Dipole symmetries from the topology of the phase space and the constraints on the low-energy spectrum

We would like to thank the referee for having read the revised version of the manuscript and recommended it for publication. Below, we address the two points that the referee draws attention to.

The first is on the discussion in Section 9, around Equation (59-60). My intuition is that somehow this dipole charge must end up trivial, because ordinarily one expects that having dipole symmetry leads to some important constraints on EFTs (changes to exponents in dynamics, Mermin-Wagner, etc...), but if I give you a generic field theory with $(\partial_0\phi)^2-V(\phi)$, I don't expect such constraints to be relevant. I think the authors should remark on this - some comments are made at the end of the paragraph but I found them a bit hard to interpret. To be direct here: is there anything useful that this "hidden" dipole symmetry tells me about the "ordinary" theory I suggested above.

Reply: The part of Sec. 9 that the referee points to revolves around the relation between local momentum conservation and the dipole conservation law in presence of the linear momentum problem (LMP). This issue is intimately related to the global structure of the target space $\mathcal M$ from which the canonical variables of the theory take values. Namely, as we show, a sufficient condition for the appearance of LMP is that the symplectic form of the theory (treated as a 2-form on $\mathcal M$) is cohomologically nontrivial. As we mention at the beginning of Sec. 2 (see also the example of antiferromagnets, discussed at the end of Sec. 4.1), for theories that have a well-defined, second-order Lagrangian description with a manifold $\mathcal N$ as the target space, the target space $\mathcal M$ of the Hamiltonian formulation is the cotangent bundle $T^*\mathcal N$. The corresponding symplectic form is exact, leading to the absence of LMP. For the type of "generic field theory" suggested by the referee, the local momentum conservation is therefore problem-free. In such cases, the dipole conservation law is literally just the curl of the local momentum conservation law, the scalar charge density being nothing but a generalized vorticity. There may still be field configurations satisfying a nontrivial boundary condition at spatial infinity for which the integral charge $Q$ is nonzero, and the momentum algebra is thus centrally extended. This however also comes with some topological constraints, similarly to the superfluids discussed in Sec. 4.3, where the presence of vortices is associated with a nontrivial first cohomology group of the vacuum manifold. Finally, there is a class of theories that do not feature LMP and where the scalar charge $Q$, if well-defined, always vanishes. This is the "default" option, likely realized for all theories with a single real-valued scalar field. For such theories, the dipole symmetry, both locally and globally, does not carry any new information beyond what follows from translation invariance alone. In the newly revised version of the manuscript, we make the distinction between these different classes of theories more explicit by expanding the discussion in the paragraph below Eq. (60).

Second and relatedly, I don't think it's accidental at all that the interesting examples described by the authors all have Lagrangians of the form Eq. (6). It seems that then the appearance of a multipole algebra is quite similar to Ref. [6], although I think the higher-dimensional generalization in this paper looks new. Anyway, the fact that the authors' model has $P_i$ playing the same role as $D_i$, and thus obtaining a multipole algebra, seems to me closely related to the Lagrangian being of the form Eq. (6). For example, if I wrote my generic simple theory from the previous paragraph in the form of Eq. (6) I would need to introduce an additional degree of freedom beyond $\phi$, and that might qualitatively change how I would interpret things such as the LMP?

Reply: In Ref. [6], a generalized vorticity conservation equivalent to the dipole conservation law is derived by taking the curl of momentum conservation. However, for us - as stressed in the conclusions - this is just a "mnemonic" since much of our paper focuses on theories with LMP where the local momentum density is ill-defined. What is new in our paper is the direct and explicit connection between the dipole conservation law, the central extension of momentum algebra, and the symplectic structure of the theory; this appeared in Ref. [6] only through specific examples and not as a universal feature of the Hamiltonian description of field theory.

It is indeed not accidental that the interesting examples in our paper have a first-order Lagrangian formulation of the form (6). This is because, as pointed out in our response to the referee's previous point, theories that have a nonsingular second-order Lagrangian formulation tend to have an exact symplectic form. Moreover, as the referee correctly states, writing such theories in the form (6) requires adding new degrees of freedom, namely the generalized momenta. This is discussed on the concrete example of antiferromagnets in Sec. 4.1.

Altogether, it seems to us that this point of the referee raises two separate issues. The first of these is the scope of the part of the field theory landscape for which our analysis has nontrivial consequences. We believe this is already addressed sufficiently by our response to the referee's first point. The second issue is the novelty of our results. In order to address this, we have expanded the last paragraph of Sec. 5 where we briefly summarize the results of the first half of the paper. In the revised text, we stress the universality of our results, whereby the local dipole conservation law only relies on translation invariance, and the central extension of the algebra of spatial translations in addition on the symplectic structure of the theory.

Attachment:

manuscript_changes.pdf

We are delighted that the referee recommends publication of our paper, and thank the referee for having invested their time in reading the revised manuscript.