Communications in Mathematical Physics

, Volume 347, Issue 3, pp 703–721 | Cite as

A Rigorous Geometric Derivation of the Chiral Anomaly in Curved Backgrounds

  • Christian BärEmail author
  • Alexander Strohmaier


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 \({\eta}\)-invariant of the Cauchy hypersurfaces.


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute for MathematicsPotsdam UniversityPotsdamGermany
  2. 2.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK

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