Loop spaces, characteristic classes, and geometric quantization

Jean-Luc Brylinski9780817636449, 0817636447

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 development in mathematical physics (e.g., in knot theory, gauge theory, and topological quantum filed 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 fiber bundles (more precisely, associated to grebes). The new 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-Cern-Simons secondary characteristics classes, and group cohomology. Finally, the last chapter deals with the Dirac monopole and Dirac’s quantizations 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 prequantizations à la Kostant-Souriau.

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Table of contents :
Table of Contents ……Page 6
Introduction ……Page 8
1. Injective resolutions and sheaf cohomology ……Page 16
2. Spectral sequences and complexes of sheaves ……Page 28
3. Cech cohomology and hypercohomology ……Page 39
4. de Rham cohomology ……Page 49
5. Deligne and Cheeger-Simons cohomologies ……Page 61
6. The Leray spectral sequence ……Page 69
1. Classification of line bundles ……Page 77
2. Line bundles with connection ……Page 85
3. Central extension of the lie algebra of hamiltonian vector fields ……Page 100
4. Central extension of a group of symplectic diffeomorphisms ……Page 109
5. Generalizations of Kostant’s central extension ……Page 118
1. The space of singular knots ……Page 125
2. Topology of the space of singular knots ……Page 130
3. Tautological principal bundles ……Page 136
4. The complex structure ……Page 141
5. The symplectic structure ……Page 150
6. The riemannian structure ……Page 159
7. The group of unimodular diffeomorphisms ……Page 167
1. Infinite-dimensional algebra bundles ……Page 173
2. Connections and curvature ……Page 183
3. Examples of projective Hilbert space bundles ……Page 190
1. Descent theory for sheaves ……Page 197
2. Sheaves of groupoids and gerbes ……Page 206
3. Differential geometry of gerbes ……Page 220
4. The canonical sheaf of groupoids on a compact Lie group ……Page 234
5. Examples of sheaves of groupoids ……Page 243
1. Holonomy of line bundles ……Page 249
2. Construction of the line bundle ……Page 251
3. The line bundle on the space of knots ……Page 258
4. Central extension of loop groups ……Page 262
5. Relation with smooth Deligne cohomology ……Page 265
6. Parallel transport for sheaves of groupoids ……Page 269
1. Dirac’s construction ……Page 272
2. The sheaf of groupoids over $S^3$ ……Page 279
3. Obstruction to $SU(2)$-equivariance ……Page 283
Bibliography ……Page 293
List of Notations ……Page 301
Index ……Page 310