Gauge equivalence of \(1+1\) Calogero–Moser–Sutherland field theory and a higher-rank trigonometric Landau–Lifshitz model

We consider the classical integrable \((1+1)\) trigonometric \(gl_N\) Landau–Lifshitz models constructed by means of quantum \(R\) -matrices that also satisfy the associative Yang–Baxter equation. It is shown that a \((1+1)\) field analogue of the trigonometric Calogero–Moser–Sutherland model is gauge equivalent to the Landau–Lifshitz model that arises from the Antonov–Hasegawa–Zabrodin trigonometric nonstandard \(R\) -matrix. The latter generalizes Cherednik’s \(7\) -vertex \(R\) -matrix in the \(GL_2\) case to the case of \(GL_N\) . An explicit change of variables between the \((1+1)\) models is obtained.