I'm reading about quantum anomalies in QFT and all the examples seem to arise in gauge theories. Is it true that theories without a local gauge invariance don't have quantum anomalies? I can't think of examples of symmetry violation in any non-gauge theories, particularly the free theories. Then again, there might be a well-known example I'm not aware of!

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    $\begingroup$ In fact, I would say that anomalies have relatively little to do with gauge theories, although they are commonly studied there. As Prof. Wen says they are about the impossibility to construct a UV complete quantum theory, because the low energy degrees of freedom cannot be quantized consistently. Therefore there must be other degrees of freedom above the cutoff. Example; there are restrictions on fermion hilbert spaces because of time reversal. The axial anonmaly keeps track of the violation of these restrictions. $\endgroup$ – BebopButUnsteady Aug 21 '13 at 18:26

There are things called sigma model anomalies, see papers listed in a sample inspire database query here.

Here, the anomaly is associated to the general coordinate invariance in the target space of the non-linear sigma model: the fields take values in a nontrivial manifold (and its associated vector bundles), rather than vector spaces. Classically, the action is independent of the coordinates used to describe the target manifold. But this independence can be lost via quantization.


A more general definition of anomaly: A QFT that has no UV completion in the same dimension is anomalous.

In other words, a QFT that has no well defined short distance regularization in the same dimension is anomalous.

Example: A 1+1D QFT with only one right moving fermion mode is anomalous.

  • $\begingroup$ Many thanks for your answer. So am I right in thinking that anomalies only arise because of regularization then? But if so, what about the chiral anomaly? That's just a result of the measure not being invariant, I think. $\endgroup$ – Edward Hughes Aug 21 '13 at 21:59
  • $\begingroup$ The UV completion definition of anomaly include the ABJ chiral anomaly (see my recent paper arXiv:1303.1803 ) $\endgroup$ – Xiao-Gang Wen Aug 21 '13 at 23:15
  • $\begingroup$ In regard to this answer and your interesting recent paper, you could perhaps be interested in writing an answer to my question physics.stackexchange.com/questions/33195/… . I would love to read it. $\endgroup$ – Diego Mazón Jan 27 '14 at 19:33
  • $\begingroup$ @drake: I am writing a related long paper. I will try to write an answer to your very interesting question after I finished the paper. $\endgroup$ – Xiao-Gang Wen Jan 28 '14 at 3:38
  • $\begingroup$ @Xiao-GangWen I look forward to it. $\endgroup$ – Diego Mazón Jan 29 '14 at 4:41

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