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 $n>\mathrm{}1\mathrm{}$ and the node number $k$ of the purely magnetic gauge field. The well-known spherically symmetric solutions have a winding number $n=1.\mathrm{}$ 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 $\gamma =0,$ with $n$ units of magnetic charge.