Hybrid classical integrable structure of squashed sigma models -- a short summary

TL;DR

This paper explores the classical integrable structure of squashed sigma models, demonstrating two equivalent descriptions—rational and trigonometric—via different Lax pairs and r/s-matrices, extending duality concepts.

Contribution

It introduces two distinct integrable descriptions of squashed sigma models and establishes their equivalence through a non-local map, extending the understanding of dualities in integrable systems.

Findings

Constructed rational and trigonometric Lax pairs.

Showed classical integrability via r/s-matrices satisfying Yang-Baxter.

Established non-local equivalence between the two descriptions.

Abstract

We give a short summary of our recent works on the classical integrable structure of two-dimensional non-linear sigma models defined on squashed three-dimensional spheres. There are two descriptions to describe the classical dynamics, 1) the rational description and 2) the trigonometric description. It is possible to construct two different types of Lax pairs depending on the descriptions, and the classical integrability is shown by computing classical r/s-matrices satisfying the extended Yang-Baxter equation in both descriptions. In the former the system is described as an integrable system of rational type. On the other hand, in the latter it is described as trigonometric type. There exists a non-local map between the two descriptions and those are equivalent. This is a non-local generalization of the left-right duality in principal chiral models.