Quantizing a solitonic string

Quite often the zero mode dynamics on solitonic vortices are described by a non-conformal effective world-sheet sigma model (WSSM). We address the problem of solitonic string quantization in this case. As well-known, only critical strings with conformal WSSMs are self-consistent in ultra-violet (UV) domain. Thus, we look for the appropriate UV completion of the low-energy non-conformal WSSM. We argue that for the solitonic strings supported in well-defined bulk theories the UV complete WSSM has a UV fixed point which can be used for string quantization. As an example, we consider BPS non- Abelian vortices supported by four-dimensional (4D) N = 2 SQCD with the gauge group U(N) and N_f quark multiplets where N_f≥ N. In addition to translational moduli the non-Abelian vortex under consideration carries orientational and size moduli. Their low- energy dynamics are described by a two-dimensional N = (2, 2) supersymmetric weighted model, namely, 𝕎ℂℙ (N, N_f− N). Given our UV completion of this WSSM we find its UV fixed point. The latter defines a superconformal WSSM. We observe two cases in which this conformal WSSM, combined with the free theory for four translational moduli, has ten-dimensional target space required for superstrings to be critical.