Harmonic Superspace , A. S. Galperin , E. A. Ivanov , V. I. Ogievetsky , and E. S. Sokatchev Cambridge U. Press, New York, 2001. $90.00 (306 pp.). ISBN 0-521-80164-8
Harmonic Superspace, by Alexander Galperin, Evgeny Ivanov, Victor Ogievetsky, and Emery Sokatchev is the first discussion, outside of the confines of the research literature, to present pedagogically the intricacies of its topic. This is a unique book.
There already exist such well known previous expositions as Julius Weiss and Jonathan Bagger’s Supersymmetry and Supergravity 2nd ed. (Princeton U. Press, 1992), Peter C. West’s Introduction to Supersymmetry and Supergravity (World Scientific, 1986), S. James Gates, Marcus T. Grusaru, Martin Roček, and Warren Siegel’s Superspace or One Thousand and One Lessons in Supersymmetry (Addison-Wesley, 1983; see also http://aps.arXiv.org/pdf/hepth/0108200), and I. L. Buchbinder and S. M. Kuzenko’s Ideas and Methods of Supersymmetry and Supergravity, or A Walk Through Superspace (rev. ed., IOP Publishing, 1998). These treat the topic of N = 1 supersymmetry and superspace extensively while giving at most a cursory treatment of the so-called extended supersymmetrical theories.
Harmonic Superspace includes N = 2 extended theories in a manner that makes manifest, for both classical and quantum calculations, the consequences of the extended supersymmetry. As the four authors are the originators of the harmonic superspace, they are eminently qualified to be its expositors.
The authors adroitly manage in twelve chapters to introduce basic concepts, compare harmonic supersymmetry and superspace to N = 1 supersymmetry and superspace, motivate the approach, discuss the harmonic space generalization of Poincaré and conformal symmetries, introduce harmonic superspace, describe N = 2 supermultiplets from scalar to supergravity, develop harmonic superspace supergraph techniques, and present applications to hyper-Kahler geometry and nonlinear sigma models. The final chapter takes the reader to the frontier of the approach, with applications beyond N = 2 supersymmetry to N = 3 supersymmetrical systems. The appendix provides the reader with all the necessary foundational material (notation, conventions, definitions, and so on) upon which to build an operational proficiency in these techniques.
This book is aimed primarily at an audience that wants to master this important subject. Normally, this audience would be individuals either already engaged in research at the frontier of theoretical particle physics or intent upon engaging in such research (graduate students, theoretical or mathematical physicists, postdoctoral researchers, and active researchers from other fields). The authors have produced a readable narrative that clearly explains the applications and achievements of the approach. To maximize the effectiveness of the book as a pedagogical tool, the reader must use it as a guide to carrying out the calculations that undergird the narrative.
Although it may not be apparent that this presentation is related to the more widely studied topics of superstring or M-theory, studies of nonperturbative (as well as perturbative) nonrenormalization theorems in Seiberg-Witten theory have been undertaken precisely by exploiting the power of the harmonic superspace formalism. Another link to modern developments that this book treats is the important four-dimensional, N = 4 supersymmetrical Yang-Mills theory, which has well known relations to the low-energy effective action of open superstrings in harmonic superspace.
This is a demanding book, for those interested in more than a passing familiarity with this technique. Because of its content, it will likely be considered more difficult than either the Weiss and Bagger or the West book, and comparable in difficulty to the other two books mentioned earlier. The prose in Harmonic Superspace is clear, with a minimum of jargon. In my assessment, it succeeds in its goals.
There are two aspects of the book that might have received additional treatment. There could have been a more complete discussion of the relationships between 4D, N = 2 vector and supergravity multiplets, both within harmonic superspace and ordinary superspace. In addition, it is within these authors’ capabilities to have included a discussion of harmonic superspace background field techniques and applications to effective actions.
The book is a worthy memorial to the late academician Ogievetsky, who died in 1996 (see Physics Today, Physics Today 0031-9228 49 11 1996 102 https://doi.org/10.1063/1.881600 November 1996, page 102 ). Although it covers only one topic on which he worked during a productive career, it does so with the same sparkling creativity and demanding attention to mathematical rigor and detail that was evident throughout his research efforts.