Solitons and Instantons with(out) Supersymmetry

TL;DR

This paper presents general, model-independent arguments showing that topological solitons and instantons satisfy bounds and self-duality equations, and explores the role of supersymmetry and central charges in various dimensions.

Contribution

It introduces a general framework for understanding topological charges and supersymmetry, including the concept of associated superfields for constructing supersymmetric extensions.

Findings

Topological solitons satisfy Bogomol'nyi bounds and self-duality equations.

In supersymmetric theories, topological charges are central charges in extended supersymmetry.

The approach is valid in nearly any spacetime dimension.

Abstract

We give model-independent arguments, valid in nearly any number of spacetime dimensions, that topological solitons and instantons satisfy Bogomol'nyi-type bounds and, when these bounds are saturated, satisfy self-duality equations. In the supersymmetric case, we also show that, in spacetime dimensions greater than two, theories with topological charges necessarily exhibit extended supersymmetry, in which the topological charge appears as the central charge. The significance of our arguments lies in their generality. In the supersymmetric case, we obtain insight into the contrast observed between topological charges in 1+1 and higher dimensional models. The centerpiece of our method is to require that the supersymmetric extension of a generic (non-supersymmetric) field theory be self-consistent. Our discussion of supersymmetric extensions is quite detailed, and introduces the notion of the "associated superfield" to construct such extensions.