  The effective gauge field actions generated by charged fermions in $QED_3$
and $QCD_3$ can be made invariant under both small and large gauge
transformations at any temperature by suitable regularization of the Dirac
operator determinant, at the price of parity anomalies. We resolve the paradox
that the perturbative expansion is not invariant, as manifested by the
temperature dependence of the induced Chern-Simons term, by showing that large
(unlike small) transformations and hence their Ward identities, are not
perturbative order-preserving. Our results are illustrated through concrete
examples of field configurations.
