In an analogy to the odd-dimensional case we define the parity anomaly as the
part of the one-loop effective action for fermions associated with spectral
asymmetry of the Dirac operator. This quantity is computed directly on
four-dimensional manifolds with boundary and related to the Chern-Simons
current on the boundary. Despite a quite unusual Chern-Simons level obtained,
the action is gauge invariant and passes all consistency checks.Comment: Revtex, 7 pp V2: added remarks and reference