Quantum corrections to mass and central charge of supersymmetric solitons

We review some recent developments in the subject of quantum corrections to soliton mass and central charge. We consider in particular approaches which use local densities for these corrections, as first discussed by Hidenaga Yamagishi. We then consider dimensional regularization of the supersymmetric kink in 1+1 dimensions and an extension of this method to a (2+1)-dimensional gauge theory with supersymmetric abelian Higgs vortices as the solitons. In the case of the supersymmetric N=1 kink, the anomalous contribution to the central charge which is required for BPS saturation can be understood as the remnant of parity violation of the (2+1)-dimensional theory where a minimally supersymmetric kink can be embedded keeping the same field content. In the case of the (2+1)-dimensional N=2 vortex, BPS saturation at the quantum level follows from explicit calculations of mass and central charge at the one-loop level. The number of fermionic zero modes in the vortex background is such as to enforce BPS saturation to all orders by multiplet shortening.