Heat Kernels and Dirac Operators (Grundlehren Text Editions) 1992nd Edition by Nicole Berline (PDF)

Description

In the first edition of this book, simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut) were presented, using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in an attractive paperback.

User’s Reviews

Editorial Reviews: Review Aus den Rezensionen:”… Das vorliegende Buch ist die zweite korrigierte und erweiterte Ausgabe eines Werkes aus dem Jahre 1992. … Ausgehend von einer Grundausbildung in klassischer Differentialgeometrie stellt das Buch alle zum Verständnis des Beweises notwendigen Voraussetzungen zur Verfügung. Dadurch eignet es sich einerseits zum Selbststudium für Studierende mit entsprechender Vorbildung … andererseits als Grundlage einer Vorlesung über dieses ergiebige Thema.”(P. Grabner, in: IMN – Internationale Mathematische Nachrichten, 2006, Issue 202, S. 45) From the Back Cover In the first edition of this book, simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut) were presented, using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in an attractive paperback. The first four chapters could be used as the text for a graduate course on the applications of linear elliptic operators in differential geometry and the only prerequisites are a familiarity with basic differential geometry. The next four chapters discuss the equivariant index theorem, and include a useful introduction to equivariant differential forms. The last two chapters give a proof, in the spirit of the book, of Bismut’s Local Family Index Theorem for Dirac operators. This book will be of interest to graduate students and researchers in differential geometry, Arakelov geometry, group representation theory and mathematical physics.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Dirac operators on Riemannian manifolds are of fundamental importance in differential geometry: they occur in situations such as Hodge theory, gauge theory, and geometric quantization.The book is based on a simple principle: 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. From this point of view, the index theorem for Dirac operators is a statement about the relationship between the heat kernel of the square of a Dirac operator and the Chern character of the associated connection. This relationship holds at the level of differential forms and not just in cohomology, and leads to think of index theory and heat kernels as a quantization of Chern-Weil theory. The importance of the heat kernel is that it interpolates between the identity operator and the projection onto the kernel of the Dirac operator. However, the authors study the heat kernel, and more particularly its restriction to the diagonal, in its own right, and not only as a tool in understanding the kernel of the Dirac operator.The authors attempt to express allof their constructions in such a way that they generalize easily to the equivariant setting, in which a compact Lie group acts on the manifold and leaves the Dirac operator invariant. They consider the most general type of Dirac operators, associated to a Clifford module over a manifold, to avoid restricting to manifolds with spin connections. They also work within Quillen’s theory of superconnections.The book is not necessarily meant to be read sequentially, and consists of four groups of chapters: (1) Chapters 1 and 7, the former giving various preliminary results in differential geometry and the latter on equivariant differential forms; they do not depend on any other chapters. (2) Chapters 2, 3, and 4 introduce the main ideas of the book, and take the reader through the main properties of Dirac operators, culminating in the local index theorem. (3) Chapters 5, 6, and 8 are on the equivariant index theorem, and may be read after the first four chapters, although Chapter 7 is needed in Chapter 8. (4) Chapters 9 and 10 are on the family index theorem, and can be read after the first four chapters, except sections 9.4 and 10.7 which have Chapter 8 as a prerequisite.The book is intended for researchers and advanced graduate students; you need a very strong background in differntial geometry, algebraic topology, harmonic analysis, and hypercomplex analysis to read it. The style is definitely French, so if you have had trouble with Bourbaki be prepared. The list of references is adequately long. Very nice printing and binding quality.

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