Gravitational anomaly

General covariance is the basis of general relativity, the classical theory of gravitation. Moreover, it is necessary for the consistency of any theory of quantum gravity, since it is required in order to cancel unphysical degrees of freedom with a negative norm, namely gravitons polarized along the time direction. Therefore, all gravitational anomalies must cancel out.

The anomaly usually appears as a Feynman diagram with a chiral fermion running in the loop (a polygon) with n external gravitons attached to the loop where ${\displaystyle n=1+D/2}$ where ${\displaystyle D}$ is the spacetime dimension.

Gravitational anomalies

Consider a classical gravitational field represented by the vielbein ${\displaystyle e_{\mu }^a}$ and a quantized Fermi field ${\displaystyle \psi }$. The generating functional for this quantum field is

$${\displaystyle Z[e_{\mu }^a]=e^{-W[e_{\mu }^a]}=\int d\overline{\psi }d\psi e^{-\int d^{4}xe{\mathcal{L}}_{\psi }},}$$

where ${\displaystyle W}$ is the quantum action and the ${\displaystyle e}$ factor before the Lagrangian is the vielbein determinant, the variation of the quantum action renders

$${\displaystyle \delta W[e_{\mu }^a]=\int d^{4}xe⟨T_a^{\mu }⟩\delta e_{\mu }^a}$$

in which we denote a mean value with respect to the path integral by the bracket ${\displaystyle ⟨⟩}$. Let us label the Lorentz, Einstein and Weyl transformations respectively by their parameters ${\displaystyle \alpha ,\xi ,\sigma }$; they spawn the following anomalies:

Lorentz anomaly

$${\displaystyle {\delta }_{\alpha }W=\int d^{4}xe{\alpha }_{ab}⟨T^{ab}⟩,}$$

which readily indicates that the energy-momentum tensor has an anti-symmetric part.

Einstein anomaly

$${\displaystyle {\delta }_{\xi }W=-\int d^{4}xe{\xi }^{\nu }\left({\mathrm{\nabla }}_{\nu }⟨T_{\nu }^{\mu }⟩-{\omega }_{ab\nu }⟨T^{ab}⟩\right),}$$

this is related to the non-conservation of the energy-momentum tensor, i.e. ${\displaystyle {\mathrm{\nabla }}_{\mu }⟨T^{\mu \nu }⟩\ne 0}$.

Weyl anomaly

$${\displaystyle {\delta }_{\sigma }W=\int d^{4}xe\sigma ⟨T_{\mu }^{\mu }⟩,}$$

which indicates that the trace is non-zero.