Introduction to Symplectic Dirac Operators Book

1 Background on Symplectic Spinors 1

1.1 Symplectic Group and Clifford Algebra 1

1.2 The Stone-von Neumann Theorem 5

1.3 Metaplectic Representation 7

1.4 Symplectic Clifford Multiplication 11

1.5 Hermite Functions 16

2 Symplectic Connections 21

2.1 Symplectic Manifolds 21

2.2 Constructions and Torsion 25

2.3 Symplectic Curvature and Ricci Tensors 29

3 Symplectic Spinor Fields 35

3.1 Metaplectic Structures 35

3.2 Symplectic Spinor Bundle 37

3.3 Splitting of the Spinor Bundle 43

4 Symplectic Dirac Operators 49

4.1 Definition of the Operators 49

4.2 Dependence on the Symplectic Connection 52

4.3 Dependence on the Metaplectic Structure 57

4.4 Dependence on the Almost Complex Structure 62

4.5 Formal Self-Adjointness 64

5 An Associated Second Order Operator 67

5.1 Definition and Ellipticity 67

5.2 A Weitzenbock Formula 68

5.3 Splitting of the Operator 74

6 The Kahler Case 81

6.1 The Operator P on Kahler Manifolds 81

6.2 Lower Bound Estimates 86

6.3 The Spectrum of P on CP¹ 87

7 Fourier Transform for Symplectic Spinors 97

7.1 Definition of the Transform 97

7.2 Basic Properties 98

7.3 Symmetry of the Spectra of D and D 99

8 Lie Derivative and Quantization 101

8.1 Lie Derivative of Symplectic Spinor Fields 101

8.2 Schrodinger Equation for Quadratic Hamiltonians 109

8.3 Lie Derivative as Quantization 111

References 115

Index 119