An Introduction to Gravitational Anomalies

Abstract

The study of anomalies in global and gauge currents in Quantum Field Theory has had a remarkable number of important applications during the 70’s. In the original version of the anomaly¹ one considers a massless fermion triangle diagram with one axial current and two vector currents. Requiring the vector currents to be conserved, one finds that the axial current is not conserved therefore leading to a breakdown of chiral symmetry in the presence of gauge fields coupled to conserved vector currents. This breakdown of chiral symmetry led to the understanding of π° decay and to the resolution of the u (1) problem.² The anomaly has also been instrumental in posing constraints to insure the mathematical consistency of gauge theories coupled to chiral currents. If one considers a theory with gauge fields coupled for instance to left handed currents, one must look at a fermion triangle diagram with V-A currents at each vertex. Again, this diagram is anomalous, and unless the anomalies cancel when summing over all the fermion species running around the loop, one finds that the V-A currents are not conserved, implying that gauge invariance is broken and thus the anomaly renders the theory inconsistent. The anomaly cancellation condition has proven to be very useful in constraining the particle content of unified gauge theories.³ More recently⁴, the anomaly has also been shown to be useful in analyzing the spectrum of massless fermions in confining theories. In the context of low energy chiral theories, the Wess-Zumino lagrangian⁵ has recently played a central role in showing that the soliton solutions of certain models⁶ can be identified with baryons.⁷ This recent development has in turn shed new light into our understanding of chiral anomalies.