Cancellation of quantum mechanical higher loop contributions to the gravitational chiral anomaly

Abstract

We give an explicit demonstration, using the rigorous Feynman rules developed in~$\0^{1}$, that the regularized trace $\tr γ_5 e^{-β\Dslash^2}$ for the gravitational chiral anomaly expressed as an appropriate quantum mechanical path integral is $β$-independent up to two-loop level. Identities and diagrammatic notations are developed to facilitate rapid evaluation of graphs given by these rules.