Heat Kernels and Dirac OperatorsThe first edition of this book presented 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), using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in an attractive softcover. The first four chapters could be used as the text for a graduate course on the applications of linear elliptic operators in differential geometry and the only prerequisites are a familiarity with basic differential geometry. The next four chapters discuss the equivariant index theorem, and include a useful introduction to equivariant differential forms. The last two chapters give a proof, in the spirit of the book, of Bismut's Local Family Index Theorem for Dirac operators. |
Contents
Background on Differential Geometry | 13 |
12 Riemannian Manifolds | 31 |
13 Superspaces | 37 |
14 Superconnections | 41 |
15 Characteristic Classes | 44 |
16 The Euler and Thorn Classes | 49 |
Asymptotic Expansion of the Heat Kernel | 61 |
21 Differential Operators | 62 |
66 The heat kernel of an equivariant vector bundle | 194 |
67 Proof of Proposition 613 | 197 |
Equivariant Differential Forms | 201 |
72 The Localization Formula | 209 |
73 Botts Formulas for Characteristic Numbers | 217 |
74 Exact Stationary Phase Approximation | 219 |
75 The Fourier Transform of Coadjoint Orbits | 221 |
76 Equivariant Cohomology and Families | 227 |
22 The Heat Kernel on Euclidean Space | 69 |
23 Heat Kernels | 71 |
24 Construction of the Heat Kernel | 74 |
25 The Formal Solution | 79 |
26 The Trace of the Heat Kernel | 85 |
27 Heat Kernels Depending on a Parameter | 95 |
Clifford Modules and Dirac Operators | 99 |
31 The Clifford Algebra | 100 |
32 Spinors | 106 |
33 Dirac Operators | 110 |
34 Index of Dirac Operators | 118 |
35 The Lichnerowicz Formula | 122 |
36 Some Examples of Clifford Modules | 123 |
Index Density of Dirac Operators | 139 |
42 Mehlers Formula | 149 |
43 Calculation of the Index Density | 153 |
The Exponential Map and the Index Density | 163 |
51 Jacobian of the Exponential Map on Principal Bundles | 164 |
52 The Heat Kernel of a Principal Bundle | 168 |
53 Calculus with Grassmann and Clifford Variables | 173 |
54 The Index of Dirac Operators | 175 |
The Equivariant Index Theorem | 179 |
62 The AtiyahBott Fixed Point Formula | 181 |
63 Asymptotic Expansion of the Equivariant Heat Kernel | 185 |
64 The Local Equivariant Index Theorem | 188 |
65 Geodesic Distance on a Principal Bundle | 192 |
77 The Bott Class | 234 |
The Kirillov Formula for the Equivariant Index | 241 |
81 The Kirillov Formula | 242 |
82 The Weyl and Kirillov Character Formulas | 246 |
83 The Heat Kernel Proof of the Kirillov Formula | 248 |
The Index Bundle | 259 |
91 The Index Bundle in Finite Dimensions | 261 |
92 The Index Bundle of a Family of Dirac Operators | 269 |
93 The Chern Character of the Index Bundle | 272 |
94 The Equivariant Index and the Index Bundle | 283 |
95 The Case of Varying Dimension | 285 |
96 The ZetaFunction of a Laplacian | 289 |
97 The Determinant Line Bundle | 294 |
The Family Index Theorem | 307 |
101 Riemannian Fibre Bundles | 310 |
102 Clifford Modules on Fibre Bundles | 315 |
103 The Bismut Superconnection | 322 |
104 The Family Index Density | 326 |
105 The Transgression Formula | 334 |
106 The Curvature of the Determinant Line Bundle | 337 |
107 The Kirillov Formula and Bismuts Index Theorem | 340 |
References | 345 |
List of Notation | 351 |
Index | 355 |
Other editions - View all
Heat Kernels and Dirac Operators Nicole Berline,Ezra Getzler,Michèle Vergne No preview available - 1992 |
Common terms and phrases
Â-genus adjoint associated asymptotic expansion AT*M Atiyah Bismut superconnection ch(A Chapter Chern character Clifford action Clifford algebra Clifford connection Clifford module coefficients compact covariant derivative defined definition denote differential operator Dirac operator element End(E End(W equal equivariant index exterior family of Dirac fibre bundle finite-dimensional fixed point formula follows G-invariant geodesic given group G heat equation heat kernel Hermitian holomorphic horizontal ind(D index bundle integral isomorphic ker(D Kirillov formula Laplacian Lemma Levi-Civita connection Lie algebra Lie group line bundle M₂ metric obtain one-form oriented orthonormal frame P₁ polynomial principal bundle Proof Proposition prove representation Rham Riemannian curvature Riemannian manifold satisfies self-adjoint smooth family smooth sections spinor bundle superbundle superconnection supertrace tangent bundle tensor trivial twisting vanishes vector bundle vector field vector space vertical VM/B Z2-graded zero
References to this book
The Semicircle Law, Free Random Variables, and Entropy Fumio Hiai,Dénes Petz No preview available - 2000 |