Recently it has been argued that the ground state of high density QCD is likely be a combination of the CFL-phase along with condensation of the K^0 field. This state spontaneously breaks a global U(1)_Y symmetry, therefore one would expect the formation of U(1)_Y global strings. We discuss the core structure of these strings and demonstrate that under some conditions the global U(1)_Y symmetry may not be restored inside the string, in contrast with the standard expectations. Instead, K^+ condensation occurs inside the core of the string if a relevant parameter costheta_K = mkzero^2/mu_eff^2 is larger than some critical value theta_K > theta_crit. If this phenomenon happens, the U(1)_Y strings become superconducting and may considerably influence the magnetic properties of dense quark matter, in particular in neutron stars.
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