  It has been shown that non-Abelian vortex string supported in four
dimensional (4D) ${\mathcal N}=2$ supersymmetric QCD (SQCD) with the U(2) gauge
group and $N_f = 4$ quark flavors becomes a critical superstring. This string
propagates in the ten dimensional space formed by a product of the flat 4D
space and an internal space given by a Calabi-Yau non-compact threefold,
namely, the conifold. The spectrum of closed string states of the associated
string theory was obtained using the equivalence between the critical string on
the conifold and the non-critical string on the semi-infinite cigar described
by SL($2, \mathbb{R}$)/U(1) Wess-Zumino-Novikov-Witten model. This spectrum was
identified with the spectrum of hadrons in 4D ${\mathcal N}=2$ SQCD. In order
to describe effective interactions of these 4D hadrons in this paper we study
correlation functions of normalizable vertex operators localized near the tip
of the SL($2, \mathbb{R}$)/U(1) cigar. We also compare our solitonic string
approach to the gauge-string duality to the AdS/CFT-type holography for little
string theories (LSTs). The latter relates off mass-shell correlation functions
on the field theory side to correlation functions of non-normalizable vertex
operators on the cigar. We show that in most channels holographic approach
works in our theory because normalizable and non-normalizable vertex operators
with the same conformal dimension are related due to the reflection from the
tip of the cigar. However, we find that holography does not work for lightest
hadrons with given baryonic charge.
