Loop spaces, characteristic classes and geometric quantization
著者
書誌事項
Loop spaces, characteristic classes and geometric quantization
（Progress in mathematics, vol. 107）
Birkhäuser, c1993
- : us
- : sz
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注記
Includes bibliographical references: p. [278]-285, list of notations: p. [286]-294 and index
内容説明・目次
- 巻冊次
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: us ISBN 9780817636449
内容説明
This book examines the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kahler geometry of the space of knots, and Cheeger--Chern--Simons secondary characteristics classes. It also covers the Dirac monopole and Dirac's quantization of the electrical charge.
目次
Complexes of Sheaves and their Hypercohomology.- Line Bundles and Central Extensions.- Kahler Geometry of the Space of Knots.- Degree 3 Cohomology: The Dixmier-Douady Theory.- Degree 3 Cohomology: Sheaves of Groupoids.- Line Bundles over Loop Spaces.- The Dirac Monopole.
- 巻冊次
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: sz ISBN 9783764336448
内容説明
This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematical physics (e.g., in knot theory, gauge theory and topological quantum field theory) have led mathematicians and physicists to look 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 fibre bundles (more precisely, associated to gerbes). The new theory is a 3-dimensional analogue 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, Kaehler geometry of the space of knots, Cheeger-Chern-Simons secondary characteristic 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 a la Kostant-Souriau.
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