  In contrast to its counterpart in a vector theory, the effective action of a
chiral gauge theory may have a non-vanishing imaginary part. It consists of the
so-called Chern-Simons form, which encodes the anomaly and a gauge invariant
piece related to the eta-invariant of some extended Dirac operator. We
investigate the imaginary part of the effective action in the 4D chiral U(1)
theory within the continuum fermion approach. Perturbative analysis indicates
in general a non-vanishing imaginary part of the effective action. However,
after anomaly cancellation we find only a small value of the imaginary part for
the investigated configurations.
