On induced CPT-odd Chern-Simons terms in the 3+1 effective action

This paper was originally designated as a Comment on the paper by R. Jackiw and V. A. Kostelecký, Phys. Rev. Lett. 82 , 3572 (1999). It gives an example of a fermionic system in which the CPT -odd Chern-Simons terms in the effective action are unambiguously induced by chiral fermions: superfluid 3 He-A. In this system the Lorentz and gauge invariances are both violated at high energy, but the behavior of the system beyond the cutoff is known. This allows one to construct a CPT -odd action which combines the conventional 3+1 CS term and the mixed axial-gravitational CS term discussed by G. E. Volovik and A. Vilenkin, http://xxx.lanl.gov/abs/hep-ph/9905460. The influence of the CS term on the dynamics of the effective gauge field has been observed experimentally in rotating 3 He-A.