Heat Kernels and Dirac Operators

by Nicole Berline

In this book, the Atiyah-Singer index theorem for Dirac operators on compact Riemannian manifolds and its more recent generalizations receive simple proofs. The main technique which is used is an explicit geometric construction of the heat kernels of a generalized Dirac operator. The first four chapters could be used at 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. Several chapters deal with other preparatory material.

Genres: Mathematics

369 pages, Hardcover

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.