Heat Kernels and Dirac Operators: Vol 298 (Die Grundlehren der Mathematischen Wissenschaften) - Tapa dura

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.

The past few years have seen the emergence of new insights into the Atiyah-Singer Index Theorem for Dirac operators. In this book, elementary proofs of this theorem, and some of its more recent generalizations, are presented. The formula for the index of the Dirac operator is obtained from the classical formula for the heat kernel of the harmonic oscillator. The only prerequisite to reading this book is a familiarity with basic differential geometry. There are several chapters of preparatory material, including a treatment of connections and Quillen's theory of superconnections, characteristic classes, the theory of the heat equation and its solution on a compact manifold, Clifford algebras, Dirac operators and equivariant differential forms. The book finishes with a treatment of the index bundle and Bismut's local version of the Atiyah-Singer Index Theorem for families. As an application, the curvature of the determinant line bundle is calculated, following Bismut and Freed.