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

Abstract

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 $${\text{G}}{{{\text{L}}}_{N}}$$ rational R-matrix, which turns into the 11-vertex R-matrix in the $$N = 2$$ 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.