There are at least two things that are called quantum anomalies in the context of quantum field theory

• anomalous action functional: the action functional (in path integral quantization) is not a globally well defined function, but instead a section of a line bundle on configuration space?;

• anomalous symmetry: a symmetry of the action functional does not extend to a symmetry of the exponentiated action times the path integral measure; or equivalently the action of a group on classical phase space? is not preserved by deformation quantization?.

anomalous action functional

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• see also Green-Schwarz mechanism

anomalous symmetry

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Examples

spinning particles and super-branes

The sigma-model for a supersymmetric fundamental brane on a target space $X$ has an anomaly coming from the nontriviality of Pfaffian line bundles associated with the fermioninc fields on the worldvolume. These anomalies disappear (i.e. these bundles are trivializable) when the structure group of the tangent bundle of $X$ has a sufficiently high lift through the Whitehead tower of $O(n)$.

• Spin structure the worldline anomaly for the spinning particle/superparticle vanishes when $X$ has Spin structure

  This is a classical result. A concrete derivation is in

  Witten, Anomalies etc. (somewhere)

• String structure the worldsheet anomaly for the spinning string/superstring in heterotic string theory? vanishes (essentially) when $X$ has String structure

  This is originally due to Killingback and Witten. A commented list of literture is here. Recently Ulrich Bunke gave the rigorous proof

  • Ulrich Bunke, String structures and trivialisations of a Pfaffian line bundle (arXiv)

• Fivebrane structure the worldvolume anomaly for the super-5-brane in dual heterotic string theory? vanishes (essentially) when $X$ has Fivebrane structure. See there.

References

A reference that very clearly identifies the mathematical nature of quantum anomalies for higher gauge theories is

• Dan Freed, Dirac charge quantization and generalized differential cohomology (arXiv)