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Book review

Author(s) Hübsch, Tristan
Title Calabi-Yau Manifold
Publisher World Scientific
Year of publication 1992
Reviewed by Paul Blaga

In the theory of superstrings naturally appears a space-time of 9+1 dimensions (noted M9+1). But the superstrings models must be consistent with the physics in the real world, which has 3+1 dimensions. Having this in mind, the physicists attempted to compactify six space-time dimensions. Actually, this means to replace M9+1 with M3+1 x N, N being a six-dimensional compact manifold. For physical reasons, N should be a Calabi-Yau (CY) manifold, namely a 3-dimensional complex Kähler manifold with vanishing first Chern characteristic class. Since 1985, when Candelas and others first pointed out the physical importance of CY manifolds, a great deal of work has been done in order to enlarge our knowledge about these objects. The book under review is (so far as I know), the first monograph ever written about this topics.

At this stage, I will try to say a couple of words about the contents of the book. There is an introductory part, reviewing some physical applications of CY manifolds to superstring theory, in order to note their properties which has been recognized as being interesting for physics. The first part of the book indicates several ways of constructing CY manifolds, by using the tools of algebraic geometry. The second part is devoted to the cohomology of CY manifolds, while the third one is dealing with the parameter spaces related to the classification of CY structures. The last part is dealing with some special application of the theory to concrete physics problems. The book ends with a lexicon (namely a glossary where are shortly explained the most significant terms which are not defined within the book). The list of references includes, separately, books and review papers, and research papers. The final point is set by the index.

Now let me say something about the preresquisites and about the level of the book. It is assumed that the reader is accustomed at the least with the basics of algebraic geometry and the geometry of complex manifolds. A good source, in the spirit of this book could be, for instance Griffiths and Harris - Principles of Algebraic Geometry, Wiley, 1978. Although the subtitle of the book is "a bestiary for physicist", the most part of the book is devoted to the mathematical construction of CY manifolds. On the other hand, if somebody would like to embark in this theory, he (or she) has to have a good enough knowledge of the physics of superstrings, in order to know which CY manifolds are important to physics. The level of the book is graduate / research.

To resume: this is a serious and solid book, written by an expert in the field and will be useful to a large class of readers. It can be recommended to workers in superstrings and to geometers interested in theoretical physics. Graduate students are, also, encouraged to purchase this monograph, written in a very pedagogical and attractive style. I have to mention, also, its excellent graphical aspect.