Opers on the projective line, Wronskian relations, and the Bethe Ansatz
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866908737952808960 |
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| author | Brinson, Ty J. Sage, Daniel S. Zeitlin, Anton M. |
| author_facet | Brinson, Ty J. Sage, Daniel S. Zeitlin, Anton M. |
| contents | It is well-known that the spectra of the Gaudin model may be described in terms of solutions of the Bethe Ansatz equations. A conceptual explanation for the appearance of the Bethe Ansatz equations is provided by appropriate $G$-opers: $G$-connections on the projective line with extra structure. In fact, solutions of the Bethe Ansatz equations are parameterized by an enhanced version of opers called Miura opers; here, the opers appearing have only regular singularities. Feigin, Frenkel, Rybnikov, and Toledano Laredo have introduced an inhomogeneous version of the Gaudin model; this model incorporates an additional twist factor, which is an element of the Lie algebra of $G$. They exhibited the Bethe Ansatz equations for this model and gave a geometric interpretation of the spectra in terms of opers with an irregular singularity. In this paper, we consider a new approach to the study of the spectra of the inhomogeneous Gaudin model in terms of a further enhancement of opers called twisted Miura-Plücker opers and a certain system of nonlinear differential equations called the $qq$-system. We show that there is a close relationship between solutions of the inhomogeneous Bethe Ansatz equations and polynomial solutions of the $qq$-system and use this fact to construct a bijection between the set of solutions of the inhomogeneous Bethe Ansatz equations and the set of nondegenerate twisted Miura-Plücker opers. We further prove that as long as certain combinatorial conditions are satisfied, nondegenerate twisted Miura-Plücker opers are in fact Miura opers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_02711 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Opers on the projective line, Wronskian relations, and the Bethe Ansatz Brinson, Ty J. Sage, Daniel S. Zeitlin, Anton M. Algebraic Geometry High Energy Physics - Theory Mathematical Physics Quantum Algebra Representation Theory It is well-known that the spectra of the Gaudin model may be described in terms of solutions of the Bethe Ansatz equations. A conceptual explanation for the appearance of the Bethe Ansatz equations is provided by appropriate $G$-opers: $G$-connections on the projective line with extra structure. In fact, solutions of the Bethe Ansatz equations are parameterized by an enhanced version of opers called Miura opers; here, the opers appearing have only regular singularities. Feigin, Frenkel, Rybnikov, and Toledano Laredo have introduced an inhomogeneous version of the Gaudin model; this model incorporates an additional twist factor, which is an element of the Lie algebra of $G$. They exhibited the Bethe Ansatz equations for this model and gave a geometric interpretation of the spectra in terms of opers with an irregular singularity. In this paper, we consider a new approach to the study of the spectra of the inhomogeneous Gaudin model in terms of a further enhancement of opers called twisted Miura-Plücker opers and a certain system of nonlinear differential equations called the $qq$-system. We show that there is a close relationship between solutions of the inhomogeneous Bethe Ansatz equations and polynomial solutions of the $qq$-system and use this fact to construct a bijection between the set of solutions of the inhomogeneous Bethe Ansatz equations and the set of nondegenerate twisted Miura-Plücker opers. We further prove that as long as certain combinatorial conditions are satisfied, nondegenerate twisted Miura-Plücker opers are in fact Miura opers. |
| title | Opers on the projective line, Wronskian relations, and the Bethe Ansatz |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2112.02711 |