ISBN: 3-540-64797-X
TITLE: Supersymmetry and Equivariant de Rham Theory
AUTHOR: Guillemin, Victor W.; Sternberg, Shlomo
TOC:

Introduction xiii 
1 Equivariant Cohomology in Topology 1 
1.1 Equivariant Cohomology via Classifying Bundles 1 
1.2 Existence of Classifying Spaces 5 
1.3 Bibliographical Notes for Chapter 1 6 
2 G* Modules 9 
2.1 Differential-Geometric Identities 9 
2.2 The Language of Superalgebra 11 
2.3 From Geometry to Algebra 17 
2.3.1 Cohomology 19 
2.3.2 Acyclicity 20 
2.3.3 Chain Homotopies 20 
2.3.4 Free Actions and the Condition (C) 23 
2.3.5 The Basic Subcomplex 26 
2.4 Equivariant Cohomology of G* Algebras 27 
2.5 The Equivariant de Rham Theorem 28 
2.6 Bibliographical Notes for Chapter 2 31 
3 The Weil Algebra 33 
3.1 The Koszul Complex 33 
3.2 The Weil Algebra 34 
3.3 Classifying Maps 37 
3.4 W* Modules 39 
3.5 Bibliographical Notes for Chapter 3 40 
4 The Weil Model and the Cartan Model 41 
4.1 The Mathai-Quillen Isomorphism 41 
4.2 The Cartan Model 44 
4.3 Equivariant Cohomology of W* Modules 46 
4.4 H ((A + E)_{bas}) does not depend on E 48 
4.5 The Characteristic Homomorphism 48 
4.6 Commuting Actions 49 
4.7 The Equivariant Cohomology of 
Homogeneous Spaces 50 
4.8 Exact Sequences 51 
4.9 Bibliographical Notes for Chapter 4 51 
5 Cartan's Formula 53 
5.1 The Cartan Model for W* Modules 54 
5.2 Cartan's Formula 57 
5.3 Bibliographical Notes for Chapter 5 59 
6 Spectral Sequences 61 
6.1 Spectral Sequences of Double Complexes 61 
6.2 The First Term 66 
6.3 The Long Exact Sequence 67 
6.4 Useful Facts for Doing Computations 68 
6.4.1 Functorial Behavior 68 
6.4.2 Gaps 68 
6.4.3 Switching Rows and Columns 69 
6.5 The Cartan Model as a Double Complex 69 
6.6 H_G(A) as an S(g*)^G-Module 71 
6.7 Morphisms of G* Modules 71 
6.8 Restricting the Group 72 
6.9 Bibliographical Notes for Chapter 6 75 
7 Fermionic Integration 77 
7.1 Definition and Elementary Properties 77 
7.1.1 Integration by Parts 78 
7.1.2 Change of Variables 78 
7.1.3 Gaussian Integrals 79 
7.1.4 Iterated Integrals 80 
7.1.5 The Fourier Transform 81 
7.2 The Mathai-Quillen Construction 85 
7.3 The Fourier Transform of the Koszul Complex 88 
7.4 Bibliographical Notes for Chapter 7 92 
8 Characteristic Classes 95 
8.1 Vector Bundles 95 
8.2 The Invariants 96 
8.2.1 G = U(n) 96 
8.2.2 G = O(n) 97 
8.2.3 G = SO(2n) 97 
8.3 Relations Between the Invariants 98 
8.3.1 Restriction from U(n) to O(n) 99 
8.3.2 Restriction from SO(2n) to U(n) 100 
8.3.3 Restriction from U(n) to U(k)  U(l) 100 
8.4 Symplectic Vector Bundles 101 
8.4.1 Consistent Complex Structures 101 
8.4.2 Characteristic Classes of Symplectic Vector Bundles 103 
8.5 Equivariant Characteristic Classes 104 
8.5.1 Equivariant Chern classes 104 
8.5.2 Equivariant Characteristic Classes of a 
Vector Bundle Over a Point 104 
8.5.3 Equivariant Characteristic Classes as Fixed Point Data 105 
8.6 The Splitting Principle in Topology 106 
8.7 Bibliographical Notes for Chapter 8 108 
9 Equivariant Symplectic Forms 111 
9.1 Equivariantly Closed Two-Forms 111 
9.2 The Case M = G 112 
9.3 Equivariantly Closed Two-Forms on 
Homogeneous Spaces 114 
9.4 The Compact Case 115 
9.5 Minimal Coupling 116 
9.6 Symplectic Reduction 117 
9.7 The Duistermaat-Heckman Theorem 120 
9.8 The Cohomology Ring of Reduced Spaces 121 
9.8.1 Flag Manifolds 124 
9.8.2 Delzant Spaces 126 
9.8.3 Reduction: The Linear Case 130 
9.9 Equivariant Duistermaat-Heckman 132 
9.10 Group Valued Moment Maps 134 
9.10.1 The Canonical Equivariant Closed Three-Form on G 135 
9.10.2 The Exponential Map 138 
9.10.3 G-Valued Moment Maps on 
Hamiltonian G-Manifolds 141 
9.10.4 Conjugacy Classes 143 
9.11 Bibliographical Notes for Chapter 9 145 
10 The Thom Class and Localization 149 
10.1 Fiber Integration of Equivariant Forms 150 
10.2 The Equivariant Normal Bundle 154 
10.3 Modifying nu 156 
10.4 Verifying that tau is a Thom Form 156 
10.5 The Thom Class and the Euler Class 158 
10.6 The Fiber Integral on Cohomology 159 
10.7 Push-Forward in General 159 
10.8 Localization 160 
10.9 The Localization for Torus Actions 163 
10.10 Bibliographical Notes for Chapter 10 168 
11 The Abstract Localization Theorem 173 
11.1 Relative Equivariant de Rham Theory 173 
11.2 Mayer-Vietoris 175 
11.3 S(g*)-Modules 175 
11.4 The Abstract Localization Theorem 176 
11.5 The Chang-Skjelbred Theorem 179 
11.6 Some Consequences of Equivariant 
Formality 180 
11.7 Two Dimensional G-Manifolds 180 
11.8 A Theorem of Goresky-Kottwitz-MacPherson 183 
11.9 Bibliographical Notes for Chapter 11 185 
Appendix 189 
Notions d'algbre diffrentielle; application aux groupes de Lie et 
aux varits o opre un groupe de Lie 
Henri Cartan 191 
La transgression dans un groupe de Lie et dans un espace fibr 
principal 
Henri Cartan 205 
Bibliography 221 
Index 227 
END
