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Topology, C * -algebras, and string duality. (English) Zbl 1208.81172
CBMS Regional Conference Series in Mathematics 111. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4922-4/pbk). viii, 110 p. $ 33.00 (2009).

This little book is based on a series of lectures given by its author at an NSF/CBMS Regional Conference in the Mathematical Sciences, 2009.

Its goal is to give a concise introduction to K-theory and the use of K-theory in the context of modern physics, in particular in string theories and their dualities. A central role in the book is played by T-duality which is presented in detail. After some introductory remarks on string theory the following topics are discussed (a collection from the table of content): A quick review of topological K-theory. K-theory and D-brane charges. K-homology and D-brane charges. A few basics of C * algebras and crossed products. Continuous trace algebras and twisted K-theory. The theory of gerbes. Connes’ Thom isomorphism. The Pimsner-Voiculescu sequence. The topology of T-duality and the Bunke-Schick construction. T-duality via crossed products. Higher-dimensional T-duality via topological methods. Higher-dimensional T-duality via C * -algebraic methods. As more advanced topics, mirror symmetry and Fourier-Mukai duality are discussed.

The book introduces the necessary concepts in a very lively manner concentrating on essential aspects of the theory. The reviewer considers the book as a highly welcome introduction to a field of ongoing mathematical research. It gives an excellent overview of the methods and results. For the reader who wants to know more, further references are given.

MSC:
81T30String and superstring theories
81T75Noncommutative geometry methods (quantum field theory)
19K99K-theory and operator algebras
46L80K-theory and operator algebras
58B34Noncommutative geometry (á la Connes)
55R10Fiber bundles
55P65Homotopy functors
55R50Stable classes of vector space bundles, K-theory
14J32Calabi-Yau manifolds
53Z05Applications of differential geometry to physics