# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 81T30 String and superstring theories 81T75 Noncommutative geometry methods (quantum field theory) 19K99 $K$-theory and operator algebras 46L80 $K$-theory and operator algebras 58B34 Noncommutative geometry (á la Connes) 55R10 Fiber bundles 55P65 Homotopy functors 55R50 Stable classes of vector space bundles, $K$-theory 14J32 Calabi-Yau manifolds 53Z05 Applications of differential geometry to physics