Non-Abelian String of a Finite Length

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.