  In a class of non-Abelian solitonic vortex strings supported in certain N=2
super-Yang-Mills theories we search for the vortex which can behave as a
critical fundamental string. We use the Polchinski-Strominger criterion of the
ultraviolet completeness. We identify an appropriate four-dimensional bulk
theory: it has the U(2) gauge group, the Fayet-Iliopoulos term and four flavor
hypermultiplets. It supports semilocal vortices with the world-sheet theory for
orientational (size) moduli described by the weighted CP(2,2) model. The latter
is superconformal. Its target space is six-dimensional. The overall Virasoro
central charge is critical. We show that the world-sheet theory on the vortex
supported in this bulk model is the bona fide critical string.
