We consider a benchmark bulk theory in four dimensions: $\mathcal{N}=2$ supersymmetric QCD with the gauge group $\mathrm{U}(N)$ and $N_f$ flavors of fundamental matter hypermultiplets (quarks). The nature of the Bogomol’nyi-Prasad-Sommerfield (BPS) strings in this benchmark theory crucially depends on $N_f$. If $N_f\ge N$ and all quark masses are equal, it supports non-Abelian BPS strings which have internal (orientational) moduli. If $N_f>N$ these strings become semilocal, developing additional moduli $\rho$ related to (unlimited) variations of their transverse size. Using the U(2) gauge group with $N_f=3$, 4 as an example, we derive an effective low-energy theory on the (two-dimensional) string world sheet. Our derivation is field theoretic, direct and explicit: we first analyze the Bogomol’nyi equations for string-geometry solitons, suggest an ansatz, and solve it at large $\rho$. Then we use this solution to obtain the world-sheet theory. In the semiclassical limit our result confirms the Hanany-Tong conjecture, which rests on brane-based arguments, that the world-sheet theory is an $\mathcal{N}=2$ supersymmetric U(1) gauge theory with $N$ positively and $N_e=N_f-N$ negatively charged matter multiplets and the Fayet-Iliopoulos term determined by the four-dimensional coupling constant. We conclude that the Higgs branch of this model is not lifted by quantum effects. As a result, such strings cannot confine. Our analysis of infrared effects, not seen in the Hanany-Tong consideration, shows that, in fact, the derivative expansion can make sense only provided that the theory under consideration is regularized in the infrared, e.g. by the quark mass differences. The world-sheet action discussed in this paper becomes a bona fide low-energy effective action only if $\Delta m_{AB}\ne 0$.

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