Gauged Sigma Models and Magnetic Skyrmions

The mains strength of this paper is to provide a mathematical description of solutions of a gauged nonlinear sigma model on an arbitrary Riemann surface using standard BPS arguments.

The case relevant for magnetic skyrmions is only sketched.

A large part of the paper is devoted to an analysis of a nonlinear sigma model (NLSM) by using a map to an SU(2) gauge theory. By itself this type of approach is known for decades and has been considered by the author himself back in 1995 [B. J. Schroers, "Bogomol’nyi solitons in a gauged O(3) sigma model", Phys. Lett. B 356, 291 (1995); https://doi.org/10.101/0370-2693(95)00833-7]. In this sense the starting point of this paper is far from being innovative. One can basically find the same equations in the mentioned paper, except for the use of differential forms, which is more appropriate for discussing general Riemannian manifolds. Thus, it seems to me that one valid criticism would be whether this submission to SciPost is sufficiently novel to justify its publication. My first impression regarding this point is that this paper constitutes mostly a more mathematically general and precise extension of the above reference by the same author. The application to magnetic skyrmions, announced in the title, abstract, and introduction, is only slightly addressed, leaving the reader somewhat frustrated. In this regard we note, for instance, that after writing the energy functional (1.1), which contains a Dzyaloshinskii-Moriya (DM) term, the author proceeds with analyzing the energy functional (2.11) [or, equivalently, its coordinate representation, Eq. (2.12)], which does not contain the DM term. When the author finally comes to the application to magnetic skyrmions in Section 3.3, we realize that most of it corresponds to a reformulation of results previously discussed in Ref. 1 of the paper, which is co-authored by the author of this SciPost submission. The author also advertises the system with impurities as a possible application of the method. It would have been nicer if the author were more specific here.

My next point touches the presentation of the mathematical formalism. Although the mathematical language of forms chosen by the author is appropriate, it is sometimes used in a way that masks obvious facts. For instance, let us consider the last term in Eq. (2.12), corresponding to the coupling of the field strength to the n field. It is easy to see that this term is just the topological charge. The author could have started the discussion by telling the significance of this term from the outset. Within a point of view more akin to Ref. 1, one writes $\vec{A}_\mu=\vec{n}\times\partial_\mu\vec{n}$, which immediately yields $\vec{F}_{12}=2(\partial_1\vec{n}\times\partial_2\vec{n})+\vec{n}\vec{n}\cdot(\partial_1\vec{n}\times\partial_2\vec{n})$, and therefore, $\vec{F}_{12}\cdot\vec{n}=3\vec{n}\cdot(\partial_1\vec{n}\times\partial_2\vec{n})$, being in this way proportional to the topological charge density. Thus, the last term in (2.12) is related to Eq. (2.20). I think it would help to alert the reader for this fact in a more explicit way than it is currently done in the paper.

Another point that it is maybe worth addressing (optional) and possibly relates to the point of gauge choices, is that the covariant derivative terms in (2.12) can also be written as something proportional to $\vec{A}^2$, assuming that $\vec{A}_\mu=\vec{n}\times\partial_\mu\vec{n}$. This would make the energy functional look like one corresponding to a massive gauge field at infinitely strong gauge coupling. I remember that such type of functional is associated to an infinite number of conserved currents. Does this fact have any impact for theory discussed in the paper?

Summarizing, I do not think that this paper should be published in its current form. However, I believe that the paper can be reconsidered if the author manages to improve the discussion and give more substance to the application to magnetic skyrmions.

The possible changes can be read off directly from the report. Being more specific, the most crucial changes would be:

1- Clearly state what are the main novelties (besides considering the model on a Riemann surface) relative to Ref. 1 and to the 1995 paper by the author mentioned in the report.

2- Improve and extend the discussion on the applications to magnetic skyrmions. It would be desirable to be more concrete with respect to the system in the presence of impurities.

This is a fine paper that will be useful to those working on the mathematical formulation and solution of magnetic Skyrmion models. It also has ramifications beyond these Skyrmions. The paper extends a formalism introduced by the author and collaborators for such Skyrmions in a plane to more general Riemann surfaces. The main idea is to introduce a gauge potential, hence the title of the paper. The main result is to obtain a solvable first-order Bogomolny equation in this context.
Suggested minor improvement: Clarify what happens to the D_z n term in (2.14). Why does this disappear going to (2.15)?
Typos: skyrmions (p.1 near start). We think (p.2). term (p.5). which we will (p.7). Maybe delete "to" in "to (3.39)" (p.12).

The paper constructs a novel energy functional in which a gauge constraint ensures that all allowed configurations extremize the energy.

It's not clear what the applications of this energy functional are, beyond those pointed out in the previous paper.

The paper is written in a differential geometric language that does little to improve the rigor of the work but makes the paper difficult to read for the condensed matter audience.

The paper studies a class of solitons in field theories in two spatial dimensions. Among these solitons are magnetic Skyrmions, which are of some interest in condensed matter systems.

The paper writes down an energy functional for an SU(2) gauge field coupled to an adjoint scalar. The gauge field appears in the energy only algebraically and so its equation of motion gives rise to a constraint on the allowed field configurations. The energy functional has the peculiar property that all solutions of this constraint also obey the equations of motion of the scalar field. This means that all solutions to the constraint are automatically extrema of the action. The configurations can be labelled by a topological charge and this determines the energy.

This is an interesting observation. But it's not clear to me what can be done with it. The set of all field configurations that obey the constraint appear to be too large to be of interest. Instead, the author suggests that one should not view the gauge field as dynamical, but instead fix it to some particular configuration. With this interpretation, the constraint need not be imposed but can instead be interpreted as a Bogomoln'yi equation of a related field theory. This is in keeping with a previous paper by the author, listed as [1] in the bibliography, in which it was pointed out that, for a particular choice of this gauge field, the model reduces to a model for magnetic Skymions, tuned to a BPS limit.

This suggests that the model could be viewed as unifying system, in which different choices of gauge field give different theories. But no other interesting model is introduced in this way. There is a suggestion in the conclusions that this may be useful to study BPS impurities, but this is not pursued. A general solution to the constraint equation is given, but the main application seems to be a recapitulation of the results of [1].

In summary, there is an interesting observation in this paper. However, the implications of this observation remain unclear, and suggestions to find a novel application are not followed through. For this reason, I do not think that this paper contains enough material to warrant publication in SciPost.