Introduction to Symplectic Dirac Operators

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1887 K. Habermann ·...

Author: Katharina Habermann Lutz Habermann

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1887

K. Habermann · L. Habermann

Introduction to Symplectic Dirac Operators

ABC

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

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A EX package Typesetting: by the authors and SPI Publisher Services using a Springer LT Cover design: design & production GmbH, Heidelberg

Printed on acid-free paper

SPIN: 11736684

V A 41/3100/ SPI

543210

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 Riemannian Dirac operator known from Riemannian spin geometry (cf. e.g. [21]). They are called symplectic because they are constructed by use of the symplectic setting of the underlying symplectic manifold. All tools which were necessary for that construction have already been known and accepted, mainly in mathematical physics. These are the symplectic Cliﬀord algebra (also known as Weyl algebra), the metaplectic group, the metaplectic representation (Segal–Shale–Weil representation) acting on L2 (Rn ), metaplectic structures, and symplectic connections. 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. There are several classical results in that direction. An example is Hodge– de Rham theory. Here, one considers the Hodge–Laplace–Beltrami operator ∆

VIII

Preface

acting on diﬀerential forms. This operator is one of the most studied operators in global Riemannian geometry and his spectrum gives important topological invariants. In particular, the dimension of the kernel of ∆ on p-forms over a closed Riemannian manifold is the p-th Betti number. Other well known and well studied operators are the Kodaira–Hodge–Laplace operator on diﬀerential forms with values in a holomorphic vector bundle or the classical Dirac operator on Riemannian manifolds. Now, symplectic spinor ﬁelds are sections in an L2 (Rn )-Hilbert space bundle over a symplectic manifold and symplectic Dirac operators, acting on symplectic spinor ﬁelds, are associated to the symplectic manifold in a very natural way. Hence they may be expected to give interesting applications in symplectic geometry and symplectic topology. It is our opinion that, besides the already stated, there are further close relations to mathematical physics. Some steps towards this direction have been made by the ﬁrst author and Andreas Klein in [28, 29, 30]. Another perspective could be the extension of Cliﬀord analysis and spin geometry to super diﬀerential geometry. According to the most used version of super geometry developed by Bertram Kostant, geometrical structures over a supermanifold consist of Z2 -graded objects and thus have an even as well as an odd part. Then one can imagine that a metric has to satisfy some kind of graded symmetry which, roughly speaking, corresponds to a symmetric object on the even part as well as to a skew symmetric object on the odd part of the supermanifold. For the ﬁrst one, we have the classical Riemannian spin geometry, whereas the second one is basically given by symplectic spin geometry. The main aspects of that idea are treated in a paper by Frank Sommen [45]. Although the construction of symplectic Dirac operators follows the same procedure as for the classical Riemannian Dirac operator, using the symplectic structure of the underlying manifold instead of the Riemannian metric, there are essential diﬀerences to the Riemannian case. These are caused by the fact that the algebraic structure of the symplectic Cliﬀord algebra is completely diﬀerent from that of Riemannian spin geometry. For the classical Cliﬀord algebra, we have the relation v2 = −v2 , whereas the algebraic structure of the symplectic Cliﬀord algebra is given by v · w − w · v = −ω0 (v, w). This implies essentially diﬀerent properties for the Cliﬀord multiplications, which enter into the deﬁnition of the Dirac operators. Moreover, the non-compactness of the symplectic group leads to analytic difﬁculties. Namely, since the typical ﬁber of the symplectic spinor bundle is the Hilbert space L2 (Rn ), we deal with operators acting on sections of a vector bundle of inﬁnite rank. For elliptic formally self-adjoint pseudo-diﬀerential operators with positive deﬁnite leading symbol acting on sections in a vector bundle of ﬁnite rank, one has a completely developed theory. So, in order to be able to apply these techniques, we are interested in equivariance properties

Preface

IX

of our operators with respect to a certain decomposition of the symplectic spinor bundle into a series of subbundles of ﬁnite rank. It turns out that an associated second order operator respects this decomposition provided that a technical assumption, which always can be realized, holds true. Let us now brieﬂy describe the content of each chapter of this book. The ﬁrst chapter is introductory. It contains preliminaries and basic material needed for our considerations. Chapter 2 is devoted to symplectic connections. In particular, we introduce a further Ricci tensor, which we will call symplectic Ricci tensor. To our knowledge, no attention has been given to this tensor in previous studies. In fact, this symplectic Ricci tensor is a new object in the case of non-vanishing torsion. To date, mostly only torsion-free symplectic connections have been considered. It turns out that, in our context, it is convenient to work also with symplectic connections with torsion and that the symplectic Ricci tensor is more suitable for our purposes. The next chapter introduces the symplectic spinor bundle and the spinor derivative and analyzes a splitting property of the spinor bundle. In Chapter 4, we give the deﬁnition of symplectic Dirac operators and describe in detail how these operators depend on the objects from which they are built. Chapter 5 is concerned with an associated second order operator of Laplace type and addresses properties of this operator. The objective of Chapter 6 is the situation for a special class of symplectic manifolds, namely K¨ ahler manifolds. Here, we also investigate the example of CP 1 . The aim of Chapter 7 is to construct a Fourier transform for symplectic spinor ﬁelds and to derive consequences for the symplectic Dirac operators. The last chapter focuses on relations to mathematical physics, in particular to quantization. This closes the circle to the beginnings by Bertram Kostant. The present text is originated in research ideas of the ﬁrst author and provides an extended version of her “Habilitationsschrift”, which was never published separately. Starting with helpful discussions from the beginning of the investigations and proposing many improvements in the selection and presentation of the material, the second author became more and more involved into the subject. We decided to write this book tree years ago and, for this book, he made a thorough revision of the material, in particular, to improve the strictness of the presentation. Then it took time to compose it in a natural and organic way. Furthermore, our working places are no longer as close to each other as before and it became diﬃcult to keep our discussions at its intensive level. Now we consider the text ready for publication and hope to present the reader a mature work. During the work on the book, we received ﬁnancial support from DFG, the German Research Foundation, contract HA 3056/1-1,2. Many thanks are due to our students Paul Rosenthal and Steﬀen Rudnick for proof-reading the LATEX-type-written manuscript.

X

Preface

We dedicate this book to our daughter Karen. She accepts our mathematical family circle and enriches it with her interest. We are grateful to being able to give her an understanding of the fascination of mathematics. Karen, you are a wonderful teenager which makes us enjoying the challenge of teaching mathematics. G¨ ottingen and Greifswald, November 2004

Katharina & Lutz Habermann

Contents

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

XII

5

Contents

An Associated Second Order Operator . . . . . . . . . . . . . . . . . . . . 67 5.1 Deﬁnition and Ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 A Weitzenb¨ ock Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.3 Splitting of the Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6

The K¨ ahler Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.1 The Operator P on K¨ ahler Manifolds . . . . . . . . . . . . . . . . . . . . . . 81 6.2 Lower Bound Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.3 The Spectrum of P on CP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7

Fourier Transform for Symplectic Spinors . . . . . . . . . . . . . . . . . . 97 7.1 Deﬁnition of the Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 . . . . . . . . . . . . . . . . . . . . . . 99 7.3 Symmetry of the Spectra of D and D

8

Lie Derivative and Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.1 Lie Derivative of Symplectic Spinor Fields . . . . . . . . . . . . . . . . . . 101 8.2 Schr¨ odinger Equation for Quadratic Hamiltonians . . . . . . . . . . . 109 8.3 Lie Derivative as Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

1 Background on Symplectic Spinors

This chapter serves two purposes. First it gives a survey on fundamental relations between the symplectic Cliﬀord algebra, the metaplectic group, and its Lie algebra. The second is to provide several elementary facts used in the later computations. Most of it is well known, but we have summarized the material in a form that makes it applicable for our considerations.

1.1 Symplectic Group and Cliﬀord Algebra We consider the 2n-dimensional real vector space R2n equipped with its standard symplectic form ω0 . We write any v ∈ R2n as v v= v with vectors v , v ∈ Rn and any 2n × 2n-matrix A as ab A= cd with n × n matrices a, b, c, d. Let I be the unit element of the linear group GL(n, R) and set 0 −I J0 = . I 0 Then

ω0 (v, w) = J0 v, w = v , w − v , w

for v, w ∈ R2n , where , denotes the Euclidean inner product on R2n as well as on Rn . Hence the standard basis {a1 , . . . , an , b1 , . . . , bn } of R2n forms a symplectic basis, which means that

2

1 Background on Symplectic Spinors

ω0 (aj , ak ) = ω0 (bj , bk ) = 0

and ω0 (aj , bk ) = δjk

(1.1.1)

for j, k = 1, . . . , n. The symplectic group Sp(n, R) is the group of all automorphisms of R2n which preserve the symplectic form ω0 . That is, Sp(n, R) is the group of those A ∈ GL(2n, R) that satisfy ω0 (Av, Aw) = ω0 (v, w) for all v, w ∈ R2n , which is equivalent to AT J0 A = J0 as well as AJ0 AT = J0 . In particular, J0 ∈ Sp(n, R). We denote the space of all symmetric real n × n-matrices by S(n) and set a 0 D(n) = : a ∈ GL(n, R) −1 0 aT

and N (n) =

Ib 0I

: b ∈ S(n) .

Obviously, D(n) and N (n) are subgroups of Sp(n, R). Moreover, we have (cf. [16]) Proposition 1.1.1 Sp(n, R) is generated by D(n) ∪ N (n) ∪ {J0 }.

Identifying R2n with Cn via v → v + iv , the Hermitian inner product of two vectors v, w ∈ R2n is v, w − iω0 (v, w) . Therefore, the intersection of Sp(n, R) with the orthogonal group O(2n) is the unitary group U(n). One can prove (cf. [16]) Proposition 1.1.2 (1) U(n) is a maximal compact subgroup of Sp(n, R). 2

(2) Sp(n, R) is homeomorphic to the product U(n) × Rn

+n

.

Corollary 1.1.3 Sp(n, R) is connected and its fundamental group is Z.

1.1 Symplectic Group and Cliﬀord Algebra

3

Proof. This follows from Proposition 1.1.2 and the corresponding properties of U(n). The symplectic Lie algebra, i.e. the Lie algebra sp(n, R) of the symplectic group Sp(n, R) is given by the space of all endomorphisms X of R2n satisfying ω0 (Xv, w) + ω0 (v, Xw) = 0 for all v, w ∈ R2n . We identify the space S 2 (R2n ) of symmetric 2-tensors of R2n with sp(n, R) by assigning to v1 v2 ∈ S 2 (R2n ) the endomorphism v ∈ R2n → ω0 (v, v1 )v2 + ω0 (v, v2 )v1 ∈ R2n . Then we have Lemma 1.1.4 For every X ∈ sp(n, R), 1 (Xaj bj − aj Xbj ) . 2 j=1 n

X=

Proof. The assertion follows from n

(ω0 (v, Xaj )bj + ω0 (v, bj )Xaj − ω0 (v, aj )Xbj − ω0 (v, Xbj )aj )

j=1

=

n

(ω0 (Xv, bj )aj − ω0 (Xv, aj )bj )

j=1

  n +X  (ω0 (v, bj )aj − ω0 (v, aj )bj ) j=1

= 2 Xv for any v ∈ R2n .

The Lie bracket of sp(n, R) now writes as [v1 v2 , v3 v4 ] = −ω0 (v2 , v4 )v1 v3 − ω0 (v2 , v3 )v1 v4 −ω0 (v1 , v4 )v2 v3 − ω0 (v1 , v3 )v2 v4 . In matrix notation, the above identiﬁcation is given by

v1 ⊗ v2 + v2 ⊗ v1 −v1 ⊗ v2 − v2 ⊗ v1 v1 v2 = . v1 ⊗ v2 + v2 ⊗ v1 −v1 ⊗ v2 − v2 ⊗ v1 Here, x ⊗ y for x, y ∈ Rn means the endomorphism

(1.1.2)

4

1 Background on Symplectic Spinors

z ∈ Rn → x, z y ∈ Rn . Clearly, the endomorphism adjoint to x ⊗ y is then (x ⊗ y)T = y ⊗ x .

(1.1.3)

Next we deﬁne the symplectic Cliﬀord algebra, which is also referred to as Weyl algebra. In contrast to the Riemannian case, this algebra is inﬁnite dimensional. Deﬁnition 1.1.5 The symplectic Cliﬀord algebra Cl(n) is the associative unital algebra over R generated by R2n with the relations v · w − w · v = −ω0 (v, w) for v, w ∈ R2n . A basis of Cl(n) is formed by β1 β2 α2 αn βn 1 aα 1 · a2 · · · · · an · b1 · b2 · · · · · bn ,

where αj , βj are non-negative integers. According to Equation (1.1.1), aj · ak = ak · aj ,

bj · bk = bk · bj ,

and aj · bk − bk · aj = −δjk

for j, k = 1, . . . , n. As usual, we set [v, w] = v · w − w · v for v, w ∈ Cl(n), which gives Cl(n) the structure of a real Lie algebra. Let a(n) denote the subspace of Cl(n) which is spanned by v · w + w · v for v, w ∈ R2n . Lemma 1.1.6 The space a(n) is a Lie subalgebra of Cl(n) which is isomorphic to the symplectic Lie algebra sp(n, R). Proof. Let v1 , v2 , v ∈ R2n and v ∈ a(n). First we observe that [v1 · v2 , v] = v1 · v2 · v − v · v1 · v2 = v1 · v · v2 + ω0 (v, v2 )v1 − v · v1 · v2 = ω0 (v, v1 )v2 + ω0 (v, v2 )v1 = (v1 v2 )v . Thus [v1 · v2 + v2 · v1 , v] = 2(v1 v2 )v and

(1.1.4)

1.2 The Stone–von Neumann Theorem

[v, v] ∈ R2n .

5

(1.1.5)

Further we have [v1 · v2 , v] = v1 · [v2 , v] + [v1 , v] · v2 and hence [v1 · v2 + v2 · v1 , v] = v1 · [v2 , v] + [v2 , v] · v1 +v2 · [v1 , v] + [v1 , v] · v2 .

(1.1.6)

Equations (1.1.5) and (1.1.6) imply that a(n) is a Lie subalgebra of Cl(n). From Equation (1.1.4) and the Jacobi identity, we get the second part of the assertion.

1.2 The Stone–von Neumann Theorem In this section, we want to recall some facts of representation theory and formulate the Stone–von Neumann theorem. This fundamental theorem will be used in the next section to construct the metaplectic representation. For details and proofs, we refer to [1, 16, 49]. Let G be a connected Lie group and let H be a separable complex Hilbert space. We endow the group GL(H) of invertible bounded linear operators on H with the strong topology. That means that a sequence (Tk ) in GL(H) converges to T0 if and only if the sequence (Tk h) converges to T0 h for all h ∈ H. Deﬁnition 1.2.1 A representation of G on H is a continuous group homomorphism r : G → GL(H). It is called unitary if it maps into the unitary group U(H) of H. Deﬁnition 1.2.2 Let r be a representation of G on H. (1) A subspace W ⊂ H is said to be r-invariant if r(a)W ⊂ W for all a ∈ G. (2) r is called irreducible if the only r-invariant closed subspaces of H are {0} and H. The following proposition is a version of Schur’s lemma.

6

1 Background on Symplectic Spinors

Proposition 1.2.3 A unitary representation r of G on H is irreducible if and only if the only bounded linear operators T : H → H such that T ◦ r(a) = r(a) ◦ T for all a ∈ G are scalar multiples of the identity.

Let r be a ﬁxed representation of G on H and let H∞ denote the space of smooth vectors of r, i.e. the set of all h ∈ H such that the map a ∈ G → r(a)h ∈ H is smooth. Theorem 1.2.4 (G˚ arding) H∞ is a dense invariant subspace of H.

The diﬀerential of r is the homomorphism r∗ : g → End(H∞ ) from the Lie algebra g of G into the endomorphism algebra End(H∞ ) of H∞ given by d r∗ (X)h = r(exp(tX))h dt t=0 for X ∈ g and h ∈ H∞ . By Theorem 1.2.4, each operator r∗ (X) can be considered as an unbounded operator on H. Deﬁnition 1.2.5 Let r1 : G → U(H1 ) and r2 : G → U(H2 ) be representations of G on Hilbert spaces H1 and H2 . Then r1 is said to be equivalent to r2 if there exists a bounded linear bijection T : H1 → H2 such that T ◦ r1 (a) = r2 (a) ◦ T for all a ∈ G. If, in addition, T can be chosen unitary, r1 is called unitary equivalent to r2 . Now we consider the Heisenberg group H(n) of R2n , i.e. H(n) = R2n × R with group multiplication given by 1 (v, s) · (w, t) = v + w, s + t + ω0 (v, w) . 2 Setting

(rS (v, s)f )(x) = e−i(s+v ,x−v ,v

/2)

f (x − v )

for (v, s) ∈ H(n), f : Rn → C and x ∈ Rn , we obtain a unitary representation rS of H(n) on the Hilbert space L2 (Rn ) of square integrable functions on Rn . This representation, which is called Schr¨ odinger representation, is irreducible. Furthermore, rS (0, s)f = e−is f for any s ∈ R and f ∈ L2 (Rn ). These properties turn out to be characteristic for rS .

1.3 Metaplectic Representation

7

Theorem 1.2.6 (Stone–von Neumann) Let r be an irreducible unitary representation of the Heisenberg group H(n) on a separable complex Hilbert space H such that r(0, s)h = e−is h for all s ∈ R and h ∈ H. Then r is unitary equivalent to the Schr¨ odinger representation rS .

1.3 Metaplectic Representation In view of Corollary 1.1.3, the symplectic group Sp(n, R) has a unique connected double cover Mp(n, R). This covering group is called metaplectic group. Let ρ : Mp(n, R) → Sp(n, R) denote the covering homomorphism. By Lemma 1.1.6, we may identify the Lie algebra mp(n, R) of Mp(n, R) with the subalgebra a(n) of Cl(n) and think the diﬀerential ρ∗ : mp(n, R) → sp(n, R) of ρ to be realized by the homomorphism given in the proof of this lemma. So ρ∗ (v)v = [v, v]

(1.3.1)

ρ∗ (v · w + w · v) = 2 v w

(1.3.2)

and for v ∈ mp(n, R) and v, w ∈ R2n . The inverse of ρ∗ is explicitly described by Lemma 1.3.1 For every X ∈ sp(n, R), 1 (bj · Xaj − aj · Xbj ) . 2 j=1 n

ρ−1 ∗ (X) =

Proof. By bj · Xaj = Xaj · bj − ω0 (bj , Xaj ) , aj · Xbj = Xbj · aj − ω0 (aj , Xbj ) and ω0 (bj , Xaj ) = ω0 (aj , Xbj ) , one has n

(bj · Xaj − aj · Xbj )

j=1

1 (Xaj · bj + bj · Xaj − aj · Xbj − Xbj · aj ) . 2 j=1 n

=

8

1 Background on Symplectic Spinors

Applying Lemma 1.1.4 and Equation (1.3.2), one gets the assertion.

In the following, we outline the construction of a unitary representation of Mp(n, R) on L2 (Rn ). Let A ∈ Sp(n, R). Then τA (v, s) = (Av, s) deﬁnes an automorphism τA of the Heisenberg group H(n). Thus, composing the Schr¨ odinger representation rS with τA , we obtain an irreducible unitary representation rA S = rS ◦ τA of H(n). Obviously, −is rA f S (0, s)f = e

for all s ∈ R and f ∈ L2 (Rn ). Therefore, by the Stone–von Neumann theorem, there exists an operator U (A) ∈ U(L2 (Rn )) such that U (A) ◦ rS (v, s) = rA S (v, s) ◦ U (A)

(1.3.3)

for all (v, s) ∈ H(n). Due to Proposition 1.2.3, U (A) is determined up to a scalar factor of modulus one. Since A ∈ Sp(n, R) → τA ∈ Aut(H(n)) is a group homomorphism, it follows that the operators U (A) give rise to a projective unitary representation of Sp(n, R), which means that U (AB) = c(A, B) U (A) ◦ U (B) for any A, B ∈ Sp(n, R) with some function c : Sp(n, R) × Sp(n, R) → S 1 . Let us compute U (A) in the cases that A is one of the generators of Sp(n, R) according to Proposition 1.1.1. Let F : L2 (Rn ) → L2 (Rn ) be the Fourier transform, i.e. that unitary operator on L2 (Rn ) which is given by −n/2 (Ff )(x) = (2π) e−ix,y f (y) dy Rn

for any f in the Schwartz space S(R ) of rapidly decreasing smooth functions on Rn . n

Lemma 1.3.2 Set U (A) = F −1 for A = J0 . Then Equation (1.3.3) is fulﬁlled. Proof. Let (v, s) ∈ H(n) and f ∈ S(Rn ). Substituting y = z − v , we conclude ((rS (v, s) ◦ F)f )(x) −n/2 −i(s+v ,x−v ,v /2)

= (2π)

e

e−ix−v

,y

f (y) dy

Rn

e−ix−v ,z−v f (z − v ) dz = (2π)−n/2 e−i(s+v ,x−v ,v /2) Rn −n/2 e−ix,z e−i(s−v ,z+v ,v /2) f (z − v ) dz = (2π) Rn

= ((F ◦ rS (J0 v, s))f )(x) .

1.3 Metaplectic Representation

9

Thus rS (v, s) ◦ F = F ◦ rS (J0 v, s) ,

which gives the assertion. Lemma 1.3.3 Let

A=

a 0 −1 0 aT

∈ D(n) .

Choose λ(a) ∈ C with λ(a)2 = det(a) and deﬁne U (A) : L2 (Rn ) → L2 (Rn ) by (U (A)f )(x) = λ(a)f aT x . Then U (A) is a unitary operator and satisﬁes Equation (1.3.3). Proof. The unitarity of U (A) follows from the transformation formula of Lebesgue integration. It remains to verify that (U (A) ◦ rS (v, s))f = (rS (Av, s) ◦ U (A))f for (v, s) ∈ H(n) and f ∈ L2 (Rn ). This can be directly calculated, using −1 T . v v , a x = av , x and v , v = av , aT

Lemma 1.3.4 Let A=

Ib 0I

∈ N (n)

and deﬁne U (A) ∈ U(L2 (Rn )) by (U (A)f )(x) = e−ibx,x/2 f (x) . Then U (A) satisﬁes Equation (1.3.3). Proof. One immediately sees that the statement is equivalent to 1 1 bx, x + v , x − v , v 2 2 1 1 = v + bv , x − v + bv , v + b(x − v ), x − v 2 2 for x, v , v ∈ Rn . The last equation reduces to bx, v = bv , x , which is true, since b ∈ S(n).

It was proved by Weil in [50] (see also [16, 32, 49]) that the projective unitary representation A ∈ Sp(n, R) → U (A) ∈ U(L2 (Rn )) described above lifts to a unitary representation of the metaplectic group Mp(n, R). More precisely, we have

10

1 Background on Symplectic Spinors

Proposition 1.3.5 There exists a unique unitary representation m : Mp(n, R) → U(L2 (Rn )) which satisﬁes m(q) ◦ rS (v, s) = rS (ρ(q)v, s) ◦ m(q) for all q ∈ Mp(n, R) and (v, s) ∈ H(n).

The unitary representation m is known as Segal–Shale–Weil representation or metaplectic representation. As we will see, it will take the role of the spinor representation of the orthogonal group in the Riemannian case. So, when one speaks of symplectic spinors, one means the elements of the representation space L2 (Rn ). We point out some properties of m (cf. e.g. [42] and the references therein). Proposition 1.3.6 (1) The metaplectic representation m is faithful. It decomposes into the sum of two inequivalent irreducible unitary representations, which are the restrictions of m to the spaces of even and odd functions, respectively. (2) The space of smooth vectors of m is precisely the Schwartz space S(Rn ). In particular, S(Rn ) is m-invariant. It turns out that, for q ∈ Mp(n, R) with ρ(q) ∈ D(n) ∪ N (n) ∪ {J0 }, the operator m(q) coincides, up to sign, with the operator U (ρ(q)) given in the above lemmas. Thus we have the following case-by-case realization of the metaplectic representation (cf. e.g. [12, 32] and Lemma 1.5.4). There, we consider N (n) as a subgroup of Mp(n, R), which may be done, since N (n) is simply connected. Proposition 1.3.7 Let q ∈ Mp(n, R) and f ∈ L2 (Rn ). (1) If ρ(q) = J0 , then

m(q)f = ±einπ/4 F −1 f .

(2) If

ρ(q) =

then

a 0 T −1 0 a

∈ D(n) ,

(m(q)f )(x) = λ(a)f aT x ,

where λ(a) is a suitably chosen root of det(a). (3) If

q=

then

Ib 0I

∈ N (n) ,

(m(q)f )(x) = e−ibx,x/2 f (x) .

1.4 Symplectic Cliﬀord Multiplication

11

1.4 Symplectic Cliﬀord Multiplication In this section, we will deﬁne Cliﬀord multiplication in the symplectic context and derive a formula expressing the diﬀerential of the metaplectic representation in terms of this multiplication. We consider the position and momentum operators Qj and Pj , j = 1, . . . , n, deﬁned by ∂f (Qj f )(x) = ixj f (x) and Pj f = ∂xj for smooth functions f : Rn → C. Generally, we will regard them as continuous operators on the Schwartz space S(Rn ). Set σ(aj ) = Qj

and σ(bj ) = Pj

for the standard basis {a1 , . . . , an , b1 , . . . , bn } of R2n . By linear continuation, this deﬁnes, for any v ∈ R2n , an operator σ(v) : S(Rn ) → S(Rn ). As one easily sees, (σ(v)f )(x) = iv , x f (x) + v , grad(f )(x) for all f ∈ S(Rn ). Lemma 1.4.1 (1) For any v, w ∈ R2n , σ(v) ◦ σ(w) − σ(w) ◦ σ(v) = −iω0 (v, w) . (2) For any v ∈ R2n and any f1 , f2 ∈ S(Rn ), σ(v)f1 , f2 = −f1 , σ(v)f2 , where , denotes the inner product of L2 (Rn ). (3) For any v ∈ R2n ,

F ◦ σ(J0 v) = σ(v) ◦ F .

Proof. Assertion (1) follows from Qj ◦ Qk = Qk ◦ Qj , Pj ◦ Pk = Pk ◦ Pj , Pk ◦ Qj = iδjk + Qj ◦ Pk . Obviously, Qj f1 , f2 = −f1 , Qj f2 and, by partial integration, Pj f1 , f2 = −f1 , Pj f2 .

12

1 Background on Symplectic Spinors

This implies (2). Assertion (3) is a reformulation of the well known relations Qj ◦ F = F ◦ Pj

and Pj ◦ F = −F ◦ Qj .

(1.4.1)

In view of Lemma 1.4.1(1), the map σ : R2n → End(S(Rn )) extends to a linear map deﬁned on the whole symplectic Cliﬀord algebra Cl(n) by setting σ(1)f = if for f ∈ S(Rn ) and σ(v1 · v2 · · · · · vm ) = σ(v1 ) ◦ σ(v2 ) ◦ · · · ◦ σ(vm ) for v1 , . . . , vm ∈ R2n . Lemma 1.4.2 (1) For any v ∈ mp(n, R) and any v ∈ R2n , σ(v) ◦ σ(v) − σ(v) ◦ σ(v) = iσ(ρ∗ (v)v) . (2) For any v ∈ mp(n, R) and any f1 , f2 ∈ S(Rn ), σ(v)f1 , f2 = f1 , σ(v)f2 . (3) The kernel of the homomorphism σ : Cl(n) → End(S(Rn )) is trivial. Proof. By Lemma 1.4.1(1), σ(v1 ) ◦ σ(v2 ) ◦ σ(v) = σ(v1 ) ◦ σ(v) ◦ σ(v2 ) − iω0 (v2 , v)σ(v1 ) = σ(v) ◦ σ(v1 ) ◦ σ(v2 ) − iω0 (v1 , v)σ(v2 ) − iω0 (v2 , v)σ(v1 ) for v1 , v2 , v ∈ R2n . It follows that σ(v1 · v2 + v2 · v1 ) ◦ σ(v) − σ(v) ◦ σ(v1 · v2 + v2 · v1 ) = 2iσ(ω0 (v, v1 )v2 + ω0 (v, v2 )v1 ) = iσ(ρ∗ (v1 · v2 + v2 · v1 )v) , where we have used Equation (1.3.2). This gives (1). Assertion (2) is obtained by applying Lemma 1.4.1(2) twice. To prove (3), let v ∈ Cl(n) and suppose that σ(v)f = 0 for all f ∈ S(Rn ). Expressing v in the basis of Cl(n) given after Deﬁnition 1.1.5 and applying σ(v) to functions f ∈ S(Rn ) which are polynomial near 0, one easily gets v = 0. We remark that, in contrast to Lemma 1.4.2(3), the Cliﬀord multiplication in Riemannian spin geometry generally has a non-trivial kernel.

1.4 Symplectic Cliﬀord Multiplication

13

For convenience, in the following, we will usually write v · f instead of σ(v)f with the convention v · w · f = v · (w · f ) for v, w ∈ Cl(n). Deﬁnition 1.4.3 The symplectic Cliﬀord multiplication is the map µ0 : R2n ⊗ S(Rn ) → S(Rn ) deﬁned by µ0 (v ⊗ f ) = v · f . Lemma 1.4.4 The Cliﬀord multiplication is Mp(n, R)-equivariant. That is, for any q ∈ Mp(n, R), v ∈ R2n , and f ∈ S(Rn ), µ0 (ρ(q)v ⊗ m(q)f ) = m(q)µ0 (v ⊗ f ) . Proof. We apply Proposition 1.3.7 and Lemma 1.4.1(3). First we assume that ρ(q) = J0 . Then µ0 (ρ(q)v ⊗ m(q)f ) = ±einπ/4 σ(J0 v) ◦ F −1 f = ±einπ/4 F −1 ◦ σ(v) f = m(q)µ0 (v ⊗ f ) . Now let

ρ(q) =

In this case,

a 0 −1 0 aT

.

grad(m(q)f )(x) = λ(a)a grad(f ) aT x .

It follows that µ0 (ρ(q)v ⊗ m(q)f )(x)

−1 = iav , x (m(q)f )(x) + aT v , grad(m(q)f )(x) = i v , aT x (m(q)f )(x) + v , a−1 grad(m(q)f )(x) = λ(a) i v , aT x f aT x + v , grad(f ) aT x = (m(q)µ0 (v ⊗ f ))(x) .

Finally, if

q=

then

Ib 0I

∈ N (n) ,

grad(m(q)f )(x) = e−ibx,x/2 (grad(f )(x) − if (x)bx)

14

1 Background on Symplectic Spinors

and hence µ0 (ρ(q)v ⊗ m(q)f )(x) = iv + bv , x (m(q)f )(x) + v , grad(m(q)f )(x) = e−ibx,x/2 (iv , x (f )(x) + v , grad(f )(x) ) = (m(q)µ0 (v ⊗ f ))(x) . Since the considered elements of Mp(n, R) generate the whole group, the lemma is proved. We now turn to the calculation of the diﬀerential m∗ : mp(n, R) → End(S(Rn )) of the metaplectic representation m. Proposition 1.4.5 For any v ∈ mp(n, R) and f ∈ S(Rn ), m∗ (v)f = −iv · f . Proof. We have to show that m∗ (v) = −iσ(v)

(1.4.2)

for all v ∈ mp(n, R). For this, we may assume that v = v · w + w · v for v, w ∈ R2n such that (a) v = w = 0, (b) v = w = 0, or (c) v = w = 0. Suppose (a). Then, by Equations (1.1.2) and (1.3.2), ρ(exp(tv)) = exp(tρ∗ (v)) = exp(2t v w) exp(2t w ⊗ v ) 0 = 0 exp(−2t v ⊗ w ) for all t ∈ R. In particular, by Equation (1.1.3), ρ(exp(tv)) ∈ D(n). Hence, according to Proposition 1.3.7, (m(exp(tv))f )(x) = det(exp(2t w ⊗ v ))f (exp(2t v ⊗ w )x) for f ∈ S(Rn ). We conclude that

1.4 Symplectic Cliﬀord Multiplication

15

d (m(exp(tv))f )(x) (m∗ (v)f )(x) = dt t=0 d = Tr(w ⊗ v )f (x) + f (exp(2t v ⊗ w )x) dt t=0 = v , w f (x) + 2v , x w , grad(f )(x) .

On the other hand, using Lemma 1.4.1(1), we obtain (σ(v)f )(x) = ((σ(v) ◦ σ(w) + σ(w) ◦ σ(v))f )(x) = iω0 (v, w)f (x) + 2((σ(v) ◦ σ(w))f )(x) = iv , w f (x) + 2iv , x w , grad(f )(x) . This implies Equation (1.4.2). Suppose (b). As above, we see 1 −2t(v ⊗ w + w ⊗ v ) ρ(exp(tv)) = ∈ N (n) . 0 1 Hence

(m(exp(tv))f )(x) = e2itv ,xw ,x f (x) . It follows that (m∗ (v)f )(x) = 2iv , x w , x f (x) = −2i((σ(v) ◦ σ(w))f )(x) = −i(σ(v)f )(x) as claimed. Suppose (c). In this case ρ(exp(tv)) = Since

1 0 2t(v ⊗ w + w ⊗ v ) 1

1 0 2t(v ⊗ w + w ⊗ v ) 1

=

J0−1

.

1 −2t(v ⊗ w + w ⊗ v ) 0 1

J0

= J0−1 ρ(exp(t (J0 v · J0 w + J0 w · J0 v)))J0 , we get m(exp(tv)) = F ◦ m(exp(t (J0 v · J0 w + J0 w · J0 v))) ◦ F −1 . By means of case (b) and Lemma 1.4.1(3), this yields

16

1 Background on Symplectic Spinors

m∗ (v) = F ◦ m∗ (J0 v · J0 w + J0 w · J0 v) ◦ F −1 = −iF ◦ (σ(J0 v) ◦ σ(J0 w) + σ(J0 w) ◦ σ(J0 v)) ◦ F −1 = −i(σ(v) ◦ σ(w) + σ(w) ◦ σ(v)) = −iσ(v) ,

so we are done.

We note that σ : Cl(n) → End(S(Rn )) is not an algebra homomorphism. However, the previous proposition implies Corollary 1.4.6 For any v, w ∈ mp(n, R), σ(v) ◦ σ(w) − σ(w) ◦ σ(v) = iσ([v, w]) . Proof. This is a consequence of Equation (1.4.2) and the fact that m∗ is a Lie algebra homomorphism. Alternatively, Corollary 1.4.6 can be obtained by a direct calculation, using Equations (1.1.6) and (1.3.1) and Lemma 1.4.2(1).

1.5 Hermite Functions Let U(n) ⊂ Mp(n, R) denote the double cover of U(n) realized as the inverse image of U(n) ⊂ Sp(n, R) under the covering map ρ : Mp(n, R) → Sp(n, R). In this section, we study the restriction u : U(n) → U(L2 (Rn )) of the metaplectic representation m to U(n). As we will see, u is strongly related to the harmonic oscillator and the Hermite functions. We start with describing the Lie algebra u(n) of U(n). First observe that the Lie algebra u(n) of U(n) ⊂ Sp(n, R) consists of those X ∈ sp(n, R) that satisfy Xv, w + v, Xw = 0 or, equivalently, J0 XJ0 = −X . Hence u(n) is spanned by the endomorphisms X − J0 XJ0 for X ∈ sp(n, R). Obviously, J0 ∈ u(n). Moreover, since J0 (X − J0 XJ0 ) = J0 X + XJ0 = (X − J0 XJ0 )J0 , J0 lies in the center of u(n). By means of

1.5 Hermite Functions

17

ω0 (v, J0 v1 )J0 v2 = −ω0 (J0 v, v1 )J0 v2 = −J0 (ω0 (J0 v, v1 )v2 ) , we see that, with the identiﬁcation sp(n, R) ≡ S 2 (R2n ), the Lie algebra u(n) u(n) = ρ−1 is spanned by v1 v2 + J0 v1 J0 v2 for v1 , v2 ∈ R2n . Hence ∗ (u(n)) is spanned by the elements v1 · v2 + v2 · v1 + J0 v1 · J0 v2 + J0 v2 · J0 v1 . In particular, aj · ak + bj · bk for 1 ≤ j ≤ k ≤ n and aj · bk − ak · bj for 1 ≤ j < k ≤ n form a basis of u(n). Let 0 = ρ−1 ∗ (J0 ). Then, by Lemma 1.3.1, 1 (aj · aj + bj · bj ) . 2 j=1 n

0 =

(1.5.1)

u(n). So Furthermore, 0 is in the center of [0 , v] = 0

(1.5.2)

for all v ∈ u(n). Now recall that the Hamilton operator of the n-dimensional harmonic oscillator is the operator H0 : S(Rn ) → S(Rn ) deﬁned by

n 1 ∂2f 2 (x) − xj f (x) . (H0 f )(x) = 2 j=1 ∂x2j Notice that, by the deﬁnition of σ and Equation (1.5.1), H0 = σ(0 ) . Equivalently,

1 (aj · aj + bj · bj ) · f . 2 j=1

(1.5.3)

n

H0 f =

(1.5.4)

Let N0 denote the set of non-negative integers and let hα ∈ S(Rn ) for multiindices α = (α1 , . . . , αn ) ∈ Nn0 be the Hermite functions on Rn . These functions are deﬁned by hα (x) = hα1 (x1 ) · · · hαn (xn ) , where hl for l ∈ N0 are the classical Hermite functions on R, i.e. 2

hl (t) = et

/2

dl −t2 e . dtl

As is well known (cf. [16]), the Hermite functions form a complete orthogonal system in L2 (Rn ) of eigenfunctions of H0 . In particular,

18

1 Background on Symplectic Spinors

n H0 hα = − |α| + hα , 2

(1.5.5)

where |α| = α1 + · · · + αn . Let Wl denote the eigenspace of H0 with eigenvalue −(l+n/2). An elementary combinatorial computation shows that the complex dimension of Wl is n+l−1 dimC Wl = . (1.5.6) l Proposition 1.5.1 For all q ∈ U(n), H0 ◦ u(q) = u(q) ◦ H0 on S(Rn ). Proof. Equations (1.4.2), (1.5.2), and (1.5.3) and Corollary 1.4.6 yield that, for all v ∈ u(n), H0 ◦ u∗ (v) − u∗ (v) ◦ H0 = −i(σ(0 ) ◦ σ(v) − σ(v) ◦ σ(0 )) = σ([0 , v]) =0. Since U(n) is connected, this implies the assertion.

Corollary 1.5.2 The spaces Wl , l ∈ N0 , form an orthogonal decomposition of L2 (Rn ) into u-invariant subspaces. Proof. The u-invariance of each Wl follows from Proposition 1.5.1. The rest is clear. We denote the restriction of the unitary representation u of U(n) to the subspace Wl by ul . For the sake of completeness, we note that each ul is irreducible (cf. [5]). We conclude this section with the following two lemmas. Lemma 1.5.3 The Fourier transform F commutes with H0 , F ◦ H 0 = H0 ◦ F . Moreover, if f ∈ Wl , then

Ff = i−l f .

1.5 Hermite Functions

19

Proof. The ﬁrst part follows from 1 2 Pj + Q2j 2 j=1 n

H0 =

and Equation (1.4.1). For the second part, it suﬃces to show that Fhα = (−i)|α| hα

(1.5.7)

for all α ∈ Nn0 . First recall that this equation is valid in case α = 0. Then check that, for j = 1, . . . , n and any α ∈ Nn0 , (iQj + Pj )hα = hα+ε(j) , where ε(j) denotes that multi-index whose only non-vanishing entry is a 1 at the j-th position. Furthermore, again by Equation (1.4.1), F ◦ (iQj + Pj ) = −i(iQj + Pj ) ◦ F . By induction, this proves the desired relation. Lemma 1.5.4 It is

π u exp = einπ/4 F −1 . 0 2

Proof. By Equations (1.4.2) and (1.5.3), we have d u(exp(t0 ))f = u∗ (0 )f = −iH0 f dt t=0 for any f ∈ S(Rn ). Applying Equation (1.5.5), it follows d n u(exp(t0 ))hα hα . = i |α| + dt 2 t=0 Thus u(exp(t0 ))hα = ei(|α|+n/2)t hα for all t ∈ R. Setting t = π/2, we obtain π u exp 0 hα = einπ/4 i|α| hα . 2 Equations (1.5.7) and (1.5.8) imply the assertion.

(1.5.8)

2 Symplectic Connections

In order to deﬁne symplectic Dirac operators, we have to ﬁx a symplectic connection on the underlying symplectic manifold. Every symplectic manifold admits symplectic connections. Unfortunately, there is no canonical condition to make a symplectic connection unique as one has in Riemannian geometry for deﬁning the Levi–Civita connection. There are systematical investigations on (almost-)symplectic connections by Tondeur [46]. Further results, in particular on curvature properties, were established by Vaisman [47]. Gelfand, Retakh, and Shubin [22, 44] studied geometrical properties of symplectic connections. Moreover, symplectic connections play a key role in Fedosov’s construction of a canonical deformation quantization (cf. [13, 14, 15]). Recently, Bourgeois and Cahen [6, 7] have introduced a variational principle for symplectic connections to single out a preferred symplectic connection. Furthermore, Cahen, Gutt, and Rawnsley [8, 9, 10] have studied the corresponding ﬁeld equation. However, these considerations were made for torsion-free connections only. Our work suggests to take into account also symplectic connections with torsion. It seems that such connections are more convenient for our purposes.

2.1 Symplectic Manifolds This section contains basic facts on symplectic manifolds. References for this are [11, 31, 39]. Let M be a manifold. Unless otherwise stated, all manifolds, all tensor ﬁelds on it, and all maps are assumed to be smooth, i.e. of class C ∞ . Moreover, we suppose that all considered manifolds are Hausdorﬀ, second countable, connected, and without boundary.

22

2 Symplectic Connections

Deﬁnition 2.1.1 A symplectic structure on M is a 2-form ω on M which is closed and non-degenerate. The pair (M, ω) is then called a symplectic manifold. Here, the non-degeneracy of ω means that, for every p ∈ M , the bilinear form ωp on the tangent space Tp M is non-degenerate. Hence M is necessarily of even dimension 2n. Moreover, since ω is non-degenerate if and only if the n-fold wedge product ωn = ω ∧ · · · ∧ ω is a volume form, M is orientable. Examples for symplectic structures are the constant form ω0 on R2n and the K¨ ahler form of a K¨ ahler manifold. Furthermore, for any manifold N , the cotangent bundle T ∗ N carries a canonical symplectic structure ωN , given as the exterior derivative of the canonical 1form on T ∗ N . Note that a symplectic structure ω on a compact manifold M cannot be an exact form. Indeed, if ω = dη for a 1-form η, then by Stokes’ theorem, ωn = dη ∧ ω n−1 = d η ∧ ω n−1 = 0 . M

M

M n

On the other hand, since ω is a volume form, ω n = 0 , M

which gives a contradiction. Therefore, a compact manifold which admits a symplectic structure ω has non-trivial second cohomology H 2 (M ; R). This implies that there are orientable even-dimensional manifolds without any symplectic structure. For example, there exist no symplectic structures on the sphere S 2n for n ≥ 2. Deﬁnition 2.1.2 A symplectomorphism between two symplectic manifolds (M1 , ω1 ) and (M2 , ω2 ) is a diﬀeomorphism F : M1 → M2 satisfying F ∗ ω2 = ω1 . (M1 , ω1 ) and (M2 , ω2 ) are then called symplectomorphic. Theorem 2.1.3 (Darboux) Every symplectic manifold (M, ω) is locally symplectomorphic to (R2n , ω0 ). In view of this theorem, the only local symplectic invariant is the dimension. Let (M, ω) be a ﬁxed symplectic manifold. For some of our later considerations, we need an additional structure.

2.1 Symplectic Manifolds

23

Deﬁnition 2.1.4 An almost complex structure J on M , i.e. an endomorphism J : T M → T M such that J 2 = −id, is called ω-compatible if g(X, Y ) = ω(X, JY ) for X, Y ∈ Γ(T M ) deﬁnes a Riemannian metric g on M . Here, Γ(T M ) means the space of smooth sections of the tangent bundle T M , i.e. the space of all vector ﬁelds on M . Lemma 2.1.5 An ω-compatible almost complex structure J preserves the symplectic structure ω as well as the associated Riemannian metric g. That means ω(JX, JY ) = ω(X, Y ) and g(JX, JY ) = g(X, Y ) for all X, Y ∈ Γ(T M ). Proof. By the skew-symmetry of ω and the symmetry of g, ω(JX, JY ) = g(JX, Y ) = g(Y, JX) = −ω(Y, X) = ω(X, Y ) and g(JX, JY ) = −ω(JX, Y ) = ω(Y, JX) = g(Y, X) = g(X, Y ) . It is not hard to show that, for any symplectic structure ω, there exists an ω-compatible almost complex structure J. Such a structure is not unique. However, one has Proposition 2.1.6 The space of all ω-compatible almost complex structures is contractible. For p ∈ M , let Rp denote the set of all symplectic basises of Tp M , i.e. the set of all 2n-tuples (e1 , . . . , en , f 1 , . . . , f n ) of tangent vectors at p satisfying ωp (ej , ek ) = ωp (f j , f k ) = 0

and ωp (ej , f k ) = δjk

for j, k = 1, . . . , n. The ﬁber bundle πR : R → M whose ﬁber over p is Rp is called the symplectic frame bundle of (M, ω). On each Rp , the symplectic group Sp(n, R) acts simply transitively. The Darboux theorem guarantees the existence of local sections of R. Thus R is locally trivial and hence a principal Sp(n, R)-bundle. We call the local sections of R symplectic frames. We write these frames as (e1 , . . . , en , f1 , . . . , fn ) and their dual frames as (e∗1 , . . . , e∗n , f1∗ , . . . , fn∗ ). Then, of course, (2.1.1) e∗j (X) = ω(X, fj ) and fj∗ (X) = −ω(X, ej )

24

2 Symplectic Connections

for j = 1, . . . , n and X ∈ Γ(T M ). It is clear that the ω-compatible almost complex structures are in 1 : 1 correspondence with the U(n)-reductions of the symplectic frame bundle. Given an ω-compatible almost complex structure J, we call the local sections of the U(n)-reduction unitary frames and write them as (e1 , . . . , e2n ). Obviously, such a frame is characterized by g(ej , ek ) = δjk ,

g(ej , en+k ) = 0 ,

and Jej = en+j

for j, k = 1, . . . , n, where g is again the Riemannian metric associated to J. Deﬁnition 2.1.7 Let u ∈ C ∞ (M ). The vector ﬁeld Xu on M determined by du(Y ) = ω(Xu , Y ) for all Y ∈ Γ(T M ) is called the Hamiltonian vector ﬁeld of the function u. The vector ﬁeld Xu is also referred to as the symplectic gradient of u. Obviously, if J is an ω-compatible almost complex structure, then JXu is the gradient of u taken with respect to the associated Riemannian metric g. Deﬁnition 2.1.8 The Poisson bracket of two functions u, v ∈ C ∞ (M ) is the function {u, v} = ω(Xu , Xv ) . The Lie bracket of Hamiltonian vector ﬁelds and the Poisson bracket are related by Proposition 2.1.9 For all u, v ∈ C ∞ (M ), [Xu , Xv ] = −X{u,v} . Proof. As is known, dη(Y1 , Y2 , Y3 ) = Y1 (η(Y2 , Y3 )) − Y2 (η(Y1 , Y3 )) + Y3 (η(Y1 , Y2 )) −η([Y1 , Y2 ], Y3 ) + η([Y1 , Y3 ], Y2 ) − η([Y2 , Y3 ], Y1 ) for any 2-form η on M and all Y1 , Y2 , Y3 ∈ Γ(T M ). Thus, since ω is closed, we have 0 = Xu (ω(Xv , Y )) − Xv (ω(Xu , Y )) + Y (ω(Xu , Xv )) −ω([Xu , Xv ], Y ) + ω([Xu , Y ], Xv ) − ω([Xv , Y ], Xu ) = Xu Y (v) − Xv Y (u) + Y ({u, v}) −ω([Xu , Xv ], Y ) − [Xu , Y ](v) + [Xv , Y ](u) = Y ({u, v}) − ω([Xu , Xv ], Y ) + Y Xu (v) − Y Xv (u)

2.2 Constructions and Torsion

25

for all u, v ∈ C ∞ (M ) and Y ∈ Γ(T M ). Applying that Xu (v) = ω(Xu , Xv ) = −{u, v} , we obtain Y ({u, v}) = −ω([Xu , Xv ], Y ) , and the proposition is proved.

2.2 Constructions and Torsion Let (M, ω) be a symplectic manifold of dimension 2n. Deﬁnition 2.2.1 A connection ∇ on M is called symplectic if ∇ω = 0, i.e. if X(ω(Y, Z)) = ω(∇X Y, Z) + ω(Y, ∇X Z) for all X, Y, Z ∈ Γ(T M ). Symplectic connections on M are in 1 : 1 correspondence with connection 1forms on the symplectic frame bundle R. So any symplectic manifold admits symplectic connections. However, as already mentioned, there is no canonically distinguished symplectic connection. Lemma 2.2.2 Let ∇ be a symplectic connection and let A be a (2, 1)-tensor ﬁeld on M . Then the connection ∇ given by ∇X Y = ∇X Y + A(X, Y ) is symplectic if and only if ω(A(X, Y ), Z) = ω(A(X, Z), Y ) for all X, Y, Z ∈ Γ(T M ). Proof. The symplecticity of ∇ is equivalent to ω(∇X Y − ∇X Y, Z) + ω(Y, ∇X Z − ∇X Z) = 0 , which gives the assertion.

Later on, we will need symplectic connections which are adapted to an ωcompatible almost complex structure J. The following proposition illustrates a method for constructing such connections.

26

2 Symplectic Connections

Proposition 2.2.3 Let J be an ω-compatible almost complex structure and let ∇ be a connection on M such that (a) ∇ g = 0 or (b) ∇ is symplectic, where g is the Riemannian metric associated to J. Then the connection ∇ on M deﬁned by 1 ∇X Y = ∇X Y + (∇X J)(JY ) 2 is symplectic and satisﬁes ∇J = 0

and

∇g = 0 .

Proof. We suppose (a). The proof for (b) proceeds analogously. Recall that

(∇X J)Y = ∇X (JY ) − J(∇X Y ) .

Using this and Lemma 2.1.5, we conclude 1 ω(∇X Y, Z) = ω(∇X Y, Z) + ω((∇X J)(JY ), Z) 2 1 = −g(∇X Y, JZ) − g((∇X J)(JY ), JZ) 2 1 1 = g(∇X (JY ), Z) − g(∇X Y, JZ) . 2 2 Hence ω(Y, ∇X Z) = −ω(∇X Z, Y ) 1 1 = g(JY, ∇X Z) − g(Y, ∇X (JZ)) . 2 2 Since ∇ g = 0 is equivalent to X(g(Y, Z)) = g(∇X Y, Z) + g(Y, ∇X Z) , it follows that 1 1 X(g(JY, Z)) − X(g(Y, JZ)) 2 2 = −X(g(Y, JZ))

ω(∇X Y, Z) + ω(Y, ∇X Z) =

= X(ω(Y, Z)) , which proves that ∇ is symplectic. From

(2.2.1)

2.2 Constructions and Torsion

27

1 ∇X (JY ) = ∇X (JY ) − (∇X J)Y 2 1 1 = ∇X (JY ) + J(∇X Y ) 2 2 and 1 J(∇X Y ) = J(∇X Y ) + J((∇X J)(JY )) 2 1 1 = J(∇X Y ) + ∇X (JY ) , 2 2 we get ∇X (JY ) − J(∇X Y ) = 0 , which shows ∇J = 0. The identity ∇g = 0 now follows from (∇X g)(Y, Z) = (∇X ω)(Y, JZ) + ω(Y, (∇X J)Z) . We recall that a connection ∇ which satisﬁes Equation (2.2.1) is called Hermitian. Given a connection ∇ on M , let T denote its torsion, which is the (2, 0)-tensor ﬁeld on M deﬁned by T(X, Y ) = ∇X Y − ∇Y X − [X, Y ] . ∇ is said to be torsion-free if T = 0. We assign to T a vector ﬁeld as follows. Deﬁnition 2.2.4 The torsion vector ﬁeld T of ∇ is the vector ﬁeld on M deﬁned by n T(ej , fj ) . T= j=1

Here and subsequently, (e1 , . . . , en , f1 , . . . , fn ) is an arbitrary symplectic frame. One easily checks that, up to the factor 1/2, the vector ﬁeld T arises from a contraction of T with respect to the symplectic structure ω and hence is well deﬁned. For a vector ﬁeld X on M , let div(X) denote the divergence of X with respect to a given connection ∇ on M , i.e. the function on M deﬁned by div(X) = Tr(Y → ∇Y X) . Lemma 2.2.5 For any X ∈ Γ(T M ), div(X) =

n ω ∇ej X, fj − ω ∇fj X, ej . j=1

28

2 Symplectic Connections

Proof. One gets the assertion from div(X) =

n

e∗j ∇ej X + fj∗ ∇fj X

j=1

and Equation (2.1.1).

In the next proposition, we use the following notations. For X ∈ Γ(T M ), let ηX be the 1-form on M given by ηX (Y ) = ω(X, Y ) , and let dM denote the volume element of (M, ω), i.e. dM =

1 n ω . n!

Proposition 2.2.6 Let ∇ be a symplectic connection on (M, ω). Then, for any X ∈ Γ(T M ), d ηX ∧ ω n−1 = (n − 1)! (div(X) + ω(X, T)) dM . Proof. By dM (e1 , f1 , . . . , en , fn ) = 1 , we have to show that d ηX ∧ ω n−1 (e1 , f1 , . . . , en , fn ) = (n − 1)! (div(X) + ω(X, T)) . Since ω is closed,

d ηX ∧ ω n−1 = dηX ∧ ω n−1 .

Consequently,

d ηX ∧ ω

n−1

(e1 , f1 , . . . , en , fn ) = (n − 1)!

n

dηX (ej , fj ) .

j=1

On the other hand, from the symplecticity of ∇ and the well known formula dηX (Y, Z) = Y (ηX (Z)) − Z(ηX (Y )) − ηX ([Y, Z]) , we derive dηX (Y, Z) = Y (ω(X, Z)) − Z(ω(X, Y )) − ω(X, [Y, Z]) = ω(∇Y X, Z) − ω(∇Z X, Y ) +ω(X, ∇Y Z) − ω(X, ∇Z Y ) − ω(X, [Y, Z]) = ω(∇Y X, Z) − ω(∇Z X, Y ) + ω(X, T(Y, Z)) .

2.3 Symplectic Curvature and Ricci Tensors

29

Using Lemma 2.2.5, it follows that n j=1

  n n ω ∇ej X, fj − ω ∇fj X, ej + ω X, dηX (ej , fj ) = T(ej , fj ) j=1

j=1

= div(X) + ω(X, T) . This proves the proposition.

Corollary 2.2.7 Let ∇ be a symplectic connection on (M, ω). Then, for any compactly supported vector ﬁeld X on M , (div(X) + ω(X, T)) dM = 0 . M

Proof. This follows from Proposition 2.2.6 by Stokes’ theorem.

2.3 Symplectic Curvature and Ricci Tensors Let again (M, ω) be a 2n-dimensional symplectic manifold and let ∇ be a connection on M. We denote the curvature of ∇ by R. Here, we use the sign convention that R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z . As already mentioned, we do not want restrict our considerations on symplectic Dirac operators to torsion-free connections. For that reason, we recall the following general relation between curvature and torsion (cf. [35]). Lemma 2.3.1 For any X, Y, Z ∈ Γ(T M ), R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = (∇X T)(Y, Z) + (∇Y T)(Z, X) + (∇Z T)(X, Y ) +T(T(X, Y ), Z) + T(T(Y, Z), X) + T(T(Z, X), Y ) . Of course, if ∇ is torsion-free, then R satisﬁes the Bianchi identity R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0 . Deﬁnition 2.3.2 The symplectic curvature tensor S of ∇ is the (4, 0)-tensor ﬁeld on M deﬁned by S(X1 , X2 , X3 , X4 ) = ω(R(X1 , X2 )X3 , X4 ) for X1 , X2 , X3 , X4 ∈ Γ(T M ).

30

2 Symplectic Connections

The symplectic curvature tensor has the following symmetries. Proposition 2.3.3 (1) For any connection ∇ on M , S(X1 , X2 , X3 , X4 ) = −S(X2 , X1 , X3 , X4 ) . (2) If ∇ is symplectic, then S(X1 , X2 , X3 , X4 ) = S(X1 , X2 , X4 , X3 ) (3) If ∇ is symplectic and torsion-free, then S(X1 , X2 , X3 , X4 ) + S(X2 , X3 , X4 , X1 ) +S(X3 , X4 , X1 , X2 ) + S(X4 , X1 , X2 , X3 ) = 0 . Proof. The ﬁrst assertion is obvious. Let ∇ be symplectic. Then X1 X2 (ω(X3 , X4 )) − X2 X1 (ω(X3 , X4 )) − [X1 , X2 ](ω(X3 , X4 )) = X1 (ω(∇X2 X3 , X4 ) + ω(X3 , ∇X2 X4 )) −X2 (ω(∇X1 X3 , X4 ) + ω(X3 , ∇X1 X4 )) −ω(∇[X1 ,X2 ] X3 , X4 ) − ω(X3 , ∇[X1 ,X2 ] X4 ) = ω(∇X1 ∇X2 X3 , X4 ) + ω(X3 , ∇X1 ∇X2 X4 ) −ω(∇X2 ∇X1 X3 , X4 ) − ω(X3 , ∇X2 ∇X1 X4 ) −ω(∇[X1 ,X2 ] X3 , X4 ) − ω(X3 , ∇[X1 ,X2 ] X4 ) = ω(R(X1 , X2 )X3 , X4 ) + ω(X3 , R(X1 , X2 )X4 ) = S(X1 , X2 , X3 , X4 ) − S(X1 , X2 , X4 , X3 ) . Since the ﬁrst expression of the equation vanishes, this gives (2). Now we additionally assume that ∇ is torsion-free. Then, by the Bianchi identity, S(X1 , X2 , X3 , X4 ) + S(X2 , X3 , X1 , X4 ) + S(X3 , X1 , X2 , X4 ) = 0 , S(X1 , X3 , X4 , X2 ) + S(X3 , X4 , X1 , X2 ) + S(X4 , X1 , X3 , X2 ) = 0 . Thus, applying (2), we get S(X1 , X2 , X3 , X4 ) + S(X2 , X3 , X4 , X1 ) = −S(X3 , X1 , X2 , X4 ) , S(X3 , X4 , X1 , X2 ) + S(X4 , X1 , X2 , X3 ) = −S(X1 , X3 , X2 , X4 ) . Together with (1), this yields (3). Moreover, we have

2.3 Symplectic Curvature and Ricci Tensors

31

Proposition 2.3.4 Let J be an ω-compatible almost complex structure and let ∇ be a symplectic connection on M that satisﬁes ∇J = 0. Then S(X1 , X2 , JX3 , JX4 ) = S(X1 , X2 , X3 , X4 ) and S(X1 , X2 , JX3 , X4 ) = −S(X1 , X2 , JX4 , X3 ) . Proof. Since ∇J = 0, R(X, Y ) ◦ J = J ◦ R(X, Y ) for any X, Y ∈ Γ(T M ). This implies S(X1 , X2 , JX3 , JX4 ) = ω(R(X1 , X2 )(JX3 ), JX4 ) = ω(J(R(X1 , X2 )X3 ), JX4 ) = ω(R(X1 , X2 )X3 , X4 ) = S(X1 , X2 , X3 , X4 ) and, further, by Proposition 2.3.3(2), S(X1 , X2 , JX3 , X4 ) = −S(X1 , X2 , X3 , JX4 ) = −S(X1 , X2 , JX4 , X3 ) . Now we are going to introduce the notion of a symplectic Ricci tensor. Let us ﬁrst recall the deﬁnition of the usual Ricci tensor. Deﬁnition 2.3.5 Let ric be the (2, 0)-tensor ﬁeld on M deﬁned by ric(X, Y ) = Tr(Z → R(Z, X)Y ) for X, Y ∈ Γ(T M ). Expressing ric in a symplectic frame (e1 , . . . , en , f1 , . . . , fn ), one gets Lemma 2.3.6 For any X, Y ∈ Γ(T M ), ric(X, Y ) =

n

(S(ej , X, Y, fj ) − S(fj , X, Y, ej )) .

j=1

Proof. By means of Equation (2.1.1), one sees

32

2 Symplectic Connections

ric(X, Y ) =

n ∗ ej (R(ej , X)Y ) + fj∗ (R(fj , X)Y ) j=1

=

=

n j=1 n

(ω(R(ej , X)Y, fj ) − ω(R(fj , X)Y, ej )) (S(ej , X, Y, fj ) − S(fj , X, Y, ej )) .

j=1

Thus the Ricci tensor ric is obtained by contracting the symplectic curvature tensor S with respect to the symplectic form ω. We use another contraction of S to deﬁne the symplectic Ricci tensor sric. As we will see later (cf. Proposition 5.3.6 and Deﬁnition 2.3.15), this notion of a Ricci tensor is more suitable for studying symplectic Dirac operators. Deﬁnition 2.3.7 We deﬁne the symplectic Ricci tensor sric to be the (2, 0)tensor ﬁeld on M given by sric(X, Y ) =

n

S(ej , fj , X, Y ) .

j=1

For completeness, we note Lemma 2.3.8 If ∇ is symplectic, then for any X, Y ∈ Γ(T M ), the endomorphism Z ∈ Γ(T M ) → R(X, Y )Z ∈ Γ(T M ) is trace-free. Proof. As in the proof of Lemma 2.3.6, one sees Tr(Z → R(X, Y )Z) =

n

(S(X, Y, fj , ej ) − S(X, Y, ej , fj )) .

j=1

Hence the assertion follows from Proposition 2.3.3(2).

Proposition 2.3.9 For any symplectic connection ∇ on M , the symplectic Ricci tensor sric is symmetric. Proof. This also follows from Proposition 2.3.3(2).

We point out that, in general, the Ricci tensor ric and the symplectic Ricci tensor sric diﬀer. In particular, also for symplectic connections, it can happen that ric is not symmetric. However, we have

2.3 Symplectic Curvature and Ricci Tensors

33

Proposition 2.3.10 If ∇ is symplectic and torsion-free, then ric = sric . Proof. Using Proposition 2.3.3, Lemma 2.3.6, and the Bianchi identity, one concludes ric(X, Y ) = −

n

(S(X, ej , fj , Y ) + S(fj , X, ej , Y ))

j=1

=

n

S(ej , fj , X, Y )

j=1

= sric(X, Y ) . Combining the previous two propositions, we obtain (cf. also [47]) Corollary 2.3.11 If ∇ is symplectic and torsion-free, then the Ricci tensor ric is symmetric. The following deﬁnition (cf. [47]) generalizes the notion of constant holomorphic sectional curvature on a K¨ ahler manifold. Deﬁnition 2.3.12 A symplectic connection ∇ on (M, ω) is called reducible if there exists a symmetric (2, 0)-tensor ﬁeld b such that S(X1 , X2 , X3 , X4 ) = ω(X1 , X3 )b(X2 , X4 ) + ω(X1 , X4 )b(X2 , X3 ) −ω(X2 , X3 )b(X1 , X4 ) − ω(X2 , X4 )b(X1 , X3 ) +2ω(X1 , X2 )b(X3 , X4 ) for all X1 , X2 , X3 , X4 ∈ Γ(T M ). Proposition 2.3.13 Let ∇ be a reducible symplectic connection on M . Then sric = 2(n + 1)b . Proof. For any X, Y ∈ Γ(T M ), one concludes sric(X, Y ) =

=

n j=1 n j=1

S(ej , fj , X, Y ) (ω(ej , X)b(fj , Y ) + ω(ej , Y )b(fj , X)

34

2 Symplectic Connections

−ω(fj , X)b(ej , Y ) − ω(fj , Y )b(ej , X) + 2ω(ej , fj )b(X, Y ))   n = b  (ω(ej , X)fj − ω(fj , X)ej ), Y  j=1

 +b 

n

 (ω(ej , Y )fj − ω(fj , Y )ej ), X  + 2nb(X, Y )

j=1

= 2(n + 1)b(X, Y ) . Remark 2.3.14 In dimension 2, i.e. for n = 1, the symplectic curvature tensor S of a connection ∇ writes as S(X1 , X2 , X3 , X4 ) = ω(X1 , X2 )sric(X3 , X4 ) . Indeed, if (e, f) is a symplectic frame, then S(e, f, X, Y ) = sric(X, Y ) = ω(e, f)sric(X, Y ) . Moreover, it is readily veriﬁed that, in this dimension, every symplectic connection is reducible. Since ω is skew-symmetric and, for symplectic connections, sric is symmetric, there is no way to intrinsically deﬁne symplectic scalar curvature. Therefore, to introduce a symplectic counterpart of scalar curvature, we need an auxiliary structure, which in our case is provided by an ω-compatible almost complex structure J. Deﬁnition 2.3.15 Let ∇ be any symplectic connection on (M, ω). We deﬁne the symplectic scalar curvature of ∇ with respect to J to be the function R on M given by 2n sric(ej , ej ) , R= j=1

where (e1 , . . . , e2n ) is an arbitrary unitary frame. Keeping in mind the deﬁnition of sric, the symplectic scalar curvature writes as 2n 1 R= S(ej , Jej , ek , ek ) . (2.3.1) 2 j,k=1

Of course, if (M, ω, J) is K¨ ahler and ∇ is the Levi–Civita connection, then R equals the Riemannian scalar curvature. In general, however, R turns out to be something complete diﬀerent.

3 Symplectic Spinor Fields

This chapter is devoted to the symplectic spinor bundle, which is the Hilbert space bundle associated to a metaplectic structure via the metaplectic representation. Since the ﬁbers of this bundle are inﬁnite-dimensional, several complications in handling diﬀerential operators acting on spinor ﬁelds occur. To circumvent some of these diﬃculties, we should like to have a suitable decomposition of the symplectic spinor bundle into a sequence of ﬁnite rank subbundles. Actually, such a decomposition is given by the irreducible components of the metaplectic representation restricted to the double cover of the unitary group.

3.1 Metaplectic Structures Metaplectic structures are the symplectic analogs of spin structures on orientable Riemannian manifolds. Deﬁnition 3.1.1 A metaplectic structure on a symplectic manifold (M, ω) is an equivariant lift of the symplectic frame bundle πR : R → M with respect to the double covering ρ : Mp(n, R) → Sp(n, R). That means, a metaplectic structure on (M, ω) is a principal Mp(n, R)-bundle πP : P → M together with a map FP : P → R such that πR ◦ FP = πP and FP (pq) = FP (p)ρ(q) for all p ∈ P and q ∈ Mp(n, R). Then, of course, P is connected and FP is a double covering, too.

36

3 Symplectic Spinor Fields

The topological obstruction to the existence of metaplectic structures is precisely the same as in Riemannian spin geometry (cf. [3, 17] and the references therein). Proposition 3.1.2 A symplectic manifold (M, ω) admits a metaplectic structure if and only if the second Stiefel–Withney class w2 (M ) ∈ H 2 (M ; Z2 ) of M vanishes. If this is the case, the isomorphy classes of metaplectic structures on (M, ω) are classiﬁed by the ﬁrst cohomology group H 1 (M ; Z2 ). In particular, because of H 1 (M ; Z2 ) ∼ = Hom(π1 (M ), Z2 ), a simply connected symplectic manifold possesses, up to isomorphism, at most one metaplectic structure. Deﬁnition 3.1.3 The ﬁrst Chern class c1 (M, ω) of a symplectic manifold (M, ω) is deﬁned to be the ﬁrst Chern class of the complex vector bundle (T M, J) for any ω-compatible almost complex structure J. In view of Proposition 2.1.6, the class c1 (M, ω) ∈ H 2 (M ; Z) does not depend on the choice of the ω-compatible almost complex structure J. The obstruction to the existence of metaplectic structures can now be formulated as a condition on c1 (M, ω) (cf. [37]). Proposition 3.1.4 A symplectic manifold (M, ω) admits a metaplectic structure if and only if the ﬁrst Chern class c1 (M, ω) is even, i.e. if there exists a class a ∈ H 2 (M ; Z) such that c1 (M, ω) = 2a. Example 3.1.5 The symplectic frame bundle R of (R2n , ω0 ) is trivial. Thus the unique isomorphy class of metaplectic structures on (R2n , ω0 ) is that of the trivial principal Mp(n, R)-bundle over R2n . Example 3.1.6 Let N be any manifold and let its cotangent bundle T ∗ N be endowed with the canonical symplectic structure ωN . If b ∈ H 1 (T ∗ N ; Z2 ) is the pull back of the ﬁrst Stiefel–Whitney class w1 (N ) ∈ H 1 (N ; Z2 ) of N , then w2 (T ∗ N ) = b2 . This implies that (T ∗ N, ωN ) carries a metaplectic structure if w1 (N ) = 0, i.e. if N is orientable. Example 3.1.7 Consider the complex projective space CPn endowed with the K¨ ahler form ωFS of the Fubini–Study metric. Then, c1 (CPn , ωFS ) coincides with the ﬁrst Chern class c1 (CPn ) of the complex tangent bundle of CPn . Since c1 (CPn ) = (n + 1)a , where a is the positive generator of H 2 (CPn ; Z), the symplectic manifold (CPn , ωFS ) has a metaplectic structure if and only if n is odd. Since CPn is simply connected, such a structure has to be unique.

3.2 Symplectic Spinor Bundle

37

3.2 Symplectic Spinor Bundle Let (M, ω) be a 2n-dimensional symplectic manifold and let P be a ﬁxed metaplectic structure on (M, ω). Deﬁnition 3.2.1 The symplectic spinor bundle Q is deﬁned to be the Hilbert space bundle Q = P ×m L2 (Rn ) associated to the metaplectic structure P via the metaplectic representation m : Mp(n, R) → U(L2 (Rn )). We recall that P ×m L2 (Rn ) is the orbit space of the Mp(n, R)-action on P × L2 (Rn ) deﬁned by (p, f )q = pq, m q −1 f for p ∈ P, f ∈ L2 (Rn ), and q ∈ Mp(n, R). As usual, we will write the equivalence class of an element (p, f ) ∈ P × L2 (Rn ) as [p, f ]. Then, by deﬁnition, [pq, f ] = [p, m(q)f ] for every q ∈ Mp(n, R). Although the bundle Q has only the structure of a topological vector bundle, it is possible to deﬁne smooth sections of Q. For this, we apply that sections ϕ of Q correspond to mappings ϕˆ : P → L2 (Rn ) that are Mp(n, R)-equivariant, i.e. satisfy ˆ . ϕ(pq) ˆ = m q −1 ϕ(p) Explicitly, this correspondence is described by ϕ(πP (p)) = [p, ϕ(p)] ˆ . Deﬁnition 3.2.2 A section ϕ of the symplectic spinor bundle Q is said to be smooth if the corresponding mapping ϕˆ : P → L2 (Rn ) is smooth. A smooth section of Q will also be referred to as a symplectic spinor ﬁeld. As for smooth bundles, we denote the space of smooth sections of Q by Γ(Q). Lemma 3.2.3 Let ϕ ∈ Γ(Q). Then ϕ(p) ˆ ∈ S(Rn ) for all p ∈ P.

38

3 Symplectic Spinor Fields

Proof. Since ϕˆ is smooth, also the map ˆ ∈ L2 (Rn ) q ∈ Mp(n, R) → ϕˆ pq −1 = m(q)ϕ(p) for any p ∈ P is smooth. Thus ϕ(p) ˆ is a smooth vector of the metaplectic representation m, which is equivalent to ϕ(p) ˆ ∈ S(Rn ) by Proposition 1.3.6(2). Besides Q, one may consider the associated vector bundle S = P ×m S(Rn ) . By the above lemma, all smooth sections of Q are in fact sections of S. In analogy with Deﬁnition 3.2.2, we say that a section ϕ of S is smooth if the corresponding mapping ϕˆ : P → S(Rn ) is smooth, where S(Rn ) is endowed with its usual local convex topology. Since S(Rn ) is continuously embedded into L2 (Rn ), the space Γ(S) of smooth sections of S is contained in Γ(Q). The symplectic Cliﬀord multiplication µ0 : R2n ⊗ S(Rn ) → S(Rn ) , deﬁned in Deﬁnition 1.4.3, gives rise to a symplectic Cliﬀord multiplication µ : TM ⊗ S → S on the bundle level as follows. First observe that T M = P ×ρ R2n .

(3.2.1)

Hence the tangent vectors to M can be written as equivalence classes [p, v] of pairs (p, v) ∈ P × R2n . We set µ([p, v] ⊗ [p, f ]) = [p, µ0 (v ⊗ f )] for [p, v] ∈ T M and [p, f ] ∈ S. Since µ0 is Mp(n, R)-equivariant by Lemma 1.4.4, this leads to a well deﬁned endomorphism µ : T M ⊗ S → S. We will write µ(X ⊗ ϕ) = X · ϕ for X ∈ Γ(T M ) and ϕ ∈ Γ(Q). Again, we will use the convention X · Y · ϕ = X · (Y · ϕ) for iterative Cliﬀord multiplication. Then, by Lemma 1.4.1(1) and the fact that ω([p, v], [p, w]) = ω0 (v, w) for [p, v], [p, w] ∈ T M , we have (X · Y − Y · X) · ϕ = −iω(X, Y )ϕ .

(3.2.2)

3.2 Symplectic Spinor Bundle

39

Remark 3.2.4 At this point, a comment is needed. By Lemma 3.2.3, for any X ∈ Γ(T M ) and ϕ ∈ Γ(Q), the Cliﬀord product X · ϕ is well deﬁned and again a section of the spinor bundle Q. This section, however, is not necessarily smooth, since the position and momentum operators Qj and Pj are not continuous on L2 (Rn ). On the other hand, if ϕ ∈ Γ(S), then, of course, also X ·ϕ ∈ Γ(S). Therefore, to avoid additional notations, we will understand ϕ ∈ Γ(Q) → X · ϕ ∈ Γ(Q) as an “unbounded” operator on Γ(Q), i.e. as an operator deﬁned on a suitable subspace of Γ(Q) containing Γ(S). The same will be done for iterative Cliﬀord multiplications and for operators involving the Cliﬀord multiplication as the spinor derivative deﬁned below and the Dirac operators introduced in the next chapter. For [p, f1 ], [p, f2 ] ∈ Q, we set [p, f1 ], [p, f2 ] = f1 , f2 , where f1 , f2 is the L2 -product of the functions f1 , f2 ∈ L2 (Rn ). Since Mp(n, R) acts by unitary operators on L2 (Rn ), this deﬁnes a Hermitian structure on Q. The symplectic Cliﬀord multiplication and the Hermitian structure on Q are related by Proposition 3.2.5 For any X ∈ Γ(T M ) and any ϕ, ψ ∈ Γ(Q), X · ϕ, ψ = −ϕ, X · ψ . Proof. This follows from Lemma 1.4.1(2).

Any symplectic connection ∇ on M induces a covariant derivative, called spinor derivative, on the symplectic spinor bundle Q. This can be described in the following way. Consider the connection 1-form Z : T R → sp(n, R) on the symplectic frame bundle R of (M, ω) which corresponds to ∇ and take the lift Z of Z to the metaplectic structure P, i.e. that connection 1-form Z : T P → mp(n, R) on P that satisﬁes

FP∗ Z = ρ∗ ◦ Z .

Then the covariant derivative on Q associated to Z is the spinor derivative induced by ∇. We denote this covariant derivative on Q also by ∇.

40

3 Symplectic Spinor Fields

Next we want to derive an expression for the spinor derivative in terms of a symplectic frame, which will be useful for our computations. For this, we ﬁx the following notations. For a symplectic frame, i.e. a local section s : U → R of R on an open subset U of M , let s : U → P denote a lift of s to P. Thus FP ◦ s = s . Furthermore, for ϕ ∈ Γ(Q), we set ϕs = ϕˆ ◦ s . Then ϕ = [s, ϕs ] on U . Proposition 3.2.6 Let ∇ be a symplectic connection on M . Then i (fj · ∇X ej − ej · ∇X fj ) · ϕ 2 j=1

(3.2.3)

i (∇X fj · ej − ∇X ej · fj ) · ϕ = [s, X(ϕs )] + 2 j=1

(3.2.4)

n

∇X ϕ = [s, X(ϕs )] −

n

for X ∈ Γ(T M ) and ϕ ∈ Γ(Q), where s = (e1 , . . . , en , f1 , . . . , fn ) is any symplectic frame. Proof. Let Z and Z be as above. Then, by the deﬁnition of the spinor derivative and Proposition 1.4.5, ∇X ϕ = s, X(ϕs ) + m∗ s∗ Z(X) ϕs = s, X(ϕs ) − i s∗ Z(X) · ϕs . (3.2.5) Furthermore, from Lemma 1.3.1 and ρ∗ ◦ s∗ Z = s∗ ρ∗ ◦ Z = s∗ FP∗ Z = s∗ Z , we conclude ∗ i s∗ Z(X) · ϕs = i ρ−1 ∗ (s Z(X)) · ϕs n i = (bj · s∗ Z(X)aj − aj · s∗ Z(X)bj ) · ϕs . 2 j=1

On the other hand, since ej = [s, aj ]

and fj = [s, bj ]

(3.2.6)

3.2 Symplectic Spinor Bundle

41

and hence ∇X ej = [s, s∗ Z(X)aj ] and ∇X fj = [s, s∗ Z(X)bj ] , we have n

(fj · ∇X ej − ej · ∇X fj ) · ϕ

j=1



= s,

n

 (bj · s∗ Z(X)aj − aj · s∗ Z(X)bj ) · ϕs  .

j=1

This proves Equation (3.2.3). Further, using Equation (3.2.2) and the symplecticity of ∇, we see (∇X fj · ej − ∇X ej · fj ) · ϕ = (ej · ∇X fj − fj · ∇X ej ) · ϕ − i(ω(∇X fj , ej ) − ω(∇X ej , fj ))ϕ = (ej · ∇X fj − fj · ∇X ej ) · ϕ + iX(ω(ej , fj ))ϕ = (ej · ∇X fj − fj · ∇X ej ) · ϕ ,

which implies Equation (3.2.4).

We next want to prove some useful properties of the spinor derivative. For this, we ﬁx a symplectic connection on M . We start with a product rule. Proposition 3.2.7 For any X, Y ∈ Γ(T M ) and ϕ ∈ Γ(Q), ∇X (Y · ϕ) = ∇X Y · ϕ + Y · ∇X ϕ . Proof. Let s : U → R be a symplectic frame. According to Equation (3.2.1), Y = [s, Ys ] for some mapping Ys : U → R2n . With this, ∇X Y = [s, X(Ys ) + s∗ Z(X)Ys ] and Y · ϕ = [s, Ys · ϕs ] . Hence, by Equation (3.2.5), ∇X Y · ϕ = [s, X(Ys ) · ϕs + (s∗ Z(X)Ys ) · ϕs ] , Y · ∇X ϕ = s, Ys · X(ϕs ) − i Ys · s∗ Z(X) · ϕs , ∇X (Y · ϕ) = s, X(Ys · ϕs ) − i s∗ Z(X) · Ys · ϕs .

42

3 Symplectic Spinor Fields

Clearly, X(Ys · ϕs ) = X(Ys ) · ϕs + Ys · X(ϕs ) . Moreover, by Lemma 1.4.2(1) and Equation (3.2.6), s∗ Z(X) · Ys · ϕs = i (s∗ Z(X)Ys ) · ϕs + Ys · s∗ Z(X) · ϕs .

Altogether, this yields the assertion. The following proposition states that the spinor derivative is Hermitian. Proposition 3.2.8 For any X ∈ Γ(T M ) and ϕ, ψ ∈ Γ(Q), Xϕ, ψ = ∇X ϕ, ψ + ϕ, ∇X ψ . Proof. With respect to a symplectic frame s, we have ∇X ϕ, ψ + ϕ, ∇X ψ = X(ϕs ), ψs + ϕs , X(ψs ) −i s∗ Z(X) · ϕs , ψs + i ϕs , s∗ Z(X) · ψs

by the deﬁnition of the Hermitian structure on Q and Equation (3.2.5). Since ∗ s Z(X) · ϕs , ψs = ϕs , s∗ Z(X) · ψs by Lemma 1.4.2(2) and Xϕ, ψ = Xϕs , ψs = X(ϕs ), ψs + ϕs , X(ψs ) ,

the proposition is proved. The curvature RQ of the spinor derivative, deﬁned by RQ (X, Y )ϕ = ∇X ∇Y ϕ − ∇Y ∇X ϕ − ∇[X,Y ] ϕ , is related to the curvature R of the underlying symplectic connection by

Proposition 3.2.9 Let s = (e1 , . . . , en , f1 , . . . , fn ) be any symplectic frame. Then i (ej · R(X, Y )fj − fj · R(X, Y )ej ) · ϕ 2 j=1

(3.2.7)

i (R(X, Y )fj · ej − R(X, Y )ej · fj ) · ϕ = 2 j=1

(3.2.8)

n

RQ (X, Y )ϕ =

n

for X, Y ∈ Γ(T M ) and ϕ ∈ Γ(Q).

3.3 Splitting of the Spinor Bundle

43

Proof. One easily checks that the right hand side of Equation (3.2.7) does not depend on the choice of the symplectic frame s. Thus, without loss of generality, we may assume that the form s∗ Z vanishes at an arbitrarily ﬁxed p ∈ M . Then, at p, R(X, Y )ej = [s, (X(s∗ Z(Y )) − Y (s∗ Z(X))aj ] , R(X, Y )fj = [s, (X(s∗ Z(Y )) − Y (s∗ Z(X))bj ] , On the other hand, again applying Equation (3.2.6), one gets RQ (X, Y )ϕ = −i s, X s∗ Z(Y ) − Y s∗ Z(X) · ϕs ∗ ∗ = −i s, ρ−1 ∗ (X(s Z(Y )) − Y (s Z(X))) · ϕs at p. Now Equation (3.2.7) follows by Lemma 1.3.1 as in the proof of Proposition 3.2.6. Equation (3.2.8) is a consequence of Proposition 2.3.3(2) and Equation (3.2.2).

3.3 Splitting of the Spinor Bundle We consider again a symplectic manifold (M, ω) of dimension 2n with a ﬁxed metaplectic structure P. Let J be an ω-compatible almost complex structure and let RJ denote the corresponding U(n)-reduction of the symplectic frame bundle R. So RJp for p ∈ M is the set of all unitary basises of Tp M . We set P J = FP−1 (RJ ) . Clearly, P J is a principal U(n)-bundle. Moreover, the symplectic spinor bunJ dle Q is associated to P via the restriction u of the metaplectic representation m to U(n), Q = P J ×u L2 (Rn ) . Therefore, the splitting of the unitary representation u into its components ul described in Section 1.5 gives rise to an orthogonal decomposition of Q into ﬁnite rank subbundles QJl . Here, QJl is the vector bundle associated to P J via ul , QJl = P J ×ul Wl . According to Equation (1.5.6), rankC QJl

=

n+l−1 l

.

The Hamilton operator H0 of the harmonic oscillator gives rise to an endomorphism of the bundle S as follows.

44

3 Symplectic Spinor Fields

Deﬁnition 3.3.1 We deﬁne HJ : S → S by HJ ([p, f ]) = [p, H0 f ] for p ∈ P J and f ∈ S(Rn ). This deﬁnition is justiﬁed by Proposition 1.5.1. Clearly, n QJl = q ∈ S : HJ (q) = − l + q . 2

(3.3.1)

For some computations, it will be useful to understand the endomorphism HJ in the following way. Lemma 3.3.2 For any unitary frame s = (e1 , . . . , e2n ) and any ϕ ∈ Γ(Q), 1 ej · ej · ϕ . 2 j=1 2n

HJ ϕ =

Proof. Writing ϕ = [s, ϕs ], we get HJ ϕ = [s, H0 ϕs ] n n 1 1 = [s, aj · aj · ϕs ] + [s, bj · bj · ϕs ] 2 j=1 2 j=1 =

1 1 ej · ej · ϕ + en+j · en+j · ϕ 2 j=1 2 j=1

=

1 ej · ej · ϕ , 2 j=1

n

n

2n

where we have used Equation (1.5.4).

Lemma 3.3.3 For any ϕ, ψ ∈ Γ(Q), HJ ϕ, ψ = ϕ, HJ ψ . Proof. This follows from H0 f1 , f2 = f1 , H0 f2 for all f1 , f2 ∈ L2 (Rn ).

Next we prove that the spinor derivative commutes with HJ provided the underlying connection on M is Hermitian.

3.3 Splitting of the Spinor Bundle

45

Proposition 3.3.4 Let ∇ be a symplectic connection on M and suppose that ∇J = 0. Then, for any ϕ ∈ Γ(Q) and X ∈ Γ(T M ), ∇X (HJ ϕ) = HJ (∇X ϕ) . Proof. Let (e1 , . . . , e2n ) be a unitary frame. From ∇X ej =

2n

ω(∇X ej , Jek )ek

k=1

and ω((∇X J)ej , ek ) = ω(∇X (Jej ) − J(∇X ej ), ek ) = ω(∇X ek , Jej ) + ω(∇X ej , Jek ) , we deduce that 2n 2n (∇X ej · ej + ej · ∇X ej ) · ϕ = ω(∇X ej , Jek )(ek · ej + ej · ek ) · ϕ j=1

j,k=1

=

2n

ω((∇X J)ej , ek )ek · ej · ϕ

j,k=1

=0. By means of Proposition 3.2.7 and Lemma 3.3.2, it follows that 1 ∇X (ej · ej · ϕ) 2 j=1 2n

∇X (HJ ϕ) =

1 (∇X ej · ej · ϕ + ej · ∇X ej · ϕ + ej · ej · ∇X ϕ) 2 j=1 2n

=

= HJ (∇X ϕ) , so we are done.

Of course, the above proposition is originated in Proposition 1.5.1 and the fact that the connection 1-form of a Hermitian connection on M reduces to the subbundle RJ . Equation (3.3.1) and Proposition 3.3.4 imply that, for a Hermitian connection, the spaces Γ(QJl ) of smooth sections of QJl are preserved by the spinor derivative. That means, we have Corollary 3.3.5 Let ∇ be a symplectic connection on M satisfying ∇J = 0. Then, for any l ∈ N0 , it holds that, if ϕ ∈ Γ(QJl ) and X ∈ Γ(T M ), then also ∇X ϕ ∈ Γ(QJl ).

46

3 Symplectic Spinor Fields

In the remaining of this section, we will give alternative descriptions of the complex line bundle QJ0 . !0 of all functions f ∈ S(Rn ) such that Lemma 3.3.6 The space W J0 v · f = iv · f for all v ∈ R2n coincides with the one-dimensional subspace W0 . !0 if and only if Proof. It is clear that f ∈ S(Rn ) belongs to W ∂f (x) = −xj f (x) ∂xj

(3.3.2)

for j = 1, . . . , n and that the Hermite function h0 ∈ W0 solves this system. On the other hand, diﬀerentiating Equation (3.3.2), we obtain ∂2f (x) = −f (x) + x2j f (x) . ∂x2j This implies

n H0 f = − f 2

and hence f ∈ W0 .

Corollary 3.3.7 The space Γ(QJ0 ) of smooth sections of QJ0 consists exactly of those symplectic spinor ﬁelds ϕ that satisfy JX · ϕ = iX · ϕ for all X ∈ Γ(T M ). Proof. One applies Lemma 3.3.6 and J([p, v]) = [p, J0 v] for p ∈ P J and v ∈ R2n . Let

T C M = T 1,0 M ⊕ T 0,1 M

be the decomposition of the complexiﬁed tangent bundle T C M given by the ±i eigenspaces of J. Since, in view of Proposition 3.1.4, the ﬁrst Chern class c1 (M, ω) is even, there exists a square root of the so-called anti-canonical line bundle KJ = Λn T 1,0 M . More explicitly, we have Proposition 3.3.8 The line bundle QJ0 is a square root of the anti-canonical line bundle KJ .

3.3 Splitting of the Spinor Bundle

Proof. We ﬁrst note that T 1,0 M ∼ = P J ×ρˆ Cn , where ρ : U(n) → GL(n, C) is the restriction of ρ to U(n) composed with a −b ι: ∈ U(n) → a + ib ∈ GL(n, C) . b a Indeed, if ΦJ : RJ × Cn → T 1,0 M is given by 1 zj (ej − iJej ) , 2 j=1 n

ΦJ ((e1 , . . . , e2n ), z) = then

[p, z] ∈ P J ×ρˆ Cn → ΦJ (FP (p), z) ∈ T 1,0 M is a well deﬁned bundle isomorphism. Hence KJ ∼ = P J ×det ◦ρˆ C . Consequently, we have proved the proposition, if we have shown, that (det ◦ ρ)(q) = (u0 (q))2 for all q ∈ U(n). For this, it suﬃces to verify that (det ◦ ρ)∗ = 2(u0 )∗ . Using Proposition 1.4.5, we compute ((u0 )∗ (aj · ak + bj · bk )h0 )(x) = (m∗ (aj · ak + bj · bk )h0 )(x) ∂ 2 h0 = i xj xk h0 (x) − (x) ∂xj ∂xk = iδjk h0 (x) and ((u0 )∗ (aj · bk − ak · bj )h0 )(x) = (m∗ (aj · bk − ak · bj )h0 )(x) ∂h0 ∂h0 = xj (x) − xk (x) ∂xk ∂xj =0. On the other hand, by means of the easily checked fact that (Tr ◦ ι∗ )(v w + J0 v J0 w) n = 2i (ω0 (v, aj )ω0 (w, aj ) + ω0 (v, bj )ω0 (w, bj )) , j=1

47

48

3 Symplectic Spinor Fields

we obtain (det ◦ ρ)∗ (aj · ak + bj · bk ) = (Tr ◦ ι∗ )(aj ak + bj bk ) = 2iδjk and (det ◦ ρ)∗ (aj · bk − ak · bj ) = (Tr ◦ ι∗ )(aj bk − ak bj ) = 0 . This concludes the proof.

4 Symplectic Dirac Operators

Having introduced all the necessary material, we now deﬁne symplectic Dirac These operators, acting on symplectic spinor ﬁelds, are operators D and D. deﬁned in a similar way as the Dirac operator on Riemannian manifolds. Ac depend on a choice of a metaplectic structure cordingly, the operators D and D as well as on a choice of a symplectic connection on the underlying manifold. These dependences can be investigated by methods developed by Friedrich for the Riemannian situation (cf. [20, 18]). To describe the dependence of the on the ω-compatible almost complex structure, we will adapt the operator D strategy used by Baum for studying the transformation of the Riemannian Dirac operator under a conformal change of the Riemannian metric (cf. [2]). At this point, we wish to emphasize that, although the whole construction follows the same procedure as for the classical Riemannian Dirac operator, using the symplectic structure on M instead of the Riemannian metric, the symplectic Dirac operators and the Riemannian Dirac operator have essentially diﬀerent properties. These are particularly caused by the completely diﬀerent structures of the Cliﬀord algebras and Cliﬀord multiplications.

4.1 Deﬁnition of the Operators Let a 2n-dimensional symplectic manifold (M, ω) be given and suppose that (M, ω) admits a metaplectic structure. After ﬁxing a metaplectic structure P and a symplectic connection ∇ on M , we may consider the symplectic spinor bundles Q and S as well as the spinor derivative ∇ : Γ(Q) → Γ(T ∗ M ⊗ Q) . In view of Lemma 3.2.3, the Cliﬀord multiplication µ : T M ⊗ S → S induces a map

50

4 Symplectic Dirac Operators

µ : Γ(T M ⊗ Q) → Γ(Q) . Further, let J be an ω-compatible almost complex structure and let g denote the associated Riemannian metric on M . Deﬁnition 4.1.1 The symplectic Dirac operator D of (M, ω) with respect to the metaplectic structure P and the symplectic connection ∇ is deﬁned as the composition ω

D = µ ◦ ∇ : Γ(Q) → Γ(T ∗ M ⊗ Q) ∼ = Γ(T M ⊗ Q) → Γ(Q) , where we identify the bundles T ∗ M and T M by putting a tangent vector into the ﬁrst argument of ω. Using the Riemannian metric g instead of ω for identifying T ∗ M and T M , we obtain a second symplectic Dirac operator g

= µ ◦ ∇ : Γ(Q) → Γ(T ∗ M ⊗ Q) ∼ D = Γ(T M ⊗ Q) → Γ(Q) . are As explained in Remark 3.2.4, the symplectic Dirac operators D and D understood as operators deﬁned on a suitable subspace of Γ(Q). Locally, D can be expressed as follows. and D Lemma 4.1.2 Let (e1 , . . . , en , f1 , . . . , fn ) be any symplectic frame. Then Dϕ =

n ej · ∇fj ϕ − fj · ∇ej ϕ j=1

and = Dϕ

n Jej · ∇fj ϕ − Jfj · ∇ej ϕ . j=1

Proof. Writing ∇ϕ =

n

e∗j ⊗ ∇ej ϕ + fj∗ ⊗ ∇fj ϕ

j=1

and using Equation (2.1.1), one has   n −fj ⊗ ∇ej ϕ + ej ⊗ ∇fj ϕ  Dϕ = µ  j=1

and



 n = µ Dϕ −Jfj ⊗ ∇ej ϕ + Jej ⊗ ∇fj ϕ  , j=1

which yields the assertion. An immediate consequence is

4.1 Deﬁnition of the Operators

51

Lemma 4.1.3 Let (e1 , . . . , e2n ) be any unitary frame. Then Dϕ = −

2n

Jej · ∇ej ϕ

= Dϕ

and

j=1

2n

ej · ∇ej ϕ .

j=1

Example 4.1.4 We consider the symplectic manifold (R2 , ω0 ). Its unique, up to isomorphism, metaplectic structure is the trivial Mp(1, R)-bundle P = R2 × Mp(1, R) . Consequently, also the bundle S is trivial. Thus a section of it is a mapping ϕ : R2 → S(R), i.e. a mapping ϕ : R3 → C such that, for any (s, t) ∈ R2 , x ∈ R → ϕ(s, t, x) ∈ C lies in S(R). Now choose the canonical ﬂat connection ∇ on R2 as symplectic connection and J0 as ω0 -compatible almost complex structure. Then, according to take the form Lemma 4.1.2, the symplectic Dirac operators D and D (Dϕ)(s, t, x) = ix

∂2ϕ ∂ϕ (s, t, x) − (s, t, x) ∂t ∂x∂s

(4.1.1)

and

∂ϕ ∂2ϕ (s, t, x) + (s, t, x) . (4.1.2) ∂s ∂x∂t Obviously, any symplectic spinor ﬁeld ϕ : R3 → C independent of (s, t) is Moreover, using the complex an element of the kernel of D as well as of D. coordinate z = s + it, Equations 4.1.1 and 4.1.2 read as ∂ϕ ∂ϕ ∂ ∂ϕ ∂ϕ (z, x) − (z, x) − i (z, x) + (z, x) (Dϕ)(z, x) = ix ∂z ∂z ∂x ∂z ∂z (Dϕ)(s, t, x) = ix

and (Dϕ)(z, x) = ix

∂ϕ ∂ϕ ∂ ∂ϕ ∂ϕ (z, x) + (z, x) + i (z, x) − (z, x) . ∂z ∂z ∂x ∂z ∂z

This implies that all mappings ϕ : C × R → C given by ϕ(z, x) = e−x

2

/2

h(z)

= 0. The for any holomorphic function h on C satisfy Dϕ = 0 and Dϕ equivalence of the last two equations for such ϕ can also be derived from the following fact. Namely, by Equations (1.4.1), (4.1.1), and (4.1.2), for any = 0, symplectic spinor ﬁeld ϕ on R2 , one has Dϕ = 0 if and only if DFϕ where Fϕ means the Fourier transformed of ϕ with respect to the variable x. In Chapter 7, we will come back to this relation in a more general setting.

52

4 Symplectic Dirac Operators

4.2 Dependence on the Symplectic Connection Here, we carry over the concepts developed for Riemannian Dirac operators deﬁned with respect to diﬀerent metric connections from the orthogonal case to the symplectic one. Let a metaplectic structure P on (M, ω) as well as an ω-compatible almost complex structure J be ﬁxed and let ∇ and ∇ be two symplectic connections on M . We deﬁne a (2, 1)-tensor ﬁeld A on M by A(X, Y ) = ∇X Y − ∇X Y . Clearly,

A(X, Y ) − A(Y, X) = (T − T)(X, Y ) ,

(4.2.1)

where T and T are the torsions of ∇ and ∇ , respectively. Let the (3, 0)-tensor deﬁned by ﬁelds B and B B(X, Y, Z) = ω(A(X, Y ), Z)

and B(X, Y, Z) = B(X, JY, JZ)

and set C(X, Y, Z) = B(X, Y, Z) + B(Y, Z, X) + B(Z, X, Y ) and

C(X, Y, Z) = B(X, Y, Z) + B(Y, Z, X) + B(Z, X, Y ) .

Lemma 4.2.1 (1) B(X, Y, Z) = B(X, Z, Y ) for any X, Y, Z ∈ Γ(T M ). are symmetric. (2) The (3, 0)-tensor ﬁelds C and C (3) B is symmetric if and only if T = T . if and only if ∇J = ∇ J. (4) B = B Proof. Assertion (1) follows from Lemma 2.2.2. Assertion (2) is a consequence of (1). Assertion (3) is obtained from Equation (4.2.1) and (1). The last assertion is seen from B(X, Y, Z) − B(X, Y, Z) = ω((∇X J)Y − (∇X J)Y, JZ) for all X, Y, Z ∈ Γ(T M ).

Let div : Γ(T M ) → C ∞ (M ) be the divergence operator with respect to ∇ ! : Γ(T M ) → C ∞ (M ) be deﬁned by and let div ! div(X) = Tr(Y → J(∇Y X)) . Then ! div(X) =

n ω ∇fj X, Jej − ω ∇ej X, Jfj , j=1

! denote where (e1 , . . . , en , f1 , . . . , fn ) is any symplectic frame. Let div and div the corresponding operators with respect to ∇ . One easily checks

4.2 Dependence on the Symplectic Connection

53

Lemma 4.2.2 For any X ∈ Γ(T M ), div (X) − div(X) =

n

(B(ej , X, fj ) − B(fj , X, ej ))

j=1

and

n

! ! (X) − div(X) = div

j , JX, fj ) − B(f j , JX, ej )) . (B(e

j=1

be the symplectic Dirac operators with respect Theorem 4.2.3 Let D and D be the corresponding operators with respect to ∇ . to ∇ and let D and D Then D and D coincide if and only if the tensor ﬁeld C identically vanishes =D if and only if and the operators div and div coincide. Analogously, D ! ! C = 0 and div = div . Proof. Let ϕ ∈ Γ(Q) and let (e1 , . . . , e2n ) be a unitary frame. Applying Proposition 3.2.6, we compute ∇X ϕ

i i − ∇X ϕ = − Jek · ∇X ek · ϕ + Jek · ∇X ek · ϕ 2 2 2n

2n

k=1

k=1

2n i =− Jek · A(X, ek ) · ϕ 2 k=1

=

i 2

2n

B(X, ek , el )Jek · Jel · ϕ

k,l=1

for the diﬀerence of the spinor derivatives induced by ∇ and ∇ . Together with Lemma 4.1.3, this implies D ϕ − Dϕ = −

2n i B(ej , ek , el )Jej · Jek · Jel · ϕ 2 j,k,l=1

=

2n i B(Jej , Jek , Jel )ej · ek · el · ϕ . 2 j,k,l=1

Next, observe that 2n

2n

B(Jek , Jel , Jej )ω(el , ej )ek = −

j,k,l=1

B(Jek , Jel , ej )ω(el , Jej )ek

j,k,l=1 2n

=−

j,k=1

=0,

B(Jek , Jej , ej )ek

54

4 Symplectic Dirac Operators

since substituting Jej for ej and Lemma 4.2.1(1) yield 2n

2n

B(Jek , Jej , ej )ek = −

j,k=1

B(Jek , ej , Jej )ek

j,k=1 2n

=−

B(Jek , Jej , ej )ek .

j,k=1

Further, 2n

B(Jek , Jel , Jej )ω(ek , ej )el = −

j,k,l=1

2n

B(ek , Jel , Jej )ω(Jek , ej )el

j,k,l=1

=

2n

B(ej , Jel , Jej )el .

j,l=1

Consequently, 2n

B(Jek , Jel , Jej )ek · el · ej · ϕ

j,k,l=1 2n

=

B(Jek , Jel , Jej )(ej · ek · el − iω(el , ej )ek − iω(ek , ej )el ) · ϕ

j,k,l=1 2n

=

B(Jek , Jel , Jej )ej · ek · el · ϕ − i

j,k,l=1

2n

B(ej , Jel , Jej )el · ϕ .

j,l=1

Analogously, 2n

B(Jel , Jej , Jek )ω(el , ej )ek =

j,k,l=1

and

2n

B(Jel , Jej , Jek )ω(el , ek )ej =

j,k,l=1

2n

B(ej , Jek , Jej )ek

j,k=1 2n

B(ek , Jej , Jek )ej ,

j,k=1

which leads to 2n

B(Jel , Jej , Jek )el · ej · ek · ϕ

j,k,l=1

=

2n

B(Jel , Jej , Jek )(ej · ek · el − iω(el , ej )ek − iω(el , ek )ej ) · ϕ

j,k,l=1

=

2n j,k,l=1

B(Jel , Jej , Jek )ej · ek · el · ϕ − 2i

2n j,k=1

B(ej , Jek , Jej )ek · ϕ .

4.2 Dependence on the Symplectic Connection

55

It follows that 2n

3

B(Jej , Jek , Jel )ej · ek · el · ϕ

j,k,l=1 2n

=

B(Jej , Jek , Jel )ej · ek · el · ϕ +

j,k,l=1

+

2n

B(Jek , Jel , Jej )ek · el · ej · ϕ

j,k,l=1

2n

B(Jel , Jej , Jek )el · ej · ek · ϕ

j,k,l=1 2n

=

C(Jej , Jek , Jel )ej · ek · el · ϕ − 3i

j,k,l=1

2n

B(ej , Jek , Jej )ek · ϕ .

j,k=1

By Lemma 4.2.2, 2n

B(ej , Jek , Jej ) = div (Jek ) − div(Jek ) .

j=1

Thus we arrive at D ϕ − Dϕ =

2n i C(Jej , Jek , Jel )ej · ek · el · ϕ 6

+

j,k,l=1 n

1 2

(div (Jek ) − div(Jek ))ek · ϕ .

k=1

Now, arranging the factors of ej · ek · el in ascending order and applying Lemma 1.4.2(3), this proves the assertion for the Dirac operators D and D . Since

2n j , ek , el )ej · ek · el · ϕ ϕ − Dϕ = i D B(e 2 j,k,l=1

and, according to Lemma 4.2.2, 2n

! ! B(Je j , ek , ej ) = div (Jek ) − div(Jek ) ,

j=1

and D . the same reasoning also yields the assertion for the operators D

Corollary 4.2.4 If ∇ and ∇ are two symplectic connections on M such that T = T , then their Dirac operators D and D coincide if and only if ∇ = ∇ . That is, diﬀerent symplectic connections having the same torsion induce diﬀerent Dirac operators D.

56

4 Symplectic Dirac Operators

Proof. By Lemma 4.2.1(3), in case T = T , the conditions C = 0 and ∇ = ∇ are equivalent. Hence the corollary is an easy consequence of the previous theorem. =D . Corollary 4.2.5 If ∇J = ∇ J, then D = D if and only if D by Lemma 4.2.1(4). This implies Proof. Suppose ∇J = ∇ J. Then B = B C = C and, by Lemma 4.2.2,

! (X) − div(X) ! div = div (JX) − div(JX) . The corollary now immediately follows from the above theorem.

To illustrate Theorem 4.2.3, we look at the following examples. Here, we restrict ourselves to the Dirac operator D. Examples explaining the relations may be found analogously. for the operator D Example 4.2.6 We consider the symplectic manifold (R2n , ω0 ) with n ≥ 2 and ﬁx any symplectic connection ∇ on it. Let Ai : R2n × R2n → R2n , i = 1, 2, 3, be deﬁned by A1 (v1 , v2 ) = ω0 (a1 , v1 )(ω0 (b1 , v2 )b2 + ω0 (b2 , v2 )b1 ) , A2 (v1 , v2 ) = ω0 (b1 , v1 )(ω0 (a1 , v2 )b2 + ω0 (b2 , v2 )a1 ) , A3 (v1 , v2 ) = ω0 (b2 , v1 )(ω0 (a1 , v2 )b1 + ω0 (b1 , v2 )a1 ) . Clearly, ω0 (Ai (v1 , v2 ), v3 ) = ω0 (Ai (v1 , v3 ), v2 )

(4.2.2)

for i = 1, 2, 3. We set A = t1 A1 + t2 A2 + t3 A3 for real numbers t1 , t2 , t3 . By Lemma 2.2.2 and Equation (4.2.2), understanding A as a constant (2, 1)-tensor ﬁeld on R2n , ∇X Y = ∇X Y + A(X, Y )

(4.2.3)

deﬁnes a further symplectic connection ∇ on R2n . One easily computes that, in this situation, C(v1 , v2 , v3 ) = (t1 + t2 + t3 ) ω0 (a1 , vς(1) )ω0 (b1 , vς(2) )ω0 (b2 , vς(3) ) , ς∈S3

where S3 denotes the symmetric group. In particular, C(b1 , a1 , a2 ) = t1 + t2 + t3 . Moreover,

4.3 Dependence on the Metaplectic Structure n

57

(B(aj , v, bj ) − B(bj , v, aj )) = (t1 − t2 )ω0 (b2 , v) .

j=1

Hence C = 0 if and only if t1 +t2 +t3 = 0, whereas, by Lemma 4.2.2, div = div if and only if t1 = t2 . Specializing the parameters such that (t1 , t2 , t3 ) = (1, t, −1 − t) , we get C = 0 for all t, but div = div only for t = 1. On the other hand, if (t1 , t2 , t3 ) = (t, t, −2) , then C = 0 only if t = 1, whereas div = div for all t. Of course, if (t1 , t2 , t3 ) = (t, 1, 1) , then C = 0 or div = div for all t. Furthermore, in all cases, ∇ = ∇. Note that, multiplying A by a suitable cut-oﬀ function and applying the Darboux theorem, the example can be carried over to any symplectic manifold of dimension at least 4. Example 4.2.7 We again consider (R2n , ω0 ), where now the case n = 1 is allowed, and deﬁne A : R2n × R2n → R2n by A(v1 , v2 ) = ω0 (a1 , v1 )ω0 (a1 , v2 )a1 . As in the above example, the connection ∇ deﬁned by Equation (4.2.3) for a given symplectic connection ∇ on R2n is symplectic, too. Moreover, we see that div = div . Thus, in contrast to the Riemannian situation, in dimension 2, there are examples of diﬀerent symplectic connections having the same divergence.

4.3 Dependence on the Metaplectic Structure We again adapt the techniques used for the classical Riemannian Dirac operator to study symplectic Dirac operators with respect to diﬀerent metaplectic structures. Let P1 and P2 be two metaplectic structures on the symplectic manifold (M, ω). We set = {(p1 , p2 ) ∈ P1 × P2 : FP (p1 ) = FP (p2 )} C 1 2 by and deﬁne an action of Z2 = {±1} on C

58

4 Symplectic Dirac Operators

(p1 , p2 )(−1) = (p1 e− , p2 e− ) . Here, e− denotes that element of Mp(n, R) that lies over the unit element 2 and let of Sp(n, R) but is not the unit element of Mp(n, R). Let C = C/Z [p1 , p2 ] ∈ C be the equivalence class of (p1 , p2 ). The group Sp(n, R) × Z2 acts on C via [p1 , p2 ](ρ(q), 1) = [p1 q, p2 q] , [p1 , p2 ](ρ(q), −1) = [p1 q, p2 qe− ] for q ∈ Mp(n, R). Deﬁning πC : C → M by πC [p1 , p2 ] = πP1 (p1 ) = πP2 (p2 ) , C gets the structure of a principal Sp(n, R)×Z2 -bundle. Moreover, the double covering FC : C → R given by FC ([p1 , p2 ]) = FP1 (p1 ) = FP2 (p2 ) satisﬁes πR ◦ FC = πC and FC ([p1 , p2 ](A, a)) = FC ([p1 , p2 ])A for any (A, a) ∈ Sp(n, R) × Z2 . Thus C together with FC is an equivariant lift of the symplectic frame bundle R with respect to the homomorphism (A, a) ∈ Sp(n, R) × Z2 → A ∈ Sp(n, R) . This lift is called the deformation of the metaplectic structures P1 and P2 . Proposition 4.3.1 The metaplectic structures P1 and P2 are isomorphic if and only if their deformation C is isomorphic to the trivial lift R × Z2 of R. Proof. Let the metaplectic structures P1 and P2 be isomorphic. That means, there is an isomorphism Φ : P1 → P2 of the principal Mp(n, R)-bundles such that (4.3.1) FP2 ◦ Φ = FP1 . In particular, Φ(p1 e− ) = Φ(p1 )e− for any p1 ∈ P1 . Therefore z(FP1 (p1 )) = [p1 , Φ(p1 )] ,

(4.3.2)

deﬁnes a section z : R → C of the covering FC . Conversely, if the deformation C is isomorphic to R × Z2 , then there exists a section z : R → C of FC such that z(rA) = z(r)(A, 1)

4.3 Dependence on the Metaplectic Structure

59

for any r ∈ R and A ∈ Sp(n, R). Let Φ : P1 → P2 be deﬁned by Equation (4.3.2). Then, clearly, Φ satisﬁes Equation (4.3.1). Moreover, for any q ∈ Mp(n, R), we have [p1 q, Φ(p1 q)] = z(FP1 (p1 q)) = z(FP1 (p1 )ρ(q)) = z(FP1 (p1 ))(ρ(q), 1) = [p1 q, Φ(p1 )q] and hence Φ(p1 q) = Φ(p1 )q . Thus Φ is an isomorphism of the metaplectic structures P1 and P2 .

Corollary 4.3.2 The metaplectic structures P1 and P2 are isomorphic if and only if their deformation C is not connected. Let E = C ×τ C be the complex line bundle associated to the deformation C via the representation τ : Sp(n, R)×Z2 → GL(C) given by the projection onto the second factor and let Q1 and Q2 denote the symplectic spinor bundles associated to P1 and P2 , respectively. Lemma 4.3.3 Setting ΦE ([[p1 , p2 ], z] ⊗ [p1 , f ]) = [p2 , zf ] , one gets a bundle isomorphism ΦE : E ⊗ Q1 → Q2 . Proof. Let q ∈ Mp(n, R). On the one hand,

[[p1 , p2 ], z] ⊗ [p1 , f ] = [[p1 q, p2 q], z] ⊗ p1 q, m q −1 f = [[p1 q, p2 qe− ], −z] ⊗ p1 q, m q −1 f .

On the other hand, since m(e− )f = −f , [p2 , zf ] = p2 q, zm q −1 f = p2 qe− , −zm q −1 f . This shows that ΦE is correctly deﬁned. Obviously, ΦE is surjective. The injectivity of ΦE can be easily checked. Consequently, if the complex line bundle E is trivial, then the symplectic spinor bundles Q1 and Q2 are isomorphic. In particular, E is trivial, if the metaplectic structures P1 and P2 are isomorphic. Since E is the complexiﬁcation of a real line bundle, one has 2c1 (E) = 0. Thus E is also trivial, if the second cohomology group H 2 (M ; Z) has no torsion of order 2. Now we ﬁx a symplectic connection ∇ on M and an ω-compatible almost complex structure J. Let Z C : T C → sp(n, R) be the lift of the connection 1-form Z : T R → sp(n, R) of ∇ to the deformation C and let ∇E denote the covariant derivative in E induced by Z C .

60

4 Symplectic Dirac Operators

Lemma 4.3.4 ∇E is ﬂat. Proof. Let C ∈ Γ(E). Since the diﬀerential τ∗ of the representation τ is trivial, writing C = [r, Cr ] for a local section r of C, we have " # ∗ C ∇E X C = r, X(Cr ) + τ∗ r Z (X) Cr = [r, X(Cr )] ,

which implies the assertion.

Let ∇1 and ∇2 denote the spinor derivatives in the symplectic spinor bundles 1 and D2 , D 2 be the corresponding Q1 and Q2 , respectively, and let D1 , D Dirac operators. Replacing the spinor derivative by the covariant derivative ∇1,E = id ⊗ ∇1 + ∇E ⊗ id : Γ(E ⊗ Q1 ) → Γ(T ∗ M ⊗ E ⊗ Q1 ) in Deﬁnition 4.1.1, we obtain the twisted Dirac operators ω

∼ Γ(T M ⊗ E ⊗ Q ) DE = µ ◦ ∇1,E : Γ(E ⊗ Q1 ) → Γ(T ∗ M ⊗ E ⊗ Q1 ) = 1 → Γ(E ⊗ Q1 ) and g

∼ Γ(T M ⊗ E ⊗ Q ) E = µ ◦ ∇1,E : Γ(E ⊗ Q1 ) → Γ(T ∗ M ⊗ E ⊗ Q1 ) = D 1 → Γ(E ⊗ Q1 ) . E are given as follows. Locally, the operators DE and D Lemma 4.3.5 Let (e1 , . . . , en , f1 , . . . , fn ) be any symplectic frame. Then DE (C ⊗ ϕ) = C ⊗ D1 ϕ +

n

E ∇E fj C ⊗ ej · ϕ − ∇ej C ⊗ fj · ϕ

j=1

and E (C ⊗ ϕ) = C ⊗ D 1 ϕ + D

n

E ∇E fj C ⊗ Jej · ϕ − ∇ej C ⊗ Jfj · ϕ

.

j=1

Proof. Proceed as for Lemma 4.1.2. A direct calculation yields E and D2 , D 2 satisfy Proposition 4.3.6 The operators DE , D ΦE ◦ DE = D2 ◦ ΦE

and

E = D 2 ◦ ΦE . ΦE ◦ D

4.3 Dependence on the Metaplectic Structure

61

To describe the relation between symplectic Dirac operators with respect to diﬀerent metaplectic structures, we choose a local non-vanishing section C : U → E. Then, by E ΦE # (ϕ) = Φ (C ⊗ ϕ) for ϕ ∈ Γ(Q1 ), an isomorphism ΦE # between the restrictions of Q1 and Q2 to U is given. Let η# be the local connection form of ∇E with respect to C, i.e. the complex-valued 1-form on U deﬁned by ∇E X C = η# (X)C . Since ∇E is ﬂat, η# is closed. Let X# be the vector ﬁeld dual to η# , i.e. the complex vector ﬁeld on U given by η# (Y ) = ω(X# , Y ) , where the symplectic structure ω is extended complex linearly. Theorem 4.3.7 Let C be any local non-vanishing section in E. Then, for all ϕ ∈ Γ(Q1 ), E D2 ◦ ΦE # (ϕ) = Φ# (D1 ϕ + X# · ϕ) and

2 ◦ ΦE (ϕ) = ΦE (D 1 ϕ + JX# · ϕ) . D # #

Proof. Let (e1 , . . . , en , f1 , . . . , fn ) be a symplectic frame. By Lemma 4.3.5, DE (C ⊗ ϕ) = C ⊗ D1 ϕ + 

n

(η# (fj )C ⊗ ej · ϕ − η# (ej )C ⊗ fj · ϕ)

j=1

= C ⊗  D1 ϕ +

n

 (ω(X# , fj )ej + ω(ej , X# )fj ) · ϕ

j=1

= C ⊗ (D1 ϕ + X# · ϕ) and E (C ⊗ ϕ) = C ⊗ D 1 ϕ + D 

n j=1

(η# (fj )C ⊗ Jej · ϕ − η# (ej )C ⊗ Jfj · ϕ) 

1 ϕ + J  = C ⊗ D

n





(ω(X# , fj )ej + ω(ej , X# )fj ) · ϕ

j=1

1 ϕ + JX# · ϕ) . = C ⊗ (D Together with Proposition 4.3.6, this gives the assertion.

Remark 4.3.8 Pay attention to the fact that Theorem 4.3.7 is a local statement depending on the choice of the local section C in E. Of course, in the case that the complex line bundle E is trivial, one can choose a global nonvanishing section in E to make the relations global.

62

4 Symplectic Dirac Operators

4.4 Dependence on the Almost Complex Structure The task of this section is to show that a change of the ω-compatible al most complex structure in the deﬁnition of the symplectic Dirac operator D corresponds to an isomorphism of the symplectic spinor bundle and a gauge transformation of the symplectic connection. Let J and K be two ω-compatible almost complex structures and set Θ = −J ◦ K . Since ω(ΘX, ΘY ) = ω(X, Y ) for all vector ﬁelds X and Y , ΦΘ (e1 , . . . , en , f 1 , . . . , f n ) = (Θe1 , . . . , Θen , Θf 1 , . . . , Θf n ) deﬁnes an isomorphism ΦΘ of the symplectic frame bundle R. Let ΨΘ : P → P be a lift of ΦΘ to the metaplectic structure P. Considering the tangent bundle T M as the associated vector bundle P ×ρ R2n , one has Θ[p, v] = [ΨΘ (p), v] . Let Ξ : Q → Q be the isomorphism of the symplectic spinor bundle Q given by Ξ([p, f ]) = [ΨΘ (p), f ] . Lemma 4.4.1 For any X ∈ Γ(T M ) and ϕ ∈ Γ(Q), Ξ(X · ϕ) = ΘX · Ξ(ϕ) . Proof. Let s : U → P be a local section of the metaplectic structure P. Writing X = [s, Xs ] and ϕ = [s, ϕs ], we have Ξ(X · ϕ) = Ξ([s, Xs ] · [s, ϕs ]) = Ξ([s, Xs · ϕs ]) = [ΨΘ ◦ s, Xs · ϕs ] and ΘX · Ξ(ϕ) = [ΨΘ ◦ s, Xs ] · [ΨΘ ◦ s, ϕs ] = [ΨΘ ◦ s, Xs · ϕs ] , which already proves the assertion.

Let again Z : T R → sp(n, R) be the connection form of the chosen symplectic connection ∇ on M and Z : T P → mp(n, R) the lift of Z to the metaplectic structure P. Then ∗ Z Z Θ = Φ−1 Θ is the connection form of a symplectic connection ∇Θ on M and the spinor derivative induced by ∇Θ , which is also denoted by ∇Θ , is associated to the connection form ∗ Θ Z = ΨΘ−1 Z on P.

4.4 Dependence on the Almost Complex Structure

63

Lemma 4.4.2 For any X ∈ Γ(T M ) and ϕ ∈ Γ(Q), Ξ(∇X ϕ) = ∇Θ X (Ξ(ϕ)) . Proof. Writing again ϕ = [s, ϕs ] with respect to a local section s of P, we have Ξ(∇X ϕ) = Ξ s, X(ϕs ) + m∗ s∗ Z(X) ϕs " # Θ = ΨΘ ◦ s, X(ϕs ) + m∗ (ΨΘ ◦ s)∗ Z (X) ϕs = ∇Θ X (Ξ(ϕ)) . Θ denote the symplectic Dirac operator which is deﬁned with respect to Let D J Θ K be the corresponding operator with respect to ∇ and ∇ and J and let D K. Θ and D K are related by Theorem 4.4.3 The symplectic Dirac operators D J K ϕ) = D Θ (Ξ(ϕ)) Ξ(D J for all ϕ ∈ Γ(Q). Proof. Let (e1 , . . . , en , f1 , . . . , fn ) be any symplectic frame. By means of Lemmas 4.1.2, 4.4.1, and 4.4.2 and the relation Θ ◦ K = −J ◦ K 2 = J , we compute K ϕ) = Ξ(D

n Ξ Kej · ∇fj ϕ − Ξ Kfj · ∇ej ϕ j=1

n Θ ◦ K(ej ) · Ξ ∇fj ϕ − Θ ◦ K(fj ) · Ξ ∇ej ϕ =

=

j=1 n

Θ Jej · ∇Θ (Ξ(ϕ)) − Jf · ∇ (Ξ(ϕ)) j fj ej

j=1

JΘ (Ξ(ϕ)) . =D

64

4 Symplectic Dirac Operators

4.5 Formal Self-Adjointness In the last section of this chapter, we want to show that, under certain con are formally self-adjoint with respect to ditions, the Dirac operators D and D 2 the L -product on Γ(Q) deﬁned by ϕ, ψ dM (ϕ, ψ) = M

for ϕ ∈ Γ(Q) and ψ ∈ Γ0 (Q). Here, Γ0 (Q) denotes the space of compactly supported smooth sections of Q. Lemma 4.5.1 For any ϕ, ψ ∈ Γ(Q), Dϕ, ψ = ϕ, Dψ + div(Y1 ) , where the vector ﬁeld Y1 on M is given by Y1 =

2n

ϕ, Jej · ψ ej

j=1

for any unitary frame (e1 , . . . , e2n ). Proof. By means of Propositions 3.2.5, 3.2.7, and 3.2.8 and Lemma 4.1.3, we see Dϕ, ψ = −

2n

Jej · ∇ej ϕ, ψ

j=1

=

2n

∇ej ϕ, Jej · ψ

j=1

=

2n

ej ϕ, Jej · ψ − ϕ, ∇ej (Jej · ψ)

j=1

= ϕ, Dψ +

2n ej ϕ, Jej · ψ − ϕ, ∇ej (Jej ) · ψ . j=1

On the other hand, we have div(Y1 ) =

2n ω ∇ej Y1 , Jej j=1

=

2n ej ϕ, Jek · ψ ω(ek , Jej ) + ϕ, Jek · ψ ω ∇ej ek , Jej j,k=1

4.5 Formal Self-Adjointness

=

65

2n ej ϕ, Jek · ψ δjk − ϕ, Jek · ψ ω ek , ∇ej (Jej ) j,k=1

=

2n

ej ϕ, Jej · ψ − ϕ, ∇ej (Jej ) · ψ ,

j=1

and the lemma is proved. Lemma 4.5.2 Suppose that ∇J = 0. Then, for any ϕ, ψ ∈ Γ(Q), ψ = ϕ, Dψ − div(Y2 ) , Dϕ, where Y2 ∈ Γ(T M ) is given by Y2 =

2n

ϕ, ej · ψ ej

j=1

for any unitary frame (e1 , . . . , e2n ). Proof. We proceed as before to derive that ψ = Dϕ,

2n

ej · ∇ej ϕ, ψ

j=1

=−

2n ∇ej ϕ, ej · ψ j=1

=

2n

ϕ, ∇ej (ej · ψ) − ej ϕ, ej · ψ

j=1

+ = ϕ, Dψ

2n

ϕ, ∇ej ej · ψ − ej ϕ, ej · ψ .

j=1

Furthermore, div(Y2 ) =

2n ej ϕ, ek · ψ ω(ek , Jej ) + ϕ, ek · ψ ω ∇ej ek , Jej j,k=1

=

2n ej ϕ, ek · ψ δjk − ϕ, ek · ψ ω ek , ∇ej (Jej ) . j,k=1

Since, by assumption, ∇J = 0, we obtain

66

4 Symplectic Dirac Operators

div(Y2 ) =

2n ej ϕ, ek · ψ δjk − ϕ, ek · ψ ω ek , J ∇ej ej j,k=1

=

2n ej ϕ, ej · ψ − ϕ, ∇ej ej · ψ . j=1

This ﬁnishes the proof.

Theorem 4.5.3 Let the symplectic connection ∇ on M be such that its torsion vector ﬁeld T vanishes. Then the Dirac operator D is formally selfadjoint, i.e., (Dϕ, ψ) = (ϕ, Dψ) for all ϕ ∈ Γ(Q) and ψ ∈ Γ0 (Q). If, in addition, ∇J = 0, then also the Dirac is formally self-adjoint. operator D Proof. If T = 0, then, by Corollary 2.2.7, div(X) dM = 0 M

for all compactly supported vector ﬁelds X on M . Hence the theorem follows from Lemmas 4.5.1 and 4.5.2.

5 An Associated Second Order Operator

In this chapter, we introduce an elliptic operator of second order which is of Laplace type and will play a central role in the further investigations. In a sense, this operator can be considered as the symplectic counterpart of the square of the Dirac operator in the Riemannian case.

5.1 Deﬁnition and Ellipticity Equation (3.2.2) suggests to study the following diﬀerential operator. Deﬁnition 5.1.1 Let P : Γ(Q) → Γ(Q) be the second order operator deﬁned by D] = i(DD − DD) . P = i[D, Let us compute the principal symbol of P. We start with Lemma 5.1.2 For any u ∈ C ∞ (M ) and ϕ ∈ Γ(Q), D(uϕ) = uDϕ + Xu · ϕ

and

+ JXu · ϕ . D(uϕ) = uDϕ

Proof. This immediately follows from the deﬁnitions of D and D.

For a covector ξ ∈ Tp∗ M , let ξ & ∈ Tp M denote the dual tangent vector, i.e. ξ = ωp (ξ & , ) = gp (Jξ & , ) . of the symplectic Dirac Lemma 5.1.3 The principal symbols σ(D) and σ(D) operators D and D are given by σ(D)(p, ξ)s = ξ & · s where ξ ∈ Tp∗ M and s ∈ Sp .

and

σ(D)(p, ξ)s = Jξ & · s ,

68

5 An Associated Second Order Operator

Proof. Let u ∈ C ∞ (M ) and ϕ ∈ Γ(Q) be such that (du)p = ξ and ϕ(p) = s. From ξ & = Xu (p) and Lemma 5.1.2, we get σ(D)(p, ξ)s = D((u − u(p))ϕ)(p) = D(uϕ)(p) − u(p)Dϕ(p) = u(p)Dϕ(p) + Xu (p) · ϕ(p) − u(p)Dϕ(p) = ξ& · s as well as σ(D)(p, ξ)s = D((u − u(p))ϕ)(p) = D(uϕ)(p) − u(p)Dϕ(p) = u(p)Dϕ(p) + JXu (p) · ϕ(p) − u(p)Dϕ(p) & = Jξ · s . Corollary 5.1.4 The principal symbol σ(P) of the operator P is given by σ(P)(p, ξ)s = −gp (ξ & , ξ & )s for ξ ∈ Tp∗ M and s ∈ Sp . In particular, P is an elliptic operator of Laplace type. Proof. By Lemma 5.1.3 and Equation (3.2.2), σ(P)(p, ξ)s = i(σ(D)(p, ξ) ◦ σ(D)(p, ξ) − σ(D)(p, ξ) ◦ σ(D)(p, ξ))s & & & & = i(Jξ · ξ · s − ξ · Jξ · s) = ωp (Jξ & , ξ & )s = −gp (ξ & , ξ & )s .

5.2 A Weitzenb¨ ock Formula Due to Corollary 5.1.4, the highest order part of the operator P is a Laplace operator. The aim of this section is to prove the corresponding Weitzenb¨ ock formula. In Section 4.5, we have deﬁned an L2 -product on Γ(Q). Similarly, we now equip Γ(T ∗ M ⊗ Q) with an L2 -product. Let Γ0 (T ∗ M ⊗ Q) denote the space of compactly supported smooth sections of T ∗ M ⊗ Q. For α ∈ Γ(T ∗ M ⊗ Q) and β ∈ Γ0 (T ∗ M ⊗ Q), we set

5.2 A Weitzenb¨ ock Formula

69

α, β dM ,

(α, β) = M

where the function α, β on M is given as α, β =

2n

α(ej ), β(ej ) .

j=1

Here and subsequently, (e1 , . . . , e2n ) is any unitary frame. Let

∇∗ : Γ(T ∗ M ⊗ Q) → Γ(Q)

be the formal L2 -adjoint operator of the spinor derivative. Using the torsion vector ﬁeld T and the divergence operator div with respect to the chosen symplectic connection on M , the operator ∇∗ can be written as follows. Lemma 5.2.1 For all α ∈ Γ(T ∗ M ⊗ Q), ∇∗ α = −

2n

∇ej (α(ej )) + div(ej )α(ej ) + α(JT) .

j=1

Proof. By deﬁnition, (∇∗ α, ψ) =

α, ∇ψ dM M

for any ψ ∈ Γ0 (Q). By means of Proposition 3.2.8 and div(uX) = X(u) + u div(X) for u ∈ C ∞ (M ) and X ∈ Γ(T M ), we derive α, ∇ψ =

2n

α(ej ), ∇ej ψ

j=1

=

2n ej α(ej ), ψ − ∇ej (α(ej )), ψ j=1

=

2n div(α(ej ), ψ ej ) − α(ej ), ψ div(ej ) − ∇ej (α(ej )), ψ j=1

= div(Y ) −

2n ∇ej (α(ej )) + div(ej )α(ej ), ψ j=1

with

70

5 An Associated Second Order Operator

Y =

2n

α(ej ), ψ ej .

j=1

Since Y is a globally deﬁned vector ﬁeld with compact support, we can apply Corollary 2.2.7 to obtain ∇∗ α = −

2n ∇ej (α(ej )) + div(ej )α(ej ) + ω(ej , T)α(ej ) , j=1

which proves the lemma. Deﬁnition 5.2.2 We call the Bochner–Laplace operator ∆Q = ∇∗ ∇ : Γ(Q) → Γ(Q) the symplectic spinor Laplacian. Proposition 5.2.3 For all ϕ ∈ Γ(Q), ∆Q ϕ = −

2n ∇ej ∇ej ϕ + div(ej )∇ej ϕ + ∇JT ϕ . j=1

Proof. This is Lemma 5.2.1 applied to α = ∇ϕ.

To prove the announced Weitzenb¨ ock formula, we proceed similarly as in the Riemannian case. For X ∈ Γ(T M ), we set P(X) =

2n

ej · ∇Jej X ,

j=1

P(X) =

2n

ej · ∇ej X ,

j=1

P(J)(X) =

2n ∇Jej J (X) · ej . j=1

Then P(X), P(X), and P(J)(X) are globally deﬁned expressions which act via symplectic Cliﬀord multiplication on symplectic spinor ﬁelds. Lemma 5.2.4 For any X ∈ Γ(T M ) and ϕ ∈ Γ(Q), D(X · ϕ) = X · Dϕ + P(X) · ϕ − i∇X ϕ , + P(X) D(X · ϕ) = X · Dϕ · ϕ + i∇JX ϕ .

5.2 A Weitzenb¨ ock Formula

71

Proof. By means of Equation (3.2.2), Proposition 3.2.7, and Lemma 4.1.3, one computes D(X · ϕ) =

2n

ej · ∇Jej (X · ϕ)

j=1

= P(X) · ϕ +

2n

ej · X · ∇Jej ϕ

j=1

= P(X) · ϕ + X · Dϕ − i

2n

ω(ej , X)∇Jej ϕ

j=1

= P(X) · ϕ + X · Dϕ − i∇X ϕ and D(X · ϕ) =

2n

ej · ∇ej (X · ϕ)

j=1

= P(X) ·ϕ+

2n

ej · X · ∇ej ϕ

j=1

+i = P(X) · ϕ + X · Dϕ

2n

ω(X, ej )∇ej ϕ

j=1

+ i∇JX ϕ . = P(X) · ϕ + X · Dϕ Using the torsion T of the ﬁxed symplectic connection ∇, we have Lemma 5.2.5 For any X ∈ Γ(T M ), P(X) + P(JX) = −P(J)(JX) − i div(X) +

2n

(ej (ω(ek , JX)) − ek (ω(ej , JX)))ej · Jek

j,k=1

−

2n

ω(T(ej , ek ) + [ej , ek ], JX)ej · Jek .

j,k=1

Proof. By Lemma 2.2.5 and Equation (3.2.2), we get P(X) = −

2n j=1

Jej · ∇ej X

72

5 An Associated Second Order Operator

=−

2n

∇ej X · Jej − i

j=1

=−

2n

2n ω ∇ej X, Jej j=1

∇ej X · Jej − i div(X) .

j=1

Further, by the symplecticity of ∇, ω ek , ∇ej (JX) = ej (ω(ek , JX)) − ω ∇ej ek , JX = ej (ω(ek , JX)) − ω(T(ej , ek ) + ∇ek ej + [ej , ek ], JX) = ej (ω(ek , JX)) − ω(T(ej , ek ) + [ej , ek ], JX) −ek (ω(ej , JX)) + ω(ej , ∇ek (JX)) , which implies P(JX) =

2n

ej · ∇ej (JX)

j=1

=

2n

ω(ek , ∇ej (JX))ej · Jek

j,k=1

=

2n

(ej (ω(ek , JX)) − ek (ω(ej , JX)))ej · Jek

j,k=1

−

2n

ω(T(ej , ek ) + [ej , ek ], JX)ej · Jek

j,k=1

−

2n

J(∇ek (JX)) · Jek .

k=1

Since 2n

2n ∇ej X + J ∇ej (JX) · Jej = − ∇ej J (JX) · Jej = P(J)(JX) ,

j=1

j=1

the assertion follows. After these preparations, we can prove Theorem 5.2.6 The operator P satisﬁes the Weitzenb¨ ock formula Pϕ = ∆Q ϕ + i

2n

Jej · ek · RQ (ej , ek )ϕ + i

j=1

j,k=1

−∇JT ϕ − i

2n j,k=1

2n

Jej · ek · ∇T(ej ,ek ) ϕ .

P(J)(Jej ) · ∇ej ϕ

5.2 A Weitzenb¨ ock Formula

Proof. By Lemmas 4.1.3 and 5.2.4, − DD)ϕ (DD =

2n

2n Jej · ∇e ϕ D D ej · ∇ej ϕ + j

j=1

j=1

2n = ej · D ∇ej ϕ + P(ej ) · ∇ej ϕ − i∇ej ∇ej ϕ j=1

+

2n ∇e ϕ + P(Je Jej · D j ) · ∇ej ϕ − i∇ej ∇ej ϕ j j=1

2n = Jej · ek · ∇ek ∇ej ϕ − ej · Jek · ∇ek ∇ej ϕ j,k=1

+

2n

(P(ej ) + P(Je j )) · ∇ej ϕ − 2i

j=1

2n

∇ej ∇ej ϕ .

j=1

Since 2n

Jej · ek · ∇ek ∇ej ϕ − ej · Jek · ∇ek ∇ej ϕ

j,k=1

=

2n

2n Jej · ek · ∇ek ∇ej ϕ − ∇ej ∇ek ϕ + i ω(ek , Jej )∇ej ∇ek ϕ

j,k=1

=−

j,k=1

2n

Jej · ek · RQ (ej , ek )ϕ −

j,k=1

2n

Jej · ek · ∇[ej ,ek ] ϕ + i

2n

∇ej ∇ej ϕ

j=1

j,k=1

and, by Lemma 5.2.5, 2n (P(ej ) + P(Je j )) · ∇ej ϕ j=1

=−

2n

P(J)(Jej ) · ∇ej ϕ − i

j=1

−

2n

div(ej )∇ej ϕ

j=1

2n

ω(T(ej , ek ) + [ej , ek ], Jel )ej · Jek · ∇el ϕ

j,k,l=1

=−

2n

P(J)(Jej ) · ∇ej ϕ − i

j=1

+

2n j,k=1

2n

div(ej )∇ej ϕ

j=1

Jej · ek · ∇T(ej ,ek ) ϕ +

2n j,k=1

Jej · ek · ∇[ej ,ek ] ϕ ,

73

74

5 An Associated Second Order Operator

we arrive at − DD)ϕ (DD =i

2n 2n ∇ej ∇ej ϕ + div(ej )∇ej ϕ + Jej · ek · RQ (ej , ek )ϕ j=1

+

2n

j,k=1 2n

P(J)(Jej ) · ∇ej ϕ −

j=1

Jej · ek · ∇T(ej ,ek ) ϕ .

j,k=1

Together with Proposition 5.2.3, this yields the claimed formula.

By Proposition 2.2.3, for any ω-compatible almost complex structure J, one can ﬁnd a Hermitian connection. For such a connection, P(J)(X) vanishes. Thus we have Corollary 5.2.7 If the symplectic connection ∇ on M satisﬁes ∇J = 0, then Pϕ = ∆Q ϕ + i

2n

Jej · ek · RQ (ej , ek )ϕ

j,k=1 2n

−∇JT ϕ − i

Jej · ek · ∇T(ej ,ek ) ϕ

j,k=1

for all ϕ ∈ Γ(T M ).

In Section 6.1, we will study the Weitzenb¨ ock formula for the case of K¨ ahler manifolds.

5.3 Splitting of the Operator We will now show that the operator P leaves the subbundles QJl of the symplectic spinor bundle Q given in Section 3.3 invariant provided the chosen connection ∇ on M is Hermitian. Thus P splits into a family of Laplace type operators Pl : Γ(QJl ) → Γ(QJl ) obtained by restricting P to Γ(QJl ). Throughout this section, it is assumed that ∇J = 0. We ﬁrst derive the and the following relations between the symplectic Dirac operators D and D endomorphism HJ . Proposition 5.3.1 For any ϕ ∈ Γ(Q), HJ (Dϕ) = D(HJ ϕ) + iDϕ

and

= D(H J ϕ) − iDϕ . HJ (Dϕ)

5.3 Splitting of the Operator

75

Proof. By means of Lemmas 3.3.2 and 4.1.3 and Proposition 3.3.4, we calculate HJ (Dϕ) = −

2n 1 ej · ej · Jek · ∇ek ϕ 2 j,k=1

=−

2n 1 (ej · Jek · ej · ∇ek ϕ − iω(ej , Jek )ej · ∇ek ϕ) 2 j,k=1

=−

2n 1 (Jek · ej · ej · ∇ek ϕ − 2iω(ej , Jek )ej · ∇ek ϕ) 2 j,k=1

=−

2n

Jek · HJ (∇ek ϕ) + i

2n

ej · ∇ej ϕ

j=1

k=1

=−

2n

Jek · ∇ek (HJ ϕ) + i

2n

ej · ∇ej ϕ

j=1

k=1

= D(HJ ϕ) + iDϕ and, analogously, 2n = 1 HJ (Dϕ) ej · ej · ek · ∇ek ϕ 2 j,k=1

=

2n 1 (ek · ej · ej · ∇ek ϕ − 2iω(ej , ek )ej · ∇ek ϕ) 2 j,k=1

=

2n

ek · HJ (∇ek ϕ) − i

k=1

=

2n

2n

ω(Jej , ek )Jej · ∇ek ϕ

j,k=1

ek · ∇ek (HJ ϕ) + i

2n

Jej · ∇ej ϕ

j=1

k=1

J ϕ) − iDϕ . = D(H Proposition 5.3.2 For all ϕ ∈ Γ(Q), HJ (Pϕ) = P(HJ ϕ) . In particular, for any l ∈ N0 , it holds that, if ϕ is a section of QJl , then so is Pϕ. Proof. By Proposition 5.3.1,

76

5 An Associated Second Order Operator

HJ (Pϕ) = iHJ (DDϕ) − iHJ (DDϕ) 2ϕ J (Dϕ)) + D2 ϕ − iD(HJ (Dϕ)) +D = iD(H J + D2 ϕ − iD(D(H J ϕ) − iDϕ) + D 2ϕ ϕ) + iDϕ) = iD(D(H J J ϕ) ϕ) − iDD(H = iDD(H

= P(HJ ϕ) . The second part of the assertion follows by Equation (3.3.1).

Remark 5.3.3 Proposition 5.3.1 and Equation (3.3.1) imply that the vector bundles QJl are not invariant under the Dirac operators D and D. In the remainder of this section, we want to describe the operator P0 in more detail. For this, we prove the following lemmas. The ﬁrst one shows that the curvature part of the Weitzenb¨ ock formula for P is related to the symplectic scalar curvature R of ∇ with respect to J. Lemma 5.3.4 If ϕ ∈ Γ(QJ0 ), then 2n

1 Q i R (Jej , ej )ϕ = Rϕ . 2 j=1 4 2n

Jej · ek · RQ (ej , ek )ϕ =

j,k=1

Proof. Let ϕ ∈ Γ(QJ0 ). Then, by Corollary 3.3.5, RQ (ej , ek )ϕ is also a section of QJ0 . Applying Corollary 3.3.7, we get 2n

2n

Jej · ek · RQ (ej , ek )ϕ =

j,k=1

ek · Jej · RQ (ej , ek )ϕ

j,k=1

=i

2n

ek · ej · RQ (ej , ek )ϕ

j,k=1

=

2n i (ej · ek − ek · ej ) · RQ (ek , ej )ϕ 2 j,k=1

=

2n 1 ω(ej , ek )RQ (ek , ej )ϕ 2 j,k=1

1 Q R (Jej , ej )ϕ . 2 j=1 2n

=

To see the second equation, ﬁrst observe that, by Proposition 3.2.9 and Corollary 3.3.7,

5.3 Splitting of the Operator

77

i ej · R(X, Y )(Jej ) · ϕ 2 j=1 2n

RQ (X, Y )ϕ =

=−

2n i S(X, Y, Jej , ek )ej · Jek · ϕ 2 j,k=1

=

1 2

2n

S(X, Y, Jej , ek )ej · ek · ϕ .

j,k=1

We use Proposition 2.3.4 and continue with RQ (X, Y )ϕ =

2n 1 S(X, Y, Jej , ek )(ej · ek − ek · ej ) · ϕ 4 j,k=1

=−

2n i ω(ej , ek )S(X, Y, Jej , ek )ϕ 4 j,k=1

= Hence

2n

i 4

2n

S(Y, X, ej , ej )ϕ .

j=1

RQ (Jej , ej )ϕ =

j=1

2n i S(ej , Jej , ek , ek )ϕ , 4 j,k=1

which, by Equation (2.3.1), gives the desired equation.

Lemma 5.3.5 If ϕ ∈ Γ(QJ0 ), then 2n

Jej · ek · ∇T(ej ,ek ) ϕ = −∇T ϕ .

j,k=1

Proof. Proceeding as in the proof of Lemma 5.3.4, one sees 2n j,k=1

1 ∇T(Jej ,ej ) ϕ = −∇T ϕ 2 j=1 2n

Jej · ek · ∇T(ej ,ek ) ϕ =

for ϕ ∈ Γ(QJ0 ). Proposition 5.3.6 For any ϕ ∈ Γ(QJ0 ), 1 Pϕ = ∆Q ϕ − Rϕ − ∇JT ϕ + i∇T ϕ . 4

Proof. This is a consequence of Corollary 5.2.7 and Lemmas 5.3.4 and 5.3.5.

78

5 An Associated Second Order Operator

Proposition 5.3.7 Let ϕ ∈ Γ(QJ0 ) be such that ∇JX ϕ = i∇X ϕ for all X ∈ Γ(T M ). Then

(5.3.1)

Pϕ = 0 .

Proof. According to Lemma 5.3.4 and Proposition 5.3.6, we have to show that i Q R (Jej , ej )ϕ . 2 j=1 2n

∆Q ϕ = −

(5.3.2)

By Proposition 5.2.3 and 2n

RQ (Jej , ej )ϕ =

j=1

2n ∇Jej ∇ej ϕ − ∇ej ∇Jej ϕ − ∇[Jej ,ej ] ϕ j=1

=−

2n

i∇Jej ∇Jej ϕ + i∇ej ∇ej ϕ + ∇[Jej ,ej ] ϕ

j=1

= −2i

2n

∇ej ∇ej ϕ −

j=1

2n

∇[Jej ,ej ] ϕ ,

j=1

Equation (5.3.2) is equivalent to 2n

i ∇[Jej ,ej ] ϕ . 2 j=1 2n

div(ej )∇ej ϕ = i∇T ϕ −

j=1

Now observe that 1 1 [Jej , ej ] = (T(ej , Jej ) − [Jej , ej ]) 2 j=1 2 j=1 2n

T−

2n

1 ∇ej (Jej ) − ∇Jej ej 2 j=1 2n

=

=

2n

∇ej (Jej )

j=1

and 2n

div(ej )ej =

j=1

2n

ω(∇ek ej , Jek )ej

j,k=1

=

2n

ω(∇ek (Jek ), ej )ej

j,k=1

=

2n k=1

J(∇ek (Jek )) .

(5.3.3)

5.3 Splitting of the Operator

79

This, together with Equation (5.3.1), yields Equation (5.3.3), and we are done. Remark 5.3.8 Equation (5.3.1) is a generalized Cauchy–Riemann equation and hence gives a notion of a “holomorphic” symplectic spinor ﬁeld. It is, however, questionable whether, for n > 1 and generic almost complex structure J, non-trivial local holomorphic spinor ﬁelds exist.

6 The K¨ ahler Case

In this chapter, we direct our considerations to K¨ ahler manifolds seen as symplectic manifolds, where the symplectic structure is the K¨ ahler form. In particular, we discuss the operator P on K¨ ahler manifolds of constant holomorphic sectional curvature and derive lower bound estimates for its eigenvalues in this situation. The chapter concludes with the computation of the spectrum of P on CP 1 . Focusing to the kernel of P, this leads to an interesting number theoretical equation.

6.1 The Operator P on K¨ ahler Manifolds As before, let (M, ω) be a symplectic manifold of dimension 2n and let J be an ω-compatible almost complex structure. Throughout this chapter, we let ∇ be the Levi–Civita connection of the Riemannian metric g associated to J and assume that ∇J = 0, which means that (M, ω, J) is K¨ ahler. Since ∇ is then also symplectic, after ﬁxating a metaplectic structure P on (M, ω), and therewith the we can construct the symplectic Dirac operators D and D operator P. Proposition 6.1.1 If (M, ω, J) is a K¨ ahler manifold and ∇ is its Levi–Civita connection, then P is formally self-adjoint. are Proof. We know from Theorem 4.5.3 that, on a K¨ ahler manifold, D and D formally self-adjoint. Hence also P has this property. As given in the next proposition, the diﬀerence between P and the spinor Laplacian ∆Q expresses in curvature terms. Proposition 6.1.2 On a K¨ ahler manifold,

82

6 The K¨ ahler Case

Pϕ = ∆Q ϕ + i

2n

Jej · ek · RQ (ej , ek )ϕ

j,k=1

holds true for all ϕ ∈ Γ(Q). Proof. This follows directly from Corollary 5.2.7.

The following equation looks very similar to the famous Weitzenb¨ ock formula for the square of the Riemannian Dirac operator (cf. [38, 43]). As mentioned in Section 2.3, in the present situation, the symplectic scalar curvature R coincides with the scalar curvature of the Riemannian metric g. Proposition 6.1.3 On a K¨ ahler manifold, 1 Pϕ = ∆Q ϕ − Rϕ 4 for any ϕ ∈ Γ(QJ0 ). Proof. This is an immediate consequence of Proposition 5.3.6.

The last proposition can also be obtained by using the following two lemmas. These lemmas are of their own interest, since they link the Ricci tensor with the symplectic spinor bundle. In Riemannian spin geometry, this is a well known fact (cf. [19]). However, in our context, it only occurs in a very special situation. Let Ric denote the Ricci tensor of g understood as a (1, 1)-tensor ﬁeld on M . Lemma 6.1.4 For any symplectic spinor ﬁeld ϕ on a K¨ ahler manifold, 2n

R (ej , Jej )ϕ = i Q

j=1

2n

Ric(ej ) · ej · ϕ .

j=1

Proof. By Lemma 2.3.6, we have ω(Ric(X), JY ) = ric(X, Y ) =

2n

S(ej , X, Y, Jej ) .

j=1

Using Propositions 2.3.3 and 2.3.4, it follows that 1 (S(ej , X, Y, Jej ) − S(Jej , X, Y, ej )) 2 j=1 2n

ω(Ric(X), JY ) =

1 (S(X, ej , Jej , Y ) + S(Jej , X, ej , Y )) 2 j=1 2n

=−

6.1 The Operator P on K¨ ahler Manifolds

=

1 S(ej , Jej , X, Y ) 2 j=1

=

1 S(ej , Jej , JX, JY ) , 2 j=1

83

2n

2n

which gives the known formula (see [36]) 1 R(ej , Jej )(JX) . 2 j=1 2n

Ric(X) =

Together with Proposition 3.2.9, this yields 2n

RQ (ej , Jej )ϕ =

j=1

2n i ek · R(ej , Jej )(Jek ) · ϕ 2 j,k=1

=

2n i R(ej , Jej )(Jek ) · ek · ϕ 2 j,k=1

=i

2n

Ric(ek ) · ek · ϕ .

k=1

Lemma 6.1.5 For any symplectic spinor ﬁeld ϕ on a K¨ ahler manifold, 2n

i Ric(ej ) · Jej · ϕ = − Rϕ . 2 j=1

Proof. By Propositions 2.3.4 and 2.3.9, ω(Ric(X), Y ) = ω(X, Ric(Y )) for all X, Y ∈ Γ(T M ). This yields 2n

Jej · Ric(ej ) · ϕ = −

j=1

2n

ω(Ric(ej ), ek )Jej · Jek · ϕ

j,k=1

=−

2n

ω(ej , Ric(ek ))Jej · Jek · ϕ

j,k=1

=−

2n k=1

Consequently,

Ric(ek ) · Jek · ϕ .

84

6 The K¨ ahler Case

iRϕ = i

2n

ω(Ric(ej ), Jej )ϕ

j=1

=

2n

Jej · Ric(ej ) · ϕ −

j=1

= −2

2n

Ric(ej ) · Jej · ϕ

j=1 2n

Ric(ej ) · Jej · ϕ ,

j=1

which concludes the proof.

We now want to specialize Proposition 6.1.2 to the case of K¨ ahler manifolds of constant holomorphic sectional curvature. Below we describe those manifolds by means of the symplectic curvature tensor S. Deﬁnition 6.1.6 Let h ∈ C ∞ (M ). A K¨ ahler manifold (M, ω, J) is said to be of constant holomorphic sectional curvature h if S(X, JX, X, X) = h (ω(X, JX))2 for all X ∈ Γ(T M ). For a proof of the next proposition, see e.g. [36]. Proposition 6.1.7 A K¨ ahler manifold (M, ω, J) is of constant holomorphic sectional curvature h if and only if S(X1 , X2 , X3 , X4 ) =

h (ω(X1 , X3 )ω(X2 , JX4 ) + ω(X1 , X4 )ω(X2 , JX3 ) 4 −ω(X2 , X3 )ω(X1 , JX4 ) − ω(X2 , X4 )ω(X1 , JX3 ) +2ω(X1 , X2 )ω(X3 , JX4 ))

for all X1 , X2 , X3 , X4 ∈ Γ(T M ).

In other words, (M, ω, J) is of constant holomorphic sectional curvature h if and only if its Levi–Civita connection ∇ is reducible in the sense of Deﬁnition 2.3.12 with h b= g. 4 If this is the case, in view of Proposition 2.3.13, (M, ω, J) is an Einstein manifold with scalar curvature R = n(n + 1)h and hence, for n ≥ 2, as for the Riemannian sectional curvature, the function h is forced to be constant.

6.1 The Operator P on K¨ ahler Manifolds

85

Proposition 6.1.8 On a K¨ ahler manifold of constant holomorphic sectional curvature h, it holds Pϕ = ∆Q ϕ +

h n(n − 1)ϕ − 2h(HJ )2 ϕ 4

(6.1.1)

for all ϕ ∈ Γ(Q). Proof. According to Propositions 3.2.9 and 6.1.7, we have i

2n

Jej · ek · RQ (ej , ek )ϕ

j,k=1

=

2n

1 2

S(ej , ek , el , Jem )Jej · ek · Jel · em · ϕ

j,k,l,m=1

=−

2n h (Jej · Jek · ej · ek + Jej · Jek · ek · ej 8 j,k=1

+ej · ek · ek · ej + ej · ek · ej · ek + 2ej · ej · ek · ek ) · ϕ =−

2n h (Jej · ej · Jek · ek + Jej · Jek · ek · ej + 4ej · ej · ek · ek 8 j,k=1

−iω(Jek , ej )Jej · ek − 3iω(ek , ej )ej · ek ) · ϕ =−

2n h (Jej · ej · Jek · ek + Jej · Jek · ek · ej + 4ej · ej · ek · ek ) · ϕ 8 j,k=1

h +i Jej · ej · ϕ . 4 j=1 2n

Using 2n

1 i (Jej · ej − ej · Jej ) · ϕ = ω(ej , Jej )ϕ = inϕ , 2 j=1 2 j=1 2n

Jej · ej · ϕ =

j=1

2n

we arrive at i

2n

Jej · ek · RQ (ej , ek )ϕ =

j,k=1

=

h h 2 2n ϕ − 16(HJ )2 ϕ − nϕ 8 4 h n(n − 1)ϕ − 2h(HJ )2 ϕ . 4

This together with Proposition 6.1.2 implies the desired equation.

Example 6.1.9 The ﬂat complex torus TCn = Cn /Z2n has vanishing holomorphic sectional curvature. Thus, by Proposition 6.1.8,

86

6 The K¨ ahler Case

P = ∆Q for any metaplectic structure on TCn , which gives that the kernel of P consists of all parallel symplectic spinor ﬁelds. Hence, with respect to the canonical metaplectic structure, this kernel is nothing else but the space S(Rn ).

6.2 Lower Bound Estimates We apply Proposition 6.1.8 to obtain estimates for the eigenvalues of P. Keep in mind that, by Proposition 6.1.1, all eigenvalues of P are real. Proposition 6.2.1 Let (M, ω, J) be a 2n-dimensional closed K¨ ahler manifold of constant holomorphic sectional curvature h and suppose that the function h is constant and non-positive. Then any eigenvalue λ of P satisﬁes λ≥

h n(n − 1) . 4

Proof. Let λ ∈ R be an eigenvalue of P and let ϕ be a non-vanishing symplectic spinor ﬁeld such that Pϕ = λϕ . Taking the L2 -product of Equation (6.1.1) with ϕ and using Lemma 3.3.3, one gets λϕ2 = (∆Q ϕ, ϕ) + = ∇ϕ2 + Thus

λ−

h n(n − 1)ϕ2 − 2h (HJ )2 ϕ, ϕ 4

h n(n − 1)ϕ2 − 2hHJ ϕ2 . 4

h n(n − 1) ϕ2 = ∇ϕ2 − 2hHJ ϕ2 . 4

Since, by assumption, h ≤ 0, this yields the asserted inequality.

Taking advantage of the splitting of Q described in Section 3.3 and the corresponding splitting of P derived in Section 5.3, the above estimate can be reﬁned in the following way. Proposition 6.2.2 Let (M, ω, J) be a 2n-dimensional closed K¨ ahler manifold of constant holomorphic sectional curvature h and suppose that h is constant. Then, for all l ∈ N0 and any eigenvalue λ of Pl , one has h λ ≥ − (n(n + 1) + 8l(l + n)) . 4

6.3 The Spectrum of P on CP 1

87

Proof. Let ϕ ∈ Γ(QJl ) be an eigensection of Pl with eigenvalue λ ∈ R. By Equation (3.3.1), n ϕ. HJ ϕ = − l + 2 Therefore, as in the proof of Proposition 6.2.1, one obtains h n 2 λϕ2 = ∇ϕ2 + n(n − 1)ϕ2 − 2h l + ϕ2 4 2 h = ∇ϕ2 − (n(n + 1) + 8l(l + n))ϕ2 , 4

which gives the assertion.

As noted in the preceding section, for n ≥ 2, the assumption in the above propositions that the function h is constant is automatically satisﬁed. The above estimates indicate that, for h > 0, in general, one can not found a common lower bound for the eigenvalues of P. That this is actually true is clariﬁed by the results in the next section.

6.3 The Spectrum of P on CP 1 For the computation of the spectrum of the operator P on the complex projective space CP 1 , we use tools of harmonic analysis. For details, we refer to [48]. First we realize CP 1 as follows. We consider the canonical action of the compact Lie group w1 −w2 2 2 SU(2) = : w1 , w2 ∈ C and |w1 | + |w2 | = 1 w2 w1 on CP 1 . This action is transitive and has U(1) as isotropy group of the point [1:0] ∈ CP 1 , where U(1) is understood as z0 : z ∈ C and |z| = 1 . U(1) = 0z Hence CP 1 is the homogeneous space SU(2)/U(1). We endow the Lie algebra su(2) of SU(2) with the inner product b deﬁned by 1 b(X, Y ) = − Tr(XY ) , 2 which is a negative multiple of the Killing form of su(2). Then the matrices i 0 0 1 0 i E0 = , E1 = , E2 = 0 −i −1 0 i 0

88

6 The K¨ ahler Case

form an orthonormal basis of su(2) with E0 ∈ u(1). Furthermore, we have the commutation relations [E0 , E1 ] = 2E2 ,

[E1 , E2 ] = 2E0 ,

[E2 , E0 ] = 2E1 .

(6.3.1)

In particular, the orthogonal complement w of u(1) ⊂ su(2) satisﬁes [w, w] = u(1) . That way we have described CP 1 as symmetric space. Let J denote the complex structure on w determined by JE1 = E2 and deﬁne a symplectic structure ω on w by ω(X, Y ) = b(JX, Y ) . Clearly, ω(E1 , E2 ) = 1 . Let J be the left-invariant almost complex structure on CP 1 induced by J and let ω be the left-invariant symplectic structure on CP 1 induced by ω. More concretely, let κ : U(1) → GL(w) be the isotropy representation of SU(2)/U(1), i.e. the restriction of the adjoint representation of SU(2) to U(1) acting on w, and identify the tangent bundle T CP 1 of CP 1 with the vector bundle SU(2) ×κ w associated to SU(2) via κ by d a exp(tX)[0 : 1] ∈ T CP 1 . [a, X] ∈ SU(2) ×κ w → dt t=0 Then J and ω are given by J[a, X] = [a, JX]

and ω([a, X], [a, Y ]) = ω(X, Y ) .

One readily sees that (ω, J) is the usual K¨ ahler structure on CP 1 . Hence the associated Riemannian metric g, which is given by g([a, X], [a, Y ]) = b(X, Y ) , is exactly the Fubini–Study metric. According to Example 3.1.7, the projective space CP 1 admits a unique metaplectic structure. We now construct this bundle. For this, we identify the real vector space w with C, seen as R2 , via the isomorphism 0 w ∈w. w ∈ C → −w 0

6.3 The Spectrum of P on CP 1

89

Then {E1 , E2 } is the standard basis and the complex structure J on w is the multiplication with i. Equation (6.3.1) implies that 0 −2 . κ∗ (E0 ) = 2 0 In particular, the image of κ is U(1) = Sp(1, R) ∩ O(2). From Section 1.5, we know that the Lie algebra u(1) of the inverse image U(1) of U(1) under the double covering ρ : Mp(1, R) → Sp(1, R) is spanned by E1 · E1 + E2 · E2 . Moreover, by Equation (1.3.2), 0 −2 . ρ∗ (E1 · E1 + E2 · E2 ) = 2 0 Thus a lift κ ˆ : U(1) → U(1) of the isotropy representation κ is determined by requiring that (6.3.2) κ ˆ ∗ (E0 ) = E1 · E1 + E2 · E2 . Since, with the above identiﬁcations, the symplectic frame bundle of CP 1 is R = SU(2) ×κ Sp(1, R) , we get that the metaplectic structure of CP 1 is P = SU(2) ×κˆ Mp(1, R) , where the covering map FP : P → R is given by FP ([a, q]) = [a, ρ(q)] . Consequently, the associated symplectic spinor bundle is Q = SU(2) ×m◦ˆκ L2 (R) and the Cliﬀord multiplication writes as [a, X] · [a, f ] = [a, X · f ] . Furthermore, the splitting of Q into the subbundles Ql , l ∈ N0 , corresponds to the decomposition of the unitary representation m ◦ κ ˆ = u◦κ ˆ into the representations ˆ : U(1) → GL(Wl ) ul ◦ κ and hence Ql = SU(2) ×ul ◦ˆκ L2 (R) . Here, u and ul are as given in Section 1.5. In particular, Wl is the onedimensional complex vector space spanned by the Hermite function hl . In view of Proposition 1.4.5 and Equations (1.5.4) and (6.3.2),

90

6 The K¨ ahler Case

(m ◦ κ ˆ )∗ (E0 )f = m∗ (E1 · E1 + E2 · E2 )f = −i(E1 · E1 + E2 · E2 ) · f

(6.3.3)

= −2iH0 f for any f ∈ S(R). Together with Equation (1.5.5), this implies (ul ◦ κ ˆ )∗ (E0 )hl = (2l + 1)ihl , which says that the weight of the representation ul ◦ κ ˆ is 2l + 1. Next observe that the Levi–Civita connection of CP 1 is induced by the canonical connection of the symmetric space SU(2)/U(1), and hence so is the spinor derivative. Thus, identifying Γ(Q) with the spaces of U(1)-equivariant maps ϕ : SU(2) → L2 (R), the spinor derivative is given by ∇[a,X] ϕ = X(ϕ)(a) for [a, X] ∈ T CP 1 . Here, X(ϕ) means the map X(ϕ) : SU(2) → L2 (R) deﬁned by d ϕ(a exp(tX)) X(ϕ)(a) = . dt t=0 on CP 1 , which are obviously For the symplectic Dirac operators D and D SU(2)-invariant, we then have Lemma 6.3.1 For any symplectic spinor ﬁeld ϕ on CP 1 , Dϕ = E1 · E2 (ϕ) − E2 · E1 (ϕ) and

= E1 · E1 (ϕ) + E2 · E2 (ϕ) . Dϕ

Proof. Since {E1 , E2 } is a unitary basis of w, this follows again from Lemma 4.1.3. We now relate our operator P to the Casimir operator Ω = E02 + E12 + E22 of su(2). Proposition 6.3.2 For any symplectic spinor ﬁeld ϕ on CP 1 , Pϕ = −Ω(ϕ) − 12 H02 ◦ ϕ .

6.3 The Spectrum of P on CP 1

91

Proof. Let ϕ : SU(2) → L2 (R) be a symplectic spinor ﬁeld on CP 1 . Then ϕ is U(1)-equivariant and it follows by Equation (6.3.3) that d ϕ(a exp(tE0 )) E0 (ϕ)(a) = dt t=0 d (m ◦ κ ˆ )(exp(−tE0 ))ϕ(a) = dt t=0 = −(m ◦ κ ˆ )∗ (E0 )ϕ(a) = 2i H0 (ϕ(a)) . Using this, Lemma 6.3.1, and X(Y · ϕ) = Y · X(ϕ) for X, Y ∈ w, we conclude that − DD)ϕ = E1 · E1 (E1 · E2 (ϕ)) + E2 · E2 (E1 · E2 (ϕ)) (DD −E1 · E1 (E2 · E1 (ϕ)) − E2 · E2 (E2 · E1 (ϕ)) −E1 · E2 (E1 · E1 (ϕ)) − E1 · E2 (E2 · E2 (ϕ)) +E2 · E1 (E1 · E1 (ϕ)) + E2 · E1 (E2 · E2 (ϕ)) = (E1 · E1 + E2 · E2 ) · (E1 (E2 (ϕ)) − E2 (E1 (ϕ))) +(E2 · E1 − E1 · E2 ) · E12 (ϕ) + E22 (ϕ) = i E12 (ϕ) + E22 (ϕ) + (E1 · E1 + E2 · E2 ) · [E1 , E2 ](ϕ) = iΩ(ϕ) − iE02 (ϕ) + 4H0 ◦ E0 (ϕ) = iΩ(ϕ) + 12i H02 ◦ ϕ , which proves the proposition.

Taking into account that the sections of Ql are identiﬁed with U(1)equivariant maps ϕ : SU(2) → Wl , we obtain Corollary 6.3.3 On CP 1 , for all l ∈ N0 and any ϕ ∈ Γ(Ql ), one has Pl ϕ = −Ω(ϕ) − 3(2l + 1)2 ϕ . Proof. This is a consequence of Proposition 6.3.2 and Equation (1.5.5).

Therefore, in order to compute the spectrum of each operator Pl , it suﬃces to determine the spectrum of the Casimir operator acting on Γ(Ql ), and this can be done by means of the Frobenius reciprocity. For convenience, we now formulate this result specialized to our situation. Let Vk for k ∈ N0 denote the complex vector space of homogeneous polynomials of order k in two complex variables z1 and z2 and let

92

6 The K¨ ahler Case

τk : SU(2) → GL(Vk ) be the representation of SU(2) on Vk induced by the canonical action of SU(2) on C2 . One checks that, for the basis {pk,0 , . . . , pk,k } of Vk deﬁned by pk,j (z1 , z2 ) = z1k−j z2j , it holds (τk )∗ (E0 )pk,j = i(2j − k)pk,j . Thus the representation τk has the weights −k, −k + 2, . . . , k − 2, k. As is well known (cf.[53]), the representations τk are irreducible and pairwise inequivalent and represent all possible irreducible representations of SU(2). Let ν : U(1) → GL(W) be a one-dimensional complex representation of U(1) and let HomU(1) (Vk , W) denote the space of all U(1)-equivariant homomorphisms φ : Vk → W. We embed Vk ⊗ HomU(1) (Vk , W) into the space Γ(SU(2) ×ν W) of sections of the line bundle SU(2) ×ν W by assigning to p ⊗ φ ∈ Vk ⊗ HomU(1) (Vk , W) the U(1)-equivarient map a ∈ SU(2) → φ τk a−1 p ∈ W . Then we have Theorem 6.3.4 (Frobenius reciprocity) (1) The space Γ(SU(2) ×ν W) decomposes into the direct sum ∞

Vk ⊗ HomU(1) (Vk , W) .

k=0

(2) If HomU(1) (Vk , W) is non-trivial, then Vk ⊗ HomU(1) (Vk , W) is an eigenspace of the Casimir operator Ω with eigenvalue 1 − (k + 1)2 . Now we can prove the following theorem. For a generalization of this result to all complex projective spaces CP n with odd n, see [52]. Theorem 6.3.5 On the complex projective space CP 1 , the operator Pl has the eigenvalues λl,j = 4(l + j + 1)2 − 3(2l + 1)2 − 1 for j = 0, 1, 2, . . ., where the multiplicity of λl,j is ml,j = 2(l + j + 1) .

6.3 The Spectrum of P on CP 1

93

Proof. We apply Theorem 6.3.4 for ν = ul ◦ κ ˆ . Thus we have to check for which k ∈ N0 the space HomU(1) (Vk , Wl ) is non-trivial. By Schur’s lemma, this is ˆ is a weight of τk , i.e., if the case if and only if the weight of ul ◦ κ 2l + 1 ∈ {−k, −k + 2, . . . , k − 2, k} . Hence HomU(1) (Vk , Wl ) is non-trivial if and only if k = 2l + 2j + 1 for j ∈ N0 . For such k, by Corollary 6.3.3 and Theorem 6.3.4, Vk ⊗ HomU(1) (Vk , Wl ) is an eigenspace of Pl with eigenvalue 4(l + j + 1)2 − 3(2l + 1)2 − 1 and dimC Vk ⊗ HomU(1) (Vk , Wl ) = dimC Vk = 2(l + j + 1) .

This yields the assertion.

The spectrum spec(P) of P is now the set of all λl,j , i.e. the set of the values of the function F (x, y) = 4(x + y + 1)2 − 3(2x + 1)2 − 1 for non-negative integer arguments. Note that diﬀerent pairs (l, j) may give the same eigenvalue λl,j .

3000 2000 1000 0 -1000 -2000 -3000 20 10 10

0 -10 -20

-15

-10

-5

0

15

20

5

As the picture of the graph of F illustrates, spec(P) is neither bounded from below nor from above. However, we see that λl,j ≥ λl,0 = −8l2 − 4l for all l, j ∈ N0 . This estimate is better than the estimate given in Proposition 6.2.2. Indeed, since the symplectic curvature tensor S of CP 1 writes as

94

6 The K¨ ahler Case

S([a, X1 ], [a, X2 ], [a, X3 ], [a, X4 ]) = b([X1 , X2 ], [X3 , JX4 ]) for [a, X1 ], [a, X2 ], [a, X3 ], [a, X4 ] ∈ T CP 1 , we have S([a, E1 ], J[a, E1 ], [a, E1 ], [a, E1 ]) = b([E1 , E2 ], [E1 , E2 ]) = 4 , which shows that CP 1 is of constant holomorphic sectional curvature 4. Consequently, Proposition 6.2.2 would only give λl,j ≥ −8l2 − 8l − 2 . In the picture below, it is visualized for which indices l and j the eigenvalues λl,j are positive and for which they are negative. l 5

*

*

4

*

*

3

*

*

*

*

2

*

*

*

*

*

*

* * positive eigenvalues

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

negative eigenvalues 1

*

*

*

*

*

*

*

*

0

*

*

*

*

*

*

*

*

0

1

2

3

4

5

6

7

k

-1

-2 -2

-1

8

Clearly, the two regions are separated by the hyperbola 4(x + y + 1)2 − 3(2x + 1)2 = 1 . The following proposition results immediately from Theorem 6.3.5. Proposition 6.3.6 On CP 1 , the kernel KerPl of Pl is non-trivial if and only if there is an non-negative integer j such that 4(l + j + 1)2 − 3(2l + 1)2 = 1 .

(6.3.4)

If this is the case, then dimC KerPl = 2(l + j + 1) .

6.3 The Spectrum of P on CP 1

95

In the remainder of this section, we solve Equation (6.3.4) within the nonnegative integers. Obviously, (l, j) = (0, 0) is a solution of this equation. Furthermore, any solution (l, j) ∈ N20 of Equation (6.3.4) yields a solution (k, m) ∈ N20 of the equation k 2 − 3m2 = 1 .

(6.3.5)

Conversely, as one readily checks, any solution (k, m) ∈ N20 of Equation (6.3.5) such that m is odd gives a solution (l, j) ∈ N20 of Equation (6.3.4). Let us consider ﬁrst Equation (6.3.5). This equation is a certain Diophantine equation called Pell’s equation. From the general theory of Pell’s equation (cf. e.g. [40]), one knows that, since the number 3 occuring in our equation is not a square, this equation has inﬁnitely many solutions. Moreover, one has Proposition 6.3.7 The solutions (ki , mi ) ∈ N20 , i = 1, 2, 3, . . ., of Equation (6.3.5) are determined by √ i √ ki + mi 3 = 2 + 3 . Remark 6.3.8 To obtain the pairs (ki , mi ) from the last equation, one has to compute the power and then to compare the rational and purely irrational terms. For example, from √ 2 √ √ k2 + m2 3 = 2 + 3 = 7 + 4 3 ,

one gets (k2 , m2 ) = (7, 4).

We now have to detect those solutions (ki , mi ) for which mi is odd. Obviously, mi is odd if and only if ki is even. From √ √ √ ki + mi 3 = ki−1 + mi−1 3 2 + 3 √ = (2ki−1 + 3mi−1 ) + (ki−1 + 2mi−1 ) 3 , we see that ki = 2ki−1 + 3mi−1 and mi = ki−1 + 2mi−1 . Therefore, mi is odd if and only if mi−1 even. Thus, since (k1 , m1 ) = (2, 1), the solutions of Equation (6.3.5) which yield a solution of Equation (6.3.4) are (k2i+1 , m2i+1 ), i = 0, 1, 2, . . ., and we obtain

96

6 The K¨ ahler Case

Proposition 6.3.9 The solutions (li , ji ) ∈ N20 , i = 0, 1, 2, . . ., of Equation (6.3.4) are determined by √ √ 2i+1 2(li + ji + 1) + (2li + 1) 3 = 2 + 3 . Proof. We relate (li , ji ) to (k2i+1 , m2i+1 ) by k2i+1 = 2(li + ji + 1)

and m2i+1 = 2li + 1 .

The assertion then follows from Proposition 6.3.7.

Corollary 6.3.10 The solutions (li , ji ) ∈ N20 , i = 0, 1, 2, . . ., of Equation (6.3.4) are given by (l0 , j0 ) = (0, 0) and (li , ji ) = (11li−1 + 4ji−1 + 7, 8li−1 + 3ji−1 + 5) for i = 1, 2, 3, . . .. Proof. From Proposition 6.3.9, we know that, for i = 1, 2, 3, . . ., √ 2(li + ji + 1) + (2li + 1) 3 √ 2 √ 2i−1 2+ 3 = 2+ 3 √ √ = 2(li−1 + ji−1 + 1) + (2li−1 + 1) 3 7 + 4 3 √ = 38li−1 + 14ji−1 + 26 + (22li−1 + 8ji−1 + 15) 3 . It follows that li + ji + 1 = 19li−1 + 7ji−1 + 13 , 2li + 1 = 22li−1 + 8ji−1 + 15 , which implies li = 11li−1 + 4ji−1 + 7 , ji = 8li−1 + 3ji−1 + 5 . Since, also by Proposition 6.3.9, (l0 , j0 ) = (0, 0), this proves the corollary.

7 Fourier Transform for Symplectic Spinors

This chapter presents a globally deﬁned Fourier transform F J for symplectic spinor ﬁelds, which was worked out by Klein in [34]. We show how this transform behaves with respect to Cliﬀord multiplication and spinor derivative. This has consequences for the symplectic Dirac operators. Applications in mathematical physics can be found in [29] and [34]. In particular, there it is explained how the quantization procedure of [41] can be expressed by means of the Fourier transform F J .

7.1 Deﬁnition of the Transform The following proposition states that the Fourier transform F on L2 (Rn ) is an intertwining operator of the restriction u of the metaplectic representation m to U(n). Proposition 7.1.1 For all q ∈ U(n), F ◦ u(q) = u(q) ◦ F . Proof. Since U(n) is connected and 0 = ρ−1 ∗ (J0 ) lies in the center of the lie algebra u(n) of U(n), exp(t0 )q = q exp(t0 ) and hence u (exp(t0 )) ◦ u(q) = u(q) ◦ u (exp(t0 )) hold true for all q ∈ U(n) and t ∈ R. Together with Lemma 1.5.4, this yields the assertion. Now, ﬁxing an ω-compatible almost complex structure J on M and writing the symplectic spinor bundle as Q = P J ×u L2 (Rn ), Proposition 7.1.1 allows to introduce a Fourier transform F J for symplectic spinors as follows.

98

7 Fourier Transform for Symplectic Spinors

Deﬁnition 7.1.2 Let F J : Q → Q be the endomorphism on Q deﬁned by F J ([p, f ]) = [p, Ff ] for p ∈ P J and f ∈ L2 (Rn ).

7.2 Basic Properties Proposition 7.2.1 For any X ∈ Γ(T M ) and any ϕ ∈ Γ(Q), F J (JX · ϕ) = X · F J ϕ .

Proof. This follows from Lemma 1.4.1(3).

Proposition 7.2.2 Let ∇ be a symplectic connection on M which satisﬁes ∇J = 0. Then the Fourier transform F J commutes with the symplectic spinor derivative, i.e. F J (∇X ϕ) = ∇X (F J ϕ) for all X ∈ Γ(T M ) and ϕ ∈ Γ(Q). Proof. Let s = (e1 , . . . , e2n ) be a unitary frame. Writing ϕ = [s, ϕs ], we have i Jej · ∇X ej · ϕ 2 j=1 2n

∇X ϕ = [s, X(ϕs )] − by Proposition 3.2.6. Obviously,

F(X(ϕs )) = X(F(ϕs )) . Using Proposition 7.2.1, we conclude i J F (Jej · ∇X ej · ϕ) 2 j=1 2n

F J (∇X ϕ) = [s, F(X(ϕs ))] − = [s, X(F(ϕs ))] −

i ej · F J (∇X ej · ϕ) 2 j=1

= [s, X(F(ϕs ))] +

i ej · J(∇X ej ) · F J ϕ . 2 j=1

2n

2n

Since, by assumption, J(∇X Y ) = ∇X (JY ), we obtain

7.3 Symmetry of the Spectra of D and D

99

i ej · ∇X (Jej ) · F J ϕ 2 j=1 2n

F J (∇X ϕ) = [s, F(X(ϕs ))] + = ∇X (F J ϕ) .

Finally, we have Proposition 7.2.3 The Fourier transform F J leaves the subbundles QJl of Q invariant, F J (QJl ) = QJl . Moreover, for any l ∈ N0 and ϕ ∈ Γ(QJl ), F J ϕ = i−l ϕ .

Proof. This is a consequence of Lemma 1.5.3.

! 7.3 Symmetry of the Spectra of D and D The next proposition shows that the Fourier transform F J is the link between the two symplectic Dirac operators D and D. are deﬁned with respect to a symplectic conProposition 7.3.1 If D and D nection ∇ satisfying ∇J = 0, then the relations , D ◦ FJ = FJ ◦ D

◦ F J = −F J ◦ D D

and

P ◦ FJ = FJ ◦ P

hold true. Proof. Applying Lemma 4.1.3 and Propositions 7.2.1 and 7.2.2, we see DF J ϕ = −

2n

Jej · ∇ej (F J ϕ)

j=1

=−

2n

Jej · F J ∇ej ϕ

j=1

=

2n

F J ej · ∇ej ϕ

j=1

= F J Dϕ and

100

7 Fourier Transform for Symplectic Spinors

Jϕ = DF

2n

ej · ∇ej (F J ϕ)

j=1

=

2n

ej · F J ∇ej ϕ

j=1

=

2n

F J Jej · ∇ej ϕ

j=1

= −F J Dϕ for any unitary frame (e1 , . . . , e2n ). The third identity follows from the ﬁrst two. Indeed, J J ϕ) PF J ϕ = i(DDF ϕ − DDF + DF J Dϕ) J Dϕ = i(DF

+ F J DDϕ) = i(−F J DDϕ = F J Pϕ . Proposition 7.3.1 immediately implies Corollary 7.3.2 If the chosen symplectic connection ∇ satisﬁes ∇J = 0, then D ◦ (F J )2 = −(F J )2 ◦ D

and

◦ (F J )2 = −(F J )2 ◦ D . D

For the spectra of the symplectic Dirac operators, we obtain have the same specCorollary 7.3.3 Suppose that ∇J = 0. Then D and D trum. Moreover, the spectrum is symmetric with respect to the origin. Proof. Apply Proposition 7.3.1.

8 Lie Derivative and Quantization

The aim of this supplementary chapter is to present new aspects of quantization theory related to symplectic spinors. For background information, we refer to the extensive literature on quantization, for instance the books [15, 51] or survey articles such as [4, 23, 33]. First we describe the construction of a Lie derivative for symplectic spinor ﬁelds in the direction of a symplectic vector ﬁeld. This clariﬁes Kostant’s remark in [37], §5.5 that Hamiltonian vector ﬁelds clearly operate as Lie differentiations on smooth symplectic spinor ﬁelds. Second we explain how a one parameter subgroup of Mp(n, R) can be assigned to any quadratic Hamiltonian on R2n such that the family of operators associated via the metaplectic representation satisﬁes the corresponding Schr¨ odinger equation. As is shown in the last section, this equation gives exactly the Lie derivative of a constant symplectic spinor ﬁeld over R2n in the direction of the Hamiltonian vector ﬁeld. That means, the Lie derivative of symplectic spinor ﬁelds gives rise to a quantization procedure.

8.1 Lie Derivative of Symplectic Spinor Fields We consider a 2n-dimensional symplectic manifold (M, ω) and suppose that a metaplectic structure P on M is ﬁxed. Let θ be a local symplectomorphism of M . Such a mapping gives rise to a local isomorphism θ˜ of the symplectic frame bundle R by ˜ 1 , . . . , en , f 1 , . . . , f n ) = (dθ(e1 ), . . . , dθ(en ), dθ(f 1 ), . . . , dθ(f n )) . θ(e Let θ¯ be a lift of θ˜ to the metaplectic structure P. Then, for any ϕ ∈ Γ(Q), the composition ϕˆ ◦ θ¯ of the Mp(n, R)-equivariant mapping ϕˆ : P → L2 (Rn ) corresponding to ϕ with the local isomorphism θ¯ is also Mp(n, R)-equivariant

102

8 Lie Derivative and Quantization

and thus deﬁnes a local symplectic spinor ﬁeld. We denote this spinor ﬁeld by (θ¯−1 )∗ ϕ. Let ∇ be a symplectic connection on M . Then it is known that, for any X, Y ∈ Γ(T M ), (8.1.1) ∇(θ−1 )∗ X (θ−1 )∗ Y = (θ−1 )∗ (∇X Y ) , where, as usual, (θ−1 )∗ X denotes the local vector ﬁeld on M given by −1 (θ )∗ X (p) = dθ−1 (X(θ(p))) . The spinor derivative and the Cliﬀord multiplication have analogous properties. Lemma 8.1.1 For any X ∈ Γ(T M ) and any ϕ ∈ Γ(Q),

and

∇(θ−1 )∗ X (θ¯−1 )∗ ϕ = (θ¯−1 )∗ (∇X ϕ)

(8.1.2)

(θ−1 )∗ X · (θ¯−1 )∗ ϕ = (θ¯−1 )∗ (X · ϕ) .

(8.1.3)

ˆ : P → R2n be the Mp(n, R)-equivariant Proof. Let X ∈ Γ(T M ) and let X mapping corresponding to X according to Equation (3.2.1). Then one easily ˆ ◦ θ¯ corresponds to the local sees that the Mp(n, R)-equivariant mapping X −1 vector ﬁeld (θ )∗ X. Using this, Equation (8.1.3) follows directly from the deﬁnition of the Cliﬀord multiplication. Equation (8.1.2) can be derived from Proposition 3.2.6 and Equations (8.1.1) and (8.1.3). For completeness, we now recall the following. Deﬁnition 8.1.2 A vector ﬁeld X on M is called symplectic if LX ω = 0 . Here, LX denotes the Lie derivative in the direction of X. Lemma 8.1.3 A vector ﬁeld X is symplectic if and only if its ﬂow (θt ) consists of local symplectomorphisms. Proof. This follows from d d ∗ θt ω = θs∗ θt∗ ω = θs∗ LX ω . dt dt t=s t=0 Lemma 8.1.4 Any Hamiltonian vector ﬁeld is symplectic.

8.1 Lie Derivative of Symplectic Spinor Fields

103

Proof. Let X be the Hamiltonian vector ﬁeld of a function u ∈ C ∞ (M ) and let iX denote the interior product of X. Then, by Deﬁnition 2.1.7, du = iX ω . Since the symplectic structure ω is a closed form and since the Lie derivative LX applied to forms satisﬁes the well known relation LX = d ◦ iX + iX ◦ d , we get LX ω = diX ω + iX dω = diX ω = ddu = 0 . Lemma 8.1.5 If X is a symplectic vector ﬁeld and ∇ is a torsion-free symplectic connection on M , then X(ω(Y, Z)) = ω([X, Y ], Z) + ω(Y, [X, Z])

(8.1.4)

ω(∇Y X, Z) = ω(∇Z X, Y )

(8.1.5)

and for all Y, Z ∈ Γ(T M ). Proof. As is known, (LX ω)(Y, Z) = X(ω(Y, Z)) − ω([X, Y ], Z) − ω(Y, [X, Z]) for arbitrary X, Y, Z ∈ Γ(T M ). This immediately implies Equation (8.1.4). The second equation is an easy consequence of the ﬁrst one. Let X be a symplectic vector ﬁeld on M . According to Lemma 8.1.3, the ﬂow (θt ) of X induces a local one parameter group (θ˜t ) of local isomorphisms of R. Let (θ¯t ) denote the lift of (θ˜t ) to P. By means of this local group, the Lie derivative of symplectic spinor ﬁelds can now be deﬁned in the usual way. Deﬁnition 8.1.6 The Lie derivative of a symplectic spinor ﬁeld ϕ in the direction of a symplectic vector ﬁeld X is the symplectic spinor ﬁeld LX ϕ given by d ¯−1 (θt )∗ ϕ LX ϕ = , dt t=0 where (θ¯t ) is as above. In the following, for brevity, we write ∇X · ϕ =

n

∇fj X · ej − ∇ej X · fj · ϕ ,

j=1

where ∇ is a connection on M and (e1 , . . . , en , f1 , . . . , fn ) is again any symplectic frame. With this, the Lie derivative LX ϕ can be expressed as follows.

104

8 Lie Derivative and Quantization

Proposition 8.1.7 Let ∇ be a torsion-free symplectic connection on M . Then, for any symplectic vector ﬁeld X on M and any ϕ ∈ Γ(Q), i LX ϕ = ∇X ϕ − ∇X · ϕ . 2 Proof. Let X be a symplectic vector ﬁeld on M and let (θt ), (θ˜t ), and (θ¯t ) again be the associated local groups. Further, let s = (e1 , . . . , en , f1 , . . . , fn ) : U → R be a symplectic frame and let s : U → P denote a lift of s to the metaplectic structure P. We set st = θ˜t−1 ◦ s ◦ θt : θt−1 (U ) → R and Clearly,

st = θ¯t−1 ◦ s ◦ θt : θt−1 (U ) → P . st = (θt−1 )∗ e1 , . . . , (θt−1 )∗ en , (θt−1 )∗ f1 , . . . , (θt−1 )∗ fn

and st is a lift of st . Let qt : U ∩ θt−1 (U ) → Mp(n, R) be deﬁned by ¯st = ¯sqt . Then At = ρ ◦ qt : U ∩ θt−1 (U ) → Sp(n, R) satisﬁes st = sAt . Thus, in the standard basis {a1 , . . . , an , b1 , . . . , bn } of R2n , At is given by At ai =

n n ω (θt−1 )∗ ei , fj aj − ω (θt−1 )∗ ei , ej bj j=1

and At bi =

n n ω (θt−1 )∗ fi , fj aj − ω (θt−1 )∗ fi , ej bj . j=1

Since

j=1

j=1

d −1 (θt )∗ Y = LX Y = [X, Y ] dt t=0

for all Y ∈ Γ(T M ), this implies n n d At ai = ω([X, ei ], fj )aj − ω([X, ei ], ej )bj dt t=0 j=1 j=1 and

8.1 Lie Derivative of Symplectic Spinor Fields

d At bi dt

= t=0

n

ω([X, fi ], fj )aj −

j=1

n

105

ω([X, fi ], ej )bj

j=1

Hence, by Lemma 1.3.1 and d d At qt = ρ∗ , dt dt t=0 t=0 we obtain n d 1 qt = (ω([X, ei ], fj )bi · aj − ω([X, ei ], ej )bi · bj dt t=0 2 i,j=1 −ω([X, fi ], fj )ai · aj + ω([X, fi ], ej )ai · bj ) . Using Equation (8.1.4), we arrive at n d 1 qt = (ω([X, ej ], fi )ai · bj − ω([X, ej ], ei )bi · bj dt t=0 2 i,j=1 −ω([X, fj ], fi )ai · aj + ω([X, fj ], ei )bi · aj ) .

(8.1.6)

Now let ϕ ∈ Γ(Q). We have ϕ = [s, ϕs ] on U , where ϕs = ϕˆ ◦ s, and, accordingly, (θ¯t−1 )∗ ϕ = [st , ϕˆ ◦ θ¯t ◦ st ] = [st , ϕs ◦ θt ] = [s, m(qt )(ϕs ◦ θt )] on U ∩ θt−1 (U ). Thus, on U , d [s, m(qt )(ϕs ◦ θt )] LX ϕ = dt t=0 % $ d d = s, ϕs ◦ θt + m(qt )ϕs dt dt t=0 t=0 % $ d qt = [s, X(ϕs )] + s, m∗ ϕs . dt t=0 Moreover, by Proposition 1.4.5 and Equation (8.1.6), we compute % $ n d i qt s, m∗ (ω([X, fj ], fi )ei · ej − ω([X, fj ], ei )fi · ej ϕs = dt t=0 2 i,j=1 −ω([X, ej ], fi )ei · fj + ω([X, ej ], ei )fi · fj ) · ϕ

106

8 Lie Derivative and Quantization

=

i ([X, fj ] · ej − [X, ej ] · fj ) · ϕ 2 j=1

=

i (∇X fj · ej − ∇X ej · fj ) · ϕ 2 j=1

n

n

i ∇fj X · ej − ∇ej X · fj · ϕ . 2 j=1 n

− Altogether, we obtain

i i (∇X fj · ej − ∇X ej · fj ) · ϕ − ∇X · ϕ . 2 j=1 2 n

LX ϕ = [s, X(ϕs )] +

Together with Proposition 3.2.6, this yields the assertion.

Recall that the Lie derivative applied to tensor ﬁelds satisﬁes L[X,Y ] = [LX , LY ] for all X, Y ∈ Γ(T M ). Since the commutator of two symplectic vector ﬁelds is again a symplectic vector ﬁeld, one may ask if this relation remains true for the Lie derivative of symplectic spinor ﬁelds. In fact, one has Proposition 8.1.8 Let X and Y be symplectic vector ﬁelds on M and let ϕ ∈ Γ(Q). Then L[X,Y ] ϕ = [LX , LY ]ϕ . In order to prove this, we ﬁrst show the following lemmas, where J is an ω-compatible almost complex structure and (e1 , . . . , e2n ) is a unitary frame. Lemma 8.1.9 Let ∇ be a symplectic connection on M . Then 2n

∇Jej X · ∇Y ej · ϕ = −

j=1

2n

∇∇Y (Jej ) X · ej · ϕ

j=1

for any X, Y ∈ Γ(T M ) and ϕ ∈ Γ(Q). Proof. By the symplecticity of ∇, we conclude 2n

∇Jej X · ∇Y ej · ϕ =

j=1

2n

ω(∇Y ej , Jei )∇Jej X · ei · ϕ

i,j=1

=

2n

ω(∇Y (Jei ), ej )∇Jej X · ei · ϕ

i,j=1

=−

2n

∇∇Y (Jei ) X · ei · ϕ .

i=1

8.1 Lie Derivative of Symplectic Spinor Fields

107

Lemma 8.1.10 Let ∇ be a torsion-free symplectic connection on M . Then, for all symplectic vector ﬁelds X and Y and any ϕ ∈ Γ(Q), ∇X · ∇Y · ϕ − ∇Y · ∇X · ϕ = 2i

2n

∇∇Jej X Y · ej · ϕ

j=1

−2i

2n

∇∇Jej Y X · ej · ϕ .

j=1

Proof. By Equation (3.2.2), ∇X · ∇Y · ϕ =

2n

∇Jei X · ei · ∇Jej Y · ej · ϕ

i,j=1

=

2n

∇Jei X · ∇Jej Y · ei · ej · ϕ

i,j=1

−i

2n

ω ei , ∇Jej Y ∇Jei X · ej · ϕ

i,j=1

=

2n

∇Jei X · ∇Jej Y · ei · ej · ϕ − i

i,j=1

2n

∇∇Jej Y X · ej · ϕ

j=1

and ∇Y · ∇X · ϕ =

2n

∇Jej Y · ∇Jei X · ej · ei · ϕ − i

i,j=1

=

2n

∇∇Jej X Y · ej · ϕ

j=1

∇Jej Y · ∇Jei X · ei · ej · ϕ − i

i,j=1

−i

2n

2n

2n

∇∇Jej X Y · ej · ϕ

j=1

∇ei Y · ∇Jei X · ϕ .

i=1

Moreover, by Equation (8.1.5), 2n

2n ω ∇Jei X, ∇Jej Y ei · ej · ϕ = ω ∇∇Jej Y X, Jei ei · ej · ϕ

i,j=1

i,j=1

=

2n j=1

and

∇∇Jej Y X · ej · ϕ

108

8 Lie Derivative and Quantization 2n

∇ei Y · ∇Jei X · ϕ =

i=1

2n

ω (∇Jei X, Jej ) ∇ei Y · ej · ϕ

i,j=1

=

2n

ω ∇Jej X, Jei ∇ei Y · ej · ϕ

i,j=1

=

2n

∇∇Jej X Y · ej · ϕ .

j=1

Hence, applying once again Equation (3.2.2), one gets the assertion.

Lemma 8.1.11 Let ∇ be a torsion-free connection on M . Then ∇X ∇Z Y − ∇Y ∇Z X − ∇Z [X, Y ] = R(X, Y )Z + ∇[X,Z] Y − ∇[Y,Z] X for all X, Y, Z ∈ Γ(T M ). Proof. It is ∇X ∇Z Y − ∇Y ∇Z X − ∇Z [X, Y ] = ∇X ∇Z Y − ∇Y ∇Z X − ∇Z ∇X Y + ∇Z ∇Y X = R(X, Z)Y + R(Z, Y )X + ∇[X,Z] Y + ∇[Z,Y ] X . By the Bianchi identity, this implies the lemma.

Proof of Proposition 8.1.8. Let ∇ be a torsion-free symplectic connection on M . Then, by Proposition 8.1.7, i i 1 LX LY ϕ = ∇X ∇Y ϕ − ∇X (∇Y · ϕ) − ∇X · ∇Y ϕ − ∇X · ∇Y · ϕ . 2 2 4 Using Proposition 3.2.7, we see that ∇X (∇Y · ϕ) =

2n ∇X ∇Jej Y · ej + ∇Jej Y · ∇X ej · ϕ + ∇Y · ∇X ϕ . j=1

Thus [LX , LY ]ϕ − L[X,Y ] ϕ i ∇X ∇Jej Y · ej + ∇Jej Y · ∇X ej 2 j=1 −∇Y ∇Jej X · ej − ∇Jej X · ∇Y ej − ∇Jej [X, Y ] · ej · ϕ 1 − (∇X · ∇Y · ϕ − ∇Y · ∇X · ϕ) . 4 2n

= RQ (X, Y )ϕ −

8.2 Schr¨ odinger Equation for Quadratic Hamiltonians

109

According to Proposition 3.2.9 and Lemma 8.1.11, i ∇X ∇Jej Y · ej − ∇Y ∇Jej X · ej − ∇Jej [X, Y ] · ej · ϕ 2 j=1 2n

i R(X, Y )(Jej ) · ej + ∇[X,Jej ] Y · ej − ∇[Y,Jej ] X · ej · ϕ 2 j=1 2n

=

i ∇[X,Jej ] Y · ej − ∇[Y,Jej ] X · ej · ϕ . 2 j=1 2n

= RQ (X, Y )ϕ +

By Lemmas 8.1.9 and 8.1.10 and the torsion-freeness of ∇, it follows that i ∇Jej Y · ∇X ej − ∇Jej X · ∇Y ej 2 j=1 +∇[X,Jej ] Y · ej − ∇[Y,Jej ] X · ej · ϕ 1 − (∇X · ∇Y · ϕ − ∇Y · ∇X · ϕ) 4 2n i ∇∇X (Jej ) Y · ej − ∇∇Y (Jej ) X · ej = 2 j=1 2n

[LX , LY ]ϕ − L[X,Y ] ϕ = −

−∇[X,Jej ] Y · ej + ∇[Y,Jej ] X · ej −∇∇Jej X Y · ej + ∇∇Jej Y X · ej · ϕ =0,

and the proposition is proved.

8.2 Schr¨ odinger Equation for Quadratic Hamiltonians We consider R2n with its standard symplectic structure ω0 and a quadratic Hamiltonian u on R2n , i.e. a function u : R2n → Rn deﬁned by u(v) = Υ v, v , where Υ is any symmetric real 2n × 2n-matrix. First we note that the Hamiltonian vector ﬁeld Xu of u is given by an element of the Lie algebra sp(n, R). Lemma 8.2.1 (1) One has Xu (v) = −2J0 Υ v for all v ∈ R2n .

110

8 Lie Derivative and Quantization

(2) J0 Υ and Υ J0 lie in sp(n, R). Proof. Assertion (1) follows from (du)v (w) = 2Υ v, w = −2ω0 (J0 Υ v, w) for any v, w ∈ R2n . To verify (2), use the fact that a real 2n × 2n-matrix A is an element of sp(n, R) if and only if AT J0 + J0 A = 0 . Let Hu denote the Hamilton operator associated to u via normal ordering quantization. That means that Hu =

n

(Υ ai , aj Qi ◦ Qj + Υ ai , bj (Qi ◦ Pj + Pj ◦ Qi )

i,j=1

+Υ bi , bj Pi ◦ Pj ) , where Qi and Pi are again the position and momentum operators. Lemma 8.2.2 One has the relation i m∗ ◦ ρ−1 ∗ (Υ J0 ) = − Hu . 2 Proof. By Lemma 1.3.1, 1 (ai · Υ ai + bi · Υ bi ) 2 i=1 n

ρ−1 ∗ (Υ J0 ) = =

n 1 (Υ ai , aj ai · aj + Υ ai , bj ai · bj 2 i,j=1

+Υ bi , aj bi · aj + Υ bi , bj bi · bj ) n 1 = (Υ ai , aj ai · aj + Υ ai , bj (ai · bj + bj · ai ) 2 i,j=1 +Υ bi , bj bi · bj ) .

Now apply Equation (1.4.2).

Let (AΥt ) be the one parameter subgroup of Sp(n, R) generated by 2Υ J0 , i.e. AΥt = exp(2tΥ J0 ) for t ∈ R, and let (qtΥ ) denote the lift of (AΥt ) to Mp(n, R).

8.3 Lie Derivative as Quantization

111

Proposition 8.2.3 For any f ∈ S(Rn ), the curve (m(qtΥ )f ) in S(Rn ) satisﬁes the Schr¨ odinger equation d m(qtΥ )f = −iHu m(qtΥ )f . dt Proof. We have d Υ d Υ d Υ ρ∗ q ρ(q ) A = = 2Υ J0 . = ds s s=0 ds s s=0 ds s s=0 Hence

d Υ q = 2 ρ−1 ∗ (Υ J0 ) . ds s s=0

By means of Lemma 8.2.2, we get

d d m(qtΥ )f = m(qsΥ )m(qtΥ )f dt ds s=0 d Υ q = m∗ m(qtΥ )f ds s s=0 Υ = 2 m∗ ρ−1 ∗ (Υ J0 ) m(qt )f = −iHu m(qtΥ )f

for any f ∈ S(Rn ).

8.3 Lie Derivative as Quantization We use the symplectic standard basis (a1 , . . . , an , b1 , . . . , bn ) understood as a global section of the symplectic frame bundle R of (R2n , ω0 ) to identify R2n × Sp(n, R) with R. That means, we assign to (v, A) ∈ R2n × Sp(n, R) the symplectic basis (Aa1 , . . . , Aan , Ab1 , . . . , Abn ) of Tv R2n = R2n . The metaplectic structure of (R2n , ω0 ) is then simply the product bundle P = R2n × Mp(n, R) , where the map FP : P → R is given by FP (v, q) = (v, ρ(q)) . We identify R2n × L2 (Rn ) with the symplectic spinor bundle Q via (v, f ) ∈ R2n × L2 (Rn ) → v, e+ , f ∈ Q , where e+ denotes the unit element of Mp(n, R). Then sections of Q are maps ϕ : R2n → L2 (Rn ). In the next lemma, we use that elements of Sp(n, R) can also be considered as symplectomorphisms of R2n .

112

8 Lie Derivative and Quantization

Lemma 8.3.1 Let B ∈ Sp(n, R) and let qB ∈ Mp(n, R) such that ρ(qB ) = B. Then one has ˜ : R → R induced by B is given by (1) The isomorphism B ˜ B(v, A) = (Bv, BA) for (v, A) ∈ R2n × Sp(n, R). ¯ : P → P deﬁned by (2) The isomorphism B ¯ B(v, q) = (Bv, qB q) ˜ for (v, q) ∈ R2n × Mp(n, R) is a lift of B. (3) It is

−1 ¯ )∗ ϕ (v) = m q −1 ϕ(Bv) (B B

for ϕ ∈ Γ(Q) and v ∈ R2n . Proof. Assertion (1) follows from (dB)v (w) = Bw for v, w ∈ R2n . Assertion (2) is easily checked. To see (3), ﬁrst observe that the Mp(n, R)-equivariant mapping ϕˆ : P → L2 (Rn ) corresponding to ϕ ∈ Γ(Q) is given by ϕ(v, ˆ q) = m q −1 ϕ(v) for v ∈ R2n and q ∈ Mp(n, R). Hence, by (2), −1 ¯ ϕ(Bv) . ϕˆ ◦ B(v, q) = m q −1 m qB This implies the stated relation.

We now give an interpretation of the Lie derivative in the direction of an Hamiltonian vector ﬁeld in the language of normal ordering quantization. For this, let f & for f ∈ S(Rn ) denote the symplectic spinor ﬁeld over R2n that is constantly equal to f . Proposition 8.3.2 Let u be a quadratic Hamiltonian on R2n . Then & LXu f & = −i F −1 ◦ Hu ◦ F(f ) for any f ∈ S(Rn ). Proof. Let Υ , (AΥt ) and (qtΥ ) be as in Section 8.2. By Lemma 8.2.1, the ﬂow (θt ) of the Hamiltonian vector ﬁeld Xu is θt = exp(−2tJ0 Υ ) = J0 AΥ−t J0−1 .

8.3 Lie Derivative as Quantization

113

By Proposition 1.3.7(1) and Lemma 8.3.1, it follows that & (θ¯t−1 )∗ f & = F −1 ◦ m(qtΥ ) ◦ F(f ) for f ∈ S(Rn ). Now, using Proposition 8.2.3, one gets the assertion.

Although the above relation cannot be carried over to arbitrary manifolds, we can prove a Heisenberg relation in the general situation. Proposition 8.3.3 Let (M, ω) be a symplectic manifold that admits a metaplectic structure. Setting q(u)ϕ = iLXu ϕ , one has [q(u), q(v)]ϕ = −iq({u, v})ϕ for all u, v ∈ C ∞ (M ) and all symplectic spinor ﬁelds ϕ over M with respect to any metaplectic structure. Proof. By Propositions 2.1.9 and 8.1.8, one concludes that [q(u), q(v)]ϕ = − [LXu , LXv ] ϕ = −L[Xu ,Xv ] ϕ = LX{u,v} ϕ = −iq({u, v})ϕ .

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Index

almost complex structure, 23 ω-compatible, 23 anti-canonical line bundle, 46 Baum, 49 Bianchi identity, 29 Bourgeois, 21 Cahen, 21 Casimir operator, 90 Cauchy–Riemann equation generalized, 79 Chern class ﬁrst, 36 complex projective space, 36, 87, 92 complex torus, 85 connection canonical, 90 Hermitian, 27 Levi–Civita, 34, 81 symplectic, 25 reducible, 33 torsion-free, 27 cotangent bundle, 22, 36 curvature, 29 of the spinor derivative, 42 Darboux theorem, 22 deformation of metaplectic structures, 58 divergence, 27 Einstein manifold, 84

Fedosov, 21 ﬂow, 102, 112 formally self-adjoint, 64, 66, 81 Fourier transform, 8 for symplectic spinors, 97 Friedrich, 49 Frobenius reciprocity, 91, 92 Fubini–Study metric, 36, 88 G˚ arding, 6 Gelfand, 21 Gutt, 21 Hamiltonian quadratic, 109 vector ﬁeld, 24 harmonic oscillator, 17 Heisenberg group, 6 relation, 113 Hermite function, 17, 89 Hermitian connection, 27 structure, 39 holomorphic sectional curvature constant, 33, 84, 94 invariant subspace, 5 K¨ ahler form, 22, 81 manifold, 22, 81 Lie derivative, 102

120

Index

of symplectic spinor ﬁelds, 103 metaplectic group, 7 representation, 10 diﬀerential of, 14 structure, 35 momentum operator, 11 normal ordering quantization, 110, 112 orthogonal group, 2 Pell’s equation, 95 Poisson bracket, 24 position operator, 11 quadratic Hamiltonian, 109 quantization, 97, 101 normal ordering, 110, 112 Rawnsley, 21 representation, 5 adjoint, 88 diﬀerential of, 6 equivalent, 6 unitary, 6 irreducible, 5 isotropy, 88 metaplectic, 10 diﬀerential of, 14 projective unitary, 8 unitary, 5 Retakh, 21 Ricci tensor, 31 symplectic, 31 scalar curvature, 34 symplectic, 34 Schr¨ odinger equation, 111 representation, 6 Schur’s lemma, 5, 93 Schwartz space, 8 sectional curvature, 84 holomorphic, 33, 84 Segal–Shale–Weil representation, 10 Shubin, 21 smooth section, 37 vector, 6 spectrum, 93, 100

spinor derivative, 39 curvature of, 42 Stiefel–Whitney class ﬁrst, 36 second, 36 Stone–von Neumann theorem, 7 symmetric space, 88 symplectic basis, 1, 23 Cliﬀord algebra, 4 multiplication, 13, 38 connection, 25 curvature tensor, 29 Dirac operator, 50 form, 1 frame, 23 frame bundle, 23 gradient, 24 group, 2 Lie algebra, 3 manifold, 22 Ricci tensor, 31 scalar curvature, 34 spinor, 10 bundle, 37 ﬁeld, 37 Laplacian, 70 structure, 22 vector ﬁeld, 102 symplectomorphic, 22 symplectomorphism, 22 Tondeur, 21 torsion, 27 vector ﬁeld, 27 unitary frame, 24 group, 2 Vaisman, 21 vector ﬁeld Hamiltonian, 24 symplectic, 102 volume element, 28 weight, 90, 92 Weil, 9 Weitzenb¨ ock formula, 72, 82 Weyl algebra, 4

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