Spin Geometry. (PMS-38)
By H. Blaine Lawson, Marie-Louise Michelsohn,
* Publisher: Princeton University Press
* Number Of Pages: 440
* Publication Date: 1990-02-01
* Sales Rank: 729404
* ISBN / ASIN: 0691085420
* EAN: 9780691085425
* Binding: Hardcover
* Manufacturer: Princeton University Press
* Studio: Princeton University Press
* Average Rating: 4.5
* Total Reviews: 2
Book Description:
This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.
Date: 2001-12-22 Rating: 5
Review:
Excellent
Who would have known that the equation discovered by P.A.M. Dirac in the 1920's would have the enormous appllications to mathematics that it currently has. This book is an excellent overview of these applications, written by two individuals who are responsible for the development of many of these. Dirac's theory of course had its origins in physics, and physicists, particularly those working in high energy physics, will find this book interesting and helpful.
The authors give a brief introduction and then move on to the representation theory of Clifford algebras and spin groups in chapter 1. The reader can see the origin of Clifford algebras and an introduction to the Pin and Spin groups. Clifford algebras are classified as matrix algebras over the real or complex numbers, and the quaternions. It is the representation theory of Clifford algebras however that has resulted in the impressive results outlined in the book Noting that the tensor product of Clifford algebras is not necessarily a Clifford algebra, the authors introduce a Z(2)-grading on a Clifford algebra, which results in a multiplicative structure in the representations of Clifford algebras. The Lie algebras of the Pin and Spin groups are discussed along with applications to geometry and Lie groups. By far the most interesting discussion though is on K-theory, which allows one to define a ring structure on vector bundles. Distinguishing a base point in the base space, relative K-groups are defined, and shown to be equal for the base space and its i-fold suspension. Bott periodicity results are stated but their proof is delayed until chapter 3. A detailed discussion is given of the Atiyah-Bott-Shapiro isomorphism and KR-theory.
The connection between spin and differential geometry is discussed in chapter 2. The first few sections is a review of standard results in the spin structure of vector bundles, such as Stiefel-Whitney classes and spin cobordism. For Riemannian vector bundles, each fiber has a quadratic form that gives rise to a Clifford algebra on the fiber. The question as to when a vector bundle over the Riemannian base space can be found that has fibers each an irreducible module over this Clifford algebra leads to a consideration of spin manifolds and spin cobordism, when the total space is chosen to be the tangent bundle. The Dirac operator acting on a bundle over this Clifford bundle allows the construction of all the standard elliptic operators such as the signature, Atiyah-Singer, and the Euler characteristic. The authors discuss these constructions in detail along with the notion of of Cl(k)-linear operators.
The Dirac operator can be viewed in Euclidean space as the square root of a Laplace operator, but over general manifolds it is the Laplacian with a correction term dependent on the curvature and Clifford multiplication. The Bochner vanishing theorems are discussed in great detail, along with the results on the existence of exotic spheres.
An entire chapter is spent on index theorems, wherein the authors present the results in terms of the approach used by Atiyah and Singer, instead of the heat kernel methods of Gilkey and Patodi. Physicists might prefer the later approach, due to its connections with applications, but the abstract K-theory approach undertaken by the authors is elegant and their presentation is excellent. The role of physics in index theorems is a fascinating one though, especially the use of supersymmetry to simplify the proofs of some of the results. The authors do not discuss this approach, but point out, interestingly, that it does not work when one is dealing with torsion elements in K-theory. These cannot be detected using cohomology nor can the modulo-two invariants appearing in the index theorems be computed from local densities.
The last chapter is a long one and discusses applications in differential topology and geometry, emphasizing index thoerems and Riemannian manifolds of positive scalar curvature. The authors outline just when the indexes are integers (the integrality theorems) and use spin geometry to discuss the immersion problem for manifolds and the vector field problem. Exotic n-spheres again make their appearance, wherein it is shown that some of these have very few symmetries and are very asymmetric objects. A short introduction to elliptic genera is given. Interestingly, C*-algebras are briefly mentioned as tools to decide whether for every compact spin manifold with positive scalar curvature all higher A-genera must be zero. Spin-c manifolds are not treated, the authors instead concentrating their attention to Kahlerian geometry. In this context the Clifford algebra multiplication has a beautiful relationship with the complex structure. A brief discussion is given of the pure spinors of Cartan and twistor spaces. The theory of holonomy and calibrations, the later due to one of the authors, is discussed in great detail. The discussion begins in the consideration of when universal covering spaces are not Riemannian manifolds and their holonomy groups have been classified. The idea of a calibration arises from the consideration of submanifolds that are homologically volume-minimizing. These become calibrations when the integrals of p-forms on them are the volumes, and these p-forms have vanishing differentials on oriented tangent p-planes on the manifold. The authors give an interesting discussion of the relation between spinors and calibrations.
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