Static axially symmetric Einstein-Yang-Mills-dilaton solutions: Regular solutions

We discuss the static axially symmetric regular solutions obtained recently in Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theory. These asymptotically flat solutions are characterized by the winding number n>1 and the node number k of the purely magnetic gauge field. The well-known spherically symmetric solutions have a winding number n=1. The axially symmetric solutions satisfy the same relations between the metric and the dilaton field as their spherically symmetric counterparts. Exhibiting a strong peak along the ρ-axis, the energy density of the matter fields of the axially symmetric solutions has a torus-like shape. For a fixed winding number n with increasing node number k, the solutions form sequences. The sequences of magnetically neutral non-Abelian axially symmetric regular solutions with winding number n tend to magnetically charged Abelian spherically symmetric limiting solutions, corresponding to “extremal” Einstein-Maxwell-dilaton solutions for finite values of γ and to extremal Reissner-Nordstrøm solutions for γ=0, with n units of magnetic charge.