# Symmetry anomaly and energy spectrum

Let us consider 't Hooft anomaly: $$\begin{eqnarray} Z[A^\lambda]=Z[A]\exp(i\alpha[A,\lambda]), \end{eqnarray}$$ where $$A$$ is the background $$G$$-gauge field and $$\lambda$$ is some $$G$$-gauge transformation.

We know that a nontrivial factor $$\exp(i\alpha[A,\lambda])\neq1$$ obstructs $$G$$-gauging and, if the symmetry $$G$$ is acted in an on-site manner, such a system cannot be realized in its own dimension(s).

My question is, how can we argue that the anomalous system cannot have a fully gapped spectrum with a unique ground state? In my understanding, a unique gapped ground state must be $$G$$-singlet, so we expect its partition function should be unambiguous. However, how can a gapless spectrum or a $$G$$-symmetric (non-symmetry-spontaneously-broken (SSB)) multiple ground states or SSB have an ambiguous partition function?

My second question is, the anomalous symmetry only implies that the symmetry cannot be on-site-ly realized. If the symmetry is non-onsite at the UV scale, we can still have a nontrivial anomaly in the IR field theory. In this case, can we argue that an IR anomaly implies that the system at the UV cannot be gapped by local $$G$$-symmetric interactions with a unique ground state?

For the first part, the most elegant way (that I know of) to see this is through anomaly inflow, where a $$d$$-dimensional anomalous QFT$$_d$$ cannot be consistently defined in $$d$$ dimensions, but rather must be accompanied by a Symmetry Protected Topological phase in one higher dimension (SPT$$_{d+1}$$) carrying edge modes which exactly compensate for the anomaly.

Now, suppose that you calculated the anomaly in the UV. Equivalently you found the invertible theory corresponding to the non-trivial SPT$$_{d+1}$$ phase which captures the anomaly. I.e. you found $$Z_{\text{SPT}}[A]$$, such that: $$Z[A^\lambda]Z_{\text{SPT}}[A^\lambda] \overset{!}{=} Z[A]Z_{\text{SPT}}[A],$$ where now $$A$$ is extended into the bulk. Keeping in mind the 't Hooft anomaly matching conditions, you flow with the RG all the way to the IR. With you flows also the SPT$$_{d+1}$$ phase. Any non-trivial SPT$$_{d+1}$$ phase can't have a unique ground state when placed on an open manifold (here it is necessarily placed on an open manifold since you want its boundary to be the original anomalous QFT$$_d$$). Translated back to QFT this means that the IR of your QFT cannot be trivially gapped. The IR should then either be gapped but non-trivial, i.e. a TQFT, or gapless, or SSB must occur.

As for the question about "how can the other possibilities have an ambiguous partition function?", notice that $$A$$ here is a background gauge-field, namely you can choose its value, tune it and not move it around. The partition function is perfectly well defined and unambiguous. The ambiguity lies in the gauge transformations, but these would only be dangerous had the gauge fields been dynamical. In the background case, when you don't look no one is going to mess with your gauge field, so everything's fine.

For your second question, if I understand correctly, what you claim is believed to be true. In particular adding $$G$$-symmetric interactions in the UV move you around the same deformation class in the words of Seiberg. Then your claim follows from Seiberg's claim [1], that: "All the theories in the same deformation class (obtained by adding such degrees of freedom and varying the parameters) have the same symmetries and anomalies". Even if I don't understand your question correctly, it is still probably very related to the aforementioned Seiberg claim.

Reference

[1] N. Seiberg, Thoughts about Quantum Field Theory, talk at Strings 2019

• Many thanks for the nice answer! I guess the fact that any nontrivial invertible phase cannot be fully gapped on open manifolds is due to that if we create a boundary, the original bulk response partition function is not gauge invariant. Thus wrongly assuming a unique gapped ground state allows us to integrate out the matter field, which must give a gauge invariant one and a contradiction. Is this understanding correct? May 21, 2020 at 1:21
• I'm not entirely sure I understand what you mean. The matter fields are already integrated out and have virtually nothing to do with the gauge non-invariance of the bulk partition function. What ruins the uniqueness and gappedness of the SPT phase ground state is that on an open manifold there are dangling edge modes May 21, 2020 at 9:39
• Yes, maybe there was some abuse in my last reply. I actually meant that integrating out the matter field results an invertible partition function, like a U(1) term depending on gauge fields. I guess my confusion has been already resolved by your answer. May 21, 2020 at 9:44