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Quantum Difference Equations for Grassmannians

Xingyu Cheng, Reese Lance, Nikhil Nagabandi, Andrey Smirnov

TL;DR

The paper resolves the quantum difference equation for the torus-equivariant quantum K-theory of Grassmannians $\mathrm{Gr}(k,n)$ by constructing explicit partition-function–based solutions, connecting the geometry to an $\mathrm{XXZ}$-type integrable system. The authors first formulate the qde as $\Psi(qz)\mathcal{O}(-1)=M(z)\Psi(z)$, derive the R-matrix from the XXZ model, and express partition functions in cohomology via factorial Schur polynomials, then extend to quantum K-theory using double Grothendieck polynomials. A geometric Satake correspondence allows the Gr$(k,n)$ solution to be built from wedge powers of Gr$(1,n)$ solutions, yielding determinant expressions for $\psi_\lambda$, whereas Bethe equations emerge as saddle-point conditions on the partition functions, reproducing the Bethe vectors and eigenvalues of the quantum multiplication operator $M(z)$. The work also covers nonequivariant limits and shows how the quantum K-theory ring aligns with the Bethe algebra of the XXZ spin chain, providing a concrete bridge between equivariant quantum geometry and integrable systems with practical computational tools. Overall, the results illuminate deep links among quantum cohomology, quantum K-theory, representation theory via Satake, and integrable models, with potential extensions to broader flag varieties and qKZ-type equations.

Abstract

We consider quantum difference equation (QDE) for equivariant quantum K-theory of the Grassmannian. In this paper we obtain a solution to the QDE and use the solution to asymptotically derive the Bethe ansatz equations. In the limit, we obtain similar results for the cohomological analogue. For both cases, we describe the nonequivariant solutions as well. As an application, we identify the quantum K-theory ring of $\mathrm{Gr}(k,n)$ with a quantum 5 vertex XXZ integrable spin chain.

Quantum Difference Equations for Grassmannians

TL;DR

The paper resolves the quantum difference equation for the torus-equivariant quantum K-theory of Grassmannians by constructing explicit partition-function–based solutions, connecting the geometry to an -type integrable system. The authors first formulate the qde as , derive the R-matrix from the XXZ model, and express partition functions in cohomology via factorial Schur polynomials, then extend to quantum K-theory using double Grothendieck polynomials. A geometric Satake correspondence allows the Gr solution to be built from wedge powers of Gr solutions, yielding determinant expressions for , whereas Bethe equations emerge as saddle-point conditions on the partition functions, reproducing the Bethe vectors and eigenvalues of the quantum multiplication operator . The work also covers nonequivariant limits and shows how the quantum K-theory ring aligns with the Bethe algebra of the XXZ spin chain, providing a concrete bridge between equivariant quantum geometry and integrable systems with practical computational tools. Overall, the results illuminate deep links among quantum cohomology, quantum K-theory, representation theory via Satake, and integrable models, with potential extensions to broader flag varieties and qKZ-type equations.

Abstract

We consider quantum difference equation (QDE) for equivariant quantum K-theory of the Grassmannian. In this paper we obtain a solution to the QDE and use the solution to asymptotically derive the Bethe ansatz equations. In the limit, we obtain similar results for the cohomological analogue. For both cases, we describe the nonequivariant solutions as well. As an application, we identify the quantum K-theory ring of with a quantum 5 vertex XXZ integrable spin chain.
Paper Structure (20 sections, 15 theorems, 131 equations, 1 figure)

This paper contains 20 sections, 15 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Layout of the paper
  3. Acknowledgments
  4. Notations and background
  5. Cohomology
  6. Factorial Schur polynomials
  7. K-theory
  8. The quantum cohomology qde
  9. The operator MC(z)
  10. Partition functions for cohomology
  11. QHT case: solution to qde
  12. QHT case: solution to qde via geometric Satake
  13. Bethe equations and Bethe vectors
  14. Nonequivariant quantum cohomology solutions
  15. Quantum K-theory
  16. ...and 5 more sections

Key Result

Theorem 1

For each partition $\lambda$ in the $k \times (n-k)$ rectangle, the following class in $\mathrm{K}_T(\mathrm{Gr}(k,n))[[z]]$ solves QDE (k-th-eqn-intro): where $\Phi$ is defined in Theorem (mainthm).

Figures (1)

  • Figure 1: The Young diagram for partition $\lambda = (4,2,1,1,0)$. The labels of the blue path's vertical steps give the $k$-subset $r = \{ 1, 3, 4, 6, 9 \}$.

Theorems & Definitions (31)

  • Theorem : Theorem \ref{['mainthm']}
  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • ...and 21 more