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Loop Spaces, Characteristic Classes and Geometric Quantization

  • Textbook
  • © 1993


  • 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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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.

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Table of contents (7 chapters)


"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

  • Department of Mathematics, The Penn State University, University Park, USA

    Jean-Luc Brylinski

Bibliographic Information

  • Book Title: Loop Spaces, Characteristic Classes and Geometric Quantization

  • Authors: Jean-Luc Brylinski

  • Series Title: Modern Birkhäuser Classics

  • DOI:

  • Publisher: Birkhäuser Boston, MA

  • 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

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