For background gauge field configurations reducible to the form $A_{\mu }=(A_3,\overrightarrow{A}(\overrightarrow{x}\left)\right)$ where $A_3$ is a constant, we provide an elementary derivation of the recently obtained result for the exact induced Chern-Simons (CS) effective action in ${\mathrm{QED}}_3$ at finite temperature. The method allows us to extend the result in several useful ways: to obtain the analogous result for the “mixed” CS term in the Dorey-Mavromatos model of parity-conserving planar superconductivity, thereby justifying their argument for flux quantization in the model; to the induced CS term for a τ-dependent flux; and to the term of second order in $\overrightarrow{A}\left(\overrightarrow{x}\right)$ (and all orders in $A_3$) in the effective action.

• Received 4 September 1997

DOI:https://doi.org/10.1103/PhysRevD.57.1171