Affine Toda Solitons and Vertex Operators

TL;DR

This paper explores the algebraic structure of soliton solutions in affine Toda theories, revealing their connection to vertex operators and Kac-Moody algebras, and deriving properties like nilpotency and fusion rules.

Contribution

It extends the nilpotency property of vertex operators to all affine Kac-Moody algebras, simplifying soliton expressions and linking their properties to algebraic structures.

Findings

Solitons are generated by exponentials of operators in affine Kac-Moody algebras.

Vertex operators exhibit nilpotency proportional to the level in all highest weight representations.

Classical Dorey's fusing rule is derived from operator product expansions.

Abstract

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.