Heat Kernels and Dirac Operators (Grundlehren Text Editions)

個数:

Heat Kernels and Dirac Operators (Grundlehren Text Editions)

  • 提携先の海外書籍取次会社に在庫がございます。通常約3週間で発送いたします。
    重要ご説明事項
    1. 納期遅延や、ご入手不能となる場合が若干ございます。
    2. 複数冊ご注文の場合、分割発送となる場合がございます。
    3. 美品のご指定は承りかねます。

  • 提携先の海外書籍取次会社に在庫がございます。通常2週間で発送いたします。
    重要ご説明事項
    1. 納期遅延や、ご入手不能となる場合が若干ございます。
    2. 複数冊ご注文の場合、分割発送となる場合がございます。
    3. 美品のご指定は承りかねます。
  • 【重要:入荷遅延について】
    各国での新型コロナウィルス感染拡大により、洋書・洋古書の入荷が不安定になっています。
    弊社サイト内で表示している標準的な納期よりもお届けまでに日数がかかる見込みでございます。
    申し訳ございませんが、あらかじめご了承くださいますようお願い申し上げます。

  • 製本 Paperback:紙装版/ペーパーバック版/ページ数 362 p.
  • 商品コード 9783540200628

基本説明

New in softcover. Hardcover is out of print. 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. Originally published as Volume 298 in the series: "Grundlehren der mathematischen Wissenschaften", 1992.

Description


(Text)
The 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.
(Table of content)
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 Grassmannand 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.
(Review)
Aus den Rezensionen:

"... Das vorliegende Buch ist die zweite korrigierte und erweiterte Ausgabe eines Werkes aus dem Jahre 1992. ... Ausgehend von einer Grundausbildung in klassischer Differentialgeometrie stellt das Buch alle zum Verständnis des Beweises notwendigen Voraussetzungen zur Verfügung. Dadurch eignet es sich einerseits zum Selbststudium für Studierende mit entsprechender Vorbildung ... andererseits als Grundlage einer Vorlesung über dieses ergiebige Thema."

(P. Grabner, in: IMN - Internationale Mathematische Nachrichten, 2006, Issue 202, S. 45)


Table of Contents

Introduction                                       1  (310)
1 Background on Differential Geometry 13 (48)
1.1 Fibre Bundles and Connections 13 (18)
1.2 Riemannian Manifolds 31 (6)
1.3 Superspaces 37 (4)
1.4 Superconnections 41 (3)
1.5 Characteristic Classes 44 (5)
1.6 The Euler and Thom Classes 49 (12)
2 Asymptotic Expansion of the Heat Kernel 61 (38)
2.1 Differential Operators 62 (7)
2.2 The Heat Kernel on Euclidean Space 69 (2)
2.3 Heat Kernels 71 (3)
2.4 Construction of the Heat Kernel 74 (5)
2.5 The Formal Solution 79 (6)
2.6 The Trace of the Heat Kernel 85 (10)
2.7 Heat Kernels Depending on a Parameter 95 (4)
3 Clifford Modules and Dirac Operators 99 (40)
3.1 The Clifford Algebra 100(6)
3.2 Spinors 106(4)
3.3 Dirac Operators 110(8)
3.4 Index of Dirac Operators 118(4)
3.5 The Lichnerowicz Formula 122(1)
3.6 Some Examples of Clifford Modules 123(16)
4 Index Density of Dirac Operators 139(24)
4.1 The Local Index Theorem 139(10)
4.2 Mehler's Formula 149(4)
4.3 Calculation of the Index Density 153(10)
5 The Exponential Map and the Index Density 163(18)
5.1 Jacobian of the Exponential Map on 164(4)
Principal Bundles
5.2 The Heat Kernel of a Principal Bundle 168(5)
5.3 Calculus with Grassmann and Clifford 173(4)
Variables
5.4 The Index of Dirac Operators 177(4)
6 The Equivariant Index Theorem 181(22)
6.1 The Equivariant Index of Dirac Operators 182(1)
6.2 The Atiyah-Bott Fixed Point Formula 183(4)
6.3 Asymptotic Expansion of the Equivariant 187(3)
Heat Kernel
6.4 The Local Equivariant Index Theorem 190(4)
6.5 Geodesic Distance on a Principal Bundle 194(2)
6.6 The heat kernel of an equivariant 196(3)
vector bundle
6.7 Proof of Proposition 6.13 199(4)
7 Equivariant Differential Forms 203(40)
7.1 Equivariant Characteristic Classes 204(7)
7.2 The Localization Formula 211(8)
7.3 Bott's Formulas for Characteristic 219(2)
Numbers
7.4 Exact Stationary Phase Approximation 221(2)
7.5 The Fourier Transform of Coadjoint 223(6)
Orbits
7.6 Equivariant Cohomology and Families 229(7)
7.7 The Bott Class 236(7)
8 The Kirillov Formula for the Equivariant 243(20)
Index
8.1 The Kirillov Formula 244(4)
8.2 The Weyl and Kirillov Character Formulas 248(4)
8.3 The Heat Kernel Proof of the Kirillov 252(11)
Formula
9 The Index Bundle 263(48)
9.1 The Index Bundle in Finite Dimensions 265(8)
9.2 The Index Bundle of a Family of Dirac 273(3)
Operators
9.3 The Chern Character of the Index Bundle 276(11)
9.4 The Equivariant Index and the Index 287(2)
Bundle
9.5 The Case of Varying Dimension 289(4)
9.6 The Zeta-Function of a Laplacian 293(5)
9.7 The Determinant Line Bundle 298(13)
10 The Family Index Theorem 311(38)
10.1 Riemannian Fibre Bundles 314(5)
10.2 Clifford Modules on Fibre Bundles 319(7)
10.3 The Bismut Superconnection 326(4)
10.4 The Family Index Density 330(8)
10.5 The Transgression Formula 338(3)
10.6 The Curvature of the Determinant Line 341(3)
Bundle
10.7 The Kirillov Formula and Bismut's 344(5)
Index Theorem
References 349(6)
List of Notation 355(4)
Index 359