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 -theory and the use of -theory in the context of modern physics, in particular in string theories and their dualities. A central role in the book is played by -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 -theory. -theory and D-brane charges. -homology and -brane charges. A few basics of algebras and crossed products. Continuous trace algebras and twisted -theory. The theory of gerbes. Connes’ Thom isomorphism. The Pimsner-Voiculescu sequence. The topology of -duality and the Bunke-Schick construction. -duality via crossed products. Higher-dimensional -duality via topological methods. Higher-dimensional -duality via -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.