Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions

The effective gauge field action due to an odd number of fermion species in three-dimensional $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theories is shown to change by $\pm \pi \mathrm{}\left|n\right|\mathrm{}$ under a homotopically nontrivial gauge transformation with winding number $n$. Gauge invariance can be restored by use of Pauli-Villars regularization, which, however, introduces parity nonconservation in the form of a parity-nonconserving, topological term in the effective action.