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.
