The Chern-Simons coefficient in the Higgs phase

We study one-loop corrections to the Chern-Simons coefficient κ{script} in abelian self-dual Chern-Simons Higgs systems and their N=2 and N=3 supersymmetric generalizations in both symmetric and asymmetric phases. One-loop corrections to the Chern-Simons coefficient of these systems turn out to be integer multiples of 1 4π in both phases. Especially in the maximally supersymmetric N=3 case, the correction in the symmetric phase vanishes and that in the asymmetric phase is κ{script}/(2π|κ{script}|). Our results suggest that nonabelian self-dual system might enjoy similar features. We also discuss various issues arising from our results.