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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.
81T30String and superstring theories
81T75Noncommutative geometry methods (quantum field theory)
19K99$K$-theory and operator algebras
46L80$K$-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