Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry

In this paper the authors proof that linearised volume preserving diffeomorphisms (VPD) agree with the so-called higher rank gauge symmetry at the linear level. In addition they compare their construction with the lowest Landau level, Wigner crystals in a magnetic field, vortex crystals, and ferromagnets. The paper is scientifically sound, and establishes a dual view in the description of such relevant systems. The manuscript has several typos the authors are probably already aware of, therefore I will not point them out here. I assume in a revise version they will correct them. On a more physical and technical level I have some questions/comments I would like the authors to address before I recommend the paper for publication.

Reply: We thank the referee for the fruitful comments, we fixed typos in the new version of our manuscript.

Q:1. In section III.A, the higher rank field is interpreted as a linear perturbation of a spatial metric around the flat one, and connect gauge transformation with linearise VPD, however the non-linear transformation of the field A_0 in section III.B somehow is postulated without much justification. The embedding of the original symmetry into VPD does not seem to be unique. Therefore, some comments on the intuition of the authors to demand such transformation would be convenient. Are the authors claiming that in general higher rank symmetries are associated to physical systems with an underlying magnetic-like field?

Reply: The transformation was motivated by symmetries of the lowest Landau level and the diff transformation of a scalar field. Nonetheless, the non-linear transformation of $A_0$ in equation (24) is the unique transformation that is the generalization of the linear transformation (16) given the following assumptions: - The non-linear transformation satisfies the area-preserving diff algebra (Eq. (23)). - In the background $(A_0,g_{ij})$, $A_0$ is a scalar, therefore the transformation $\delta_{\lambda} A_0$ should be a scalar. - The transformation of $A_0$ is at most linear in $A_0$.
- Rotational symmetry is preserved. We added a comment under equation (24) to emphasize these points. We also added an Appendix to sketch the argument on the uniqueness of the non-linear transformation (24).

Q: 2. Eqs. 27 and 28 relate the fields A_0 and h_{ij} as the sources of a conserved charge \rho and the stress tensor T^{ij}, and claim that since \rho is conserved a usual gauge field A_i can be introduced . However, the continuity equation 28, relates the operator that couples to A_i with T^{ij}, it seems this two sources (h_{ij}, A_i) are not independent. In fact Eq. 43 supports this idea. Is not obvious the necessity of introducing A_i. I would appreciate if the authors could comment on that, it is a bit confusing to me.

Reply: In fact, we have two independent conservation laws (and two Ward’s identities consequently). One is the charge conservation as the usual U(1) in equation (40), and one is for momentum conservation which is equation (41). In the lowest Landau level limit, one can think of equation (41) as the force balance equation from which one can derive current density (or $\vec{\nabla}\vec{j}$) in terms of derivative of stress tensor $T^{ij}$ as in equation (43) as the referee noticed. Then the conservation law in equation (44) is the linear combination of (40) and (41). However, since we begin with two Ward’s identities, we should end up with two conservation laws after manipulations. The second one is the original charge conservation, equation (40). From (44) and (40), one can again relate the derivative of stress tensor $T^{ij}$ and current $j^i$. In summary, the conservation law (28) and the conservation of charge are independent. This is also true in the Lowest Landau level limit that we discussed in Section 4. We added a few comments at the end of Section 4.1 to clarify this point.

Q: 3. In section IV, and related with my previous comment, I wonder what is the physical interpretation of keeping a gauge field fixed under a gauge transformation. This section is dedicated to connect with quantum Hall, I would expect demanding transformations that keep B fixed, but not A_i.

Reply: The answer to this question is partly addressed above. One can indeed consider the transformation that keeps $B$ fixed instead of $A_i$ (The transformation of $A_i$ can be cancelled by a U(1) gauge transformation). Under this transformation, the conservation law should be modified in which it should include charge density $\rho$, current density $j^i$ and stress tensor $T^{ij}$. However, one can eliminate $j^i$ in the new conservation law using the charge conservation law, so the final conservation law includes only charge density and stress tensor. This is exactly the consequence of the gauge transformation that keeps $A_i$ unchanged instead of $B$.

Q: 4. Section V discusses the generalisation of the proposal to 3+1 D, and uses a Kalb-Ramond field as a natural generalisation of the U(1) gauge field A_\mu. This model is intriguing since corresponds to a new class of vector charge higher-rank theory, the continuity equation contains two space derivatives on the current, therefore the cit is a three indices tensor. The usual vector charge systems I am aware of [2], contain a symmetric second rank current, and a continuity equation analogue to the momentum conservation one. Since the authors successfully relate the model with ferromagnetic system, I think more emphasis on their finding, and a comparison of the differences of these two classes vector charge systems would be convenient.

Reply: In fact, the conservation law (70) looks like the conservation law of a vector charge theory. However, since our gauge transformation is distinct from the gauge transformation that generates the vector charge version of symmetric tensor gauge theory in Ref[2], our conservation law differs from one in Ref [2]. Moreover, the definitions of gauge-invariant electric field and magnetic field should be different. We added some comments at the end of section 5 in the revised version to distinguish the differences.

Q: 5. What do the authors mean with the curly bracket notation in Eqs. 72, 74, 75?.

Reply: The curly bracket means symmetrizing over the indices. We added the definition under equation 74. We also added footnote [2] to define the explicit definition in the case of the curly brackets with 2 and 3 indices.

Q: 6. I understand all subsections in VI basically show known results and systems, however for self consistency it is necessary some details on the definition of the objects. I believe this paper is of high interest to a broad audience, and some readers may be familiar with certain systems but not with others. I will enumerate a list a variables that are not properly introduced in the manuscript.

• The quantity b in equation 86

In the revised manuscript, we defined $b$ under equation 86 (equation 93 in the new version).

-The acronym NLS in Eq. 88?

We added the explanation for NLS and the citation to Fradkin’s book for more details above equation 88 (equation 95 in the new version).

• In Eqs. 89 and 93, I think there is a typo, and it should be either J or f^2 in the last terms

Thank you for noticing this typo. We changed the coefficient in equation 93 (equation 100 in the new version) to $J$ for consistency.

• In subsection VI.D when discussing the 3+1D ferromagnets, the authors start with the set of fields n^a, B_{\mu\nu} and g_{ij}, then they define the fields z^\dagger and z however they end up with (z^\dagger,z,g^{ij}, a_i), unfortunately, I didn't find the definition of a_i. In equation 104 should S_{A_0} be S_{B_{\mu\nu}}?

Thank you for the comments. This is the extra section devoted to formulating Ferromagnet in 2+1D using $\mathbb{CP}^1$ parametrization. Therefore it should be $S_{A_0}$ in equation (104) instead of $S_{B_{\mu\nu}}$. To avoid confusion, we separate this section VI.D into two subsections, now 6.4.1 and 6.4.2.

We added the definition of new emergent gauge field $a_i$

We transferred the discussion on $\mathbb{CP}^1$ parametrization to subsection 6.4.1 (Ferromagnets in 2+1D).

Comment: To conclude, I find the paper interesting and with potential impact. I think the results presented here should be published. However, given the questions and comments above I do not recommend it for publication in the present form.

Reply: We thank the referee for expressing interest in our manuscript. We hope that we addressed the issues/comments satisfactorily that the referee raised.