Heat Kernels and Dirac Operators

As an application the curvature of the determinant line bundle is calculated, following Bismut and Freed. This book will be of interest to graduate students and researchers in differential geometry, Arakelov geometry, group representation, and also in theoretical physics.

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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, due to the authors and J.-M. Bismut, 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 prerequisites to reading this book are a familiarity with basic differential geometry. There are several chapters of preparatory material, including a treatment of connections and Quillens 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 Bismuts local version of the Atiyah-Singer Index Theorem for families.