Affine Toda Solitons and Vertex Operators

Affine Toda theories with imaginary couplings associate with any simple Lie algebra ${\bf g}$ generalisations of Sine Gordon theory which are likewise integrable and possess soliton solutions. The solitons are \lq\lq created" by exponentials of quantities $\hat F^i(z)$ which lie in the untwisted affine Kac-Moody algebra ${\bf\hat g}$ and ad-diagonalise the principal Heisenberg subalgebra. When ${\bf g}$ is simply-laced and highest weight irreducible representations at level one are considered, $\hat F^i(z)$ can be expressed as a vertex operator whose square vanishes. This nilpotency property is extended to all highest weight representations of all affine untwisted Kac-Moody algebras in the sense that the highest non vanishing power becomes proportional to the level. As a consequence, the exponential series mentioned terminates and the soliton solutions have a relatively simple algebraic expression whose properties can be studied in a general way. This means that various physical properties of the soliton solutions 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 $\hat F$'s, at least when ${\bf g}$ is simply laced. This adds to the list of resemblances of the solitons with respect to the particles which are the quantum excitations of the fields.