A long-standing puzzle concerns the gauge invariance 
of induced Chern-Simons (CS) terms at finite temperature.General 
arguments imply that the coefficient of the CS term must be quantised,
even at finite T,if the action is to be invariant under 
topologically non-trivial gauge transformations.On the other hand,simple
perturbative calculations contradict this quantisation requirement 
(see for example I.J.R.Aitchison and J.A.Zuk, Ann.Phys. 242, 77 (1995)).
Some exact (non-perturbative) results have been obtained for 
special gauge field configurations,which show the mechanism whereby 
gauge invariance is in fact preserved.An elementary derivation is 
given in I.J.R.Aitchison and C.D.Fosco, Phys.Rev. D 57,1171 (1998).