Parity anomaly in four dimensions
It is known for more that 30 years that quantum Dirac fermions in odd dimensional spaces possess the so called parity anomaly related to the spectral asymmentry of the Dirac operator. As a result, the Chern-Simons action with the level k=1/2 is induced in 3D. We have studied Dirac fermions on four-dimensional manifolds with with boundaries and demonstrated that a similar phenomenon take place: spectral assymetry of the Dirac operator leads to a boundary Chern-Simons term. We found a somewhat unusual Chern-Simons level k=1/4, which is however consistent with the counting of boudnary degrees of freedom. Applications to the condensed matter physics and holography will be briefly discussed.