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Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero-Moser model

A. Zotov

TL;DR

This work constructs 1+1 field analogues of elliptic Calogero-Moser systems and proves that their Lax connections obey the classical non-ultralocal Maillet bracket, with the same elliptic $r$-matrix structure as in the finite-dimensional models, now with field-valued coordinates $q_i(x)$. It extends the formalism to spin and multispin realizations, detailing unreduced and reduced phase spaces, the corresponding $r$-matrices, and the Nekrasov multi-spin generalization, demonstrating how reductions yield standard spinless dynamics. The paper further develops the 1+1 field theory version of these models via a Zakharov-Shabat framework, provides explicit U- and V-matrices, and derives both the (modified) classical $r$-matrices and the IRF-Vertex connections in the field setting, illustrating gauge relations between dynamical Felder-type and non-dynamical Belavin-Drinfeld structures. Finally, it articulates the continuous IRF-Vertex relation for field theories, showing how the intertwiner from the IRF-Vertex correspondence implements a gauge transformation that maps dynamical $r$-matrices to non-dynamical elliptic $r$-matrices in the field context. The results deepen the correspondence between dynamical and non-dynamical $r$-matrices in integrable field theories and lay groundwork for further connections to gauge-theoretic and string-theoretic frameworks.

Abstract

We consider 1+1 field generalization of the elliptic Calogero-Moser model. It is shown that the Lax connection satisfies the classical non-ultralocal $r$-matrix structure of Maillet type. Next, we consider 1+1 field analogue of the spin Calogero-Moser model and its multipole (or multispin) extension. Finally, we discuss the field analogue of the classical IRF-Vertex correspondence, which relates utralocal and non-ultralocal $r$-matrix structures.

Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero-Moser model

TL;DR

This work constructs 1+1 field analogues of elliptic Calogero-Moser systems and proves that their Lax connections obey the classical non-ultralocal Maillet bracket, with the same elliptic -matrix structure as in the finite-dimensional models, now with field-valued coordinates . It extends the formalism to spin and multispin realizations, detailing unreduced and reduced phase spaces, the corresponding -matrices, and the Nekrasov multi-spin generalization, demonstrating how reductions yield standard spinless dynamics. The paper further develops the 1+1 field theory version of these models via a Zakharov-Shabat framework, provides explicit U- and V-matrices, and derives both the (modified) classical -matrices and the IRF-Vertex connections in the field setting, illustrating gauge relations between dynamical Felder-type and non-dynamical Belavin-Drinfeld structures. Finally, it articulates the continuous IRF-Vertex relation for field theories, showing how the intertwiner from the IRF-Vertex correspondence implements a gauge transformation that maps dynamical -matrices to non-dynamical elliptic -matrices in the field context. The results deepen the correspondence between dynamical and non-dynamical -matrices in integrable field theories and lay groundwork for further connections to gauge-theoretic and string-theoretic frameworks.

Abstract

We consider 1+1 field generalization of the elliptic Calogero-Moser model. It is shown that the Lax connection satisfies the classical non-ultralocal -matrix structure of Maillet type. Next, we consider 1+1 field analogue of the spin Calogero-Moser model and its multipole (or multispin) extension. Finally, we discuss the field analogue of the classical IRF-Vertex correspondence, which relates utralocal and non-ultralocal -matrix structures.
Paper Structure (28 sections, 4 theorems, 168 equations)

This paper contains 28 sections, 4 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Purpose of the paper.
  3. Classical mechanics
  4. Spinless Calogero-Moser model
  5. Spin and multispin Calogero-Moser model
  6. Classical spin Calogero-Moser model: Lax equations.
  7. Spin Calogero-Moser model: classical $r$-matrix.
  8. Reduction to the spinless model.
  9. Nekrasov's multispin Calogero-Moser model.
  10. 1+1 field theories
  11. 1+1 Calogero-Moser field theory
  12. Equations of motion.
  13. Zakharov-Shabat equation.
  14. Classical $r$-matrix.
  15. Modified classical $r$-matrix.
  16. ...and 13 more sections

Key Result

Theorem 1

The $U$-matrix (e61) satisfies the Maillet $r$-matrix structure (e11) with the $r$-matrix for the finite-dimensional Calogero-Moser model (e26), where positions of particles $q_i$ are replaced with the fields $q_i(x)$:

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Theorem 3