Brief lectures on duality, integrability and deformations

TL;DR

This paper offers a pedagogical overview of integrability, dualities, and deformations in low-dimensional physical systems, highlighting the E-model framework and presenting new conditions for strong integrability.

Contribution

It introduces a comprehensive pedagogical review of dualities and integrability, and presents original results on conditions for strong integrability of E-models.

Findings

Analysis of T-duality and Ruijsenaars duality in physical models

Review of Lax integrability and Yang-Baxter deformations

New conditions for strong integrability of E-models

Abstract

We provide a pedagogical introduction to some aspects of integrability, dualities and deformations of physical systems in 0+1 and in 1+1 dimensions. In particular, we concentrate on the T-duality of point particles and strings as well as on the Ruijsenaars duality of finite many-body integrable models, we review the concept of the integrability and, in particular, of the Lax integrability and we analyze the basic examples of the Yang-Baxter deformations of non-linear sigma-models. The central mathematical structure which we describe in detail is the E-model which is the dynamical system exhibiting all those three phenomena simultaneously. The last part of the paper contains original results, in particular a formulation of sufficient conditions for strong integrability of non-degenerate E-models.