Parity anomaly in four dimensions

In an analogy to the odd-dimensional case we define the parity anomaly as the part of the one-loop effective action for fermions associated with spectral asymmetry of the Dirac operator. This quantity is computed directly on four-dimensional manifolds with a boundary and related to the Chern-Simons current on the boundary. Despite a quite unusual Chern-Simons level obtained, the action is gauge invariant and passes all consistency checks.

Figure 1

The geometry for testing the Laughlin argument. $B$ is the magnetic field piercing the whole of the torus. $-n$ is the outward pointing normal to the surface torus at different points. We remind the reader that $n$ is the inward pointing normal vector in our notations.