Loop Spaces, Characteristic Classes and Geometric Quantization (Progress in Mathematics)

In chapter 1, the author overviews the language of sheaf theory and how to construct complexes of sheaves. Although the presentation is somewhat abstract, the author does give some examples of the constructions, such as the exponential exact sequence of sheaves. Using an injective resolution of a sheaf, the sheaf cohomology groups are defined and then shown to be independent of the injective resolution. Using the idea of a double complex, spectral sequences are introduced, along with the concept of sheaf hypercohomology. The later is constructed using an injective resolution corresponding to a sheaf complex. Most interestingly, the author shows how the hypercohomology of sheaves is related to the Cech cohomology. The later is more concrete from an applications point of view, and is one that can be more readily understood by physicists, as well as de Rham cohomology that is introduced later, and is shown to be a resolution of the constant sheaf of a smooth manifold. The Cech cohomology groups are shown to be canonically isomorphic to the de Rham cohomology groups.

A cohomology theory not so familiar to most is the Deligne cohomology, which is also introduced in chapter 1. This is also called Cheeger-Simons cohomology by some, and has applications in conformal field theory. The presentation here is actually quite good, as the author shows how Deligne cohomology is related to ordinary cohomology via a few examples, and how Deligne cohomology can be used to compare Cech cohomology classes with de Rham cohomology classes. The chapter ends with an overview of the famous Leray spectral sequence.

In chapter 2, the author goes into the classification of line bundles, basically using the Weil-Kostant theory. When the line bundle has a connection, the author shows that the isomorphism classes of line bundles with connections is related to the second Deligne cohomology group. The Kostant central extensions of the group of symplectic diffeomorphims is also considered, and the author shows how this acts on sections of line bundles. In chapter 3, the author considers first the topology on the space of singular knots in a smooth three-dimensional manifold, which is shown to great surprise to be a Kahler manifold. Not only that, the author further shows it to have a symplectic, complex, and a Riemannian structure.

The discussion gets considerably more interesting in chapter 4, wherein the author discusses how to generalize the classical result that the second integral cohomology group of a manifold is the group of isomorphism classes of line bundles over the manifold. The goal is to characterize the third integral cohomology group, and the author does this by using the theory of C*-algebras. The result of Dixmier-Douady relating the algebra of compact operators on a separable Hilbert space is shown to give the geometric description of the third integral cohomology group. The section on connections and curvature in this chapter is especially well written because the author explains and motivates well the eventual identification of the Hilbert space as the space of infinitely differentiable functions on START TRANSACTION WITH CONSISTENT SNAPSHOT; /* 2205 = 6ad597fd78069098fa7763baa62e7534

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