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I've seen a 't Hooft anomaly defined in two ways. Roughly, a theory has a 't Hooft anomaly when

  1. Once the theory is coupled to a background gauge field $A$ (so study eg the partition function $Z[A]$), the symmetry of the theory is violated precisely when $A\neq 0$. So this definition manifestly tells us that we cannot gauge the symmetry

Or

  1. The global symmetry survives and the Ward identities hold. However, the correlation functions are symmetric only at separated points, but contact terms can be symmetry violating. See eg p42 of [1], or [2].

How are these equivalent, if indeed they are? (I feel like I am probably missing something very simple!)


[1] https://www.sissa.it/tpp/phdsection/OnlineResources/16/SISSA_AdS_CFT_course2018.pdf

[2] arxiv 1702.07035

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  • $\begingroup$ Can you give a reference for your second definition of a 't Hooft anomaly? Ward-Takahashi identities generally only hold "up to contact terms", so I'm a bit unclear on where exactly the difference to a non-anomalous symmetry is supposed to like here. $\endgroup$
    – ACuriousMind
    Mar 1, 2020 at 10:56
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    $\begingroup$ never seen your 2. $\endgroup$
    – user21299
    Mar 1, 2020 at 16:18
  • $\begingroup$ Link to abstract pages? $\endgroup$
    – Qmechanic
    Mar 1, 2020 at 17:35
  • $\begingroup$ OK, Alexarvanitakis, but I trust the authors of [2]! $\endgroup$ Mar 2, 2020 at 10:38
  • $\begingroup$ @SvenForkbeard sure so do I --- I don't think it's too simple that 1. and 2. are equivalent! $\endgroup$
    – user21299
    Mar 2, 2020 at 16:51

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