AFFINE TODA SOLITONS AND VERTEX OPERATORS

Affine Toda theories with imaginary couplings associate with any simpl e Lie algebra g generalisations of sine-Gordon theory which are likewi se integrable and possess soliton solutions. The solitons are ''create d'' by exponentials of quantities F(i)(z) which lie in the untwisted a ffine Kac-Moody algebra g and ad-diagonalise the principal Heisenberg subalgebra. When g is simply laced and highest-weight irreducible repr esentations at level one are considered, F(i)(z) can be expressed as a vertex operator whose square vanishes. This nilpotency property is ex tended to all highest-weight representations of all affine untwisted K ac-Moody algebras in the sense that the highest non-vanishing power be comes proportional to the level. As a consequence, the exponential ser ies mentioned terminates and the soliton solutions have a relatively s imple algebraic expression whose properties can be studied in a genera l way. This means that various physical properties of the soliton solu tions can be directly related to the algebraic structure. For example, a classical version of Dorey's fusing rule follows from the operator product expansion of two F's, at least when g is simply laced. This ad ds to the list of resemblances of the solitons with respect to the par ticles which are the quantum excitations of the fields.