Non-Abelian vortices of higher winding numbers

We make a detailed study of the moduli space of winding number two ($k=2$) axially symmetric vortices (or equivalently, of coaxial composite of two fundamental vortices), occurring in $U(2)$ gauge theory with two flavors in the Higgs phase, recently discussed by Hashimoto and Tong and by Auzzi, Shifman, and Yung. We find that it is a weighted projective space WCP(2,1,1)²CP²/Z₂. This manifold contains an A₁-type (Z₂) orbifold singularity even though the full moduli space including the relative position moduli is smooth. The $SU(2)$ transformation properties of such vortices are studied. Our results are then generalized to $U(N)$ gauge theory with N flavors, where the internal moduli space of $k=2$ axially symmetric vortices is found to be a weighted Grassmannian manifold. It contains singularities along a submanifold.