Supersymmetry and Equivariant De Rham Theory

by Guillemin, Victor W.; Sternberg, Shlomo

• ISBN13: 9783540647973
• ISBN10: 354064797X
• Format: Hardcover
• Copyright: 1999-07-01
• Publisher: Springer Verlag

Summary

Equivariant cohomology on smooth manifolds is the subject of this book which is part of a collection of volumes edited by J. Brüning 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.

Author Biography

From the reviews:MATHEMATICAL 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 ideasGǪThere are GÇÿgifts' for the more advanced readers as well, in the form of many refreshing modern points of view proposed by the authorsGǪThe 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."

Table of Contents

Introduction	xiii
1 Equivariant Cohomology in Topology	1	(8)
1.1 Equivariant Cohomology via Classifying Bundles	1	(4)
1.2 Existence of Classifying Spaces	5	(1)
1.3 Bibliographical Notes for Chapter 1	6	(3)
2 G* Modules	9	(24)
2.1 Differential-Geometric Identities	9	(2)
2.2 The Language of Superalgebra	11	(6)
2.3 From Geometry to Algebra	17	(10)
2.3.1 Cohomology	19	(1)
2.3.2 Acyclicity	20	(1)
2.3.3 Chain Homotopies	20	(3)
2.3.4 Free Actions and the Condition (C)	23	(3)
2.3.5 The Basic Subcomplex	26	(1)
2.4 Equivariant Cohomology of G* Algebras	27	(1)
2.5 The Equivariant de Rham Theorem	28	(3)
2.6 Bibliographical Notes for Chapter 2	31	(2)
3 The Weil Algebra	33	(8)
3.1 The Koszul Complex	33	(1)
3.2 The Weil Algebra	34	(3)
3.3 Classifying Maps	37	(2)
3.4 W* Modules	39	(1)
3.5 Bibliographical Notes for Chapter 3	40	(1)
4 The Weil Model and the Cartan Model	41	(12)
4.1 The Mathai-Quillen Isomorphism	41	(3)
4.2 The Cartan Model	44	(2)
4.3 Equivariant Cohomology of W* Modules	46	(2)
4.4 H ((A(xxx)E)(bas)) does not depend on E	48	(1)
4.5 The Characteristic Homomorphism	48	(1)
4.6 Commuting Actions	49	(1)
4.7 The Equivariant Cohomology of Homogeneous Spaces	50	(1)
4.8 Exact Sequences	51	(1)
4.9 Bibliographical Notes for Chapter 4	51	(2)
5 Cartan's Formula	53	(8)
5.1 The Cartan Model for W* Modules	54	(3)
5.2 Cartan's Formula	57	(2)
5.3 Bibliographical Notes for Chapter 5	59	(2)
6 Spectral Sequences	61	(16)
6.1 Spectral Sequences of Double Complexes	61	(5)
6.2 The First Term	66	(1)
6.3 The Long Exact Sequence	67	(1)
6.4 Useful Facts for Doing Computations	68	(1)
6.4.1 Functorial Behavior	68	(1)
6.4.2 Gaps	68	(1)
6.4.3 Switching Rows and Columns	69	(1)
6.5 The Cartan Model as a Double Complex	69	(2)
6.6 H(G)(A) as an S(g*)^(G)-Module	71	(1)
6.7 Morphisms of G* Modules	71	(1)
6.8 Restricting the Group	72	(3)
6.9 Bibliographical Notes for Chapter 6	75	(2)
7 Fermionic Integration	77	(18)
7.1 Definition and Elementary Properties	77	(8)
7.1.1 Integration by Parts	78	(1)
7.1.2 Change of Variables	78	(1)
7.1.3 Gaussian Integrals	79	(1)
7.1.4 Iterated Integrals	80	(1)
7.1.5 The Fourier Transform	81	(4)
7.2 The Mathai-Quillen Construction	85	(3)
7.3 The Fourier Transform of the Koszul Complex	88	(4)
7.4 Bibliographical Notes for Chapter 7	92	(3)
8 Characteristic Classes	95	(16)
8.1 Vector Bundles	95	(1)
8.2 The Invariants	96	(2)
8.2.1 G = U(n)	96	(1)
8.2.2 G = O(n)	97	(1)
8.2.3 G = SO(2n)	97	(1)
8.3 Relations Between the Invariants	98	(3)
8.3.1 Restriction from U(n) to O(n)	99	(1)
8.3.2 Restriction from SO(2n) to U(n)	100	(1)
8.3.3 Restriction from U(n) to U(k) x U(l)	100	(1)
8.4 Symplectic Vector Bundles	101	(3)
8.4.1 Consistent Complex Structures	101	(2)
8.4.2 Characteristic Classes of Symplectic Vector Bundles	103	(1)
8.5 Equivariant Characteristic Classes	104	(2)
8.5.1 Equivariant Chern classes	104	(1)
8.5.2 Equivariant Characteristic Classes of a Vector Bundle Over a Point	104	(1)
8.5.3 Equivariant Characteristic Classes as Fixed Point Data	105	(1)
8.6 The Splitting Principle in Topology	106	(2)
8.7 Bibliographical Notes for Chapter 8	108	(3)
9 Equivariant Symplectic Forms	111	(38)
9.1 Equivariantly Closed Two-Forms	111	(1)
9.2 The Case M = G	112	(2)
9.3 Equivariantly Closed Two-Forms on Homogeneous Spaces	114	(1)
9.4 The Compact Case	115	(1)
9.5 Minimal Coupling	116	(1)
9.6 Symplectic Reduction	117	(3)
9.7 The Duistermaat-Heckman Theorem	120	(1)
9.8 The Cohomology Ring of Reduced Spaces	121	(11)
9.8.1 Flag Manifolds	124	(2)
9.8.2 Delzant Spaces	126	(4)
9.8.3 Reduction: The Linear Case	130	(2)
9.9 Equivariant Duistermaat-Heckman	132	(2)
9.10 Group Valued Moment Maps	134	(11)
9.10.1 The Canonical Equivariant Closed Three-Form on G	135	(3)
9.10.2 The Exponential Map	138	(3)
9.10.3 G-Valued Moment Maps on Hamiltonian G-Manifolds	141	(2)
9.10.4 Conjugacy Classes	143	(2)
9.11 Bibliographical Notes for Chapter 9	145	(4)
10 The Thom Class and Localization	149	(24)
10.1 Fiber Integration of Equivariant Forms	150	(4)
10.2 The Equivariant Normal Bundle	154	(2)
10.3 Modifying Nu	156	(1)
10.4 Verifying that Tan is a Thom Form	156	(2)
10.5 The Thom Class and the Euler Class	158	(1)
10.6 The Fiber Integral on Cohomology	159	(1)
10.7 Push-Forward in General	159	(1)
10.8 Localization	160	(3)
10.9 The Localization for Torus Actions	163	(5)
10.10 Bibliographical Notes for Chapter 10	168	(5)
11 The Abstract Localization Theorem	173	(16)
11.1 Relative Equivariant de Rham Theory	173	(2)
11.2 Mayer-Vietoris	175	(1)
11.3 S(g*)-Modules	175	(1)
11.4 The Abstract Localization Theorem	176	(3)
11.5 The Chang-Skjelbred Theorem	179	(1)
11.6 Some Consequences of Equivariant Formality	180	(1)
11.7 Two Dimensional G-Manifolds	180	(3)
11.8 A Theorem of Goresky-Kottwitz-MacPherson	183	(2)
11.9 Bibliographical Notes for Chapter 11	185	(4)
Appendix	189	(32)
Notions d'algebre differentielle; application aux groupes de Lie et aux varietes ou opere un groupe de Lie	191	(14)
Henri Cartan
La transgression dans un groupe de Lie et dans un espace fibre principal	205	(16)
Henri Cartan
Bibliography	221	(6)
Index	227