  We consider world-sheet theories for non-Abelian strings assuming
compactification on a cylinder with a finite circumference $L$ and periodic
boundary conditions. The dynamics of the orientational modes is described by
two-dimensional CP$(N-1)$ model. We analyze both non-supersymmetric (bosonic)
model and ${\mathcal N}=(2,2)$ supersymmetric CP$(N-1)$ emerging in the case of
1/2-BPS saturated strings in \ntwo supersymmetric QCD with $N_f=N$. The
non-supersymmetric case was studied previously; technically our results agree
with those obtained previously, although our interpretation is totally
different. In the large-$N$ limit we detect a phase transition at $L\sim
\Lambda_{\rm CP}^{-1}$ (which is expected to become a rapid crossover at finite
$N$). If at large $L$ the CP$(N-1)$ model develops a mass gap and is in the
Coulomb/confinement phase, with exponentially suppressed finite-$L$ effects, at
small $L$ it is in the deconfinement phase, and the orientational modes
contribute to the L\"usher term. The latter becomes dependent on the rank of
the bulk gauge group.
  In the supersymmetric CP$(N-1)$ models at finite $L$ we find a large-$N$
solution which was not known previously. We observe a single phase
independently of the value of $L\Lambda_{\rm CP}$. For any value of this
parameter a mass gap develops and supersymmetry remains unbroken. So does the
$SU(N)$ symmetry of the target space. The mass gap turns out to be independent
of the string length. The L\"uscher term is absent due to supersymmetry.
