Modern Birkhäuser Classics

Loop Spaces, Characteristic Classes and Geometric Quantization

Authors: Brylinski, Jean-Luc

  • Discusses open problems and further research extensions
  • Written by a distinguished and well-published researcher in the field
  • Includes many examples and figures
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  • ISBN 978-0-8176-4731-5
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  • Immediate eBook download after purchase
Softcover $49.99
price for USA
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About this Textbook

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


Table of contents (7 chapters)

  • Complexes of Sheaves and their Hypercohomology

    Brylinski, Jean-Luc

    Pages 1-61

  • Line Bundles and Central Extensions

    Brylinski, Jean-Luc

    Pages 62-109

  • Kähler Geometry of the Space of Knots

    Brylinski, Jean-Luc

    Pages 110-157

  • Degree 3 Cohomology: The Dixmier-Douady Theory

    Brylinski, Jean-Luc

    Pages 158-181

  • Degree 3 Cohomology: Sheaves of Groupoids

    Brylinski, Jean-Luc

    Pages 182-233

Buy this book

eBook $39.99
price for USA (gross)
  • ISBN 978-0-8176-4731-5
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.99
price for USA
  • ISBN 978-0-8176-4730-8
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Loop Spaces, Characteristic Classes and Geometric Quantization
Authors
Series Title
Modern Birkhäuser Classics
Copyright
1993
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Boston
eBook ISBN
978-0-8176-4731-5
DOI
10.1007/978-0-8176-4731-5
Softcover ISBN
978-0-8176-4730-8
Series ISSN
2197-1803
Edition Number
1
Number of Pages
XVI, 302
Additional Information
Originally published as volume 107 in the series: Progress in Mathematics
Topics