THE TOPOLOGICAL CHARGES OF THE $a_n^{(1)}$ AFFINE TODA SOLITONS

Abstract

The topological charges of the [Formula: see text] 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. Bach 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.

Inverse scattering and solitons in An − 1 affine Toda field theories II

“…they have the same time delays [4] when they scatter with other solitons. However not all the missing charges are found by this construction, in particular no new charges are found for A 3 .…”

“…We now discuss the actual values of the topological charges which have been obtained. We note that for the given value of Q ′ 3 , and Q 1 =Q 1 = Q 5 =Q 5 = 0, we have the topological charge [3], where α i are the simple roots:…”

The R-matrix of the Uq(d4(3)) algebra and g2(1) affine Toda field theory

“…In particular, it would be interesting to see more evidence for the validity of the approach based on the quantum symmetry algebra. It is not obvious that this approach is correct and the main issue is that classically the soliton solutions fail to fill up the affine algebra multiplets completely [26], while the quantum symmetry approach takes this for granted at the quantum level. It would also be interesting to see how a unitary restriction can emerge from a theory which is strongly nonunitary, taking into account especially the instability of classical solitonic solutions [6] and the strong unitarity violation at the quantum level that may persist even after an RSOS restriction [27].…”

Integrability of generalised type II defects in affine Toda field theory

“…These were first shown to have zero curvature (and so an infinite number of conserved quantities) [17,18] and later shown to be integrable [19,20] using the method of the Lax pair and r-matrix. All ATFTs have solitons as solutions [21][22][23]. As well as being integrable solitons (stable by virtue of a cancellation of nonlinear and dispersive forces) these solitons are also topological (stable due to possessing some topological charge, in this case the difference between the field as x → ±∞).…”