  We study non-Abelian vortex strings in four-dimensional (4D) $\mathcal{N} =
2$ supersymmetric QCD with U$(N=2)$ gauge group and $N_f=4$ flavors of quark
hypermultiplets. It has been recently shown that these vortices behave as
critical superstrings. The spectrum of closed string states in the associated
string theory was found and interpreted as a spectrum of hadrons in 4D
$\mathcal{N} = 2$ supersymmetric QCD. In particular, the lowest string state
appears to be a massless BPS "baryon." Here we show the occurrence of this
stringy baryon using a purely field-theoretic method. To this end we study the
conformal world-sheet theory on the non-Abelian string -- the so called
weighted $\mathcal{N} = (2,2)$ supersymmetric $\mathbb{CP}$ model. Its target
space is given by the six-dimensional non-compact Calabi-Yau space $Y_6$, the
conifold. We use mirror description of the model to study the BPS kink spectrum
and its transformations on curves (walls) of marginal stability. Then we use
the 2D-4D correspondence to show that the deformation of the complex structure
of the conifold is associated with the emergence of a non-perturbative Higgs
branch in 4D theory which opens up at strong coupling. The modulus parameter on
this Higgs branch is the vacuum expectation value of the massless BPS "baryon"
previously found in string theory.
