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

We investigate the existence of black hole solutions of four dimensional 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. The work can be divided into two sections. In the first half, we use theorems of Wang's to derive a new topologically general su(N)-invariant one-form connection which may serve as the ansatz for our gauge potential. The second half is devoted to proving the existence of non-trivial solutions to the field equations for any integer N, with N - 1 gauge degrees of freedom. Specifically, we prove existence in two separate regimes: for fixed values of the initial parameters and as vertical bar Lambda vertical bar -> infinity; and 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.