Higher-loop anomalies in chiral gravities

The one-loop anomalies for chiral W3 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 h2 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 Q2 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 Q2, 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. (orig.)