Topological order in a color-flavor locked phase of $(3+1)$-dimensional $U(N)$ gauge-Higgs system

We study a $(3+1)$-dimensional $U(N)$ gauge theory with $N$-flavor fundamental scalar fields, whose color-flavor locked (CFL) phase has topologically stable non-Abelian vortices. The $U(1)$ charge of the scalar fields must be $Nk+1$ for some integer $k$ in order for them to be in the representation of $U(N)$ gauge group. This theory has a $mathbb{Z}_{Nk+1}$ one-form symmetry, and it is spontaneously broken in the CFL phase, i.e., the CFL phase is topologically ordered if $k ot=0$. We also find that the world sheet of topologically stable vortices in CFL phase can generate this one-form symmetry.