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 nonlinear σ-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.

中文翻译：

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关于对偶性、可积性和变形的简短讲座

我们对物理系统的可积性、对偶性和变形的某些方面进行了教学介绍0 + 1并且在1 + 1方面。特别是，我们专注于点粒子和弦的 T 对偶以及有限多体可积模型的 Ruijsenaars 对偶，我们回顾了可积性的概念，特别是 Lax 可积性的概念，并分析了非线性的杨-巴克斯特变形的基本例子σ-楷模。我们详细描述的中心数学结构是ℰ-模型是同时表现出所有这三种现象的动力系统。论文的最后部分包含原始结果，特别是对非退化强可积性的充分条件的表述ℰ-楷模。