Modulated symmetries from generalized Lieb-Schultz-Mattis anomalies

Authors study modulated symmetries obtained by gauging a non-anomalous internal subgroup in a model with LSM-like anomaly. The anomaly appears as a projective phase in the algebra of symmetry operators that is dependent on the system size. The manuscript focuses mainly on two-dimensional space where such a projective phase can appear in a higher group setting with various 0-form and 1-form symmetries. The results are novel as the duality between the modulated symmetries and (0-form) internal symmetries with LSM anomalies have been explicitly shown in 1D previously, while they are not unexpected as the gauging-induced dualities in the lattice models are well-known at this point. I think the manuscript has room for improvement and accordingly I have the following comments.

1) It appears that the manuscript is mostly centered around gauging symmetries by minimally coupling particular models with these symmetries. However, the gauging induced dualities should apply to all Hamiltonians with the appropriate symmetries subjected to projection onto the appropriate sub-Hilbert space. Authors should add more explanation from this point of view, i.e., gauging-induced dualities as isomorphisms between algebras of symmetric local operators.

2) In relation to the above point, it can be beneficial to discuss what kind of phases can be realized in systems with the LSM-like anomalies the authors considered and what these phases are mapped to under the gauging map the authors considered. For example, when an LSM-like anomaly is present, there cannot be non-degenerate gapped ground state that is simulatenaously symmetric.

3) In the prelude of Sec. 3, the authors make the claim that the phase factor depending on the area of the system somewhat is beyond what is conventionally called as LSM anomaly. This is not correct as the generalization of LSM anomalies with 0-form symmetries to higher than 1D predates generalization to those with higher-form symmetries. For example, an important paper authors should cite is Yao, Oshikawa, PhysRevLett.126.217201. I find this paragraph very misleading.

4) The equation on page 3 is not precise as the projective phase between p- and q-form symmetries should also depent on some linking number in general.

5) Authors use operators with hat and without hat to denote different things. I found this notation confusing.

6) In the label of Fig. 2, it should read "...defined in (14), that respect...". I also do not understand the sublabel (b) which I think should be rewritten.

7) Above Eq. (32) it should read "...0-form modulated symmetry..."

8) I do not understand what authors mean by "...to make the gauged theory dynamically trivial."

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1-, The illustrations for the interactions are very beautiful and easily understood.

This paper discusses the relation between dipole symmetries and LSM anomalies, in terms of discrete gauging. It extends the relations to higher-form dipole symmetries and generalized anomalies in models of higher spacetime dimensional and provides concrete examples of lattice models in (2+1)d and (3+1)d. It also conjectures the relation in arbitrary spacetime dimensions and more general modulated symmetries. I think it meets the criteria of this journal to publish, once the authors are able to answer the questions mentioned below.

1-, In page 16, the first paragraph, the last sentence, $Q_I[\Sigma_{-p}]$ should be $Q_I[\Sigma_{d-p}]$?
2-, In page 20, the next to last paragraph, the last sentence, "a" should be "an".
3-, In page 24, formula 89, the operators in the second term should be $Z$'s not $X$'s.
4-, In page 32, around formula 116, $N$ should already be abelian when the authors state that formula 116 is a central extension that is characterized by $H^2(K,N)$.

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This paper studies the connection between modulated symmetries and Lieb-Schultz-Matthis (LSM) anomaly via gauging. It extends earlier observations in one-dimensional spin chains to higher dimensions, showing how gauging connects modulated higher-form symmetry and LSM type anomalies. It provides both explicit lattice constructions and field theoretic descriptions.

The referee recommends the publication of this manuscript in SciPost once the authors address the questions and comments raised by the referee.

1. In the abstract, “In one-dimensional a spin chain” -> “In a one-dimensional spin chain”

2. In (21), it was stated that $T_y\xi^Z_{x,y} T_y^{-1}=\xi_2^{Z\dagger} \xi_{x,y}^Z$. This equation holds only in the ground state subspace, where $B_{l_y}=1$ and is not true on the full tensor-factorized Hilbert space. The authors should clarify this point.

3. In (32), there are two zero-form symmetry listed after gauging, while we have only one zero-form symmetry $U_Z^{(0)}$ before gauging other than the $U_X^{(0)}$ symmetry that were gauged. It would be helpful if the authors can clarify why the symmetry operators are doubled. One explanation the referee sees is that the $U_Z^{(0)}$ does not commute with the Gauss law in (25) but it can be made gauge invariant in two different ways, which then lead to the two symmetry operators listed in (32).

4. Related to point 3, in the field theory analysis in section 4.3 (which is related to the lattice model discussed section 3.1), there is only one zero-form symmetry after gauging the LSM anomaly, while on the lattice, there are two zero-form symmetry after gauging as listed in (32). Can the author give a field theoretic explanation on this point?

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