Non-Abelian string junctions as confined monopoles

Abstract

Various dynamical regimes associated with confined monopoles in the Higgs phase of $\mathcal{N}=2$ two-flavor QCD are studied. The microscopic model we deal with has the $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge group, with a Fayet--Iliopoulos term of the U(1) factor, and large and (nearly) degenerate mass terms of the matter hypermultiplets. We present a complete quasiclassical treatment of the BPS sector of this model, including the full set of the first-order equations, derivations of all relevant zero modes, and derivation of an effective low-energy theory for the corresponding collective coordinates. The macroscopic description is provided by a ${\mathrm{CP}}^{1}$ model with or without twisted mass. The confined monopoles---string junctions of the microscopic theory---are mapped onto BPS kinks of the ${\mathrm{CP}}^{1}$ model. The string junction is $\frac{1}{4}$ BPS. Masses and other characteristics of the confined monopoles are matched with those of the ${\mathrm{CP}}^{1}$-model kinks. The matching demonstrates the occurrence of an anomaly in the monopole central charge in 4D Yang--Mills theory. We study what becomes of the confined monopole in the bona fide non-Abelian limit of degenerate mass terms where a global SU(2) symmetry is restored. The solution of the macroscopic model is known, e.g., from the mirror description of the ${\mathrm{CP}}^{1}$ model. The monopoles, aka ${\mathrm{CP}}^{1}$-model kinks, are stabilized by nonperturbative dynamics of the ${\mathrm{CP}}^{1}$ model. We explain an earlier rather puzzling observation of a correspondence between the BPS kink spectrum in the ${\mathrm{CP}}^{1}$ model and the Seiberg--Witten solution.