On R-matrix valued Lax pairs for Calogero-Moser models

TL;DR

This paper develops R-matrix-valued Lax pairs for elliptic Calogero-Moser models, linking classical integrable systems with quantum R-matrices and Hitchin systems, revealing conditions for their existence and extensions.

Contribution

It constructs R-matrix-valued Lax pairs for Calogero-Moser models associated with classical root systems, generalizing D'Hoker-Phong pairs and exploring their quantum extensions.

Findings

Lax pairs exist only in special cases depending on coupling constants.

Some classical Lax pairs can be extended to the quantum case.

The models relate to Hitchin systems on SL_{N ilde N} bundles with nontrivial characteristic classes.

Abstract

The article is devoted to the study of $R$-matrix-valued Lax pairs for $N$-body (elliptic) Calogero-Moser models. Their matrix elements are given by quantum ${\rm GL}_{\tilde N}$ $R$-matrices of Baxter-Belavin type. For $\tilde N=1$ the widely known Krichever's Lax pair with spectral parameter is reproduced. First, we construct the $R$-matrix-valued Lax pairs for Calogero-Moser models associated with classical root systems. For this purpose we study generalizations of the D'Hoker-Phong Lax pairs. It appeared that in the $R$-matrix-valued case the Lax pairs exist in special cases only. The number of quantum spaces (on which $R$-matrices act) and their dimension depend on the values of coupling constants. Some of the obtained classical Lax pairs admit straightforward extension to the quantum case. In the end we describe a relationship of the $R$-matrix-valued Lax pairs to Hitchin systems defined on ${\rm SL}_{N\tilde N}$ bundles with nontrivial characteristic classes over elliptic curve. We show that the classical analogue of the anisotropic spin exchange operator entering the $R$-matrix-valued Lax equations is reproduced in these models.