Title:

Exact Smatrices for quantum affine Toda solitons and their bound states

The primary aim of this dissertation is to construct and study exact Smatrices for (1+1)dimensional integrable field theories with quantum affine symmetry. We introduce a general scheme of how to use trigonometric Rmatrices for the construction of exact Smatrices. This scheme is then applied to a specific class of relativistic field theories defined in (1+1)dimensional Minkowski space, the socalled affine Toda field theories with imaginary coupling constant. The most important feature of these theories is the fact that their classical equations of motion admit soliton solutions. As a step towards the consistent quantisation of these theories we attempt to construct Smatrices for the quantum scattering of affine Toda solitons. Apart from the solitons there are also bound states of solitons in affine Toda field theories. By using the bootstrap principle we derive the Smatrices for the scattering of bound states. We focus in particular on the scalar bound states which are the analogues of the breathers in SineGordon theory, and show that the Smatrices for the lowest breathers in the theory are identical to the Smatrices for the fundamental quantum particles. We also provide evidence for the consistency of the conjectured Smatrices through a detailed examination of their pole structure. We find that a large number of poles can be explained in terms of higher order diagrams, many of which involve a generalised ColemanThun mechanism. The layout of this thesis is as follows. In the introduction the axioms of analytic Smatrix theory are reviewed and some of the main features of affine Toda field theories are introduced. In chapter 2 we provide an introduction to the theory of quantised universal enveloping algebras and their Rmatrices, where there are trigonometric solutions of the YangBaxter equation. We also describe the main features of quantum affine symmetries in twodimensional field theories. Chapters 3, 4 and 5 deal with the detailed discussion of soliton Smatrices and bound states in a_{n}^{(1)}, d_{n}^{(2)}_{+}_{1, }b_{n}^{(1) }and a_{2n}^{(2) }affine Toda field theories. Special attention is given to the two cases of a_{2}^{(1) }and d_{3}^{(2)}, and the pole structures of their proposed Smatrices are examined in great detail. We also provide a conjecture of the complete quantum spectrum of these theories. The dissertation concludes with a summary of results and some remarks about open questions and unsolved problems. In the appendices we provide detailed proofs and calculations omitted from the main part of the thesis. We also attempt to construct integral representations of Smatrix scalar factors and give tables containing relevant data for affine Lie algebras.
