The purpose of this thesis is to examine soliton-like solutions of various 2 + 1 dimensional models, including the Ginzburg-Landau theory of superconductivity [1]. Among the models studied are also some with a Chern-Simons term, which may play a role in high temperature superconductivity [2], Our study is relevant to planar physics and phenomena with cylindrical symmetry. Different models are studied to find generic features and pinpoint the differences. The most important property of the models we study is that there are extended structures in each of them. The models share many other properties which are investigated in this thesis.
The study of extended objects in gauge theories goes back to t ’Hooft [3] and Polyakov [4] and their work on magnetic monopoles. Since then numerous papers have been written on static extended objects in gauge theories. In recent years the dynamics of soliton-like objects in (2 + 1) and (3 + 1) dimensions has also been studied. The analytic studies were mainly based on the geodesic approximation [5], whose validity has recently been proved for vortices [6] and monopoles [7], and were mainly concerned with the scattering of two objects. There are, however, some interesting analytic results for the scattering of more than two extended objects, (see for example Refs [8] [9] [10] ). Our work will add to this body of knowledge.
In Chapters two to four various time independent solutions are exhibited and their interaction energies studied as a function of the relative strength of matter self coupling and electromagnetic coupling. The result is that the vortices have phases of attraction and repulsion. There is no interaction between the vortices when the topological lower bound on the energy is saturated. Then vortices satisfy the corresponding first order partial differential
equations. It will be shown that for the critical value of the coupling constant in the potential a lower bound on the energy of an n—vortex configuration can be attained. Since the vortex number is a topological invariant, any configuration which achieves this bound will be a minimum of the energy and thus a solution of the Euler-Lagrange equations.
Several models will be studied analytically and numerically in this way. In particular, in Chapter 3 the gauged 0(3) sigma model with Maxwell term and in Chapter 4 the gauged 0(3) sigma model with Chern-Simons term are studied. In Chapters 5 and 6 the Abelian Chern-Simons-Higgs vortices and the Abelian Yang-Mills-Higgs vortices are studied respectively. To find the static solutions a similar approach is taken, thus highlighting the generic features of the models.
The scattering properties of the vortices in the Abelian Yang-Mills-Higgs model are examined in the second half of this thesis. Given suitable initial data we prove that a global time dependent solution exists for the Abelian Higgs model. We show, using a formal method of solution to the equations, that symmetry properties of the initial data are retained by the solution. Considering transformations of the initial data which leave the energy density invariant, it is shown that the vortices scatter at a tt/ n angle.
When the vortices attain their minimal energy, the vortices do not interact and the diagonal components of the stress tensor vanish. Then the vortices can be placed at arbitrary positions in the plane. We exploit this fact to set up an initial value problem with initial boosts which does not cost any potential energy. The scattering properties can be examined for these processes.
Our aim is to study generic features of a whole range of vortex models, and to demonstrate how to progress from the statics to the dynamics of vortices in one of the models.