Lectures on classical Affine Gaudin models

TL;DR

This paper introduces classical Affine Gaudin models, a framework for constructing and analyzing integrable 2D field theories, emphasizing their symmetries, conserved charges, and connections to sigma-models.

Contribution

It provides a detailed Hamiltonian formulation of Affine Gaudin models, including the construction of conserved charges and explicit examples linking to integrable sigma-models.

Findings

Construction of infinite families of conserved charges

Explicit examples connecting to integrable sigma-models

Discussion on potential quantisation approaches

Abstract

These lecture notes present an introduction to classical Affine Gaudin models, which provide a general framework for the systematic construction and study of a large class of integrable two-dimensional field theories. A key role is played by Kac-Moody currents, which are fields satisfying a particular Poisson bracket. After reviewing this notion, we discuss in detail the construction of Affine Gaudin models in the language of Hamiltonian field theories. Special emphasis is placed on their symmetries and conserved quantities, including the construction of infinite families of local and non-local Poisson-commuting charges in terms of Kac-Moody currents. Moreover, we study explicit examples of affine Gaudin models, making the link with the realm of integrable sigma-models. Finally, we mention briefly various perspectives concerning these theories, including the question of their quantisation. Minimal prerequisites on classical Hamiltonian field theories and integrability are required to follow the presentation and a brief reminder of these notions is given at the beginning of the notes. Moreover, various exercises are included throughout the document. These notes were prepared for the Young Researchers Integrability School and Workshop held in Durham from 17 to 21 July 2023.