Topological solitons of two-field scalar theories in rotationally symmetric backgrounds

TL;DR

This paper develops a framework for analyzing topological solitons in scalar field theories with nontrivial vacua across various symmetric backgrounds, providing exact solutions and insights into their stability and structure.

Contribution

It introduces a Bogomol'nyi framework for symmetric scalar theories, derives integrable orbit equations, and maps solutions to a one-dimensional BPS theory, enabling complete analysis of solitons in diverse geometries.

Findings

Exact solutions in multiple backgrounds including Minkowski and Schwarzschild.

Stability of solitons achieved via explicit radial potential dependence.

Target space orbits are shared across different backgrounds, but solutions are not.

Abstract

This work concerns scalar field theories with topologically nontrivial vacuum manifold in rotationally symmetric backgrounds of arbitrary dimension. Lagrangians with canonical and generalized kinetic terms are considered, and a Bogomol'nyi framework is developed for the symmetric restriction of the theory. Localized topological solutions are found. Their stability, which would normally be prevented in higher dimensions due to scaling instability, is made possible by the presence of an explicit radial dependence on the potential. The first-order equations give rise to an integrable orbit equation which can be used to solve the problem completely. It is shown that target space orbits - but not the solutions themselves - are shared between analogous systems defined in different backgrounds. Moreover, the first-order equations can be mapped into a one-dimensional BPS theory through a transformation encoded by a function $\xi(r)$. The internal structure, size and existence of defects follows from the properties and range of this mapping. We use these tools to evaluate the effect of geometry on confinement, existence, and structure of solitons. Exact solutions are provided in Minkowski, Schwarzschild, de Sitter, Schwarzschild de Sitter and conformally flat backgrounds.