Topology, \(C^*\)-algebras, and string duality. (English) Zbl 1208.81172

Rosenberg, Jonathan

CBMS Regional Conference Series in Mathematics 111. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4922-4/pbk). viii, 110 p. (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.

Reviewer: Martin Schlichenmaier (Luxembourg)

MSC:

81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T75	Noncommutative geometry methods in quantum field theory
19K99	\(K\)-theory and operator algebras
46L80	\(K\)-theory and operator algebras (including cyclic theory)
58B34	Noncommutative geometry (à la Connes)
55R10	Fiber bundles in algebraic topology
55P65	Homotopy functors in algebraic topology
55R50	Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
14J32	Calabi-Yau manifolds (algebro-geometric aspects)
53Z05	Applications of differential geometry to physics

Keywords:

\(K\)-theory; string theory; dualities; crossed products; mirror symmetry