© 1993

Loop Spaces, Characteristic Classes and Geometric Quantization


Part of the Progress in Mathematics book series (MBC)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Jean-Luc Brylinski
    Pages 1-61
  3. Jean-Luc Brylinski
    Pages 62-109
  4. Jean-Luc Brylinski
    Pages 110-157
  5. Jean-Luc Brylinski
    Pages 158-181
  6. Jean-Luc Brylinski
    Pages 182-233
  7. Jean-Luc Brylinski
    Pages 234-256
  8. Jean-Luc Brylinski
    Pages 257-277
  9. Back Matter
    Pages 278-302

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.


Algebra Category theory Knot theory Kähler geometry curvature differential geometry fiber bundles group cohomology loop groups manifold

Authors and affiliations

  1. 1.Department of MathematicsThe Penn State UniversityUniversity ParkUSA

Bibliographic information

  • Book Title Loop Spaces, Characteristic Classes and Geometric Quantization
  • Authors Jean-Luc Brylinski
  • Series Title Progress in Mathematics
  • DOI
  • Copyright Information Birkhäuser Boston 1993
  • Publisher Name Birkhäuser, Boston, MA
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-8176-3644-9
  • Softcover ISBN 978-0-8176-4730-8
  • eBook ISBN 978-0-8176-4731-5
  • Edition Number 1
  • Number of Pages XVI, 302
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Additional Information Originally published as volume 107 in the series: Progress in Mathematics
  • Topics Differential Geometry
    Category Theory, Homological Algebra
  • Buy this book on publisher's site


"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