Existence of topological hairy dyons and dyonic black holes in anti-de Sitter SU(N) Einstein-Yang-Mills theory

TL;DR

This paper proves the existence of non-trivial dyonic black hole and soliton solutions in four-dimensional SU(N) Einstein-Yang-Mills theory with a negative cosmological constant, expanding understanding of such solutions in anti-de Sitter space.

Contribution

It demonstrates the existence of global dyonic solutions with non-zero magnetic gauge fields in SU(N) Einstein-Yang-Mills theory, for any N, with implications for stability.

Findings

Solutions exist locally at infinity and the horizon/origin.

Solutions can be extended globally into the asymptotic regime.

Non-trivial dyonic solutions with magnetic gauge fields having no zeroes are found.

Abstract

We investigate dyonic black hole and dyon solutions of four-dimensional $SU(N)$ Einstein-Yang-Mills theory with a negative cosmological constant. We derive a set of field equations in this case, and prove the existence of non-trivial solutions to these equations for any integer $N$, with $2N-2$ gauge degrees of freedom. We do this by showing that solutions exist locally at infinity, and at the event horizon for black holes and the origin for solitons. We then prove that we can patch these solutions together regularly into global solutions that can be integrated arbitrarily far into the asymptotic regime. Our main result is to show that dyonic solutions exist in open sets in the parameter space, and hence that we can find non-trivial dyonic solutions in a number of regimes whose magnetic gauge fields have no zeroes, which is likely important to the stability of the solutions.