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Ideas and Methods of Supersymmetry and Supergravity: Or a Walk Through Superspace (Studies in High Energy Physics, Cosmology and Gravitation)

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Among the various books available in the literature on these topics, this book succeeds by being a precise, organic and self-contained introduction to four dimensional simple supersymmetry (SUSY) and supergravity (SUGRA) with a focus on the use of superspace techniques. It is perfect for both graduate students who are starting to learn about supersymmetry and experts undertaking current research, providing a precise reference to the treated subjects. I know well the book. I used it many times for my research, and it has always been a very helpful reference.

The aim of the book being to provide an introduction to SUSY and SUGRA, the
authors, as explained in the preface, have decided to focus on 4D N=1 supersymmetry. This turns out to be a pedagogical choice for students, since the reader is given a solid understanding of the principles and techniques through a complete presentation of topics which are very well developed and understood in literature. After studying the book, the student will be able to start his own research in the field and have a good
foundation for exploration of questions in extended SUSY, supersymmetry and SUGRA in dimensions different from 4 and other more advanced topics not treated in the book.

Any reader with a pretty basic knowledge of QFT, differential geometry and gravity will have no great difficulty studying this book. Prior to introducing new topics, the authors provide brief reviews of necessary background material, setting up with care the conventions that they use. Even though the book clearly focuses on superspace techniques, the authors usefully choose to include a description in terms of component fields for many results.

The authors start the book with a long chapter of mathematical background. There, they provide a review of the Poincare' group, the conformal group, elements of differential geometry and gravity, and they give a nice introduction to Grassmann algebras, superspaces and analysis with supernumbers. The first chapter also turns out to be a good collection of the main notation and of useful formulae used throughout the book.

The second chapter is where supersymmetry is introduced. The authors define the concept of superalgebras and super Lie groups, then introduce the Poincare' supergroup and its unitary representations. After that, 4D superspace is defined and on-shell superfield representations of the Poincare' supergroup are studied in detail.

Chapter three is devoted to the study of globally supersymmetric field theories. In this chapter the use of 4D N=1 superspace to define manifestly off-shell supersymmetric field theories is described in detail. The authors introduce and study at the classical level many SUSY field theories, including the Wess-Zumino model, vector multiplets and SUSY gauge theories, Kaehler nonlinear sigma-models and they explain the classical equivalence/duality of some matter theories with different multiplets.

In chapter four the quantization of supersymmetric field theories is given. The authors explain how to define superfunctional methods and then introduce superfield quantization via path integrals with care and clarity. Feynman rules are given for matter and Yang-Mills SUSY theories and the non-renormalization theorem is explained. They provide examples of one and two loop quantum computation, also giving a nice introduction to the use of Schwinger proper time techniques.

Chapter five is devoted to the introduction of the more involved topic of the superspace geometry of 4D N=1 supergravity. I think that this is probably the best chapter in the book. It gives a detailed overview of the different techniques developed to study supergravity in superspace. The chapter starts by defining the curved superspace and introduces the Ogievetsky-Sokatchev formulation of supergravity; it then comes through the Wess-Zumino covariant formulation of SUGRA; it studies conformal supergravity in superspace and the authors describe the solution of the Bianchi identities. They then discuss the Gates-Siegel prepotential theory (whose description continues in chapter 6), finishing by explaining the reduction to components of 4D simple SUGRA theories in a Wess-Zumino gauge.

Chapter six continues the study of SUGRA in superspace. In particular it focuses on dynamics in supergravity. It starts by giving the action principle for different 4D N=1 SUGRA theories. The authors then describe the formulation of supergravity-matter systems and study various properties of curved superspaces. The chapter continues with a description of non-minimal and new-minimal supergravity and finishes with the study of free massless higher superspin field theories.

The authors end the book with a final chapter devoted to the computation of the quantum effective action in curved superspace. This chapter goes beyond the pedagogical line of the book but it provides very nontrivial examples of the use of the techniques introduced in the previous chapters.

I believe that this book is probably going to become a new classic in the literature on supersymmetry, particularly for those interested in superspace techniques. Let me now compare it to other well known classic books on the subject. For the topics also covered in the books of Wess-Bagger and West, the Buchbinder-Kuzenko (BK) book is definitely more extended and detailed. In some respects, it is closer to another great classic "SUPERSPACE Or One Thousand and One Lessons in Supersymmetry" by Gates-Grisaru-Rocek-Siegel, even if I believe that the BK book results more homogeneous and pedagogical. With respect to the previously cited classics, I also believe that the BK book results more detailed and clear in the presentation of 4D N=1 superfield supergravity. As a possible weakness of the BK book I would note that the authors could have added a chapter on some aspects of supersymmetry breaking which is present in all the other books mentioned, even if this topic is still very much a topic of current research. The book does not really cover the phenomenological applications of 4D supersymmetry and supergravity, or more recent non-perturbative results in supersymmetric theories; for these topics one could look at other books, such as the Weinberg-3 or the Dine book, or many, very good recent reviews that can be found on arXiv. But I have to say that what is lost in the topics not treated is won back by the detail of the presentation.

In comparing with other books, it is also worth to note that there are topics which only appears in the BK book like: novel approach to the realization of the superconformal group in superspace; classification of on-shell supermultiplets; off-shell higher spin supermultiplets; the Ogievetsky-Sokatchev approach to supergravity; the Schwinger-DeWitt techniques in supergravity.

To conclude, I recommend this book because it is an extremely solid starting point for any student wishing to study SUSY and SUGRA. It is also a great superspace reference for any expert.