Einstein–Yang-Mills–Chern-Simons solutions inD=2n+1dimensions

Abstract

We investigate finite energy solutions of the Einstein--Yang-Mills--Chern-Simons system in odd spacetime dimensions, $D=2n+1$, with $n>1$. Our configurations are static and spherically symmetric, approaching at infinity a Minkowski spacetime background. In contrast with the Abelian case, the contribution of the Chern-Simons term is nontrivial already in the static, spherically symmetric limit. Both globally regular, particlelike solutions and black holes are constructed numerically for several values of $D$. These solutions carry a nonzero electric charge and have finite mass. For globally regular solutions, the value of the electric charge is fixed by the Chern-Simons coupling constant. The black holes can be thought of as nonlinear superpositions of Reissner-Nordstr\"om and non-Abelian configurations. A systematic discussion of the solutions is given for $D=5$, in which case the Reissner-Nordstr\"om black hole becomes unstable and develops non-Abelian hair. We show that some of these non-Abelian configurations are stable under linear, spherically symmetric perturbations. A detailed discussion of an exact $D=5$ solution describing extremal black holes and solitons is also provided.