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Quantum Deformation of the Effective Theory on Non-Abelian string and 2D-4D correspondence

We explore non-Abelian strings in the r=N-1 vacuum of N=2 supersymmetric QCD with the gauge group U(N) and N_f flavors of quarks (N_f\geq N), where r is the number of condensed quarks. N=2 supersymmetry is broken down to N=1 by a small mass term for the adjoint matter. We discover that the low-energy two-dimensional theory on the string world-sheet receives nonperturbative corrections from the bulk, through the bulk gaugino condensate. This is in contradistinction with the r=N vacuum situation, in which nonperturbative effects on the world sheet are determined by internal dynamics of the world-sheet theory. The 2D-4D correspondence (the coincidence of spectra of two-dimensional kinks and four-dimensional monopoles) remains valid in the BPS sector. Nonperturbative bulk effects deforming the weighted CP model on the world sheet are found by virtue of the method of resolvents suggested by Gaiotto, Gukov and Seiberg for surface defects. In the r=N vacuum the gaugino condensate in the bulk vanishes, and there are no "outside" nonperturbative corrections on the world sheet.