On 2D and 3D Localized Solutions with Nontrivial Topology

Localized solutions of nonlinear field models with nontrivial topological properties are discussed. Existence of various systems of definitions of the topological objects, developed in this area of research historically, can potentially lead to the wrong conclusions about existence of such solutions. The classification allowing to define accurately and differentiate objects with different topological properties is proposed, which prevents from inferring wrong conclusions. Such classification is especially important for multidimensional solutions. Such solutions are divided into 2 classes: the topological solitons (TS) and topological defects (TD). Solutions of both types describe the localized distributions of field energy, but they differ in topological properties. We exemplify and compare stationary TSs and TDs in 2 and 3 spatial dimensions. Examples of TSs are: solitons in Heisenberg magnets, Belavin-Polyakov solitons/instantons, Skyrmions, “baby-skyrmions”. Examples of TDs are: sine-Gordon kinks, Nielsen-Olesen strings-vortices in the Abelian Higgs (AHM) model, ’t Hooft-Polyakov hedgehog-monopoles in the Georgi-Glashow model. We note some technical problems with TDs, which are not met in the case of TSs. Soliton analogs of Nielsen-Olesen TDs in the AHM have been found: they are TSs in the A3M model. We have started search for TSs in the SU(2)-Higgs model which is currently in progress.

topological charge, solitons, defects, mapping degree, Abelian-Higgs model, Yang-Mills field, Heisenberg antiferromagnet, Georgi-Glashow and Weinberg-Salam models, hedgehog ansatz