On integrability of the Yang-Baxter $\si$-model

TL;DR

The paper proves the integrability of the Yang-Baxter σ-model, a Poisson-Lie deformation of the principal chiral model, and provides an explicit Lax pair. It constructs a one-to-one map between ordinary PCM solutions and Yang-Baxter model solutions via extended solutions and Iwasawa decomposition, enabling direct transfer of PCM solution methods such as dressing to the deformed context. The work clarifies how the spectral parameter can be interpreted as the Poisson-Lie deformation parameter and extends these ideas to a bi-Yang-Baxter model, laying groundwork for Poisson-Lie T-duality and further integrability analyses. Overall, it integrates deformation theory with classical integrability to expand the toolbox for generating and relating solutions across deformed sigma-models.

Abstract

We prove the integrability of the Yang-Baxter $\si$-model which is the Poisson-Lie deformation of the principal chiral model. We find also an explicit one-to-one map transforming every solution of the principal chiral model into a solution of the deformed model. With the help of this map, the standard procedure of the dressing of the principal chiral solutions can be directly transferred into the deformed Yang-Baxter context.