  Recently it was shown that the quantum vacuum effects of massless chiral
fermion field in curved space-time leads to the parity-violating Pontryagin
density term, which appears in the trace anomaly with imaginary coefficient. In
the present work the anomaly-induced effective action with the parity-violating
term is derived. The result is similar to the Chern-Simons modified general
relativity, which was extensively studied in the last decade, but with the
kinetic terms for the scalar different from those considered previously in the
literature. The parity-breaking term makes no effect on the zero-order
cosmology, but it is expected to be relevant in the black hole solutions and in
the cosmological perturbations, especially gravitational waves.
