I am a first year PhD student in theoretical physics with a background in QFT (up until relativistic fields, path integrals and gauge theories and anomalies) and some algebraic topology but my understanding of anomalies is weak and I am looking for references for discrete anomalies or anomalies in general. Any textbooks/papers that could be useful are appreciated.

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    $\begingroup$ Anomaly means the symmetry of the classical lagrangian does not persist at the quantum level, because the regularization (and renormalization) procedure doesn’t preserve the symmetry. $\endgroup$
    – QuantumDot
    Dec 31, 2019 at 3:28
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    $\begingroup$ A good introduction might be this lecture series by Bilal. arxiv.org/abs/0802.0634 $\endgroup$
    – nilai
    Dec 31, 2019 at 4:42
  • $\begingroup$ Thanks so much! $\endgroup$ Jan 2, 2020 at 23:07

2 Answers 2


Anomalies is very popular and fast growing theme, so I will try collect main terminology and give major references.

Anomalies with chiral fermions:

David Tong: Lectures on Gauge Theory

Jeffrey A. Harvey: TASI 2003 Lectures on Anomalies

Adel Bilal: Lectures on Anomalies

• An ABJ anomaly implies an explicit violation of the global symmetry. It is as bad as breaking a symmetry by turning on a mass term etc. In 4d it is tested by triangle diagrams with two dynamical fields and one background. Example is famous chiral anomaly: have give rise to novel physical effects, like pion decay.

• A ’t Hooft anomaly has to do purely with background fields, and does not point to any breaking of symmetry (in a trivial background). It has to match between UV and IR for instance (while ABJ anomalies don’t need to match since the symmetry is not existent at all). In 4d it is tested by triangle diagrams with three background fields. (Give us restrictions on the low-energy dynamics of our theory)

• While anomalies in global symmetries are physically interesting, anomalies in gauge symmetries kill all the physics completely: they render the theory mathematically inconsistent! This is because “gauge symmetries” are not really symmetries at all, but redundancies in our description of the theory. It we wish to build a consistent theory, we must ensure that all gauge anomalies vanish.

The SU(2) Anomaly or non-perturbative anomaly (also for $Sp(N)$ group). This was first discovered by Witten and, unlike our previous anomalies, cannot be seen in perturbation theory. It is a non-perturbative anomaly. An SU(2) gauge theory with a single Weyl fermion in the fundamental representation is mathematically inconsistent. Furthermore, an SU(2) gauge theory with any odd number of Weyl fermions is inconsistent. To make sense of the theory, Weyl fermions must come in pairs.

Anomalies in Discrete Symmetries

David Tong: Lectures on Gauge Theory, 3.6

Yuji Tachikawa: Anomalies and topological phases

Finite group symmetry (both internal and spacetime) can have an anomaly. These will have nothing to do with chiral fermions, or ultra-violet divergences in quantum field theory. There is a mixed ’t Hooft anomaly, but it is between rather different symmetries, known as generalised symmetries. Using this technique dynamics of non-supersymmetric pure Yang-Mills at $\theta=\pi$ was analyzed in detail.

A conformal anomaly, scale anomaly, or Weyl anomaly

Zohar Komargodski: Aspects of Renormalization Group Flows

Zohar Komargodski: Lectures Notes

David Tong: Lectures on String Theory: 4.2.2, 5

Anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory.

Famous example from string theory: In string theory, conformal symmetry on the worldsheet is a local Weyl symmetry and the anomaly must therefore cancel if the theory is to be consistent. The required cancellation implies that the spacetime dimensionality must be equal to the critical dimension which is either 26 in the case of bosonic string theory or 10 in the case of superstring theory.

Gravitational Anomalies

Marco Serone: Anomalies in Quantum Field Theory: 8,9

L. Alvarez-Gaume and E. Witten: Gravitational Anomalies

Beside Spin 1/2 also for Spin 3/2 and Self-Dual Tensors. Pure gravity anomaly exist only in $d=4n+2$.

I also recommend to read Samuel Monnier: A Modern Point of View on Anomalies

  • $\begingroup$ That was very helpful, thank you very much! $\endgroup$ Jan 2, 2020 at 23:08

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