Twenty Years of the Weyl Anomaly

TL;DR

Twenty Years of the Weyl Anomaly surveys the birth and maturation of quantum Weyl (trace) anomalies in curved spacetime, tracing Capper and Duff’s initial discovery and the subsequent empirical and conceptual debates. It details the structure of the 2D and 4D anomalies, including the heat-kernel/b4 framework and explicit coefficient formulas, and shows how these anomalies influence cosmology, supersymmetry, and string theory. The article also discusses nonlocal effective actions like the Polyakov action, and surveys ongoing methods and open questions surrounding higher-dimensional generalizations, dualities, and the precise nature of anomaly matching. Overall, it highlights the anomaly’s central role across gravity, quantum field theory, and high-energy theory, while acknowledging unsettled issues that motivate current research.

Abstract

In 1973 two Salam protégés (Derek Capper and the author) discovered that the conformal invariance under Weyl rescalings of the metric tensor $g_{μν}(x)\rightarrowΩ^2(x)g_{μν}(x)$ displayed by classical massless field systems in interaction with gravity no longer survives in the quantum theory. Since then these Weyl anomalies have found a variety of applications in black hole physics, cosmology, string theory and statistical mechanics. We give a nostalgic review. (Talk given at the {\it Salamfest}, ICTP, Trieste, March 1993.)