On the stability of soliton and hairy black hole solutions of $\mathfrak{su}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant

Abstract

We investigate the stability of spherically symmetric, purely magnetic, soliton and black hole solutions of four-dimensional $\mathfrak{su}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant $\Lambda$. These solutions are described by $N-1$ magnetic gauge field functions ${\omega }_j$. We consider linear, spherically symmetric, perturbations of these solutions. The perturbations decouple into two sectors, known as the sphaleronic and gravitational sectors. For any $N$, there are no instabilities in the sphaleronic sector if all the magnetic gauge field functions ${\omega }_j$ have no zeros, and satisfy a set of $N-1$ inequalities. In the gravitational sector, we are able to prove that there are solutions which have no instabilities in a neighbourhood of stable embedded $\mathfrak{su}(2)$ solutions, provided the magnitude of the cosmological constant $\left|\Lambda \right|$ is sufficiently large.

43 pages, 4 figures, minor changes, discussion expanded and references updated. Matches version accepted for publication in J. Math. Phys

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