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Gravitating sphalerons in the Skyrme model

We construct self-gravitating axially symmetric sphaleron solutions of the 3+1 dimensional Skyrme model coupled to Einstein gravity. The solutions are static and asymptotically flat, they are characterized by two integers n and m, where n is the winding numbers of the constituents and the second integer m defines type of the solution. These configuration correspond to the chains of charge n Skyrmions and charge -n anti-Skyrmions placed along the axis of symmetry in alternating order. We investigate the dependency of the masses of the gravitating sphalerons on the gravitational coupling. We find new chains of self-gravitating |n| = 1 Skyrmions-anti-Skyrmions (S-A) which emerge at some critical non-zero value of the gravitational coupling and do not have flat space limit. In contrast, the branches of self-gravitating |n| $\ge$ 2 S-A chains emerge from the corresponding flat space configurations. In both cases these branches merge at some maximal value of the effective gravitational coupling the branches of different type. The branch of gravitating S-A pair extends all the way back to the limit of vanishing coupling constant where solutions approach the corresponding generalised Bartnik-McKinnon solutions. The upper branch of gravitating S-A-S chain exist up to some critical value of the gravitational coupling at which the chain becomes broken. We further find that 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 outer Skyrmions which are separating from the inner configuration.