Loop Spaces, Characteristic Classes and Geometric Quantization

by Brylinski, Jean-Luc

• ISBN13: 9780817647308
• ISBN10: 0817647309
• Edition: Reprint
• Format: Paperback
• Copyright: 2007-11-05
• Publisher: Birkhauser

Summary

This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics.

Table of Contents

Introduction	p. ix
Complexes of Sheaves and Their Cohomology
Injective resolutions and sheaf cohomology	p. 1
Spectral sequences and complexes of sheaves	p. 13
&Cbreve;ech cohomology and hypercohomology	p. 24
de Rham cohomology	p. 34
Deligne and Cheeger-Simons cohomologies	p. 46
The Leray spectral sequence	p. 54
Line Bundles and Geometric Quantization
Classification of line bundles	p. 62
Line bundles with connection	p. 70
Central extension of the Lie algebra of hamiltonian vector fields	p. 85
Central extension of a group of symplectic diffeomorphisms	p. 94
Generalizations of Kostant's central extension	p. 103
Kähler Geometry of the Space of Knots
The space of singular knots	p. 110
Topology of the space of singular knots	p. 115
Tautological principal bundles	p. 121
The complex structure	p. 126
The symplectic structure	p. 135
The riemannian structure	p. 144
The group of unimodular diffeomorphisms	p. 152
Degree 3 Cohomology: The Dixmier-Douady Theory
Infinite-dimensional algebra bundles	p. 158
Connections and curvature	p. 168
Examples of projective Hilbert space bundles	p. 175
Degree 3 Cohomology: Sheaves of Groupoids
Descent theory for sheaves	p. 182
Sheaves of groupoids and gerbes	p. 191
Differential geometry of gerbes	p. 205
The canonical sheaf of groupoids on a compact Lie group	p. 219
Examples of sheaves of groupoids	p. 228
Line Bundles over Loop Spaces
Holonomy of line bundles	p. 234
Construction of the line bundle	p. 236
The line bundle on the space of knots	p. 243
Central extension of loop groups	p. 247
Relation with smooth Deligne cohomology	p. 250
Parallel transport for sheaves of groupoids	p. 254
The Dirac Monopole
Dirac's construction	p. 257
The sheaf of groupoids over S2+3	p. 264
Obstruction to SU(2)-equivariance	p. 268
Bibliography	p. 278
List of Notations	p. 286
Index	p. 295
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