In quantum field theory, a global symmetry group that can't be gauged is said to have an 't Hooft anomaly. One of the most familiar examples is the free massless Dirac fermion in $3+1$ dimensional spacetime: it has a global $(U(1)_V\times U(1)_A)/\mathbb{Z}_2$ symmetry, and we can gauge the $U(1)_V$ part to get quantum electrodynamics, but then we can't gauge the rest of it.

That's a nice example because it starts with a non-interacting quantum field theory. Many different types of 't Hooft anomaly are known from other quantum field theories, many of them not involving chiral fermions, and non-interacting examples of those would also be nice to have — but slide $13$ in [1] seems to say that such examples don't exist! Here it is in the author's words:

For free theories or theories which are free in the UV, all 't Hooft anomalies arise from chiral fermions...$^\dagger$

The statement is probably true (the author is an expert), but I don't know how to deduce it. I'm not even sure I understand what it means, because pages $19$-$22$ in the same presentation review what seems to be an exception, namely the 't Hooft anomaly in the combination of the electric and magnetic $1$-form symmetries of the free electromagnetic field. Isn't that an example of an 't Hooft anomaly in a non-interacting theory that doesn't arise from chiral fermions? What am I missing? Does the previous statement about chiral fermions only apply to conventional ($0$-form) global symmetries? Or is there a sense in which the electromagnetism example isn't "free" (like the lattice version with a compact gauge group)?

Question: What exactly are the condition(s) that make the highlighted statement true?

The paper [2] has the same title and includes the same author. I haven't finished studying it yet (that will take me a while), but I searched through it and didn't find an answer.


$^\dagger$ The Dirac fermion example involves "chiral fermions" in the sense that the global symmetry with the anomaly acts independently on the two chiral parts.


[1] Kapustin, slides titled "Generalized Global Symmetries" (http://physics.berkeley.edu/sites/default/files/_/PDF/kapustin.pdf)

[2] Gaiotto et al, "Generalized Global Symmetries", https://arxiv.org/abs/1412.5148


1 Answer 1


In my understanding, what the author means in the slides is that the anomalies can be carried by chiral fermions. Namely, that for free theories the 't Hooft anomaly can be realised by inflow through a Chern-Simons term, and so since chiral fermions have the same inflow it can be cancelled by putting chiral fermions in the theory. Reversing the logic if one is interested just in the anomaly they can study it by studying the chiral fermions (see also [1]).

This is a special case of the belief that anomalies can always be saturated by symmetry preserving gapless phases [2] (an aside to that, the main claim of [2] was that they can't be always saturated by symmetry preserving gapped phases)

[1] L. Alvarez-Gaume and P. H. Ginsparg, The Structure of Gauge and Gravitational Anomalies, Annals Phys. 161 (1985) 423.

[2] C. Córdova and K. Ohmori, Anomaly Obstructions to Symmetry Preserving Gapped Phases, [arXiv:1910.04962].

  • $\begingroup$ That makes sense. Thank you! To make sure I understand the scope: does this include the anomaly in the mixed $1$-form symmetry in the pure-electromagnetism example I mentioned? I mean, is that anomaly the same as the anomaly of some chiral-fermion theory, in the sense that both can be realised by inflow from the same $5$-dimensional Chern-Simons theory? Or, equivalently, that the anomaly in the pure-electromagnetism theory could be cancelled by adding chiral fermions? $\endgroup$ Commented Apr 27, 2020 at 12:32
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    $\begingroup$ Yes, that's the claim. See also the discussion around page 38 of arxiv.org/pdf/hep-th/0509097.pdf. Of course, higher-form symmetries weren't a thing yet in 2005 (at least weren't formalised), but weirdly enough Harvey discusses higher-form Chern-Simons inflow. $\endgroup$ Commented Apr 27, 2020 at 13:09

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