熱核與狄拉克算子（簡體書）

《熱核與狄拉克算子(英文版)》講述了：This book, which began as a seminar in 1985 at MIT, contains complete proofs of thelocal index theorem for Dirac operators using the heat kernel approach, together withits generalizations to equivariant Dirac operators and families of Dirac operators, aswell as background material on superconnections and equivariant differential forms.
Since the publication of the first edition, the subjects treated here have contin-ued to find new applications. Equivariant cohomology plays an important role in thestudy of symplectic reduction, and Bismut superconnections and the local index the-orem for families have had many applications, through the construction of higheranalytic torsion forms and currents. （For a survey of some of these developments,we recommend reading Bismuts talk at the Berlin International Congress of Mathe-maticians, reference
Although this book lacks some of the usual attributes of a textbook （such asexercises）, it has been widely used in advanced courses in differential geometry;for many of the topics discussed here, there are no other treatments available inmonograph form. Because of the continuing demand from students for the book,we were very

Introduction
1 Background on Differential Geometry
1.1 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 Dirae 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 Mehlers Formula
4.3 Calculation of the Index Density
5The Exponential Map and the Index Density
5,1 Iacobian 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 Proposition6.13

7 Equivariant Differential Forms
7.1 Equivariant Characteristic Classes
7.2 The Localization Formula
7.3 Botts Formulas for Characteristic Numbers
7.4 Exact Stationary Phase Approximation
5 The Fourier Transform of Coadjoint Orbits
7.6 Equivariant Cohomology and Families
7.7 The Bott Class

8 The Kirillov 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
9.1 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 Laplacian
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 Bismuts Index Theorem