Can someone give a simple derivation of the gravitational contribution to the fermion chiral anomaly in 3+1 dimension using the Fujikawa method? Can someone suggest a simple derivation of the gravitational contribution to the fermion chiral anomaly in 3+1 dimension using the Fujikawa method?
$\epsilon_{\mu\nu\rho\lambda}R^{\mu\nu}_{\gamma\delta}R^{\rho\lambda\gamma\delta}$
 A: I trust you are already comfortable with the Fujikawa method applied to the chiral gauge anomaly, path integral-style. For gravity instead, Wick rotate to Euclidean space and build up the covariant derivative from the $\mathrm{SO}(d)$ generators:
$$
J_{ab}=\frac14[\gamma^a, \gamma^b]
\\D_\mu=\partial_\mu+\frac12\omega^{ab}_\mu J_{ab}
\\\gamma^a = e^a{}_\mu\gamma^\mu, \qquad e(x)=\det e^a{}_\mu
\\\mathcal L=e\ \bar\psi\left(ie^\mu{}_a\gamma^aD_\mu-m\right)\psi
$$
The chiral transformation $\psi\overset{\delta^{\gamma_5}_\epsilon}\longrightarrow e^{i\epsilon\gamma_5}\psi$ induces an anomalous transformation of the path integral measure, and we proceed with heat kernel regularisation
$$
\not\!\!D\chi_n=\lambda_n\chi_n
\\\phi=\{A_\mu, \psi, \bar\psi\}
\\\mathcal D\phi\overset{\delta^{\gamma_5}_\epsilon}\longrightarrow\mathcal D\phi\ \exp\left(-2i\int\epsilon(x)\sum_n e\ \chi_n^\dagger\gamma_5\chi_n\right)
\\\mathcal A(x)\equiv\sum_n e\ \chi_n^\dagger\gamma_5\chi_n\sim\lim_{M\to\infty}\mathrm{Tr}(\gamma_5 e^{-\not{D}^2/M^2})
\\\mathcal A(x)=\lim_{M\to\infty}\lim_{x'\to x}\mathrm{Tr}\left\{\gamma_5\ e\ \exp\left[-\frac{1}{M^2}\left(D^2_x+\frac R4\right)\right]\ e^{-1}\delta(x-x')\right\}
$$
At this stage, I will refrain from reproducing Fujikawa's original, rather long but straightforward derivation in its entirety, advising you to consult the original paper: Path integral for gauge theories with fermions. However, I will highlight the main points of difficulty here:

*

*The completeness relation for the eigenvectors involves the vielbein determinant, hence the trailing $e^{-1}$


*The intermediate plane wave representation of the delta function does not immediately work, you instead need to introduce a "geodesic biscalar" $\sigma(x, x')$ which is like a curved space Green's function of the operator $D_\mu D^\nu$. Then $\delta(x-x')=\int\frac{\mathrm d^4k}{(2\pi)^4}\exp(ik_\mu D^\mu)\sigma(x, x')$.
[If, however, you assume regularisation independence (say, using the gauge anomaly as a hint), then you can transform to Riemann normal coordinates and use the usual plane wave parameterisation of the delta function]
Finally, after much tedious algebra, we obtain the following, which we then expand in powers of $1/M$:
$$
\mathcal A(x)=\lim_{M\to\infty}\lim_{x'\to x}\int\frac{\mathrm d^4k}{(2\pi)^4}\mathrm{Tr}\gamma_5\exp\left[-\frac{1}{M^2}\left((ik_\alpha D^\mu D^\alpha\sigma+D^\mu)(ik_\alpha D_\mu D^\alpha\sigma+D_\mu)+\frac R4\right)\right]
$$$$
{}^\text{First order in }\big\Downarrow{}^{1/M\text{ (exact)}}
$$$$
=\frac{e}{192\pi^2}\mathrm{Tr}(\gamma_5[D^\mu, D^\nu][D_\mu, D_\nu])
\\=-\frac{1}{384\pi^2}\frac12\epsilon^{\alpha\beta\gamma\delta}R^{\mu\nu}{}_{\alpha\beta}R_{\mu\nu\gamma\delta}
$$

However, you can be utterly slick using the Atiyah-Singer index theorem formulation of mixed gauge-gravitational anomalies. In 3+1D these are the only contributors since there are no pure gravitational anomalies unless $D=4k+2$. The anomalous Ward identity is given by
$$
\langle\partial_\mu J_5^\mu\rangle=2i\ \widehat{A}(\mathcal M)\mathrm{ch}_\mathcal R(F)\big|_4
$$
where $\widehat{A}(\mathcal M)$ is the A-roof (or Dirac) genus of the manifold (specifically, it is a function of its curvature), $\mathrm{ch}_\mathcal R(F)=\mathrm{tr}_\mathcal R e^{\frac{iF}{2\pi}}$ is the Chern (characteristic) class of the gauge field evaluated for a fermion transforming in representation $\mathcal R$, and $|_4$ denotes that we take the 4-form component of this object. This evaluates to
$$
2\widehat{A}(\mathcal M)\mathrm{ch}_\mathcal R(F)\big|_4=\frac{i}{24\pi^3}\mathrm{tr}_\mathcal R F^2-\frac{i\dim\mathcal R}{192\pi^2}\mathrm{tr}R^2
$$
which agrees with the path integral approach.
