Gravitating monopole-antimonopole chains and vortex rings

We construct monopole-antimonopole chain and vortex solutions in Yang-Mills-Higgs theory coupled to Einstein gravity. The solutions are static, axially symmetric, and asymptotically flat. They are characterized by two integers $(m,n)$ where $m$ is related to the polar angle and $n$ to the azimuthal angle. Solutions with $n=1$ and $n=2$ correspond to chains of $m$ monopoles and antimonopoles. Here the Higgs field vanishes at $m$ isolated points along the symmetry axis. Larger values of $n$ give rise to vortex solutions, where the Higgs field vanishes on one or more rings, centered around the symmetry axis. When gravity is coupled to the flat space solutions, a branch of gravitating monopole-antimonopole chain or vortex solutions arises and merges at a maximal value of the coupling constant with a second branch of solutions. This upper branch has no flat space limit. Instead in the limit of vanishing coupling constant it either connects to a Bartnik-McKinnon or generalized Bartnik-McKinnon solution, or, for $m>4$, $n>4$, it connects to a new Einstein-Yang-Mills solution. In this latter case further branches of solutions appear. For small values of the coupling constant on the upper branches, the solutions correspond to composite systems, consisting of a scaled inner Einstein-Yang-Mills solution and an outer Yang-Mills-Higgs solution.

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