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

This thesis is concerned with the possible sums of perturbation series in mass- less, renormalizable field theories. It shows that, given a free choice of scheme, the limit of the sequence of approximants is arbitrary. Restricting the choice to finite schemes, in particular "zero schemes", yields a perturbatively unique limit to the sequence of approximants. An operational method for calculating perturbative expansions in the class of zero schemes is discussed. A comparison of various optimization schemes is given for a few phenomenological examples in QCD and QED.

Thesis (Ph.D.)--University of Delaware, 2007. / Principal faculty advisor: Qaisar Shafi, Dept. of Physics & Astronomy. Includes bibliographical references.

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Thesis (Ph. D.)--Northeastern University, 2009. / Title from title page (viewed June 22, 2009). Graduate School of Arts and Sciences, Dept. of Physics. Includes bibliographical references (p. 163-178).

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In this thesis, we extend the numerical S-matrix bootstrap program to 1+1d theories with a boundary, where we bootstrap the 1-to-1 reflection matrix (R-matrix). We review the constraints that a physical R-matrix must obey, namely unitarity, analyticiy and crossing symmetry. We then carve out the allowed space of 2d R-matrices with the O(N) nonlinear sigma model and the periodic Yang Baxter solution in the bulk. We find a variety of integrable R-matrices along the boundary of the allowed space both with and without free parameters. The integrable models without a free parameter appear at vertices of the allowed space, while those with a free parameter occupy the whole boundary. We also introduce the extended analyticity constraint where we increase the domain of analyticity beyond the physical region. In some cases, the allowed space of R-matrices shrinks drastically and we observe new vertices which correspond to integrable theories. We also find a new integrable R-matrix through our numerics, which we later obtained by solving the boundary Yang--Baxter equation. Finally, we derive the dual to the extended analyticity problem and find that the formalism allows for R-matrices which do not saturate unitarity to lie on the boundary of the allowed region.

By formulating the six dimensional (0,2) superconformal field theory X[j] on a Riemann surface decorated with certain codimension two defects, a multitude of four dimensional N=2 supersymmetric field theories can be constructed. In this dissertation, various aspects of this construction are investigated in detail for j=A,D,E. This includes, in particular, an exposition of the various partial descriptions of the codimension two defects that become available under dimensional reductions and the relationships between them. Also investigated is a particular observable of this class of four dimensional theories, namely the partition function on the four sphere and its relationship to correlation functions in a class of two dimensional non-rational conformal field theories called Toda theories. It is pointed out that the scale factor that captures the Euler anomaly of the four dimensional theory has an interpretation in the two dimensional language, thereby adding one of the basic observables of the 4d theory to the 4d/2d dictionary. / text

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