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

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. The work can be divided into two sections. In the first half, we use theorems of Wang's to derive a new topologically general $$\mathfrak {su}(N)$$ -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 $$|\varLambda |\rightarrow \infty $$ ; and for any $$\varLambda <0$$ , 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. [ABSTRACT FROM AUTHOR]