This book, which began as a seminar in 1985 at MIT, contains complete proofs of the local index theorem for Dirac operators using the heat kernel approach, together with its generalizations to equivariant Dirac operators and families of Dirac operators, as well as background material on superconnections and equivariant differential forms.

Introduction

I Background on Differential Geometry

 1.I Fibre Bundles and Connections

 1.2 Riemannian Manifolds

 1.3 Superspaces

 1.4 Superconnections

 1.5 Characteristic Classes

 1.6 The Euler and Thorn Classes

2 Asymptotic Expansion of the Heat Kernel

 2.1 Differential Operators

 2.2 The Heat Kernel on Euclidean Space

 2.3 Heat Kernels

 2.4 Construction of the Heat Kernel

 2.5 The Formal Solution

 2.6 The Trace of the Heat Kernel

 2.7 Heat Kernels Depending on a Parameter

3 Clifford Modules and Dirac Operators

 3.1 The Clifford Algebra

 3.2 Spinors

 3.3 Dirac Operators

 3.4 Index of Dirac Operators

 3.5 The Lichnerowicz Formula

 3.6 Some Examples of Clifford Modules

4 Index Density of Dirac Operators

 4.1 The Local Index Theorem

 4.2 Mehler's Formula

 4.3 Calculation of the Index Density

5 The Exponential Map and the Index Density

 5.1 Jacobian of the Exponential Map on Principal Bundles

 5.2 The Heat Kernel of a Principal Bundle

 5.3 Calculus with Grassmann and Clifford Variables

 5.4 The Index of Dirac Operators

6 The Equivariant Index Theorem

 6.1 The Equivariant Index of Dirac Operators

 6.2 The Atiyah-Bott Fixed Point Formula

 6.3 Asymptotic Expansion of the Equivariant Heat Kernel

 6.4 The Local Equivariant Index Theorem

 6.5 Geodesic Distance on a Principal Bundle

 6.6 The heat kernel of an equivariant vector bundle

 6.7 Proof of Proposition 6.13

7 Equivariant Differential Forms

 7.1 Equivariant Characteristic Classes

 7.2 The Localization Formula

 7.3 Bott's Formulas for Characteristic Numbers

 7.4 Exact Stationary Phase Approximation

 7.5 The Fourier Transform of Coadjoint Orbits

 7.6 Equivariant Cohomology and Families

 7.7 The Bott Class

8 The Kiriliov Formula for the Equivariant Index

 8.1 The Kirillov Formula

 8.2 The Weyl and Kirillov Character Formulas

 8.3 The Heat Kernel Proof of the Kirillov Formula

9 The Index Bundle

 9.l The Index Bundle in Finite Dimensions

 9.2 The Index Bundle of a Family of Dirac Operators

 9.3 The Chern Character of the Index Bundle

 9.4 The Equivariant Index and the Index Bundle

 9.5 The Case of Varying Dimension

 9.6 The Zeta-Function of a Laplaeian

 9.7 The Determinant Line Bundle

10 The Family Index Theorem

 10.1 Riemannian Fibre Bundles

 10.2 Clifford Modules on Fibre Bundles

 10.3 The Bismut Superconnection

 10.4 The Family Index Density

 10.5 The Transgression Formula

 10.6 The Curvature of the Determinant Line Bundle

 10.7 The Kirillov Formula and Bismut's Index Theorem

References

List of Notation

Index