Gravitational anomaly

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

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

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

${\displaystyle \delta W[e_{\;\mu }^{a}]=\int d^{4}x\;e\langle T_{\;a}^{\mu }\rangle \delta e_{\;\mu }^{a}}$

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

Lorentz anomaly

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

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

Einstein anomaly

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

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

Weyl anomaly

${\displaystyle \delta _{\sigma }W=\int d^{4}xe\,\sigma \langle T_{\;\mu }^{\mu }\rangle }$,

which indicates that the trace is non-zero.

• Mixed anomaly
• Green–Schwarz mechanism
• Gravitational instanton

• Luis Álvarez-Gaumé; Edward Witten (1984). "Gravitational Anomalies". Nucl. Phys. B. 234 (2): 269. Bibcode:1984NuPhB.234..269A. doi:10.1016/0550-3213(84)90066-X.
• Witten, Edward (1985). "Global gravitational anomalies". Commun. Math. Phys. 100 (2): 197–229. Bibcode:1985CMaPh.100..197W. doi:10.1007/BF01212448. S2CID 9145165.