Higher-loop anomalies in chiral gravities

TL;DR

This paper investigates higher-loop anomalies in chiral $W_{3}$ gravity, deriving one- and two-loop anomalies using Fujikawa regularisation, verifying their consistency with known results, and exploring their implications for BRST symmetry and reparametrisation invariance.

Contribution

It provides a detailed derivation of higher-loop anomalies in chiral gravity, confirming their consistency with previous calculations and analyzing their relation to BRST Ward identities.

Findings

One-loop anomalies derived via Fujikawa method.

Two-loop anomalies obtained from Wess-Zumino consistency.

Verification of anomalies satisfying Lam's theorem and BRST identities.

Abstract

The one-loop anomalies for chiral $W_{3}$ gravity are derived using the Fujikawa regularisation method. The expected two-loop anomalies are then obtained by imposing the Wess-Zumino consistency conditions on the one-loop results. The anomalies found in this way agree with those already known from explicit Feynman diagram calculations. We then directly verify that the order $\hbar^2$ non-local BRST Ward identity anomalies, arising from the ``dressing'' of the one-loop results, satisfy Lam's theorem. It is also shown that in a rigorous calculation of $Q^2$ anomaly for the BRST charge, one recovers both the non-local as well as the local anomalies. We further verify that, in chiral gravities, the non-local anomalies in the BRST Ward identity can be obtained by the application of the anomalous operator $Q^2$, calculated using operator products, to an appropriately defined gauge fermion. Finally, we give arguments to show why this relation should hold generally in reparametrisation-invariant theories.