The content discusses the quantum K-ring of partial flag varieties, which is a deformation of the usual K-ring using K-theoretic Gromov-Witten invariants. There are two sets of predictions for the structure of this quantum K-ring:
From the study of a 3D gauged linear sigma model (GLSM), a conjectural description of the relations was given in terms of the Coulomb Branch equations.
From integrable systems, a conjecture was made that the quantum K-theory of the Nakajima quiver variety (the cotangent bundle of the flag variety) is isomorphic to the Bethe algebra of a certain Yang-Baxter algebra. In the compact limit, this was expected to describe the quantum K-theory of the flag variety itself.
The main results are:
The author provides a geometric interpretation of the Bethe Ansatz equations by relating them to the quantum K-theory of the abelianization of the flag variety. This allows the author to prove the conjecture from the GLSM side.
The author shows that the stable map quantum K-ring of the flag variety is isomorphic to the Bethe algebra, identifying the quantum tautological bundles.
The key steps are:
This provides a direct geometric explanation for the coincidence between the Coulomb Branch equations and the Bethe Ansatz, and resolves the conjecture on the structure of the quantum K-ring of partial flag varieties.
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arxiv.org
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by Irit Huq-Kur... at arxiv.org 10-01-2024
https://arxiv.org/pdf/2409.15575.pdfDeeper Inquiries