Lectures on Anomalies

TL;DR

The notes provide a comprehensive, self-contained treatment of anomalies in quantum field theory, starting with explicit derivations of the abelian and non-abelian gauge anomalies in four dimensions via fermion measure transformations and triangle diagrams. They then develop a formal differential-geometric framework—characteristic classes, descent equations, and BRST cohomology—to characterize anomalies in arbitrary even dimensions and connect them to index theorems. The text also discusses locality, finiteness, and the cancellation of anomalies in the Standard Model, and extends the discussion to gravitational and mixed gauge–gravitational anomalies, including explicit applications to ten-dimensional string theories. Together, these lectures illuminate how topological invariants control quantum violations of classical symmetries and how such violations are either canceled or accommodated in consistent theories with gravity or strings. The material provides essential tools for analyzing anomaly cancellation, topological terms, and the deep link between quantum anomalies and index theory.

Abstract

These lectures on anomalies are relatively self-contained and intended for graduate students who are familiar with the basics of quantum field theory. We begin with several derivations of the abelian anomaly: anomalous transformation of the measure, explicit computation of the triangle Feynman diagram, relation to the index of the Euclidean Dirac operator. The chiral (non-abelian) gauge anomaly is derived by evaluating the anomalous triangle diagram with three non-abelian gauge fields coupled to a chiral fermion. We discuss in detail the relation between anomaly, current non-conservation and non-invariance of the effective action, with special emphasis on the derivation of the anomalous Slavnov-Taylor/Ward identities. We show why anomalies always are finite and local. A general characterization is given of gauge groups and fermion representations which may lead to anomalies in four dimensions, and the issue of anomaly cancellation is discussed, in particular the classical example of the standard model. Then, we move to more formal developments and arbitrary even dimensions. After introducing a few basic notions of differential geometry, in particular characteristic classes, we derive the descent equations. We prove the Wess-Zumino consistency condition and show that relevant anomalies correspond to BRST cohomologies at ghost number one. We discuss why and how anomalies are related to characteristic classes in two more dimensions and outline their computation in terms of the index of an appropriate Dirac operator. Finally we derive the gauge and gravitational anomalies in arbitrary even dimensions from the appropriate index and explain the anomaly cancellations in ten-dimensional IIB supergravity and in type I and heterotic superstrings.

Figures (10)

• Figure 1: The triangle diagram for a fermion of charge $q_j$ contributing to the abelian anomaly
• Figure 2: The two triangle diagrams contributing to the abelian anomaly
• Figure 3: The two triangle diagrams contributing to the non-abelian chiral anomaly
• Figure 4: In four dimensions, considering only fermion loops, the box diagram on the left is divergent, while the pentagon and hexagon diagrams in the middle and on the right are convergent.
• Figure 5: The triangle diagram coupling to gauge bosons of the gauge groups $G$, $G'$ and $G"$.