A rigorous geometric derivation of the chiral anomaly in curved backgrounds

This is the authors' abstract. We don't add key points for this paper.

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the $η$-invariant of the Cauchy hypersurfaces.