We consider non-Abelian strings in $\mathcal{N}=2$ supersymmetric quantum chromodynamics (QCD) with the $\mathrm{U}(N)$ gauge group and $N_f=N$ quark flavors deformed by a mass term for the adjoint matter. This deformation breaks $\mathcal{N}=2$ supersymmetry down to $\mathcal{N}=1$. Dynamics of orientational zero modes on the string world sheet are described then by $CP(N-1)$ model with $\mathcal{N}=(0,2)$ supersymmetry. We study the string of a finite length $L$ assuming compactification on a cylinder (periodic boundary conditions). The world-sheet theory is solved in the large-$N$ approximation. At $N=\infty$ we find a rich phase structure in the $(L,u)$ plane where $u$ is a deformation parameter. At large $L$ and intermediate $u$ we find a phase with broken $Z_{2N}$ symmetry, $N$ vacua and a mass gap. At large values of $L$ and $u$ still larger we have the $Z_{2N}$-symmetric phase with a single vacuum and massless fermions. In both phases $\mathcal{N}=(0,2)$ supersymmetry is spontaneously broken. We also observe a phase with would-be broken $\mathrm{SU}(N)$ symmetry at small $L$ (it is broken only for $N=\infty$). In the latter phase the mass gap vanishes and the vacuum energy is zero in the leading $1/N$ approximation. We expect that at large but finite $N$ corrections $O(1/N)$ will break $\mathcal{N}=(0,2)$ supersymmetry. Simultaneously, the phase transitions will become rapid crossovers. Finally we discuss how the observed rich phase structure matches the $\mathcal{N}=(2,2)$ limit in which the world-sheet theory has a single phase with the mass gap independent of $L$.

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• Non-Abelian gauge theories
• Quantum chromodynamics
• String phenomenology
• Supersymmetric field theories