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  • Textbook
  • © 1993

Loop Spaces, Characteristic Classes and Geometric Quantization

Birkhäuser

Authors:

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  • 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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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xvi
  2. Complexes of Sheaves and their Hypercohomology

    • Jean-Luc Brylinski
    Pages 1-61
  3. Line Bundles and Central Extensions

    • Jean-Luc Brylinski
    Pages 62-109
  4. Kähler Geometry of the Space of Knots

    • Jean-Luc Brylinski
    Pages 110-157
  5. Degree 3 Cohomology: The Dixmier-Douady Theory

    • Jean-Luc Brylinski
    Pages 158-181
  6. Degree 3 Cohomology: Sheaves of Groupoids

    • Jean-Luc Brylinski
    Pages 182-233
  7. Line Bundles over Loop Spaces

    • Jean-Luc Brylinski
    Pages 234-256
  8. The Dirac Monopole

    • 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.

Keywords

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

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

  • 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

  • DOI: https://doi.org/10.1007/978-0-8176-4731-5

  • Publisher: Birkhäuser Boston, MA

  • eBook Packages: Springer Book Archive

  • Copyright Information: Birkhäuser Boston 1993

  • Softcover ISBN: 978-0-8176-4730-8

  • eBook ISBN: 978-0-8176-4731-5

  • 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

Buying options

eBook
USD 64.99
Price excludes VAT (USA)
  • ISBN: 978-0-8176-4731-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 84.99
Price excludes VAT (USA)