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Quantum integrable model for the quantum cohomology/K-theory of flag varieties and the double $β$-Grothendieck polynomials

Jirui Guo

TL;DR

The paper constructs a GL$(n)$ quantum integrable model via the $R$-matrix $R^{(n)}(x,y)$ with deformation parameter $\beta$ to encode the ring relations of the equivariant quantum cohomology and quantum K-theory of flag varieties. By applying an algebraic Bethe ansatz, it derives Bethe equations that reproduce Whitney-type relations for the tautological bundles, linking the roots $\sigma^{(m)}_a$ to Chern roots of the flag's tautological bundles in cohomology ($\beta=0$) and to $1-\sigma^{(m)}_a$ in K-theory ($\beta=-1$). The Bethe states are shown to generate the double $\beta$-Grothendieck polynomials $\mathcal{G}^{(\beta)}_w$, providing explicit expansions of quantum states in terms of Schubert/Grothendieck bases; this yields a Bethe/Gauge correspondence compatible with known GLSM Coulomb-branch vacua. Together, these results establish a unifying framework linking quantum integrable systems, flag variety (co)homology theories, and double $\beta$-Grothendieck polynomials, with clear specializations to quantum cohomology and quantum K-theory. The construction offers a path to computing quantum (co)homology via integrable-model techniques and connects to lattice-model representations of the Grothendieck polynomials.

Abstract

A GL$(n)$ quantum integrable system generalizing the asymmetric five vertex spin chain is shown to encode the ring relations of the equivariant quantum cohomology and equivariant quantum K-theory ring of flag varieties. We also show that the Bethe ansatz states of this system generate the double $β$-Grothendieck polynomials.

Quantum integrable model for the quantum cohomology/K-theory of flag varieties and the double $β$-Grothendieck polynomials

TL;DR

The paper constructs a GL quantum integrable model via the -matrix with deformation parameter to encode the ring relations of the equivariant quantum cohomology and quantum K-theory of flag varieties. By applying an algebraic Bethe ansatz, it derives Bethe equations that reproduce Whitney-type relations for the tautological bundles, linking the roots to Chern roots of the flag's tautological bundles in cohomology () and to in K-theory (). The Bethe states are shown to generate the double -Grothendieck polynomials , providing explicit expansions of quantum states in terms of Schubert/Grothendieck bases; this yields a Bethe/Gauge correspondence compatible with known GLSM Coulomb-branch vacua. Together, these results establish a unifying framework linking quantum integrable systems, flag variety (co)homology theories, and double -Grothendieck polynomials, with clear specializations to quantum cohomology and quantum K-theory. The construction offers a path to computing quantum (co)homology via integrable-model techniques and connects to lattice-model representations of the Grothendieck polynomials.

Abstract

A GL quantum integrable system generalizing the asymmetric five vertex spin chain is shown to encode the ring relations of the equivariant quantum cohomology and equivariant quantum K-theory ring of flag varieties. We also show that the Bethe ansatz states of this system generate the double -Grothendieck polynomials.

Paper Structure

This paper contains 5 sections, 15 theorems, 149 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The GL$(n)$ asymmetric five vertex model
  3. Bethe ansatz equations and the equivariant quantum cohomology/K-theory ring of flag varieties
  4. Relationship with the double $\beta$-Grothendieck polynomials
  5. Quantum cohomology/K-theory of flag varieties and the double $\beta$-Grothendieck polynomials

Key Result

Theorem 2.1

The R-matrix defined by Eq.Rmatrix satisfies the Yang-Baxter equation for all $n >1$, where $R^{(n)}_{ab}$ acts on the $a$th and $b$th factor of $V \otimes V \otimes V$.

Figures (2)

  • Figure 1: The left action of $B_k(u)$ on $\ket{n_1,n_2,\cdots,n_N}$ for $N=12$ and a specific choice of $(m_1,\cdots,m_{N-1})$ in the expansion \ref{['exdR']}.
  • Figure 2: The right action of $B_k(u)$ on $\bra{n_1,n_2,\cdots,n_N}$ for $N=12$ and a specific choice of $(m_2,\cdots,m_N)$ in the expansion \ref{['exdL']}.

Theorems & Definitions (37)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 1
  • Remark 2
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 27 more