Bogomolnyi Bound with a Cosmological Constant

TL;DR

This paper constructs a Bogomolnyi-type bound for topological solitons in an O(3) sigma model coupled to gravity with a negative cosmological constant, revealing new solutions and conditions for vortex saturation.

Contribution

It introduces a Bogomolnyi bound in a gravitational setting with a cosmological constant and explores new self-dual solutions and their properties.

Findings

Self-dual solutions form G"{o}del-type universes.

Stationary solutions for a spinless particle do not violate causality.

Vortices saturate the bound only when the cosmological constant is zero.

Abstract

Bogomolnyi-type bound is constructed for the topological solitons in O(3) nonlinear $\sigma$ model coupled to gravity with a negative cosmological constant. Spacetimes made by self-dual solutions form a class of G\"{o}del-type universe. In the limit of a spinless massive point particle, the obtained stationary metric does not violate the causality and it is a new point particle solution different from the known static hyperboloid and black hole. We also showed that static Nielsen-Olesen vortices saturate Bogomolnyi-type bound only when the cosmological constant vanishes.