I'm learning about anomalies and I'm a bit confused about their relationships to 2-cocycles and 3-cocycles (in the group cohomology $H^{\bullet}(G, U(1))$). The below might only apply to 't Hooft anomalies.

I read in some places (e.g. this answer) that an anomaly corresponds to a nontrivial 2-cocycle, i.e. when the Hilbert space transforms under a projective representation.

Elsewhere (e.g. p4 of this paper) 't Hooft anomalies in a $D$-dimensional theory correspond instead to non-trivial (D+1)-cocycles; later in s2.2 the author refers to a `nontrivial extension with a trivial anomaly'. I asked a prof and he said the anomaly is the obstruction to uplifting the projective representation to a genuine representation. But I was under the impression that you could always uplift to a genuine representation of a central extension.

Would appreciate any clarification/classification/pointers to references where the two things are discussed together - and of course corrections for anything I have misunderstood.

  • $\begingroup$ Due diligence. $\endgroup$ Dec 14, 2022 at 15:23
  • $\begingroup$ @CosmasZachos I'm not sure what you mean. Could you indicate which section might be helpful? The word cocycle does not appear, and the cohomology bits I can find seem to be about specific scenarios and are quite involved. Do you know something that addresses my question a bit more directly? $\endgroup$
    – quixot
    Dec 14, 2022 at 15:35
  • $\begingroup$ Quicker. Your must slug through the entire Bilal review. Cocycles are implicit there. $\endgroup$ Dec 14, 2022 at 15:51
  • $\begingroup$ sorry. what is your question, exactly? $\endgroup$ Dec 15, 2022 at 17:51
  • $\begingroup$ I would like to know which cocycles are considered a diagnostic of an anomaly, and which anomalies are associated with which cocycle degree. And how the different degrees interact - e.g. if the 2-cocycle is only relevant when the higher cocycle is trivial. $\endgroup$
    – quixot
    Dec 15, 2022 at 20:25


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