Introduction to Symplectic Dirac Operators
One of the basic ideas in differential geometry is that the study of analytic properties of certain differential operators acting on sections of vector bundles yields geometric and topological properties of the underlying base manifold. Symplectic spinor
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Katharina Habermann Lutz Habermann
Introduction to Symplectic Dirac Operators
1887
K. Habermann · L. Habermann
Introduction to Symplectic Dirac Operators
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Authors Katharina Habermann State and University Library Göttingen Platz der Göttinger Sieben 1 37073 Göttingen Germany e-mail: [email protected] Lutz Habermann Department of Mathematics University of Hannover Welfengarten 1 30167 Hannover Germany e-mail: [email protected]
Library of Congress Control Number: 2006924i23 Mathematics Subject Classification (2000): 53-02, 53Dxx, 58-02, 58Jxx ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-33420-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-33420-0 Springer Berlin Heidelberg New York DOI 10.1007/b138212
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To Karen
Preface
This book aims to give a systematic and self-contained introduction to the theory of symplectic Dirac operators and to reflect the current state of the subject. At the same time, it is intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology, which have become important fields and very active areas of mathematical research. The basic idea of symplectic spin geometry goes back to the early 1970s, when Bertram Kostant introduced symplectic spinors in order to give the construction of the half-form bundle and the half-form pairings in the context of geometric quantization [37]. During the next two decades, however, almost no attention has been given to a closer study of symplectic spin geometry itself. In 1995, the first author introduced symplectic Dirac operators [24] and started a systematical investigation [25, 26, 27]. These symplectic Dirac operators are called Dirac operators, since they are defined in an analogous way as the classical R
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