Integrable sigma models from affine Gaudin models

Abstract

In this thesis we describe recent results obtained in the area of integrable field theories. In particular, we present the construction of two new broad classes of integrable sigma models in the framework of affine Gaudin models. Firstly, we focus on integrable deformations of a class of theories defined on the direct product of N copies of a Lie group. More precisely, for N = 1 the corresponding models coincide with the Yang-Baxter or lambda-deformations of the principal chiral model, while for general N they consist of arbitrary combinations of these deformed models. We describe both the Hamiltonian and Lagrangian formulation of models with general N and give explicit expressions of their action and Lax connection. The second class of theories is defined on a coset of the direct product of N copies of a Lie group over some diagonal subgroup, generalising the well-known symmetric space sigma model corresponding to N = 1. Specifying the construction to the case of two copies of the group SU(2), we obtain a new three-parametric integrable sigma model on the manifold T^{1,1}. We comment on the connection of our results with the ones existing in the literature.