A Pedagogical Introduction to Anomalies in Quantum Field Theory
Abstract
These notes are aimed at providing a relatively concise and self-contained introduction to anomalies in quantum field theory. A first part reviews the necessary notions of quantum field theory, in particular its functional integral formulation which we use throughout. Then we discuss in some detail how and why anomalies arise as non-invariance of the functional integral measure or through the absence of a regulator preserving all symmetries, and how they manifest themselves as current non-conservation in anomalous loop diagrams and anomalous Ward identities, or equivalently as a non-invariance of the effective action. Specifically, we deal with the so-called abelian anomaly under chiral transformations, and the so-called chiral anomaly which is a violation of non-abelian gauge invariance in the presence of chiral fermions. We perform a detailed triangle Feynman diagram computation for both cases. Then we move on to provide a brief overview of some important properties of anomalies (locality, finiteness, one-loop exactness), and how they are characterized in general by the Wess-Zumino consistency condition. We show how this condition can be reformulated in terms of BRST cohomology and how it can be solved, for arbitrary (even) space-time dimensions, by the descent equations. Finally, we briefly discuss the general relation between anomalies in arbitrary even dimensions 2n and index theorems in 2n+2 dimensions, generalizing also to gravitational anomalies. We conclude with the exhibition of the highly non-trivial anomaly cancellation in ten-dimensional IIB supergravity.
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BibTeX
@misc{cubukcu2026pedagogical,
title = {{A Pedagogical Introduction to Anomalies in Quantum Field Theory}},
author = {Ada Cubukcu and Adel Bilal},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28835},
eprint = {2609.28835},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05781793v1},
}
RIS
TY - UNPB
TI - A Pedagogical Introduction to Anomalies in Quantum Field Theory
AU - Ada Cubukcu
AU - Adel Bilal
PY - 2026
DA - 2026/09/23
PB - HAL
DO - 10.48550/arxiv.2609.28835
UR - https://hal.science/hal-05781793v1
AB - These notes are aimed at providing a relatively concise and self-contained introduction to anomalies in quantum field theory. A first part reviews the necessary notions of quantum field theory, in particular its functional integral formulation which we use throughout. Then we discuss in some detail how and why anomalies arise as non-invariance of the functional integral measure or through the absence of a regulator preserving all symmetries, and how they manifest themselves as current non-conservation in anomalous loop diagrams and anomalous Ward identities, or equivalently as a non-invariance of the effective action. Specifically, we deal with the so-called abelian anomaly under chiral transformations, and the so-called chiral anomaly which is a violation of non-abelian gauge invariance in the presence of chiral fermions. We perform a detailed triangle Feynman diagram computation for both cases. Then we move on to provide a brief overview of some important properties of anomalies (locality, finiteness, one-loop exactness), and how they are characterized in general by the Wess-Zumino consistency condition. We show how this condition can be reformulated in terms of BRST cohomology and how it can be solved, for arbitrary (even) space-time dimensions, by the descent equations. Finally, we briefly discuss the general relation between anomalies in arbitrary even dimensions 2n and index theorems in 2n+2 dimensions, generalizing also to gravitational anomalies. We conclude with the exhibition of the highly non-trivial anomaly cancellation in ten-dimensional IIB supergravity.
ER -