Topological Effects in Medium

Developing Fockʹs ideas, we consider here the topological effects in the gauge field theory. Two closely related topological phenomena are studied at finite density and temperature. These are chiral anomaly and the Chern-Simons term. It occurs that the chiral anomaly does not depend on density and temperature. The Chern-Simons term appearance in even dimensions is studied under two types of constraints: the chiral and usual charges conservation. In odd dimensions, by using different methods, it is shown that mo^2 = m^2 is the crucial point for Chern-Simons at zero temperature. So, when mo^2 < m^2,(mo) influence disappears and we get the usual Chern-Simons term. On the other hand, when mo^2 > m^2, the Chern-Simons term vanishes because of nonzero density of background fermions. The connection between parity anomalous Chern-Simons in odd dimension and chiral anomaly in even dimension is established at arbitrary density and temperature. These results hold in any dimension both in Abelian and in non-Abelian cases.