We study Bogomol'nyi-Prasad-Sommerfield (BPS) non-Abelian semilocal vortices in U(N(C)) gauge theory with N(F) flavors, N(F)> N(C), in the Higgs phase. The moduli space for an arbitrary winding number is described using the moduli matrix formalism. We find a relation between the moduli spaces of the semilocal vortices in Seiberg-like dual pairs of theories, U(N(C)) and U(N(F)-N(C)). They are two alternative regularizations of a parent non-Hausdorff space, which tend to the same moduli space of sigma model lumps in the infinite gauge coupling limits. We examine the normalizability of the zero-modes and find the somewhat surprising phenomenon that the number of normalizable zero-modes, dynamical fields in the effective action, depends on the point of the moduli space we are considering. We find, in the lump limit, an effective action on the vortex world sheet, which we compare to that found by Shifman and Yung.
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