A rigorous geometric derivation of the chiral anomaly in curved backgrounds

TL;DR

This paper provides a rigorous geometric derivation of the gravitational chiral anomaly for Weyl fields in curved spacetime, linking it to an index theorem involving the Dirac operator and eta-invariants.

Contribution

It introduces a novel index theorem for the Lorentzian Dirac operator to rigorously describe the gravitational chiral anomaly in curved backgrounds.

Findings

Derived a formula for total charge generated by gravitational and gauge fields.

Connected the anomaly to the Atiyah-Singer index theorem and eta-invariants.

Provided a mathematically rigorous framework for gravitational chiral anomaly analysis.

Abstract

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 $\eta$-invariant of the Cauchy hypersurfaces.