On the existence of topological hairy black holes in \(\mathfrak{su}(N)\) EYM theory with a negative cosmological constant

Abstract

We investigate the existence of black hole solutions of four dimensional \(\mathfrak{su}(N)\) EYM theory with a negative cosmological constant. Our analysis differs from previous works in that we generalise the field equations to certain non-spherically symmetric spacetimes. We prove the existence of non-trivial solutions for any integer \(N\), with \(N-1\) gauge degrees of freedom. Specifically, we prove two results: existence of solutions for fixed values of the initial parameters and as \(|\Lambda|\rightarrow\infty\), and existence of solutions for any \(\Lambda<0\) in some neighbourhood of existing trivial solutions. In both cases we can prove the existence of `nodeless' solutions, i.e. such that all gauge field functions have no zeroes; this fact is of interest as we anticipate that some of them may be stable.