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Author Duistermaat, J. J. (Johannes Jisse), 1942-2010.
Title The heat kernel Lefschetz fixed point formula for the spin-c dirac operator / J.J. Duistermaat.
Imprint Boston : Birkhauser, 2011.

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LOCATION CALL # STATUS MESSAGE
 OHIOLINK SPRINGER EBOOKS    ONLINE  
View online
Author Duistermaat, J. J. (Johannes Jisse), 1942-2010.
Series Modern Birkhauser Classics
Modern Birkhauser classics.
Subject Almost complex manifolds.
Operator theory.
Dirac equation.
Differential topology.
Mathematical physics.
Mathematics.
Description 1 online resource (vii, 247 pages)
Contents 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-Khler 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 Khler 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 Khler Case -- 13. Characteristic Classes -- 13.1. Weils 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 Cartans Theorem -- 16.4. Proof of Weils Theorem -- 16.5. General Actions.
Summary Interest in the spin-c Dirac operator originally came about from the study of complex analytic manifolds, where in the non-Kahler casethe Dolbeault operator is no longer suitable for getting local formulas for the Riemann-Roch number or the holomorphic Lefschetz number. However, every symplectic manifold (phase space in classical mechanics) also carries an almost complex structure and hence a corresponding spin-c Dirac operator. Using the heat kernels theory of Berline, Getzler, and Vergne, this workrevisits some fundamental concepts of the theory, and presents the application to symplectic geometry. J.J. Duistermaat was well known for his beautiful and concise expositions of seemingly familiar concepts, and this classic studyis certainly no exception. Reprinted as it was originally published, this workis as an affordable textthat will be of interest to a range of researchers in geometric analysis and mathematical physics. Overall this is a carefully written, highly readable book on a very beautiful subject.-Mathematical Reviews The book of J.J. Duistermaat is a nice introduction to analysis related[to the]spin-c Dirac operator. ... The book is almost self contained, [is] readable, and will be useful for anybody who is interested in the topic.-EMS Newsletter The author's book is a marvelous introduction to [these] objects and theories.-Zentralblatt MATH.
ISBN 9780817682477 (electronic bk.)
0817682473 (electronic bk.)
9781461253440 (electronic bk.)
1461253446 (electronic bk.)
9781461253464 (print)
1461253462 (print)
9780817682460
0817682465
ISBN/ISSN 10.1007/978-0-8176-8247-7
10.1007/978-1-4612-5344-0
OCLC # 748519240
Additional Format Printed edition: 9780817682460



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