Gravitational Landau levels and the chiral anomaly

A popular physical picture of the mechanism behind the four-dimensional chiral anomaly is provided by the massless Dirac equation in the presence of constant electric and magnetic background fields. The magnetic field creates highly degenerate Landau levels, the lowest of which is gapless. Any parallel component of the electric field drives a spectral flow in the gapless mode that causes particles to emerge from, or disappear into, the Dirac sea. Seeking a similar picture for the gravitational contribution to the chiral anomaly, we consider the massless Dirac equation in a background spacetime that creates gravitational Landau levels. We find that in this case the resulting spectral flow, with its explicit particle production, accounts for only a small part of the anomalous creation of chiral charge. The balance is provided by the vacuum expectation charge arising from the spectral asymmetry.