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Introduction to symplectic Dirac operators. (English) Zbl 1102.53032
Lecture Notes in Mathematics 1887. Berlin: Springer (ISBN 3-540-33420-3/pbk). xii, 120 p. (2006).
The present book is the first one giving a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflecting the current state of the subject. It is also intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in both symplectic geometry and symplectic topology.
The book starts with an introductory chapter which contains the background on symplectic spinors, needed in the next chapters. In Chapter 2 the basic properties of symplectic connections are presented. Here the authors stress on symplectic connections with torsion, and introduce the symplectic Ricci tensor which is considered more suitable for their purposes. Chapter 3 is devoted to the symplectic spinor bundle, which is the Hilbert space bundle associated to a metaplectic structure via the metaplectic representation.
The main topic of the book, that is, symplectic Dirac operators, are introduced in Chapter 4. The authors describe how these operators depend on the symplectic connection, the metaplectic structure and the almost complex structure on the underlying manifold. In the next chapter an elliptic operator of second order is introduced which can be considered as the symplectic counterpart of the square of the Dirac operator in the Riemannian case. In Chapter 6 the special case of Kähler manifolds is considered. The purpose of Chapter 7 is to present a globally defined Fourier transform for symplectic spinor fields, and to derive consequences for symplectic Dirac operators. Finally, the last chapter is devoted to the study of relations with mathematical physics, in particular to quantization.
The book will be of interest to researchers working on symplectic geometry and/or symplectic topology, and in physicists interested to quantization.

MSC:
53C27 Spin and Spin\({}^c\) geometry
53D05 Symplectic manifolds (general theory)
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C80 Applications of global differential geometry to the sciences
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