Affine Toda solitons and vertex operators

Abstract

Affine Toda theories with imaginary couplings associate with any simple Lie Algebra g generalisations of sine-Gordon theory which are likewise integrable and posses soliton solutions. The solitons are “created” by exponentials of quantities F ̂ i(z) which lie in the untwisted affine Kac-Moody algebra ĝ and ad-diagonalise the principal Heisenberg subalgebra. When g is simply laced and highest-weight irreducible representations at level one are considered, 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 F ̂ ' s , at least when 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.