Weyl anomalies on conformal manifolds and moduli spaces

Abstract

A Weyl (conformal) anomaly signals a subtle quantum breaking of classical conformal invariance in conformal field theory. Over the years, Weyl anomalies have been used to characterize nonperturbative properties of conformal field theory. Anomalies associated with the energy–momentum tensor, like the coefficients [Formula: see text] and [Formula: see text] in four space–time dimensions, are generic and have been studied extensively. More generally, in even dimensions, there are also conformal anomalies associated with any primary operator that has integer scaling dimension. Some of the most interesting features of Weyl anomalies have to do with their behavior under continuous deformations or in vacua with spontaneously broken conformal symmetry. In this review, we summarize the defining properties of conformal anomalies, their classification into A- and B-type, and their implications on the structure of correlation functions. We point out that type-B anomalies can exhibit complicated dynamics and review recent progress in the study of this dynamics with special focus on four-dimensional [Formula: see text] superconformal field theories. We emphasize two applications of type-B anomalies in this context: potential constraints on the holonomy of superconformal manifolds and the deconstruction of anomalies in higher dimensions from anomalies in broken phases of lower-dimensional conformal field theories.