Hjem

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.- 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.