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

TL;DR

This paper establishes a geometric link between quantum K-theory of flag varieties, Coulomb branch equations, and Bethe Ansatz equations, proving explicit presentations of the quantum K-ring and isomorphisms with Bethe algebras.

Contribution

It provides the first geometric proof of the coincidence between Coulomb branch equations and Bethe Ansatz equations, and resolves conjectures on quantum K-rings of flag varieties.

Findings

Proved explicit presentations for the quantum K-ring of flag varieties.

Established isomorphism between stable map and quasimap K-theory of partial flag varieties.

Connected quantum tautological bundles with Bethe algebra interpretations.

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.