Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond

We thank the referee for their accurate summary of our paper and their positive recommendation for our paper to be published in SciPost Physics.

The referee writes:

page 4: maybe you should mention above eq 1.1 that you will assume for simplicity that not only $Q_L$ but also that $A_L$, $G_L$ are all abelian groups, as that seems to be the assumption later in the paper.

Our response:

We have added such a remark after Eq. 1.1 that we consider examples whose groups $A_{\mathcal{L}}$ and $Q_{\mathcal{L}}$ are abelian. However, the group $G_{\mathcal{L}}$ under these conditions need not be abelian, and we consider examples with $G_{\mathcal{L}}$ non-Abelian. For example, $G_{\mathcal{L}}$ is non-Abelian whenever $A_{\mathcal{L}}$ describes a modulated symmetry since then ${G_{\mathcal{L}} = A_{\mathcal{L}} \rtimes Q_{\mathcal{L}}}$.

The referee writes:

page 7: "Generalizations to non-invertible symmetries are conceptually straightforward" why is it conceptually straightforward? Much of the paper seems to get the SymTFT by considering an inflow SPT and then gauging the internal part of the symmetry. In the general non-invertible case this seems difficult to implement because there is no nice inflow SPT to begin with? For non-invertible it seems that you might instead want some direct generalization of the Drinfeld center construction to include spatial symmetries and outputs the topological defects of an SET?

Our response:

The aspects we were referring to, which seem "conceptually straightforward" were unclear in our original sentence, and we thank the referee for pointing this out to us through their question. For simplicity, we have removed the sentence ``Generalizations to non-invertible symmetries are conceptually straightforward.'' from the manuscript.

What we originally meant by the sentence was that one can explore interplays between spacetime symmetries and general fusion category symmetries by enriching non-Abelian topological orders with spacetime symmetries. This seems like a conceptually straightforward generalization to studying SETs described by symmetry-enriched finite gauge theory. We agree that whether such general SETs can be constructed by gauging 0-form symmetries of some SPT is not at all clear.

The referee writes:

top line of page 11: "No anyons in $A$ carry fractional $Q$ charge" might be better wording.

Our response:

This phrasing is much nicer than our original; we have adopted it!

The referee writes:

can you move figure 4 to where it is first referenced (page 13)?

Our response:

Good point, thank you! We have moved figure 4 (now figure 3) to appear where it is first referenced.

The referee writes:

Footnote 15 seems conceptually important. Consider putting it in the main text.

Our response:

We agree that footnote 15 is conceptually important. However, for the sake of the text's flow, we kept it as a footnote.

The referee writes:

Regarding footnote 16, I thought that we should consider the SymTFT to be the inflow SPT after gauging $N$. Then if we in addition gauge $Q$, we get the usual SymTFT (if $Q$ is finite, internal, etc.). In this case, there should be no ambiguity of the defectification class? In particular, there should still be signatures of self-anomalies of $Q$ from the symmetry fluxes of the SymTFT even before gauging $Q$, which are pieces of SET data? In principle this should be part of the data describing boundaries of the SET, so that even when the underlying topological order is trivial (i.e. the SET is just an SPT) one can see that there is no gapped, symmetric boundary.

Our response:

Our understanding of the defectification class for an internal finite $Q$ symmetry is that it describes the different $Q$ SPTs that can be stacked onto the ${2+1}$D $Q$-SET. This would make the defectification class the $2+1$D signatures of $Q$'s possible self-anomaly in $1+1$D without having to gauge $Q$ in $2+1$D. The referee is correct that the defectification class can be detected before gauging $Q$, but it would not manifest through a nontrivial anyon automorphism or symmetry fractionalization class. After gauging $Q$, the defectification class would indeed affect the resulting SymTFT, appearing as a type of $Q$ Dijkgraaf-Witten term for the SymTFT.

The referee writes:

Around eq 2.22 maybe you should mention that $[\nu]$ is describes a "type III cocycle" and $[\eta]$ describes a "type II cocycle"? Because you mention these types later (on page 49) without definition. Then on page 49 you can refer back to around eq 2.22.

Our response:

We agree that when $A = \mathbb{Z}_{N_1} $ $\times$ $\mathbb{Z}_{N_2} $ and $Q = \mathbb{Z}_{N_3} $, the total anomalies described by Eqs. 2.25 and 2.26 are type III and type II anomalies, respectively. We are not sure if this terminology holds for more general $A$ and $Q$, so we will play it safe and not use "type II" and "type III" cocycles. However, we thank the referee for their suggestion!

The referee writes:

page 20: notation $\to$ notion?

Our response:

Great catch, thank you!

The referee writes:

page 21: why is the SPT trivial? can the modulated symmetry be anomalous?

Our response:

We assumed throughout section 3 that each modulated symmetry is anomaly-free. This assumption was implicitly made when we specialized to modulated symmetries whose symmetry operators took the onsite form of Eq. (3.1). However, modulated symmetries can have anomalies, in which case the SPT mentioned at the beginning of Section 3.1 would be non-trivial. We have expanded footnote 22 to explicitly emphasize why the SPT is trivial and that it would not be if the modulated symmetry were anomalous.

The referee writes:

eq 3.47-48 I think you forgot some brackets here i.e. $Rr \to R(r)$

Our response:

Indeed, thank you for noticing this!

The referee writes:

In general, there are also examples of anomalies, phases, etc. involving spacetime symmetries alone (i.e. just time-reversal). How would one approach such symmetries?

Our response:

There are indeed phases and anomalies characterized solely by spacetime symmetries. Within our formalism, the SymTFT for such a scenario would be a trivial topological order enriched by spacetime symmetries --- a spacetime SPT. This applies to both discrete and continuous spacetime symmetries. Getting the full SymTFT from this SPT would require gauging spacetime symmetries. However, we note that some progress has recently been made in gauging continuous spacetime symmetries in arXiv:2509.07965.

The referee writes:

can one also construct a symTFT that diagnoses things like filling fraction/sector anomalies? These types of anomalies can even occur when the internal symmetry is finite.

Our response:

We thank the referee for this interesting question. It is something we have thought about. These filling constraints and their generalizations are anomalies that arise after projecting the Hilbert space to a fixed symmetry sector. Within this symmetry sector, there is no longer a symmetry operator, but still symmetry defects. To include filling constraints into the SymTFT formalism, one would not only have to capture the interplay of symmetry defects with discrete translations (as done in our paper) but also generalize the SymTFT to classify phases within fixed symmetry sectors. We believe that this additional step causes this interesting question to fall outside the scope of this current paper and justifies an independent investigation. We are currently interested in pursuing this as a follow-up problem to this paper.

We thank the referee for their accurate summary of our paper and their positive recommendation for our paper to be published in SciPost Physics.

The referee writes:

The set of “uncondensed” anyons $A_{\mathcal{L}}$ should be described as the quotient $A/\mathcal{L}$, rather than as a subgroup of $A$, as written above Equation (2.4).

Our response:

We thank the referee for pointing out this imprecise statement made in the paper. We have changed the corresponding text to refer to $A_{\mathcal{L}}$ without explicit reference to $A$.

The referee writes:

It would be very helpful if the authors could also summarize some of the examples and results from Sections 4 and 5, similar to the illustrative example already included for Section 3.

Our response:

We thank the referee for this suggestion! We have added a new paragraph at the end of the summary section to include more details from the example worked out in Section 4.