Parity violation and gauge noninvariance of the effective gauge field action in three dimensions

The effective gauge field action due to fermions coupled to $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge fields in three dimensions is found to change by $\pm \pi \mathrm{}\left|n\right|\mathrm{}$ under a homotopically nontrivial gauge transformation with winding number $n$. This gauge noninvariance can be eliminated by adding a parity-violating topological term to the action, or by regulating the theory in a way which produces this term automatically in the effective action. The Euler-Heisenberg effective action is calculated in the SU(2) theory and in QED.