Understanding Solitons Algebraically

Abstract

We describe techniques we have developed to analyze the physics of topological solitons in a model-independent way. Our central result is to show that topological solitons in generic field theories exhibit Bogomol’nyi bounds and Bogomol’nyi equations. Our methods turn the derivation of these Bogomol’nyi relationships into algebraic calculations and do not depend on the particular equations of motion. We present a discussion of the O(3) nonlinear σ-model as an example of our techniques.

Supersymmetry and Bogomol'nyi equations in the Abelian Higgs model

“…In the next section, the reasons why condition (11) ensuring N = 2 supersymmetry is also needed in order to attain the Bogomol'nyi bound will be clear at the light of Hlousek-Spector approach [9]- [10].…”

“…Although the connection between vortex solutions and supersymmetry in the Abelian Higgs model was afterwards thoroughly studied [8], the reasons behind the overlap of these two apparently divorced matters (supersymmetry and topological bounds) were not investigated till very recently [9]- [10]. Much was understood in these last works on the connection by analysing several models but the case in which the coincidence was first stressed, i.e.…”

“…They could be interpreted as solitons (Nielsen-Olesen vortices [11]). In the approach of Hlousek-Spector [9]- [10], the identification of the topological charge for the vortex configuration (its magnetic flux) with the N = 2 supersymmetry central charge plays a central rôle. Since in the Abelian Higgs model the topological charge is defined in 2-dimensional space, we shall first work, taking advantage of axial symmetry, in (2+1) space-time dimensions so that charges are given by two-dimensional integrals.…”

“…Although the general analysis of refs. [9]- [10] does not apply to 2-dimensional models, we shall also study the Abelian Higgs model as a bidimensional model showing that again in this case the arguments connecting supersymmetry and Bogomol'nyi equations can be applied.…”

Solitons and instantons with(out) supersymmetry

“…This is the Bogomol'nyi bound. Second, this in turn implies that any theory, supersymmetric or not, with topological solitons exhibits Bogomol'nyi relationships [7] [9]. We review the idea briefly here.…”