Higher-loop anomalies in chiral gravities

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 ℏ 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.