Topology and Energy of Time Dependent Unitons

TL;DR

This paper studies time-dependent multi-soliton solutions in a 2+1 dimensional integrable chiral model, revealing a topological quantization of their energy linked to the third homotopy class of extended solutions.

Contribution

It introduces a topological framework explaining classical energy quantization of moving solitons in the U(N) integrable chiral model.

Findings

Total energy is quantized and determined by the third homotopy class.

Extended solutions have a pole with arbitrary multiplicity in the spectral parameter.

First example of a topological mechanism for classical energy quantization.

Abstract

We consider a class of time dependent finite energy multi-soliton solutions of the U(N) integrable chiral model in $(2+1)$ dimensions. The corresponding extended solutions of the associated linear problem have a pole with arbitrary multiplicity in the complex plane of the spectral parameter. Restrictions of these extended solutions to any spacelike plane in $\R^{2,1}$ have trivial monodromy and give rise to maps from a three sphere to U(N). We demonstrate that the total energy of each multi-soliton is quantised at the classical level and given by the third homotopy class of the extended solution. This is the first example of a topological mechanism explaining classical energy quantisation of moving solitons.