Integrable deformations of principal chiral model from solutions of associative Yang–Baxter equation

D. A. Domanevsky^a, A. M. Levin^bc, M. A. Olshanetsky^cd, A. V. Zotov<sup>ec

a</sup> Institute for Theoretical and Mathematical Physics of Lomonosov Moscow State University
^b National Research University Higher School of Economics, Moscow
^c National Research Centre "Kurchatov Institute", Moscow
^d Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
^e Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: We describe deformations of the classical principal chiral model and the $(1+1)$-dimensional Gaudin model related to the Lie group $\mathrm{GL}_N$. The deformations are generated by $R$-matrices satisfying the associative Yang–Baxter equation. Using the coefficients of the expansion for these $R$-matrices we derive equations of motion based on a certain ansatz for $U$–$V$ pair satisfying the Zakharov–Shabat equation. Another deformation comes from the twist function, which we identify with the cocentral charge in the affine Higgs bundle underlying the Hitchin approach to $2d$ integrable models.

Keywords: integrable field theory, Hitchin system, soliton equation.

MSC: Primary 14H70, 37K10, 81R12; Secondary 37K10, 81R12

Received: 16.01.2025
Revised: 03.03.2025

DOI: 10.4213/im9717

 English version: Izvestiya: Mathematics, 2026, 90:1, 109–143