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Supersymmetry and Equivariant de Rham Theory - Shlomo Sternberg

Supersymmetry and Equivariant de Rham Theory


Published: 4th May 1999
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Equivariant cohomology on smooth manifolds is the subject of this book which is part of a collection of volumes edited by J. Bruning and V.W. Guillemin. The point of departure are two relatively short but very remarkable papers be Henry Cartan, published in 1950 in the Proceedings of the "Colloque de Topologie". These papers are reproduced here, together with a modern introduction to the subject, written by two of the leading experts in the field. This "introduction" comes as a textbook of its own, though, presenting the first full treatment of equivariant cohomology in the de Rahm setting. The well known topological approach is linked with the differential form aspect through the equivariant de Rahm theorem. The systematic use of supersymmetry simplifies considerably the ensuing development of the basic technical tools which are then applied to a variety of subjects, leading up to the localization theorems and other very recent results.

From the reviews:


"The authors are very generous to the reader, and explain all the basics in a very clear and efficient manner. The understanding is enhanced by appealing to concepts which developed after Cartan's seminal work, which also help to place things in a broader context. This approach sheds light on many of Cartan's motivations, and helps the reader appreciate the beauty and the simplicity of his ideas...There are `gifts' for the more advanced readers as well, in the form of many refreshing modern points of view proposed by the authors...The second part of the book is in my view a very convincing argument for the usefulness and versatility of this theory, and can also serve as a very good invitation to more detailed investigation. I learned a lot from this book, which is rich in new ideas. I liked the style and the respect the authors have for the readers. I also appreciated very much the bibliographical and historical comments at the end of each chapter. To conclude, I believe this book is a must have for any mathematician/physicist remotely interested in this subject."

Introductionp. xiii
Equivariant Cohomology in Topologyp. 1
Equivariant Cohomology via Classifying Bundlesp. 1
Existence of Classifying Spacesp. 5
Bibliographical Notes for Chapter 1p. 6
G* Modulesp. 9
Differential-Geometric Identitiesp. 9
The Language of Superalgebrap. 11
From Geometry to Algebrap. 17
Cohomologyp. 19
Acyclicityp. 20
Chain Homotopiesp. 20
Free Actions and the Condition (C)p. 23
The Basic Subcomplexp. 26
Equivariant Cohomology of G* Algebrasp. 27
The Equivariant de Rham Theoremp. 28
Bibliographical Notes for Chapter 2p. 31
The Weil Algebrap. 33
The Koszul Complexp. 33
The Weil Algebrap. 34
Classifying Mapsp. 37
W* Modulesp. 39
Bibliographical Notes for Chapter 3p. 40
The Weil Model and the Cartan Modelp. 41
The Mathai-Quillen Isomorphismp. 41
The Cartan Modelp. 44
Equivariant Cohomology of W* Modulesp. 46
H ((A ⊗ E)bas) does not depend on Ep. 48
The Characteristic Homomorphismp. 48
Commuting Actionsp. 49
The Equivariant Cohomology of Homogeneous Spacesp. 50
Exact Sequencesp. 51
Bibliographical Notes for Chapter 4p. 51
Cartan's Formulap. 53
The Cartan Model for W* Modulesp. 54
Cartan's Formulap. 57
Bibliographical Notes for Chapter 5p. 59
Spectral Sequencesp. 61
Spectral Sequences of Double Complexesp. 61
The First Termp. 66
The Long Exact Sequencep. 67
Useful Facts for Doing Computationsp. 68
Functorial Behaviorp. 68
Gapsp. 68
Switching Rows and Columnsp. 69
The Cartan Model as a Double Complexp. 69
HG(A) as an S(g*)G-Modulep. 71
Morphisms of G* Modulesp. 71
Restricting the Groupp. 72
Bibliographical Notes for Chapter 6p. 75
Fermionic Integrationp. 77
Definition and Elementary Propertiesp. 77
Integration by Partsp. 78
Change of Variablesp. 78
Gaussian Integralsp. 79
Iterated Integralsp. 80
The Fourier Transformp. 81
The Mathai-Quillen Constructionp. 85
The Fourier Transform of the Koszul Complexp. 88
Bibliographical Notes for Chapter 7p. 92
Characteristic Classesp. 95
Vector Bundlesp. 95
The Invariantsp. 96
G = U(n)p. 96
G = O(n)p. 97
G = SO(2n)p. 97
Relations Between the Invariantsp. 98
Restriction from U(n) to O(n)p. 99
Restriction from SO(2n) to U(n)p. 100
Restriction from U(n) to U(k) × U(ℓ)p. 100
Symplectic Vector Bundlesp. 101
Consistent Complex Structuresp. 101
Characteristic Classes of Symplectic Vector Bundlesp. 103
Equivariant Characteristic Classesp. 104
Equivariant Chern classesp. 104
Equivariant Characteristic Classes of a Vector Bundle Over a Pointp. 104
Equivariant Characteristic Classes as Fixed Point Datap. 105
The Splitting Principle in Topologyp. 106
Bibliographical Notes for Chapter 8p. 108
Equivariant Symplectic Formsp. 111
Equivariantly Closed Two-Formsp. 111
The Case M = Gp. 112
Equivariantly Closed Two-Forms on Homogeneous Spacesp. 114
The Compact Casep. 115
Minimal Couplingp. 116
Symplectic Reductionp. 117
The Duistermaat-Heckman Theoremp. 120
The Cohomology Ring of Reduced Spacesp. 121
Flag Manifoldsp. 124
Delzant Spacesp. 126
Reduction: The Linear Casep. 130
Equivariant Duistermaat-Heckmanp. 132
Group Valued Moment Mapsp. 134
The Canonical Equivariant Closed Three-Form on Gp. 135
The Exponential Mapp. 138
G-Valued Moment Maps on Hamiltonian G-Manifoldsp. 141
Conjugacy Classesp. 143
Bibliographical Notes for Chapter 9p. 145
The Thom Class and Localizationp. 149
Fiber Integration of Equivariant Formsp. 150
The Equivariant Normal Bundlep. 154
Modifying ¿p. 156
Verifying that ¿ is a Thom Formp. 156
The Thom Class and the Euler Classp. 158
The Fiber Integral on Cohomologyp. 159
Push-Forward in Generalp. 159
Localizationp. 160
The Localization for Torus Actionsp. 163
Bibliographical Notes for Chapter 10p. 168
The Abstract Localization Theoremp. 173
Relative Equivariant de Rham Theoryp. 173
Mayer-Vietorisp. 175
S(g*) Modulesp. 175
The Abstract Localization Theoremp. 176
The Chang-Skjelbred Theoremp. 179
Some Consequences of Equivariant Formalityp. 180
Two Dimensional G-Manifoldsp. 180
A Theorem of Goresky-Kottwitz-Mac Phersonp. 183
Bibliographical Notes for Chapter 11p. 185
Appendixp. 189
Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Liep. 191
La transgression dans un groupe de Lie et dans un espace fibré principalp. 205
Bibliographyp. 221
Indexp. 227
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540647973
ISBN-10: 354064797X
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 232
Published: 4th May 1999
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 24.13 x 15.88  x 1.91
Weight (kg): 0.5