Revisiting critical vortices in three-dimensional supersymmetric QED

Abstract

We consider renormalization of the central charge and the mass of the $\mathcal{N}=2$ supersymmetric Abelian vortices in $2+1$ dimensions. We obtain $\mathcal{N}=2$ supersymmetric theory in $2+1$ dimensions by dimensionally reducing the $\mathcal{N}=1$ supersymmetric QED in $3+1$ dimensions with two chiral fields carrying opposite charges. Then we introduce a mass for one of the matter multiplets without breaking $\mathcal{N}=2$ supersymmetry. This massive multiplet is viewed as a regulator in the large mass limit. We show that the mass and the central charge of the vortex get the same nonvanishing quantum corrections, which preserves Bogomol'nyi-Prasad-Sommerfield saturation at the quantum level. Comparison with the operator form of the central extension exhibits fractionalization of a global U(1) charge; it becomes $\ifmmode\pm\else\textpm\fi{}1/2$ for the minimal vortex. The very fact of the mass and charge renormalization is due to a ``reflection'' of an unbalanced number of the fermion and boson zero modes on the vortex in the regulator sector.