  We use the moduli matrix approach to study the moduli space of 1/4 BPS kinks
supported by vortices in the Higgs phase of N = 2 supersymmetric U(N) gauge
theories when non-zero masses for the matter hypermultiplets are introduced. We
focus on the case of degenerate masses. In these special cases vortices acquire
new orientational degrees of freedom, and become "non-Abelian". Kinks acquire
new degrees of freedom too, and we will refer to them as "non-Abelian". As
already noticed for the Abelian case, non-Abelian kinks must correspond to
non-Abelian monopoles of the unbroken phase of SU(N) Yang-Mills. We show, in
some special cases, that the moduli spaces of the two objects are in one-to-one
correspondence. We argue that the corre- spondence holds in the most general
case. The consequence of our result is two-fold. First, it gives an alternative
way to construct non-Abelian monopoles, in addition to other well- known
techniques (Nahm transform, spectral curves, rational maps). Second, it opens
the way to the study of the quantum physics of non- Abelian monopoles, by
considering the simpler non-Abelian kinks.
