Gauge Equivalence Between 1 + 1 Rational Calogero–Moser Field Theory and Higher Rank Landau–Lifshitz Equation

In this paper we study 1 + 1 field generalization of the rational N-body Calogero–Moser model. We show that this model is gauge equivalent to some special higher rank matrix Landau–Lifshitz equation. The latter equation is described in terms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{G}}{{{\text{L}}}_{N}}$$\end{document} rational R-matrix, which turns into the 11-vertex R-matrix in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N = 2$$\end{document} case. The rational R-matrix satisfies the associative Yang–Baxter equation, which underlies construction of the Lax pair for the Zakharov–Shabat equation. The field analogue of the IRF-Vertex transformation is proposed. It allows to compute explicit change of variables between the field Calogero–Moser model and the Landau–Lifshitz equation.


Calogero-Moser field theory
The 1+1 field generalization1 of the Calogero-Moser model was proposed in [1,2], see also [3].The Hamiltonian is given by the following expression2 : where x is the (space) field variable.It is a coordinate on a unit circle.Dynamical variables are the (C-valued) fields p i = p i (x), q i = q i (x), i = 1, ..., N , and the lower index "x" means derivative with respect to x.For instance, q jxx = ∂ 2 x q j (x).The parameter c ∈ C is a coupling constant and k ∈ C is an auxiliary parameter, which can be fixed as k = 1 but we keep it as it is.The momenta p i and coordinates q j are canonically conjugated fields: Equations of motion (the Hamiltonian equations ḟ = {f, H}) take the following form: The model (1.1) is integrable in the sense that it has algebro-geometric solutions and equations of motion are represented in the Zakharov-Shabat (or Lax or zero curvature) form: where U -V pair is a pair U 2dCM (z), V 2dCM (z) of matrix valued functions of the fields p j (x), q j (x), j = 1, ..., N and their derivatives.They also depend on the spectral parameter z, and (1.5) holds true identically in z (on-shell equations of motion).Explicit expression for U -V pair is as follows: where and In what follows we assume the center of mass frame : Notice that in our previous paper on this topic [4] we used slightly different normalization coefficients and the gauge choice for U -V pair, which was more convenient for the case N = 2 when q 1 = −q 2 .
Limit to 0+1 mechanics.The finite-dimensional classical mechanics appears in the limit k → 0. All the fields become independent of x, and the field Poisson brackets turn into the ordinary Poisson brackets for mechanical N -body system: The Hamiltonian density (1.1) in this limit provides the ordinary Calogero-Moser model [5,6]: where | k=0 in the l.h.s.assumes also transition to x-independent variables.Similarly, the Zakharov-Shabat equation (1.5) reduces to the Lax equation: (1.13) Purpose of the paper.The 1+1 field generalizations under consideration are widely known for the Toda chains [7].For the relativistic models of Ruijsenaars-Schneider type the field generalizations were proposed recently in [8].In [3] the results of [1,2] were extended to (multi)spin generalizations of the Calogero-Moser model.It was also explained (using modification of bundles and the symplectic Hecke correspondence) that the field Calogero-Moser system should be gauge equivalent to some model of Landau-Lifshitz type.That is, there exist a gauge transformation G(z) ∈ Mat(N, C), which transforms U -V pair for the field Calogero-Moser model to the one for some Landau-Lifshitz type model: (1.14) For the N = 2 case explicit construction of the matrix G(z) and the change of variables was derived in our paper [4], and the Landau-Lifshitz model for GL 2 rational R-matrix was derived in [9].The goal of this article is to define the gauge transformation in gl N case, describe the corresponding Landau-Lifshitz type model and find explicit change of variables using relation (1.14).

Rational top and Landau-Lifshitz equation
Rational integrable top.In order to explain what kind of Landau-Lifshitz model is expected in (1.14) we first consider its 0+1 mechanical analogue.The mechanical version of (1.14) is as follows: where L top (z) is the Lax matrix of some integrable top like model.It is the model, which was introduced in [10] and called the rational top.Equations of motion for top like models are of the form: where S is a matrix of dynamical variables (E ij is the standard matrix basis), c ∈ C is a constant and J(S) is some special linear map (see [10]).The Hamiltonian is quadratic, and the Poisson brackets are given by the Poisson-Lie structure on gl * N Lie coalgebra: 3) It was shown in [10] that in the special case rk(S) = 1 (and tr(S) = c) this model is gauge equivalent (2.1) to the rational Calogero-Moser model.Namely, it was proved by direct evaluation that the expression in the r.h.s. of (2.1) is represented in the form: where S ij = S ij (p 1 , ..., p N , q 1 , ..., q N , c), r 12 (z) is some classical non-dynamical r-matrix (satisfying the classical Yang-Baxter equation), 1 N is the identity N × N matrix and tr 2 means trace over the second tensor component in Mat(N, C) ⊗2 .The gauge equivalence means that the Hamiltonians H top (2.3) and H CM (1.12) coincide under a certain change of variables, which will be given below in (2.15).
Description through R-matrix.In [11] a construction of Lax pairs with spectral parameter was suggested based on (skew-symmetric and unitary) solution of the associative Yang-Baxter equation [12,13]: In fact, a skew-symmetric and unitary solution of (2.5) in the fundamental representation of GL N Lie group is a quantum R-matrix, i.e. it satisfies also the quantum Yang-Baxter equation R 12 R 13 R 23 = R 23 R 13 R 12 .Consider the classical limit expansion of such R-matrix: Then the Lax pair can be written as follows: It generates the Euler-Arnold equation (2.2) with Rational R-matrix.In this paper we will use the rational R-matrix calculated in [14].In the N = 2 case it reproduces the 11-vertex R-matrix found by I. Cherednik [15]: (2.9) For N > 2 all its properties, different possible forms and explicit expressions for the coefficients of expansions (2.6) and (2.18) can be found in [16].
Landau-Lifshitz equation.Recently the 1+1 field generalization of the Lax pair (2.7) to U -V pair was suggested in [18].In the field case the Poisson brackets (2.3) are replaced with {S ij (x), S kl (y)} = 1 N S il (x)δ kj − S kj (x)δ il δ(x − y) . (2.16) The construction of U -V pair is again based on R-matrix satisfying the associative Yang-Baxter equation (2.5).For this purpose the following relation is used (it can be deduced from (2.5)):