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arXiv · 2409.15575

Quantum K-Rings of Partial Flag Varieties, Coulomb Branches, and the Bethe Ansatz

Irit Huq-Kuruvilla
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Abstract

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.

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Irit Huq-Kuruvilla. 2024-09-23. Quantum K-Rings of Partial Flag Varieties, Coulomb Branches, and the Bethe Ansatz. https://arxiv.org/abs/2409.15575

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