We examine the claim that the effective action of four-dimensional $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_L$ gauge theory at high and low temperature contains a three-dimensional Chern-Simons term which has the chemical potential for baryon number as its coefficent. The four-dimensional theory has a two-dimensional analogue in which exact calculations can be performed. These calculations demonstrate that the existence of the Chern-Simons term in four dimensions may be rather subtle.

• Received 14 October 1997

DOI:https://doi.org/10.1103/PhysRevD.58.085007