On the topological charges of the affine Toda solitons
This thesis investigates the two dimensional, integrable field theories known as the affine Toda field theories, which are based on the Kac-Moody algebras with zero central extension. In particular, the construction of static solitons in these theories and their topological charges are considered. Following a general overview of the affine Toda theories and the Kac-Moody structure which underlies them, the construction of solitons in the a(_n)((^1)) theory using Hirota's method, originally used by HoUowood, is generalized and extended to the remaining theories. The soliton masses are calculated and general expressions presented for the twisted as well as the untwisted theories. The major results of this work concern the calculation of topological charge, one of the infinite number of conserved quantities that each theory possesses. Firstly, the a(_n)((^1)) model is considered. An expression for the number of charges associated with each soliton, as well as a general expression for the charges themselves, is constructed. The previously alluded to connection between the charges and the associated fundamental representations is proven showing that the charges are, in general, a subset of the weights lying in these representations. For the a(_n)((^1)) theory, the charges associated with each soliton can be derived from just one by making use of the cyclic symmetry of the model's extended Dynkin diagram. Further, the action of this symmetry on the set of charges is synonymous with the action of a Coxeter element. It is found that the ordering of the Weyl reflections which make up this element is important (except when the end-point solitons are considered) - the familiar "black-white" ordering doesn't work. The multisolitons of the theory are considered and it is shown that when the individual solitons are sufficiently well separated their topological charges simply add together. Multi-solitons can be constructed having topological charge equal to each of the simple roots, and can therefore be used to construct further solitons filling the entire weight lattice. Next, the topological charges of the remaining aflttne Toda theories are investigated. For the infinite series of algebras the number of topological charges and expressions for the charges themselves are derived. For the remaining cases, the charges are calculated explicitly. This thesis concludes with some comments on more recent work into the theory of quantum solitons and considers further lines of enquiry.