BAER, C. and STROHMAIER, A., 2016. A rigorous geometric derivation of the chiral anomaly in curved backgrounds. Communications in Mathematical Physics, 347 (3), pp. 703-721.
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 directly in Lorentzian
signature and 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.