Notes

Interest in the spin-c Dirac operator originally came about from the study of complex analytic manifolds, where in the non-K hler case the Dolbeault operator is no longer suitable for getting local formulas fot 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 ReviewsThe 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 NewsletterThe author's book is a marvelous introduction to [these] objects and theories. "Zentralblatt