Heat kernels and Dirac operators. (English) Zbl 0744.58001

Berline, Nicole; Getzler, Ezra; Vergne, Michèle

Grundlehren der Mathematischen Wissenschaften. 298. Berlin etc.: Springer-Verlag. vii, 369 p. (1992).

The book is devoted to the investigation of Dirac operators, mainly to the index theorem for them. The reader can see some details of the content of the book from the list of the chapters below, so we describe only some important points of the presentation.
Dirac operators on Riemannian manifolds were introduced by Atiyah and Singer as well as by Lichnerowicz. It is not necessary to say much about their importance because they occur in Hodge theory, gauge theory, geometric quantization and so on, most of first order linear differential operators of geometric origin are Dirac operators, etc. The authors’ aim as they feel it, is to write a book in which the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its more recent generalizations would receive elementary proofs.
The book is based on a principle learned from D. Quillen: Dirac operators are a quantization of the theory of connections, and the supertrace of the heat kernel of the square of a Dirac operator is the quantization of the Chern character of the corresponding connection. This leads the authors to think of index theory and the heat kernels as a quantization of Chern-Weyl theory.
The main technique in the book is an explicit geometric construction of the kernel of the heat operator \(e^{-tD^ 2}\) associated to the square of the Dirac operator \(D\). Note that in the proofs of many theorems due to J. M. Bismut, the original use of probability theory is replaced by the applications of classical asymptotic expansion methods. The constructions are expressed in the form such that they generalize easily to the equivariant setting. The authors consider the most general type of Dirac operators, associated to a Clifford module over a manifold, to avoid restricting themselves to manifolds with spin structures. The Quillen’s theory of superconnections is also in work.
List of Chapters: Introduction, 1. Background on Differential Geometry, 2. Asymptotic Expansion of Heat Kernel, 3. Clifford Modules and Dirac Operators, 4. Index Density of Dirac Operators, 5. The Exponential Map and the Index Density, 6. The Equivariant Index Theorem, 7. Equivariant Differential Forms, 8. The Kirillov Formula for the Equivariant Index, 9. The Index Bundle, 10. The Family Index Theorem.

Reviewer: Yu.E.Gliklikh (Voronezh)

MSC:

58-02	Research exposition (monographs, survey articles) pertaining to global analysis
58J05	Elliptic equations on manifolds, general theory
58J20	Index theory and related fixed-point theorems on manifolds
58J35	Heat and other parabolic equation methods for PDEs on manifolds

Keywords:

heat kernels; Dirac operators; Atiyah-Singer theorem