Supersymmetry and Equivariant de Rham Theory

242 Pages·1999·9.389 MB·English

by  Victor W. Guillemin, Shlomo Sternberg, Jochen Brüning (auth.)

Supersymmetry and Equivariant de Rham Theory
Springer-Verlag Berlin Heidelberg GmbH
Victor W. Guillemin
Shlomo Sternberg
Su persym metry
and Equivariant
de Rham Theory
, Springer
Victor W. Guillemin Shlomo Sternberg
Department of Mathematics Department of Mathematics
Massachusetts Institute Harvard University
of Technology One Oxford Street
77, Massachusetts Avenue Cambridge, MA 02138
Cambridge, MA 02139 USA
USA
Jochen Brüning
Institut für Mathematik
Mathematisch -Naturwissen
schaftliche Fakultät 11
Humboldt -Universität Berlin
Unter den Linden 6
D-I01l7 Berlin
Germany
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Guillemin, Vietor W.:
Supersymmetry and equivariant de Rham theory I Victor W. Guillemin; Shlomo
Sternberg.
ISBN 978-3-642-08433-1 ISBN 978-3-662-03992-2 (eBook)
DOI 10.1007/978-3-662-03992-2
Mathematics Subject Classification (1991): 58-XX
ISBN 978-3-642-08433-1
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© Springer-Verlag Berlin Heidelberg 1999
Originally published by Springer-Verlag Berlin Heidelberg NewYork in 1999
Softcover reprint of the hardcover 1st edition 1999
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SPIN 10689157 44/3143-543210 -Printed on acid-freepaper
En hommage Cl Henri Cartan
Preface
This is the second volume of the Springer collection Mathematics Past and
Present. In the first volume, we republished Hörmander's fundamental papers
Fourier integral operators together with abrief introduction written from
the perspective of 1991. The composition of the second volume is somewhat
different: the two papers of Cartan which are reproduced here have a total
length of less than thirty pages, and the 220 page introduction which precedes
them is intended not only as a commentary on these papers but as a textbook
of its own, on a fascinating area of mathematics in which a lot of exciting
innovations have occurred in the last few years. Thus, in this second volume
the roles of the reprinted text and its commentary are reversed. The seminal
ideas outlined in Cartan's two papers are taken as the point of departure for
a fuH modern treatment of equivariant de Rham theory which does not yet
exist in the literature.
We envisage that future volumes in this collection will represent both vari
ants of the interplay between past and present mathematics: we will publish
classical texts, still of vital interest, either reinterpreted against the back
ground of fuHy developed theories or taken as the inspiration for original
developments.
Contents
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 Bibliographieal Notes for Chapter 1 . . . . . . . . 6
2 G* Modules 9
2.1 Differential-Geometrie Identities . 9
2.2 The Language of Superalgebra . 11
2.3 Prom Geometry to Algebra. 17
2.3.1 Cohomology .... 19
2.3.2 Acyclicity . . . . . . 20
2.3.3 Chain Homotopies 20
2.3.4 Pree Actions and the Condition (C) . 23
2.3.5 The Basie Subcomplex . . . . . . 26
2.4 Equivariant Cohomology of G* Aigebras 27
2.5 The Equivariant de Rham Theorem . 28
2.6 Bibliographieal 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 Bibliographieal 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
x Contents
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 HG(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) x U(i) . 100
Contents xi
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 Data105
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 v . . . . . . . . . . . . . . 156
10.4 Verifying that T 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
xii Contents
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'algebre differentielle; application aux groupes de Lie et
aux varietes Oll opere un groupe de Lie
Henri Cartan . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191
La transgression dans un ,graupe de Lie et dans un espace fibre
principal
Henri (Jartan . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 205
Bibliography 221
Index 227