mathfrak{g}}$ where $^L\widehat{\mathfrak{g}}$ is the Langlands dual (twisted) affine algebra. A key element in this $q$DE/IM correspondence is the $QQ$-system that has appeared previously in the study of the ODE/IM correspondence and the Grothendieck ring of the category ${\mathcal O}$ of the relevant quantum affine algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_07344 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | q-Opers, QQ-Systems, and Bethe Ansatz Frenkel, Edward Koroteev, Peter Sage, Daniel S. Zeitlin, Anton M. Algebraic Geometry High Energy Physics - Theory Mathematical Physics Quantum Algebra Representation Theory We introduce the notions of $(G,q)$-opers and Miura $(G,q)$-opers, where $G$ is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of $(G,q)$-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. This may be viewed as a $q$DE/IM correspondence between the spectra of a quantum integrable model (IM) and classical geometric objects ($q$-differential equations). If $\mathfrak{g}$ is simply-laced, the Bethe Ansatz equations we obtain coincide with the equations that appear in the quantum integrable model of XXZ-type associated to the quantum affine algebra $U_q \widehat{\mathfrak{g}}$. However, if $\mathfrak{g}$ is non-simply laced, then these equations correspond to a different integrable model, associated to $U_q {}^L\widehat{\mathfrak{g}}$ where $^L\widehat{\mathfrak{g}}$ is the Langlands dual (twisted) affine algebra. A key element in this $q$DE/IM correspondence is the $QQ$-system that has appeared previously in the study of the ODE/IM correspondence and the Grothendieck ring of the category ${\mathcal O}$ of the relevant quantum affine algebra. |
| title | q-Opers, QQ-Systems, and Bethe Ansatz |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2002.07344 |