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arXiv:2404.01898 · Journal of Physics A Mathematical and Theoretical

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

Andrei Vladimirovich Zotov

Published Jul 3, 2024• 7 citations
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Abstract

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.

DOI
10.1088/1751-8121/ad5ee1

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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.
Abstract We describe the most general GL NM classical elliptic finite-dimensional integrable system, which Lax matrix has n simple poles on elliptic curve. For M = 1 it reproduces the classical inhomogeneous spin chain, for N = 1 it is the Gaudin type (multispin) extension of the spin Ruijsenaars–Schneider model, and for n = 1 the model of M interacting relativistic GL N tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on GL-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars–Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general GL NM model is also described through R -matrices satisfying associative Yang–Baxter equation. This description provides the trigonometric and rational analogues of GL NM models.
In this paper we discuss R-matrix-valued Lax pairs for slN Calogero–Moser model and their relation to integrable quantum long-range spin chains of the Haldane–Shastry–Inozemtsev type. First, we construct the R-matrix-valued Lax pairs for the third flow of the classical Calogero–Moser model. Then we notice that the scalar parts (in the auxiliary space) of the M-matrices corresponding to the second and third flows have form of special spin exchange operators. The freezing trick restricts them to quantum Hamiltonians of long-range spin chains. We show that for a special choice of the R-matrix these Hamiltonians reproduce those for the Inozemtsev chain. In the general case related to the Baxter's elliptic R-matrix we obtain a natural anisotropic extension of the Inozemtsev chain. Commutativity of the Hamiltonians is verified numerically. Trigonometric limits lead to the Haldane–Shastry chains and their anisotropic generalizations.