Quantum correction to the Chern-Simons coefficient in self-dual SU(n) Chern-Simons Higgs theories

We calculate explicitly the quantum correction to the Chern-Simons coefficient in self-dual Chern-Simons Higgs theories with matter fields in the adjoint representation of SU(n)- It turns out that all the quantum corrections in various phases satisfy the quantization condition as long as there are remaining gauge symmetries in the system. In particular, the correction is still quantized even if the remaining symmetry is U(1), although there is no need in principle for it to satisfy the quantization condition in such a case.