Gauged Sigma Models and Magnetic Skyrmions

In response to the comments by referees 1 and 3 I have made major changes to the paper, detailed and justified below. I hope that they address the referees' concerns. I thank referee 2 for their positive remarks, and for pointing out typos, which I corrected.

1. The introduction is substantially re-written and extended to bring out clearly that this paper goes far beyond the earlier paper [8] (in the new ordering of references), both in terms of applications (in deals with the most general DM interaction for magnetic skyrmions and includes impurities) and in terms of the underlying geometry (the underlying mathematical reason for solvability was not clear in [8] but is fully understood in this paper). The difference between this model and gauged non-linear sigma models studied elsewhere in the literature is also clarified.

2. I have made minor changes to Sections 2 (some more details on boundary terms in 2.4) and 3 (a new final section 3.3 to highlight the general formula for solutions). Despite the reservations the referees have expressed about the use of differential forms I have not replaced them. I find them by far the most efficient tools for the calculations in these sections, and rely on them to bring out the mathematical structures which underpin the model. This would be harder and less clear in traditional vector calculus notation. I would also make the case that this paper has a strong interdisciplinary element, connecting condensed matter physics with differential geometry and mathematical physics, and that some compromise in the notation is therefore inevitable. Finally, I have made only very minor use of differential forms in the new Section 4 which deals with applications.

3. I have introduced an entirely new Section 4 which spells out the application to magnetic skyrmions and impurities in much more detail then the previous version. This should bring out clearly the range of applications and also the extent to which the gauged sigma models provide a unifying picture for magnetic skyrmions and impurities.

4. The new version uses the SciPost style file, and this automatically adds a table of contents.