Loop Spaces, Characteristic Classes and Geometric Quantization [Paperback]

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, K?hler geometry of the space of knots, and Cheeger--Chern--Simons secondary characteristics classes. It also covers the Dirac monopole and Diracs quantization of the electrical charge.

This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics.Complexes of Sheaves and their Hypercohomology.- Line Bundles and Central Extensions.- K?hler 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.

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

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 ló%