Heat Kernels and Dirac Operators

by Nicole Berline, Ezra Getzler, Michèle Vergne

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.

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.- Bibliography.- List of Notations.- Index.

Genres: Mathematics

372 pages, Paperback

First published May 28, 1996

Reviews

[Andrew]

If you see the Atiyah-Singer index theorem as an organizing principle for mathematics, this book gives a version which we "know" should exist but which Atiyah, Bott, Patodi, and Singer never write about: a G-equivariant heat kernel version for families. The generalization is nontrivial and requires a key insight from Quillen: superconnections and their Chern characters, promoting notions from equivariant cohomology to the level of differential forms.