At some point in our physics education, we begin to accumulate a bunch of slogans related to anomalies. At some (later, in my case) point, we learn that actually there were two different kinds of anomalies all along: 't Hooft and ABJ anomalies. We then realize we need to go back and disentangle things; which slogans pertain to which kinds of anomalies? Broadly speaking, I appreciate that the difference is mainly that a global symmetry with a 't Hooft anomaly is still a symmetry in some sense, just one that cannot be gauged, while a global symmetry with an ABJ anomaly is simply not a symmetry of the quantum theory. But I would like some more details. At first I felt a bit embarrassed about asking this question, but after talking with various people, my sense is that the confusion about this issue is widespread enough that others besides just me would benefit from having a coherent answer in one place. So below, I offer a list of slogans, and my guesses about which kind of anomaly they are meant to be describing, and I was hoping someone could help me put each in the correct column, so to speak. Many of the slogans also have accompanying questions. I apologize in advance for the length!
The first definition you encounter is that an anomaly is a symmetry of a classical theory which does not survive the transition to the quantum theory. I expect this slogan applies to both kinds of anomalies, but that one is referring to slightly different things which are not surviving the transition (to be elaborated upon throughout the rest of this Stack Exchange post). But already, I have a sort of complaint about this definition. It makes reference to a classical theory, whereas my impression is that an anomaly is a property intrinsic to the quantum theory, and can be/often is (at least in certain settings) formulated intrinsically in quantum terms. (One place where this seems to happen is in CFTs, where a 't Hooft anomaly is characterized by certain universal terms appearing in the current-current OPEs, without any reference to a classical theory). It seems both practically and conceptually important to be able to do this in general because there are many examples of theories with no known Lagrangian/classical limits (e.g. many entrants in the zoo of 4d $\mathcal{N}=2$ SCFTs, though it's my understanding that what one means by a "non-Lagrangian" theory in this setting is a moving goal post).
In the case of a continuous symmetry, one often says that it is anomalous if the corresponding current (in the presence of a background gauge field) does not satisfy $\partial_\mu j^\mu = 0$ in the quantum theory. I know this is true of the ABJ anomaly. Can you have a 't Hooft anomalous continuous global symmetry for which $\partial_\mu j^\mu =0$? (I think yes, but the entry two items down in the list seems to be in tension with the answer being yes).
The above definition is unavailable in the case of discrete symmetries: I know that there are discrete 't Hooft anomalies, but are there discrete anomalies of ABJ-type? What would one compute?
Anomalies are computed by triangle diagrams. Again, I know this is true of the ABJ anomaly, but I heard recently from someone that it's also true of 't Hooft. If it is true, how is the procedure modified?
An anomaly arises because the path integral measure does not respect the symmetry. This at least applies to ABJ; does it apply to 't Hooft? Again, there is this issue of non-Lagrangian theories.
The Hilbert space doesn't organize into representations of the symmetry group. If this is true, I think it refers to the ABJ anomaly. But in the case of 't Hooft, one says that the symmetry is still a fine symmetry of the quantum theory so I would think that there is still an action of the symmetry group on the Hilbert space.
I know there is some relationship between projective representations and anomalies; this must be referring to 't Hooft. Let me start with 't Hooft anomalies in quantum mechanics (i.e. 0+1-dimensional QFT). In appendix D of this well-known paper, the claim is made that 't Hooft anomalies in QM manifest themselves as projective representations of the classical symmetry group. If one was just handed the quantum theory, they can't really tell whether $G$ is being realized projectively or a central extension of $G$ is being realized honestly, so can I really determine whether a symmetry is anomalous in quantum mechanics without having some notion of what the classical theory is? I suspect this issue is related to the idea that what one means by a symmetry in quantum mechanics is a little finicky, because if you mean "all unitary operators which commute with the Hamiltonian" for example, then this collection of operators is certainly far too large; you typically eliminate most of those operators by demanding that symmetries are realized in a not too horrendous way on the fields which appear in the classical Lagrangian.
Continuing with the the above item, if we graduate to higher dimensional QFT, certainly one can still talk about the symmetry group being realized projectively on the Hilbert space. But does this necessarily mean there is still a 't Hooft anomaly if it is? On the one hand, I feel that the answer is probably no because in general, many anomalies are classified by the cohomology group $H^{d+1}(G)$; for $d=1$ (QM), this cohomology group indeed classifies the projective representation theory of $G$, recovering the statement in the paper I cite above, but for $d>1$ it corresponds to something else. On the other hand, I suspect the answer is in some sense yes, because e.g. the Weyl anomaly in 2d CFT corresponds to a central extension of the Witt algebra (i.e. the central charge of the Virasoro algebra), which presumably means you can think about this as the non-centrally extended algebra is being realized projectively. What is the resolution?
A standard way to detect an anomaly is to couple to background gauge fields (discrete or continuous) and do a gauge transformation, checking whether the theory is invariant. Surely, this should apply to both ABJ and 't Hooft, since this condition is describing an obstruction to gauging, and both kinds of anomaly pose such an obstruction. So when I do this computation, how do I know whether I have a 't Hooft anomaly or an ABJ anomaly? I have the vague impression that this failure of gauge invariance is supposed to occur in a "controlled" way for 't Hooft anomalies, but perhaps someone could make this more precise.
Here are a few slogans all of which I believe apply to 't Hooft anomalies. An anomaly can be characterized by a classical theory in one dimension higher, which cancels the anomaly on its associated boundary theory via anomaly inflow (e.g. Chern-Simons in the bulk and fermions on the boundary). The bulk theory is called an SPT phase; these are classified (to a large extent, but perhaps not entirely) by group cohomology.
Is any of the discussion regarding 't Hooft anomalies modified if the symmetry is spontaneously broken?
Thanks in advance for any insights anyone can offer.
