TEMPERATURE-DEPENDENCE OF THE INDUCED CHERN-SIMONS TERM IN 2+1 DIMENSIONS

We study the temperature dependence of the,radiatively generated parit y-breaking (Chern-Simons) term for massive fermions coupled to an exte rnal Abelian gauge held in 2+1 dimensions, using the imaginary time fo rmalism. We allow the gauge field to be rapidly varying in space (as i n a vortex, for example), and we also consider arbitrary discrete freq uency components of the field. We thus go beyond the usual ''low momen tum'' and ''adiabatic'' approximations. We give a simple analytical ex pression that is exact at both large and small momenta and provides a good interpolation at intermediate values. We also show under which co nditions the usual adiabatic approximation is accurate, providing a si mple physical picture for its prediction. Finally, we consider the sam e-problem for massless fermions coupled to a non-Abelian gauge held.