Quantum K-Rings of Partial Flag Varieties, Coulomb Branches, and the Bethe Ansatz
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916949360902144 |
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| author | Huq-Kuruvilla, Irit |
| author_facet | Huq-Kuruvilla, Irit |
| contents | We give a purely geometric explanation of the coincidence between the Coulomb Branch equations for the 3D GLSM describing the quantum $K$-theory of a flag variety, and the Bethe Ansatz equations of the 5-vertex lattice model. In doing so, we prove two explicit presentations for the quantum $K$-ring of the flag variety, resolving conjectures of Gu-Sharpe-Mihalcea-Xu-Zhang-Zou and Rimanyi-Tarasov-Varchenko. We also prove that the stable map and quasimap $K$-theory of the partial flag varieties are isomorphic, using the work of Koroteev-Pushkar-Smirnov-Zeitlin identifying the latter ring with the Bethe algebra of the 5-vertex lattice model. Our isomorphism gives an additional interpretation of the quantum tautological bundles described in the quasimap ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15575 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum K-Rings of Partial Flag Varieties, Coulomb Branches, and the Bethe Ansatz Huq-Kuruvilla, Irit Algebraic Geometry High Energy Physics - Theory Mathematical Physics Representation Theory 14N35 (Primary), 81T30, 20C35 (Secondary) We give a purely geometric explanation of the coincidence between the Coulomb Branch equations for the 3D GLSM describing the quantum $K$-theory of a flag variety, and the Bethe Ansatz equations of the 5-vertex lattice model. In doing so, we prove two explicit presentations for the quantum $K$-ring of the flag variety, resolving conjectures of Gu-Sharpe-Mihalcea-Xu-Zhang-Zou and Rimanyi-Tarasov-Varchenko. We also prove that the stable map and quasimap $K$-theory of the partial flag varieties are isomorphic, using the work of Koroteev-Pushkar-Smirnov-Zeitlin identifying the latter ring with the Bethe algebra of the 5-vertex lattice model. Our isomorphism gives an additional interpretation of the quantum tautological bundles described in the quasimap ring. |
| title | Quantum K-Rings of Partial Flag Varieties, Coulomb Branches, and the Bethe Ansatz |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics Representation Theory 14N35 (Primary), 81T30, 20C35 (Secondary) |
| url | https://arxiv.org/abs/2409.15575 |