First, a disclaimer. Horatiu Nastase was my Ph.D. advisor, and is a very good friend, so this review may not be utterly impartial, but I must also say that I’ve tried to be extremely neutral.
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The book gives a pedagogical introduction to AdS/CFT, a very active research area of physics. Nastase has been an active researcher in this area, and amongst his collaborators is collaborator of Juan Maldacena, the father of the AdS/CFT correspondence. Together with David Berenstein, Nastase and Maldacena made important contributions to the so called BMN limit. The first nice thing about this book is that it is really interesting to see what he has to say about the correspondence, the fundamental aspects, and how the subject evolved in 20 years.
When I tried to learn the AdS/CFT correspondence for the first time I followed the classical review of MAGOO. Today I love this review, but in my first readings I wanted to do some calculations and see things working. After some familiarity, I could return (and I did) to the physical and philosophical aspects of the duality. The first think that I love this book is that it avoids long verbose discussions, it has explicit calculations, intuitive explanations, good exercises and some useful references. For me, it is a big plus.
My general impressions are the following:
PART I is the background material. Horatiu tries to explain the necessary tools of quantum field theory, general relativity and string theory. Given that each of these topics can be a one-year course, and that the book is an introduction to the AdS/CFT, it should be evident that you will not find an in-depth explanation of the mentioned topics.
In the book’s preface Nastase wrote that the book tries to “introduce AdS/CFT without assuming anything beyond an introductory course in quantum field theory“. Based on the book, I assume that by “introductory course in QFT” he meant the following topics:
- Feynman rules and path integrals
- Gauge theories
- S-matrix and correlation functions
- Some notions on anomalies
- Notions on renormalization
The standard reference of introductory QFT courses has been the book of Peskin and Schroeder. Look, instead, Ryder’s book, which undoubtedly easier than Peskin and Schroeder, and you will see that it addresses all topics above. So, I think that it fair to assume the topics above as introductory. But evidently, we need to know if the topics above are enough for a good understanding of the AdS/CFT correspondence.
I think that any reader could understand the bulk of the book (let’s say ~ 95% of it) knowing the topics above, then I agree with the author when he says that you can understand the AdS/CFT correspondence with the tools he provides, and the book is, indeed, self-contained.
It is quite remarkable that he managed to write an advanced book in which most of the background material fits in 1/3 of the book length. Personally I should say that when I first started learning the material, I didn’t know most of what this part covers, and if I had had just this first part, my life would have been way easier.
PART II starts discussing the famous correspondence between AdS5xS5 and N=4 SYM. This is, by far, the most interesting part of the book. It is precise and intuitive in the right proportion, the sections has some explicit calculations, and he explains many results that are not addressed in other reviews on the AdS/CFT. Nastase, together with Maldacena and Berenstein, has made important contributions to the so called BMN limit, and he explains this topic in chapter 17, and it is particularly illuminating. I really liked the second part of the book, but chapters 16, 17 and 18 are absolutely elegant.
PART III discusses some recent material and results. Chapters 21 and 26 have important and modern aspects of the AdS/CFT. They are very short, but these, together with ch. 17, are my favorites chapters of the whole book. As someone interested in research, it is clear that there are many open problems along these directions, and Chap 21 and 26 are very inspiring.
In summary, it is a great book, and I think that it is a must for any serious student or researcher interested in AdS/CFT.
Thiago Araujo 