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Heat Kernels and Dirac Operators

In the first edition of this book, simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut) were presented, using an explicit geometric construction of the heat kernel of a generalized Dirac ope...

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Bibliographic Details
Main Authors: Berline, Nicole, Getzler, Ezra (Author), Vergne, Michèle (Author)
Format: eBook
Language:English
Published: Berlin, Heidelberg Springer Berlin Heidelberg 2004, 2004
Edition:1st ed. 2004
Series:Grundlehren Text Editions
Subjects:
Group Theory And Generalizations
Geometry, Differential
Group Theory
Mathematical Physics
Differential Geometry
Theoretical, Mathematical And Computational Physics
Mathematical Methods In Physics
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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 Bismut’s Index Theorem
  • References
  • List of Notation
  • 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 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 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

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