The anti–de Sitter/conformal field theory (AdS/CFT) correspondence, also known as holographic duality, is a conjectured relation between quantum field theory and a higher-dimensional gravity theory. It was proposed in 1997 by Juan Maldacena in the context of string theory.
Since then, the correspondence has been extensively checked and generalized. It serves as a strong–weak coupling relation that maps a strongly coupled problem in quantum field theory to a weakly coupled problem in classical gravity. Some calculations that are hard on the field-theory side of the duality are easy on the gravity side. The AdS/CFT duality establishes an unexpected relation between high-energy particle physics and condensed-matter physics on the one hand, and general relativity on the other, and it is a new manifestation of the fundamental unity underlying the different subfields of physics. A pedagogical book on this sophisticated and rapidly developing subject is clearly welcome.
Horaţiu Năstase’s very readable Introduction to the AdS/CFT Correspondence is such a book. Written by a well-known expert on the AdS/CFT correspondence and its applications, the book is intended for graduate students and researchers who want to be acquainted with the new holographic techniques. Readers are assumed to understand basic notions of quantum field theory and have some familiarity with general relativity.
The book is organized into three parts. In the first, Năstase introduces the necessary concepts to formulate the correspondence. In the second, he gives the formulation and applies the duality to maximally supersymmetric gauge theories in four spacetime dimensions. The last part of the book deals with advanced topics and applications. That natural organization is, not surprisingly, similar to the one adopted by Martin Ammon and Johanna Erdmenger in their text Gauge/ Gravity Duality: Foundations and Applications (Cambridge University Press, 2015).
One of the most difficult aspects of writing a book on the AdS/CFT correspondence is deciding how much string-theory background to include. Clearly, minimizing the string-theory content would help to reach a wider audience. Too much string theory and you may lose readers. Less string theory will keep the readers but may cause them to miss the depth of the subject and see the holographic rules as arbitrary and unjustified.
Part 1 of Introduction to the AdS/CFT Correspondence achieves that balance, which makes it suitable for teaching the subject starting from well-defined string theory solutions. The opposite, phenomenological approach has been followed by Jan Zaanen, Yan Liu, Ya-Wen Sun, and Koenraad Schalm in Holographic Duality in Condensed Matter Physics (Cambridge University Press, 2016); that text focuses on the application of AdS/CFT to condensed matter. An intermediate point of view can be found in the monograph Gauge/String Duality, Hot QCD and Heavy Ion Collisions (Cambridge University Press, 2014) by Jorge Casalderrey-Solana and coauthors, which is more centered on the themes mentioned in the title.
Part 2 of the book contains an account of the original formulation of the correspondence; part 3 deals with the extensions of the holographic duality to models in different numbers of spacetime dimensions or with fewer symmetries. The content of that third part reflects Năstase’s preferences and will not please some advanced readers. I appreciated the sections devoted to a particular model—the so-called ABJM model— developed by Maldacena and colleagues; many details presented by Năstase are not available in existing books. Unfortunately, the book only briefly covers (in chapter 21) the addition of flavor fields to the correspondence; a better treatment can be found in the text by Ammon and Erdmenger and in the monograph by Casalderrey-Solana and company.
Năstase’s book is written in a student-friendly style with detailed explanations and many worked-out calculations. Each chapter ends with a summary of important concepts and a set of exercises. It will certainly be of great help both to those teaching an advanced graduate course on the AdS/CFT correspondence and to students or researchers planning to learn it by themselves. Introduction to AdS/CFT Correspondence is a valuable addition to the small set of existing books on the topic.
Alfonso V. Ramallo is professor of theoretical physics at the University of Santiago de Compostela in Spain. He works on quantum field theory and the holographic duality in string theory.
