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

The effective gauge field action due to fermions coupled to SU (N) gauge fields in three dimensions is found to change by |n| 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.