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Classical r-matrix structure for elliptic Ruijsenaars chain and 1+1 field analogue of Ruijsenaars-Schneider model

D. Murinov, A. Zotov

TL;DR

This work develops a comprehensive classical $r$-matrix framework for the periodic elliptic Ruijsenaars chain and its 1+1 field analogue, establishing Liouville integrability via explicit Poisson brackets of the monodromy and demonstrating how the chain reduces to the elliptic Calogero–Moser system in the non-relativistic limit. It then extends the construction to field theories, formulating 1+1 Ruijsenaars–Schneider and Calogero–Moser field theories with Zakharov–Shabat pairs and Maillet-type non-ultralocal $r$-matrices, and proves that the non-relativistic limit reproduces the well-known Maillet brackets. The paper provides detailed derivations of the finite-chain $r$-matrix relations, the field-theoretic $r$-matrices, and their limits, supported by Appendices on elliptic functions and the NKSR formulation. Overall, the results illuminate the link between discrete integrable chains and their continuous field limits, clarifying how non-ultralocality arises and is preserved under canonical limits.

Abstract

The classical dynamical $r$-matrix structure for the periodic elliptic Ruijsenaars chain is described. The Poisson brackets for the monodromy matrix are calculated as well, thus providing Liouville integrability of the model. Next, we study its continuous non-relativistic limit and reproduce the Maillet type non-ultralocal $r$-matrix structure for the field analogue of the elliptic Calogero-Moser model.

Classical r-matrix structure for elliptic Ruijsenaars chain and 1+1 field analogue of Ruijsenaars-Schneider model

TL;DR

This work develops a comprehensive classical -matrix framework for the periodic elliptic Ruijsenaars chain and its 1+1 field analogue, establishing Liouville integrability via explicit Poisson brackets of the monodromy and demonstrating how the chain reduces to the elliptic Calogero–Moser system in the non-relativistic limit. It then extends the construction to field theories, formulating 1+1 Ruijsenaars–Schneider and Calogero–Moser field theories with Zakharov–Shabat pairs and Maillet-type non-ultralocal -matrices, and proves that the non-relativistic limit reproduces the well-known Maillet brackets. The paper provides detailed derivations of the finite-chain -matrix relations, the field-theoretic -matrices, and their limits, supported by Appendices on elliptic functions and the NKSR formulation. Overall, the results illuminate the link between discrete integrable chains and their continuous field limits, clarifying how non-ultralocality arises and is preserved under canonical limits.

Abstract

The classical dynamical -matrix structure for the periodic elliptic Ruijsenaars chain is described. The Poisson brackets for the monodromy matrix are calculated as well, thus providing Liouville integrability of the model. Next, we study its continuous non-relativistic limit and reproduce the Maillet type non-ultralocal -matrix structure for the field analogue of the elliptic Calogero-Moser model.

Paper Structure

This paper contains 33 sections, 4 theorems, 181 equations.

Table of Contents

  1. Introduction and overview
  2. Many-body systems.
  3. Chains.
  4. Field theories.
  5. Purpose of the paper.
  6. Elliptic Ruijsenaars-Schneider model
  7. Hamiltonian description and Lax pair
  8. Classical $r$-matrix
  9. Non-relativistic limit to Calogero-Moser model
  10. The limit in the Lax matrix.
  11. The limit in $r$-matrix structure.
  12. Finite-dimensional case: Ruijsenaars chain
  13. Description of the model
  14. Classical $r$-matrix structure
  15. Field theory
  16. ...and 18 more sections

Key Result

Proposition 2.1

For the matrix $L^{\hbox{\tiny{RS}}}(z)$ (q101) the following quadratic $r$-matrix relation holds: where with and In the last definition $"\partial_{\eta}L^{\hbox{\tiny{RS}}}(z)"$ means that the derivative with respect to $\eta$ acts on the function $\phi$ as in (a08) but does not act on $b_j$: That is and similarly

Theorems & Definitions (4)

  • Proposition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.1