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