Rings and Balls

TL;DR

This paper investigates the topology of linked solitons in nonlinear sigma models, revealing topological terms that quantify linking numbers in different dimensions and constructing explicit models for these phenomena.

Contribution

It introduces topological terms for linking numbers of solitons in 2+1 and 3+1 dimensions and generalizes them beyond strict non-overlap conditions.

Findings

Existence of topological linking terms in 2+1 and 3+1 dimensions.

Explicit construction of linking terms for specific models.

Generalization of linking terms to overlapping soliton configurations.

Abstract

We examine the various linkings in space-time of ``ball-like'' and ``ring-like'' topological solitons in certain nonlinear sigma models in 2+1 and 3+1 dimensions. By going to theories where soliton overlaps are forbidden, these linkings become homotopically nontrivial and can be studied by investigating the topology of the corresponding configuration spaces. Our analysis reveals the existence of topological terms which give the linking number of the world-tubes of distinct species of ball solitons in 2+1 dimensions, or which in 3+1 dimensions count the number of times a ball or ring soliton threads through the center of a ring of a different species. We explicitly construct these terms for our models, and generalize them to cases where soliton overlaps are no longer strictly forbidden so the terms are no longer purely topological. One of the (3+1)-dimensional theories we consider also has topological solitons which consist of two rings (of distinct species) linked in space.