Unitary models in two dimensions

Abstract

Unitary models in two dimensions are classes of low dimensional theories which provide us with a convenient theoretical laboratory for studying various aspects of the theory of elementary particles. In this thesis, purely bosonic U(N) sigma models with the Wess-Zuraino-Witten (WZW) term in two-dimensional Euclidean space and the supersymmetric (Susy) U(N) σ models with and without this term are discussed. Particular attention is paid to the classical solutions of the equations of motion of these models. Due to the integrabihty of these models, we can associate with them a Lax-pair formalism. We observe that solutions of the Lax-pair equations of the U(N) a model provide us with solutions of the U(N) a model with the WZW-term. This is also the case for solutions of the Susy U(N) a model with the WZW-term which can be constructed from solutions of the Lax-pair equations of the Susy U(N) σ model. We present also some explicit solutions of the Susy U(N) a model without the WZW-term. Many properties of the constructed solutions for both the purely bosonic and Susy models are explored. In particular, we calculate the values of the action for some solutions and study the stability properties of these solutions and find that all the constructed solutions of these models correspond to the saddle points of the action. Finally we consider the hnearized fermion equations in the fixed background of a bosonic field. Special attention is paid to the case when the background field is given by a solution of the U(N) σ model with and/or without the WZW-term. Some classes of solutions of this problem are presented and their properties are discussed. We observe that a class of these solutions is related to the components of the energy-momentum tensor of the purely bosonic σ model and prove that some of these solutions are traceless.