On duality between Cosserat elasticity and fractons

This manuscript formulates and discusses a duality for the so-called Cosserat theory of elasticity. To set the stage, the authors first review the duality between ordinary (i.e. symmetric) two-dimensional elasticity theory and a tensor gauge theory for fractons, which has been discussed in recent literature. Then, following the same lines, they formulate the duality transformation for the case of Cosserat elasticity. They find that the dual theory is a non-symmetric tensor tensor gauge theory with one additional massive degree of freedom with respect to the symmetric case.

The paper is well organised and well written. The results are sound and interesting and the topic is timely. I think the paper deserves publication. I do have some comments, though, that the authors should consider before publication.

1- Conventions: the authors do not seem to follow usual conventions (see, e.g., Landau book): they omit a factor $1/2$ in the definition of the strain tensor and a factor $1/2$ in the definition of the elastic action. Is there a special reason for that? If not, I would encourage them to use the standard conventions.

2- It should be specified at the beginning of section 2 that it is assumed that the elastic action is isotropic. What happens to the dual theory if the elastic action is not isotropic?

3- In Sec. 2 it would be useful to have the expression of the stress tensor in terms of the strain tensor.

4- In Eq. 4, the inverse of the elastic moduli tensor appear. In the appendix A we are told that this tensor is only partially invertible. Then, what is the precise meaning of the inverse C appearing in Eq. 4?

5- In Eq. 5 the effective action, after integration over the smooth part of the displacement field, still depends on its singular part. The notation should reflect this.

6- In Eq. 11 the symbol $\xi^i$ is used, which appears also in Eq. 35 (without index) with a completely different meaning. This should be avoided.

7- It would be useful to extend the introduction of the section on Cosserat elasticity theory. This theory is probably unfamiliar to most readers and the authors should explain a bit more extensively why it is physically interesting and relevant. For example, they could mention concrete examples of materials whose elastic properties are described by this theory.

8- In Eq. 35 the symbol $\xi$ should be defined.

9- I am a bit confused by the form of the action in Eq. 35. If, as usual, the tensor of elastic moduli $C_{ijkl}$ is symmetric under exchange of i and j (and of k and l), the term $\epsilon_{ij}\theta$ in the strain tensor $\gamma_{ij}$ does not contribute to the action term $C_{ijkl}\gamma_{ij}\gamma_{kl}$ and as consequence the field $\theta$ should be decoupled from the field $u_i$. If, on the other side, the tensor $C$ is not symmetric, then in the limit $\theta=0$, the action would depend on both the symmetric and antisymmetric part of the strain tensor, so the theory would not reduce to the symmetric one for $\theta=0$. The authors should clarify this point.

10- This is related to point 7: At the beginning of section 3.3, the authors introduce the "integrability conditions" in the Cosserat theory. The physical explanation and justification should be given here.

11- This is also related to point 7: While it is well-known from classical elasticity theory that singular configurations of $u_i$ are related to disclinations, it is much less know what is the physical meaning of a singular configuration of the field $\theta$. When introducing it, the authors should briefly explain what is it.