Brief lectures on duality, integrability and deformations

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.

43 pages, the origin (URL) of the pictures is specified

References in corpus (7)

• On deformations of AdS_n x S^n supercosets
• Abelian Yang-Baxter Deformations and TsT transformations
• Yang-Baxter $\sigma$-model with WZNW term as $\mathcal{E}$-model
• On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models
• Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one integrable sigma-model
• On integrability of 2-dimensional $\sigma$-models of Poisson-Lie type
• T-duality and T-folds for point particles

Cited by in corpus (14)

• Integrable Deformations of Sigma Models
• Infinite Family of Integrable Sigma Models Using Auxiliary Fields
• Integrable degenerate $\mathcal{E}$-models from 4d Chern-Simons theory
• Homogeneous Yang-Baxter deformations as undeformed yet twisted models
• On strong integrability of the dressing cosets
• Deformed WZW Models and Hodge Theory -- Part I
• Improvement of a conserved current density versus adding a total derivative to a Lagrangian density
• Integrable sigma models on Riemann surfaces
• On a class of conformal $\mathcal{E}$-models and their chiral Poisson algebras
• Poisson-Lie T-duality defects and target space fusion
• Yang-Baxter structure of the extended space
• Constraining all possible Korteweg-de Vries type hierarchies
• Novel aspects of integrability for NLSMs in symmetric spaces
• Beyond Noether: A Covariant Study of Poisson-Lie Symmetries in Low Dimensional Field Theory