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AuthorDuistermaat, J. J. author
TitleThe Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator [electronic resource] / by J. J. Duistermaat
ImprintBoston, MA : Birkhรคuser Boston, 1996
Connect tohttp://dx.doi.org/10.1007/978-1-4612-5344-0
Descript VIII, 247 p. online resource

SUMMARY

When visiting M.I.T. for two weeks in October 1994, Victor Guillemin made me enthusiastic about a problem in symplectic geometry which involved the use of the so-called spin-c Dirac operator. Back in Berkeley, where I had l spent a sabbatical semester , I tried to understand the basic facts about this operator: its definition, the main theorems about it, and their proofs. This book is an outgrowth of the notes in which I worked this out. For me this was a great learning experience because of the many beautiful mathematical structures which are involved. I thank the Editorial Board of Birkhauser, especially Haim Brezis, for sugยญ gesting the publication of these notes as a book. I am also very grateful for the suggestions by the referees, which have led to substantial improvements in the presentation. Finally I would like to express special thanks to Ann Kostant for her help and her prodding me, in her charming way, into the right direction. J.J. Duistermaat Utrecht, October 16, 1995


CONTENT

1 Introduction -- 1.1 The Holomorphic Lefschetz Fixed Point Formula -- 1.2 The Heat Kernel -- 1.3 The Results -- 2 The Dolbeault-Dirac Operator -- 2.1 The Dolbeault Complex -- 2.2 The Dolbeault-Dirac Operator -- 3 Clifford Modules -- 3.1 The Non-Kรคhler Case -- 3.2 The Clifford Algebra -- 3.3 The Supertrace -- 3.4 The Clifford Bundle -- 4 The Spin Group and the Spin-c Group -- 4.1 The Spin Group -- 4.2 The Spin-c Group -- 4.3 Proof of a Formula for the Supertrace -- 5 The Spin-c Dirac Operator -- 5.1 The Spin-c Frame Bundle and Connections -- 5.2 Definition of the Spin-c Dirac Operator -- 6 Its Square -- 6.1 Its Square -- 6.2 Dirac Operators on Spinor Bundles -- 6.3 The Kรคhler Case -- 7 The Heat Kernel Method -- 7.1 Traces -- 7.2 The Heat Diffusion Operator -- 8 The Heat Kernel Expansion -- 8.1 The Laplace Operator -- 8.2 Construction of the Heat Kernel -- 8.3 The Square of the Geodesic Distance -- 8.4 The Expansion -- 9 The Heat Kernel on a Principal Bundle -- 9.1 Introduction -- 9.2 The Laplace Operator on P -- 9.3 The Zero Order Term -- 9.4 The Heat Kernel -- 9.5 The Expansion -- 10 The Automorphism -- 10.1 Assumptions -- 10.2 An Estimate for Geodesies in P -- 10.3 The Expansion -- 11 The Hirzebruch-Riemann-Roch Integrand -- 11.1 Introduction -- 11.2 Computations in the Exterior Algebra -- 11.3 The Short Time Limit of the Supertrace -- 12 The Local Lefschetz Fixed Point Formula -- 12.1 The Element g0 of the Structure Group -- 12.2 The Short Time Limit -- 12.3 The Kรคhler Case -- 13 Characteristic Classes -- 13.1 Weilโ{128}{153}s Homomorphism -- 13.2 The Chern Matrix and the Riemann-Roch Formula -- 13.3 The Lefschetz Formula -- 13.4 A Simple Example -- 14 The Orbifold Version -- 14.1 Orbifolds -- 14.2 The Virtual Character -- 14.3 The Heat Kernel Method -- 14.4 The Fixed Point Orbifolds -- 14.5 The Normal Eigenbundles -- 14.6 The Lefschetz Formula -- 15 Application to Symplectic Geometry -- 15.1 Symplectic Manifolds -- 15.2 Hamiltonian Group Actions and Reduction -- 15.3 The Complex Line Bundle -- 15.4 Lifting the Action -- 15.5 The Spin-c Dirac Operator -- 16 Appendix: Equivariant Forms -- 16.1 Equivariant Cohomology -- 16.2 Existence of a Connection Form -- 16.3 Henri Cartanโ{128}{153}s Theorem -- 16.4 Proof of Weilโ{128}{153}s Theorem -- 16.5 General Actions


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