Inverse scattering and solitons in An−1A_{n-1} affine Toda field theories II

New single soliton solutions to the affine Toda field theories are constructed, exhibiting previously unobserved topological charges. This goes some of the way in filling the weights of the fundamental representations, but nevertheless holes in the representations remain. We use the group doublecross product form of the inverse scattering method, and restrict ourselves to the rank one solutions.Comment: 19 pages, latex, 12 fig

Gravity induced from quantum spacetime

We show that tensoriality constraints in noncommutative Riemannian geometry in the 2-dimensional bicrossproduct model quantum spacetime algebra [x,t]=\lambda x drastically reduce the moduli of possible metrics g up to normalisation to a single real parameter which we interpret as a time in the past from which all timelike geodesics emerge and a corresponding time in the future at which they all converge. Our analysis also implies a reduction of moduli in n-dimensions and we study the suggested spherically symmetric classical geometry in n=4 in detail, identifying two 1-parameter subcases where the Einstein tensor matches that of a perfect fluid for (a) positive pressure, zero density and (b) negative pressure and positive density with ratio w_Q=-{1\over 2}. The classical geometry is conformally flat and its geodesics motivate new coordinates which we extend to the quantum case as a new description of the quantum spacetime model as a quadratic algebra. The noncommutative Riemannian geometry is fully solved for $n=2$ and includes the quantum Levi-Civita connection and a second, nonperturbative, Levi-Civita connection which blows up as \lambda\to 0. We also propose a `quantum Einstein tensor' which is identically zero for the main part of the moduli space of connections (as classically in 2D). However, when the quantum Ricci tensor and metric are viewed as deformations of their classical counterparts there would be an O(\lambda^2) correction to the classical Einstein tensor and an O(\lambda) correction to the classical metric.Comment: 42 pages LATEX, 4 figures; expanded on the physical significanc

Inverse scattering and solitons in An−1A_{n-1} affine Toda field theories

We implement the inverse scattering method in the case of the $A_n$ affine Toda field theories, by studying the space-time evolution of simple poles in the underlying loop group. We find the known single soliton solutions, as well as additional solutions with non-linear modes of oscillation around the standard solution, by studying the particularly simple case where the residue at the pole is a rank one projection. We show that these solutions with extra modes have the same mass and topological charges as the standard solutions, so we do not shed any light on the missing topological charge problem in these models. We also show that the integrated energy-momentum density can be calculated from the central extension of the loop group.Comment: 28 pages, Latex, 4 figs include