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The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator (Progress in Nonlinear Differential Equations and Their Applica
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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 sug 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.
- Sales Rank: #6285853 in Books
- Published on: 1996-02-01
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x .63" w x 6.14" l, 1.17 pounds
- Binding: Hardcover
- 152 pages
Review
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
From the Back Cover
Interest in the spin-c Dirac operator originally came about from the study of complex analytic manifolds, where in the non-Kähler case the 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 work revisits 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 study is certainly no exception. Reprinted as it was originally published, this work is as an affordable text that 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
About the Author
Hans Duistermaat was a geometric analyst, who unexpectedly passed away in March 2010. His research encompassed many different areas in mathematics: ordinary differential equations, classical mechanics, discrete integrable systems, Fourier integral operators and their application to partial differential equations and spectral problems, singularities of mappings, harmonic analysis on semisimple Lie groups, symplectic differential geometry, and algebraic geometry. He was (co-)author of eleven books.
Duistermaat was affiliated to the Mathematical Institute of Utrecht University since 1974 as a Professor of Pure and Applied Mathematics. During the last five years he was honored with a special professorship at Utrecht University endowed by the Royal Netherlands Academy of Arts and Sciences. He was also a member of the Academy since 1982. He had 23 PhD students.
Most helpful customer reviews
0 of 0 people found the following review helpful.
Short but pretty good
By Dr. Lee D. Carlson
This book, of great interest to both mathematicians and physicists, is a good overview of the analysis and topology of the spin-c Dirac operator and how it is connected with the Lefschetz fixed point formula. Readers familiar with algebraic topology will see the similarities between the case discussed in this book and the one there where the formula involves computing the alternating sum of cohomology groups of a mapping acting on a compact manifold. If the number computed by this sum is non-zero, the mapping has a fixed point.
After a brief overview of what is ahead in the book the author introduces the formalism behind the Dolbeault-Dirac operator. The object of interest is the Dolbeault complex and the existence of Hermitian structures on the tangent bundle and in fibers of holomorphic vector bundles over a given complex analytic manifold. The Dolbeault-Dirac operator is elliptic and has finite-dimensional kernel if the manifold is compact. The goal is to get explicit formulas for the Riemann-Roch number and its generalization, the holomorphic Lefschetz number. When the manifold is not Kahler, the author shows that it is more straightforward to calculate these numbers using the spin-c Dirac operator. This is because the Levi-Civita connection on the tangent bundle does not leave the almost complex structure invariant. The difference of the diagonal heat kernels converges as time approaches 0 from above, and the limit can be calculated. The price one pays for using the spin-c Dirac operator is that its kernel does not have holomorphic sections, and its square only preserves the degrees of the differential forms modulo two.
The main goal of chapter 3 is to introduce the Clifford bundle, with its main properties sketched by the author. Then after a discussion of the spin and spin-c groups, the spin-c Dirac operator is defined in Chapter 6. The author proves that the spin-c Dirac operator is self-adjoint, and its principal symbol equal to the principal symbol of the Dolbeault-Dirac operator.
The author then proves in chapter 6 that the square of the spin-c Dirac operator is equal to the Laplace operator with a zero order term involving curvature expressions. When a spin structure is present, it is also shown that the formulas for the square of the spin-c Dirac operator are equal. When the manifold is Kahler, the square satisfies the Bochner-Kodaira formula. This result can be used to prove the Kodaira vanishing theorem, and the author gives a reference for this.
Taking the square of the spin-c Dirac operator gives the heat diffusion operator, and the author shows how to get the formula for the index using heat kernel methods in the next chapter, and discusses the heat kernel expansion in the chapter after that. These results should be very familiar to physicists working in quantum field theory.
In chapter 9, it is assumed that the heat diffusion operator acts on sections of a vector bundle associated to a principal spin bundle. The "zeroth-order" term which is added to the Laplacian is characterized explicitly. The heat kernel expansion is give and then generalized in chapter 10 to the case where an automorphism acts on the manifold. The Hirzebruch-Riemann-Roch integrand, which is the constant term in the expansion, is calculated in chapter 11, and then generalized to the case where an automorphism is present in chapter 12, via the local Lefschetz fixed point formula.
Finally, in chapter 13, the results are connected to the theory of characteristic classes, and then the (orbifold) spin-c Dirac operator and the corresponding heat kernel studied on orbifolds in chapter 14. The virtual character is computed via the heat kernel method and the Lefschetz formula is proven.
The next chapter is the most interesting in the book and discusses how one can apply these results over a symplectic manifold. The symplectic form has to satisfy an integrality condition that allows it to be a Chern form of a connection of a complex line bundle over the manifold. The author briefly reviews the theory of symplectic geometry, including Hamiltonian group actions and reduction and discusses geometric quantization. If the de Rham cohomology class of the symplectic form is integral, then one can put an almost complex structure on the symplectic manifold, and using the action of a compact and connected Lie group, one can obtain the necessary structures for the definition of the spin-c Dirac operator. The author gives an explicit example using toric varieties. If the almost complex structure is integrable, then the symplectic manifold can be viewed as a projective complex algebraic variety, and studied via algebraic geometry.
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