The Topological Charges of the $a_n^{(1)}$ Affine Toda Solitons

TL;DR

This paper analyzes the topological charges of affine Toda solitons, providing formulas for their number and structure, and explores their invariance properties and behavior in multisoliton configurations.

Contribution

It introduces a general formula for the topological charges of {a_n^{(1)}} affine Toda solitons and characterizes their representation-theoretic properties.

Findings

Charges lie in fundamental representations, not necessarily filled

Charges are invariant under Coxeter group actions

Multisolitons fill remaining weights in the representation

Abstract

The topological charges of the \an affine Toda solitons are considered. A general formula is presented for the number of charges associated with each soliton, as well as an expression for the charges themselves. For each soliton the charges are found to lie in the corresponding fundamental representation, though in general these representations are not filled. Each soliton's topological charges are invariant under cyclic permutations of the simple roots plus the extended root or equivalently, under the action of the Coxeter element (with a particular ordering). Multisolitons are considered and are found to have topological charges filling the remainder of the fundamental representations as well as the entire weight lattice. The article concludes with a discussion of some of the other affine Toda theories.