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Geometric realizations of the Bethe ansatz equations

Anton M. Zeitlin

Abstract

These lecture notes are devoted to the recent progress in the geometric aspects of quantum integrable systems based on quantum groups solved using the Bethe ansatz technique. One part is devoted to their enumerative geometry realization through the quantum K-theory of Nakajima quiver varieties. The other part describes a recently studied $q$-deformation of the correspondence between oper connections and Gaudin models. The notes are based on a minicourse at C.I.M.E. Summer School ``Enumerative geometry, quantisation and moduli spaces," September 04-08, 2023.

Geometric realizations of the Bethe ansatz equations

Abstract

These lecture notes are devoted to the recent progress in the geometric aspects of quantum integrable systems based on quantum groups solved using the Bethe ansatz technique. One part is devoted to their enumerative geometry realization through the quantum K-theory of Nakajima quiver varieties. The other part describes a recently studied -deformation of the correspondence between oper connections and Gaudin models. The notes are based on a minicourse at C.I.M.E. Summer School ``Enumerative geometry, quantisation and moduli spaces," September 04-08, 2023.

Paper Structure

This paper contains 35 sections, 20 theorems, 176 equations.

Table of Contents

  1. Introduction: geometric wonders of integrable systems
  2. Algebraic Bethe ansatz and Baxter $Q$-operators
  3. Quantum Integrable systems of spin chain type
  4. The quantum algebra ${\mathcal{U}}_{\hbar}(\widehat{\frak{sl}}_{2})$
  5. Bethe ansatz for XXZ spin chain
  6. Baxter Q-operator and prefundamental representations
  7. Modern approaches to Bethe ansatz
  8. Quantum Knizhnik-Zamolodchikov equations
  9. $QQ$-systems and Baxter operators
  10. Quantum K-theory of Nakajima quiver varieties
  11. Nakajima quiver varieties
  12. ${\rm T^* Gr}_{k,n}$ and its equivariant K-theory
  13. Geometric quantum group action
  14. Quasimaps, nonsingular and relative conditions
  15. Capped tensors and the gluing formula
  16. ...and 20 more sections

Key Result

Theorem 2.1

Vectors $\{B(v_1)\dots B(v_k)\Omega_{+}\}$ are the eigenvectors of $tr_{\pi_1(1)}(\mathcal{M}^Z(u))$ with eigenvalues so that are the eigenvalues of $A(u)$ and $D(u)$ on $\Omega_+$ and parameters $v_i$ are subject to Bethe equations: ii) Vectors $\{B(v_1)\dots B(v_k)\Omega_{+}\}$ indexed by the solutions of Bethe equations, span the weight subspace $\mathscr{H}_k\subset\mathscr{H}$ correspondin

Theorems & Definitions (22)

  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 12 more