Heat Kernels and Dirac Operators

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.

Written for:
Graduate students and researchers in differential geometry, Arakelov geometry, group representation theory and mathematical physics

Contents

• Introduction
• Background on Differential Geometry
• Asymptotic Expansion of the Heat Kernel
• Clifford Modules and Dirac Operators
• Index Density of Dirac Operators
• The Exponential Map and the Index Density
• The Equivariant Index Theorem
• Equivariant Differential Forms
• The Kirillov Formula for the Equivariant Index
• The Index Bundle
• The Family Index Theorem

• References
• List of Notation
• Index