Gauge Invariance, Finite Temperature, and Parity Anomaly in $\mathit{D}=3\mathrm{}$

The effective gauge field actions generated by charged fermions in ${\mathrm{QED}}_3$ and ${\mathrm{QCD}}_3$ 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, where the interplay between gauge and parity anomalies is also exhibited.