Supersymmetry and Equivariant de Rham Theory
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1 Supersymmetry and Equivariant de Rham Theory Bearbeitet von Victor W Guillemin, Shlomo Sternberg, Jochen Brüning 1. Auflage Buch. xxiii, 232 S. Hardcover ISBN Format (B x L): 15,5 x 23,5 cm Gewicht: 1190 g Weitere Fachgebiete > Physik, Astronomie > Quantenphysik > Teilchenphysik schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.
2 Contents Introduction xiii 1 Equivariant Cohomology in Topology Equivariant Cohomology via Classifying Bundles Existence of Classifying Spaces Bibliographical Notes for Chapter G Modules Differential-Geometric Identities The Language of Superalgebra From Geometry to Algebra Cohomology Acyclicity Chain Homotopies Free Actions and the Condition (C) The Basic Subcomplex Equivariant Cohomology of G Algebras The Equivariant de Rham Theorem Bibliographical Notes for Chapter The Weil Algebra The Koszul Complex The Weil Algebra Classifying Maps W Modules Bibliographical Notes for Chapter The Weil Model and the Cartan Model The Mathai-Quillen Isomorphism The Cartan Model Equivariant Cohomology of W Modules H ((A E) bas ) does not depend on E The Characteristic Homomorphism Commuting Actions ix
3 x Contents 4.7 The Equivariant Cohomology of Homogeneous Spaces Exact Sequences Bibliographical Notes for Chapter Cartan s Formula The Cartan Model for W Modules Cartan s Formula Bibliographical Notes for Chapter Spectral Sequences Spectral Sequences of Double Complexes The First Term The Long Exact Sequence Useful Facts for Doing Computations Functorial Behavior Gaps Switching Rows and Columns The Cartan Model as a Double Complex H G (A) asans(g ) G -Module Morphisms of G Modules Restricting the Group Bibliographical Notes for Chapter Fermionic Integration Definition and Elementary Properties Integration by Parts Change of Variables Gaussian Integrals Iterated Integrals The Fourier Transform The Mathai-Quillen Construction The Fourier Transform of the Koszul Complex Bibliographical Notes for Chapter Characteristic Classes Vector Bundles The Invariants G = U(n) G = O(n) G = SO(2n) Relations Between the Invariants Restriction from U(n) too(n) Restriction from SO(2n) tou(n) Restriction from U(n) tou(k) U(l)
4 Contents xi 8.4 Symplectic Vector Bundles Consistent Complex Structures Characteristic Classes of Symplectic Vector Bundles Equivariant Characteristic Classes Equivariant Chern classes Equivariant Characteristic Classes of a Vector Bundle Over a Point Equivariant Characteristic Classes as Fixed Point Data The Splitting Principle in Topology Bibliographical Notes for Chapter Equivariant Symplectic Forms Equivariantly Closed Two-Forms The Case M = G Equivariantly Closed Two-Forms on Homogeneous Spaces The Compact Case Minimal Coupling Symplectic Reduction The Duistermaat-Heckman Theorem The Cohomology Ring of Reduced Spaces Flag Manifolds Delzant Spaces Reduction: The Linear Case Equivariant Duistermaat-Heckman Group Valued Moment Maps The Canonical Equivariant Closed Three-Form on G The Exponential Map G-Valued Moment Maps on Hamiltonian G-Manifolds Conjugacy Classes Bibliographical Notes for Chapter The Thom Class and Localization Fiber Integration of Equivariant Forms The Equivariant Normal Bundle Modifying ν Verifying that τ is a Thom Form The Thom Class and the Euler Class The Fiber Integral on Cohomology Push-Forward in General Localization The Localization for Torus Actions Bibliographical Notes for Chapter
5 xii Contents 11 The Abstract Localization Theorem Relative Equivariant de Rham Theory Mayer-Vietoris S(g )-Modules The Abstract Localization Theorem The Chang-Skjelbred Theorem Some Consequences of Equivariant Formality Two Dimensional G-Manifolds A Theorem of Goresky-Kottwitz-MacPherson Bibliographical Notes for Chapter Appendix 189 Notions d algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie Henri Cartan La transgression dans un groupe de Lie et dans un espace fibré principal Henri Cartan Bibliography 221 Index 227
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