Classical solutions of sigma models in (2+1) dimensions

Abstract

This work is concerned with the large class of nonlinear scalar field theories known as σ-models, and in particular with their classical solutions. It is shown how the σ-models can admit solitons in (2+1) dimensions; and how, in many cases, these solitons can be classified topologically. For the Kähler c-models, the instanton (i.e. static soliton) solutions are derived explicitly via the Bogomolny equations. The main part of the thesis looks at the behaviour of solitons under the influence of small perturbations, and at their (classical) interactions. Attention is confined to the O(3) a-model and its close relatives. A recurring theme is the ability of solitons to change in size as they evolve, a feature which is attributed to the conformal invariance of the theory. There seem to be three possible approaches. In some special cases, the theory is integrable, in the sense that one can write down explicit time-dependent solutions. More often, one must resort to a numerical simulation, or else some sort of approximation. For theories that possess a topological lower bound on the energy, there is a useful approximation in which the kinetic energy is assumed to remain small. All three of these approaches are used at various stages of the thesis. Chapter IIIdeals with the properties of wave-like solitons in an integrable theory, and reveals some hitherto unseen behaviour. Chapters IV and V develop a numerical simulation based on topological arguments, which is then used in a study of soliton stability in the pure O(3) model. The conclusion is that the solitons are unstable to small perturbations, in the sense that their size is subject to large changes, even though their energy remains roughly constant. Chapter VI uses the slow-motion approximation to investigate soliton interactions in the O(3) model, and uncovers a plethora of possibilities. Finally, some suggestions are made regarding possible directions for future research. In particular, attention is focussed on ways of modifying the O(3) model in an attemptto stabilize its solitons against changes in size