Gravitating superconducting solitons in the (3+1)-dimensional Einstein
  gauged non-linear sigma-model
  In this paper, we construct the first analytic examples of (3+1)-dimensional
self-gravitating regular cosmic tube solutions which are superconducting, free
of curvature singularities and with non-trivial topological charge in the
Einstein-SU(2) non-linear sigma-model. These gravitating topological solitons
at a large distance from the axis look like a (boosted) cosmic string with an
angular defect given by the parameters of the theory, and near the axis, the
parameters of the solutions can be chosen so that the metric is singularity
free and without angular defect. The curvature is concentrated on a tube around
the axis. These solutions are similar to the Cohen-Kaplan global string but
regular everywhere, and the non-linear sigma-model regularizes the gravitating
global string in a similar way as a non-Abelian field regularizes the Dirac
monopole. Also, these solutions can be promoted to those of the fully coupled
Einstein-Maxwell non-linear sigma-model in which the non-linear sigma-model is
minimally coupled both to the U(1) gauge field and to General Relativity. The
analysis shows that these solutions behave as superconductors as they carry a
persistent current even when the U(1) field vanishes. Such persistent current
cannot be continuously deformed to zero as it is tied to the topological charge
of the solutions themselves. The peculiar features of the gravitational lensing
of these gravitating solitons are shortly discussed.
