The unitary representations of the Poincar\'e group in any spacetime dimension

1- Clear and well structured introduction to unitary representation of the Poincaré group in any dimensions (pedagogical introductions usually only deal with the D=4 case);
2- Useful review of tensorial representations, including remarks on branching rules that often do not appear in introductory texts;
3- Precise references allow to complete the steps that are not discussed in detail in the text.

1- some additional remarks on 3D infinite-spin representations could have been added in Appendix B, in order to develop this case, whose analysis is rather recent, as the other ones reviewed in the text.

The manuscript is an improved version of a review posted on the ArXiv by the authors as hep-th/0611263. The main text contains only minor modifications with respect to the preprint: for instance, a couple of remarks have been added at pag. 6, the notation at the beginning of section 4.4 has been clarified and a reference on a review on the recent developments on infinite-spin representations has been added to section 5.3.2. The main addition are the two appendices: Appendix A reviews a method to build massless representations of the Poincaré group in any dimensions developed by Siegel and Zwiebach, implementing the condition that the translation generators of the little group act trivially on physical states by means of a covariant constraint. Appendix B reviews instead the classification of unitary representations of the Poincaré group in D=3.
The original preprint was already an excellent review, introducing in a complete although pedagogical way the unitary representations of the Poincaré group in D>3, while also gathering a lot of useful material on tensorial representations. Appendix B is an important addition to the original text, including references on recent developments on fractional and infinite spin representations. Given the peculiarities of the 3D setup, it doesn't hurt if the material has been collected in an appendix rather than being included in the main text. Appendix A is an interesting bonus, recalling a not-so-well-known method that is not mentioned in any of the textbooks I am familiar with. All in all, I think that the manuscript is an excellent review and, given that it only appeared in the form of ArXiv preprint, I suggest its publication in SciPost.

This is an important review paper devoted to field realisations of the unitary representations of the Poincar\'e group in diverse dimensions. Its first version appeared in the hep-th archive, hep-th/0611263. Since then it has generated almost 100 citations, which shows that many members of the higher-spin community have used this review in their research. As compared with the 2006 preprint, the current manuscript contains two new appendices A and B, as well as additional references. These additions are natural and useful.

I do not hesitate to recommend this review for publication. However, I have a few minor suggestions that the authors might consider to implement or not. (All suggestions are optional.)

The review is well organized, systematic and clearly written.

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The review by Bekaert and Boulanger is slightly extended version of their
article which appeared many years ago in arxives (Useful appendices A and B added). By now this review is well known to experts in field and I'm happy to provide report on this nice review.

First, in Secs 2, 3, and 4 the Authors provide review of UIR of the Poincare group, linear, and orthogonal groups. In those sections the interested
reader will find many interesting facts which are relevant for discussion of relativistic equations. In Sec. 5, the Authors review relativistic field equations. They provide the systematic step by step procedure for building covariant equations of motion associated massive and massless UIR of the Poincare group. The equations are given in (60), (61). The equations are given in terms of field strengths (and therefore does not require discussion of gauge symmetries). Though in the Abstract of the review the Authors mention UIR only with non-negative mass-squared, they present in Sec. 5 their interesting results for so called continuous spin field (massless and massive). To my opinion these results for equations (and constraints) of continuous spin field are very important for future development of the this theme.

The review is well organized, systematic and clearly written. To my opinion all relevant references are mentioned in the list of References.

I strongly recommend the article by Bekaert and Boulanger for publication in SciPost.

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