Topological black holes in ${\mathfrak {su}}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant

TL;DR

This paper explores the phase space of topological black hole solutions in ${\mathfrak {su}}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant, revealing simpler structures than spherical cases and identifying stable solutions with zero or non-zero gauge functions.

Contribution

It provides a detailed analysis of topological black hole solutions in higher gauge groups, highlighting differences from spherically symmetric cases and stability properties.

Findings

Phase space is simpler than for spherical black holes.

Existence of solutions with gauge functions having zeros for N>2.

Some solutions with no zeros are linearly stable.

Abstract

We investigate the phase space of topological black hole solutions of ${\mathfrak {su}}(N)$ Einstein-Yang-Mills theory in anti-de Sitter space with a purely magnetic gauge potential. The gauge field is described by $N-1$ magnetic gauge field functions $\omega_{j}$, $j=1,\ldots , N-1$. For ${\mathfrak {su}}(2)$ gauge group, the function $\omega_{1}$ has no zeros. This is no longer the case when we consider a larger gauge group. The phase space of topological black holes is considerably simpler than for the corresponding spherically symmetric black holes, but for $N>2$ and a flat event horizon, there exist solutions where at least one of the $\omega_{j}$ functions has one or more zeros. For most of the solutions, all the $\omega_{j}$ functions have no zeros, and at least some of these are linearly stable.