Anomaly-induced effective action and Chern-Simons modification of   general relativity
  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. 
Sebasti\~ao Mauro and Ilya L. Shapiro
gr-qc hep-ph hep-th