Integrable deformations of principal chiral model from solutions of associative Yang-Baxter equation
D. Domanevsky, A. Levin, M. Olshanetsky, A. Zotov
TL;DR
The paper develops integrable deformations of the 1+1 principal chiral model and related Gaudin-type systems from solutions of the associative Yang-Baxter equation (AYBE) for ${ m GL}_N$. It builds a Lax framework via affine Higgs bundles, where the Lax operator $U(z)$ is constructed from AYBE-derived $r$- and $m$-matrices and residues $S^a$, ensuring a classical $r$-matrix structure and Zakharov-Shabat integrability. Two deformation channels are analyzed: (i) a twist function $k(z)$ acting as a cocentral charge that introduces additional poles and modifies the $r$-matrix data, and (ii) AYBE-based deformations that generate higher-rank Landau-Lifshitz, PCM, and Gaudin-type models with multi-point marked configurations. The work supplies explicit constructions for elliptic, trigonometric, and rational $R$-matrices, derives first and second flow equations for PCM and Gaudin models, and situates the formalism within the Hitchin/affine-Higgs bundle framework, including detailed appendices on the underlying geometric structure. These results unify ultralocal Lax descriptions with twist-induced deformations, expanding the toolkit for 2d integrable field theories and their finite-dimensional reductions.
Abstract
We describe deformations of the classical principle chiral model and 1+1 Gaudin model related to ${\rm GL}_N$ Lie group. 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.