Gravitational anomalies
Consider a classical gravitational field represented by the vielbein e^a_{\;\mu} and a quantized Fermi field \psi. The generating functional for this quantum field is Z[e^a_{\;\mu}]=e^{-W[e^a_{\;\mu}]}=\int d\bar{\psi}d\psi\;\; e^{-\int d^4x e \mathcal{L}_{\psi}}, where W is the quantum action and the e factor before the Lagrangian is the vielbein determinant, the variation of the quantum action renders \delta W[e^a_{\;\mu}]=\int d^4x \; e \langle T^\mu_{\;a}\rangle \delta e^a_{\;\mu} in which we denote a mean value with respect to the path integral by the bracket \langle\;\;\; \rangle. Let us label the Lorentz, Einstein and Weyl transformations respectively by their parameters \alpha,\, \xi,\, \sigma; they spawn the following anomalies: Lorentz anomaly \delta_\alpha W=\int d^4x e \, \alpha_{ab}\langle T^{ab} \rangle, which readily indicates that the energy-momentum tensor has an anti-symmetric part. Einstein anomaly \delta_\xi W=-\int d^4x e \, \xi^\nu \left(\nabla_\nu\langle T^\mu_{\;\nu}\rangle-\omega_{ab\nu}\langle T^{ab}\rangle\right), this is related to the non-conservation of the energy-momentum tensor, i.e. \nabla_\mu\langle T^{\mu\nu}\rangle \neq 0. Weyl anomaly \delta_\sigma W=\int d^4x e \, \sigma\langle T^\mu_{\;\mu}\rangle, which indicates that the trace is non-zero. == See also ==