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Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains

Nicolai Reshetikhin, Jasper Stokman

TL;DR

This work generalizes the asymptotic boundary KZB framework to all noncompact real connected semisimple groups $G$ with finite center and links it to quantum Calogero-Moser spin chains. By developing coordinate radial component maps and leveraging Harish-Chandra radial theory, it constructs commuting first-order boundary Hamiltonians on the abelian subspace $\mathfrak{a}$ and derives the associated Schrödinger operator with sinh$^{-2}$ interactions tied to the restricted root system. The main results show that the asymptotic boundary KZB operators satisfy coupled classical dynamical Yang-Baxter–reflection equations, and that in the real split case these equations decouple, reflecting the underlying symmetry. The paper also provides an explicit SU$(p,r)$ example with BC-type restricted roots, illustrating the folding of Felder-type dynamical $r$-matrices and the resulting local factors and gauged Hamiltonians. Overall, the work bridges harmonic analysis on symmetric spaces, boundary WZWN CFT, and quantum integrable spin chains, yielding a rich class of exactly solvable quantum systems with concrete algebraic and analytic structures.

Abstract

Asymptotic boundary KZB equations describe the consistency conditions of degenerations of correlation functions for boundary Wess-Zumino-Witten-Novikov conformal field theory on a cylinder. In the first part of the paper we define asymptotic boundary KZB operators for connected real semisimple Lie groups G with finite center. We prove their main properties algebraically using coordinate versions of Harish-Chandra's radial component map. We show that their commutativity is governed by a system of equations involving coupled versions of classical dynamical Yang-Baxter equations and reflection equations. We use the coordinate radial components maps to introduce a new class of quantum superintegrable systems, called quantum Calogero-Moser spin chains. A quantum Calogero-Moser spin chain is a mixture of a quantum spin Calogero-Moser system associated to the restricted root system of G and an one-dimensional spin chain with two-sided reflecting boundaries. The asymptotic boundary KZB operators provide explicit expressions for its first order quantum Hamiltonians. We also explicitly describe the Schrödinger operator.

Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains

TL;DR

This work generalizes the asymptotic boundary KZB framework to all noncompact real connected semisimple groups with finite center and links it to quantum Calogero-Moser spin chains. By developing coordinate radial component maps and leveraging Harish-Chandra radial theory, it constructs commuting first-order boundary Hamiltonians on the abelian subspace and derives the associated Schrödinger operator with sinh interactions tied to the restricted root system. The main results show that the asymptotic boundary KZB operators satisfy coupled classical dynamical Yang-Baxter–reflection equations, and that in the real split case these equations decouple, reflecting the underlying symmetry. The paper also provides an explicit SU example with BC-type restricted roots, illustrating the folding of Felder-type dynamical -matrices and the resulting local factors and gauged Hamiltonians. Overall, the work bridges harmonic analysis on symmetric spaces, boundary WZWN CFT, and quantum integrable spin chains, yielding a rich class of exactly solvable quantum systems with concrete algebraic and analytic structures.

Abstract

Asymptotic boundary KZB equations describe the consistency conditions of degenerations of correlation functions for boundary Wess-Zumino-Witten-Novikov conformal field theory on a cylinder. In the first part of the paper we define asymptotic boundary KZB operators for connected real semisimple Lie groups G with finite center. We prove their main properties algebraically using coordinate versions of Harish-Chandra's radial component map. We show that their commutativity is governed by a system of equations involving coupled versions of classical dynamical Yang-Baxter equations and reflection equations. We use the coordinate radial components maps to introduce a new class of quantum superintegrable systems, called quantum Calogero-Moser spin chains. A quantum Calogero-Moser spin chain is a mixture of a quantum spin Calogero-Moser system associated to the restricted root system of G and an one-dimensional spin chain with two-sided reflecting boundaries. The asymptotic boundary KZB operators provide explicit expressions for its first order quantum Hamiltonians. We also explicitly describe the Schrödinger operator.

Paper Structure

This paper contains 15 sections, 32 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. $n$-Point spherical functions
  3. Structure theory of real semisimple Lie groups
  4. Generalised radial component maps
  5. Differential operators
  6. Radial component maps
  7. Coordinate radial component maps
  8. The quantum Calogero-Moser spin chain
  9. A dynamical factorisation of the Casimir element
  10. The radial component of the action of the Casimir element
  11. The Schrödinger operator
  12. The asymptotic boundary KZB operators
  13. The derivation of the explicit expression
  14. Coupled classical dynamical Yang-Baxter-reflection equations
  15. Example: $\textup{SU}(p,r)$.

Key Result

Lemma 2.4

The linear map $f\mapsto f^\flat$ restricts to a linear isomorphism The preimage of $f^\flat\in C_{\sigma_{\ell;n},\sigma_r}^\infty(G)$ is the function $f\in C_{\sigma_\ell,\underline{\tau},\sigma_r}^\infty(G^{\times (n+1)})$ defined by for $\mathbf{g}\in G^{\times (n+1)}$.

Theorems & Definitions (90)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 80 more