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

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.