Overview
- Discusses open problems and further research extensions
- Written by a distinguished and well-published researcher in the field
- Includes many examples and figures
Part of the book series: Modern Birkhäuser Classics (MBC)
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Table of contents (7 chapters)
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Front Matter
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Back Matter
Keywords
About this book
This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics.
Various developments in mathematical physics (e.g., in knot theory, gauge theory, and topological quantum field theory) have led mathematicians and physicists to search for new geometric structures on manifolds and to seek a synthesis of ideas from geometry, topology and category theory. In this spirit, this book develops the differential geometry associated to the topology and obstruction theory of certain fiber bundles (more precisely, associated to grebes). The theory is a 3-dimensional analog of the familiar Kostant--Weil theory of line bundles. In particular the curvature now becomes a 3-form.
Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kähler geometry of the space of knots, Cheeger--Chern--Simons secondary characteristics classes, and group cohomology. Finally, the last chapter deals with the Dirac monopole and Dirac’s quantization of the electrical charge.
The book will be of interest to topologists, geometers, Lie theorists and mathematical physicists, as well as to operator algebraists. It is written for graduate students and researchers, and will be an excellent textbook. It has a self-contained introduction to the theory of sheaves and their cohomology, line bundles and geometric prequantization à la Kostant--Souriau.
Reviews
"The book is not only a well-written and thorough exposition of....abstract ideas, but it also treats geometric applications throughout... In addition, the book contains a nice exposition of various aspects of Cech, de Rham, and Deligne cohomology and an exposition of Grothendieck's decent theory for sheaves."
--Mathematical Reviews
Authors and Affiliations
Bibliographic Information
Book Title: Loop Spaces, Characteristic Classes and Geometric Quantization
Authors: Jean-Luc Brylinski
Series Title: Modern Birkhäuser Classics
DOI: https://doi.org/10.1007/978-0-8176-4731-5
Publisher: Birkhäuser Boston, MA
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eBook Packages: Springer Book Archive
Copyright Information: Birkhäuser Boston 1993
Softcover ISBN: 978-0-8176-4730-8Published: 15 November 2007
eBook ISBN: 978-0-8176-4731-5Published: 30 December 2009
Series ISSN: 2197-1803
Series E-ISSN: 2197-1811
Edition Number: 1
Number of Pages: XVI, 302
Additional Information: Originally published as volume 107 in the series: Progress in Mathematics
Topics: Differential Geometry, Algebra, Topology, Category Theory, Homological Algebra