**MR2218620** **(2007m:58001)** Giachetta, Giovanni (I-CAM); Mangiarotti, Luigi (I-CAM); Sardanashvily, Gennadi (RS-MOSC)

Geometric and algebraic topological methods in quantum mechanics. *World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,* 2005. x+703 pp. ISBN: 981-256-129-3 58-02 (37J05 53D55 81-02 81R10 81R60 81S10)

Representation theory, functional analysis, differential geometry, and other classical mathematical concepts have proven their relevance to the formulation and understanding of models in theoretical and mathematical physics. These theories might nowadays be common knowledge for physicists working in these fields. Within the last 20 years in quantum theory new ideas have been developed, e.g. super- and BRST symmetries, geometric and deformation quantization, topological field theories, quantization of conformal field theories, non-commutativity, strings, branes, etc. These developments triggered the use and sometimes even the development of more advanced mathematics related to geometry, to algebraic geometry, and to algebra. All these techniques have a certain algebraic flavor. It is the passage from the commutative world to the noncommutative world (either by the quantization itself or by considering field theory over noncommutative space) which forces us to replace the category of "usual'' geometric objects by its dual category, the category of function algebras with certain additional structures. The dual category might admit an extension (e.g. a deformation) into the noncommutative world.

It is the goal of the authors of the book under review to introduce the mathematical definitions of these (mainly algebraic) objects, to collect some of the most relevant facts, and to give a guide to the literature. The book has the following chapters. 1. Commutative geometry, including homology of complexes, groups and algebras, algebraic varieties. 2. Classical Hamiltonian systems, including the cohomology of Kähler manifolds, Poisson manifolds, groupoids. 3. Algebraic quantization, including GNS construction. 4. Geometry of algebraic quantization, including Berezin's quantization, Banach and Hilbert manifolds. 5. Geometric quantization. 6. Supergeometry, including graded manifolds, BRST complex of constrained systems, superconnections. 7. Deformation quantization, including the relevant cohomology, Fedosov's and Kontsevich's construction. 8. Non-commutative geometry, including *C** algebras, noncommutative differential calculus, Connes' noncommutative geometry, Morita equivalence, *K*- and *KK*-theory. 9. Geometry of quantum groups, including the differential calculus for Hopf algebras, quantum principal bundles. After a general appendix recalling some more basic material assumed in the book, such as categories, Hopf algebras, groupoids, algebroids, measures on non-compact spaces, and more information on fibre bundles, the book closes with a very useful extensive bibliography (452 items). Note that, in the above, this is only a selection of the topics of subsections.

With respect to a prospective reader having a reasonably good background in mathematics, the notions, concepts, etc. are introduced in a self-contained but condensed manner. In most cases, proofs are not supplied for the results presented. But in any case reference to the literature is given. The book is not a mathematical "textbook'' in the usual sense. Also, because of the number of covered subjects and hence necessarily condensed style, there is not enough space for internal mathematical motivation for the introduced concepts.

The book gives a very helpful supply of mathematical tools needed by a theoretical or mathematical physicist to effect entry into some of the new directions in theoretical physics. Also, a mathematician might appreciate the condensed presentation of definitions and results in one of the modern fields of mathematics for which one may be seeking an overview. Clearly, as the main goal of the book is to present the more algebraic background of modern geometry, certain other geometric methods of importance in modern theoretical physics are beyond the intended scope of the book. One example is the notion of moduli spaces of geometric structures and their generalisation.

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