  Color superconductivity is expected to be formed in high density quark matter
where color symmetry is spontaneously broken in the presence of di-quark
condensate. Stable non-Abelian vortices or color magnetic flux tubes exist in
the color-flavor locked phase at asymptotically high density. $\mathbb{C}P^2$
Nambu-Goldstone(NG) bosons and Majorana fermions belonging to the triplet
representation are localized around a non-Abelian vortex. We discuss the zero
mode analysis and the low-energy effective world sheet theory of a non-Abelian
vortex. We determine the interactions of these bosonic and fermionic modes by
using the nonlinear realization method. We also discuss the Aharanov-Bohm (AB)
phases of charged particles, such as, electrons, muons, and color-flavor locked
mesons made of tetra-quarks encircling around a non-Abelian vortex in the
presence of electro-magnetic fields. This is a review based on our recent works
arXiv:1602.01677, arXiv:1512.06603, arXiv:1612.09419.
