Spinning $\sigma$-model solitons in $2+1$ Anti-de Sitter space

TL;DR

This paper numerically constructs rotating topological solitons in the nonlinear sigma-model within 2+1 Anti-de Sitter space, revealing solutions with and without naked singularities and multiple bifurcations.

Contribution

It introduces new numerical solutions for rotating sigma-model solitons in AdS space, including singularity-free and naked singularity solutions, with bifurcation phenomena.

Findings

Type i solutions have naked singularities despite bounded curvature scalars.

Type ii solutions are singularity-free and exhibit bifurcation phenomena.

No horizons are observed in any of the solutions.

Abstract

We obtain numerical solutions for rotating topological solitons of the nonlinear $\sigma$-model in three-dimensional Anti-de Sitter space. Two types of solutions, $i)$ and $ii)$, are found. The $\sigma$-model fields are everywhere well defined for both types of solutions, but they differ in their space-time domains. Any time slice of the space-time for the type $i)$ solution has a causal singularity, despite the fact that all scalars constructed the curvature tensor are bounded functions. No evidence of a horizon is seen for any of the solutions, and therefore the type $i)$ solutions have naked singularities. On the other hand, the space-time domain, along with the fields, for the type $ii)$ solutions are singularity free. Multiple families of solutions exhibiting bifurcation phenomena are found for this case.