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

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 radial Bogomol’nyi framework is developed for the symmetric restriction of the theory. A weak formalism is developed for the second-order radial equations, capable of accommodating the inclusion of weaker singularities introduced by the metric or generalized functions. Next, localized topological solutions are explicitly found. In canonical models, stability of static solitonic solutions is precluded by scaling arguments, but these concerns can be circumvented with the use of a potential with explicit radial dependence. The stability equations are also derived for general perturbations depending on all coordinates, and their properties are briefly discussed under appropriate regularity and self-adjointness assumptions. 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 $$\xi (r).$$ ξ ( 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.