Selected Titles in This Series

25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000

24 He lmut Koch, Number theory: Algebraic numbers and functions, 2000

23 A lberto Candel and Lawrence Conlon, Foliations I, 2000

22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov

dimension, 2000

21 John B. Conway, A course in operator theory, 2000

20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999

19 Lawrence C . Evans, Partial differential equations, 1998

18 Winfried Just and Mart in Weese , Discovering modern set theory. II: Set-theoretic

tools for every mathematician, 1997

17 Henryk Iwaniec, Topics in classical automorphic forms, 1997

16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator

algebras. Volume II: Advanced theory, 1997

15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator

algebras. Volume I: Elementary theory, 1997

14 Ell iott H. Lieb and Michael Loss, Analysis, 1997

13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996

12 N . V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996

11 Jacques Dixmier , Enveloping algebras, 1996 Printing

10 Barry Simon, Representations of finite and compact groups, 1996

9 D ino Lorenzini, An invitation to arithmetic geometry, 1996

8 Winfried Just and Mart in Weese , Discovering modern set theory. I: The basics, 1996

7 Gerald J. Janusz, Algebraic number fields, second edition, 1996

6 Jens Carsten Jantzen, Lectures on quantum groups, 1996

5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995

4 Russel l A . Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 Wi l l iam W . A d a m s and Phi l ippe Loustaunau, An introduction to Grobner bases,

1994 2 Jack Graver, Brig i t te Servatius, and H e r m a n Servatius, Combinatorial rigidity,

1993

1 E than Akin, The general topology of dynamical systems, 1993

http://dx.doi.org/10.1090/gsm/025

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Dira c Operator s in Riemannia n Geometr y

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Dira c Operator s in Riemannia n Geometr y

Thomas Friedrich

Translated by

Andreas Nestke

Graduate Studies

in Mathematics

Volume 25

IK

Providence , Rhod e Islan d ^ ™ ^ America n Mathematica l Societ y

Editorial Board

James Humphreys (Chair) David Saltman David Sattinger

Ronald Stern

2000 Mathematics Subject Classification. P r imary 58Jxx; Secondary 53C27, 53C28, 57R57, 58J05, 58J20, 58J50, 81R25.

Originally published in the German language by Priedr. Vieweg &; Sohn Verlagsge-sellschaft mbH, D-65189 Wiesbaden, Germany, as "Thomas Priedrich: Dirac-Operatoren in der Riemannschen Geometr ie . 1. Auflage (1st edit ion)" © by Priedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1997

Transla ted from the German by Andreas Nestke

ABSTRACT. This text examines the Dirac operator on Riemannian manifolds, especially its con­nection with the underlying geometry and topology of the manifold. The presentation includes a review of preliminary material, including spin and spinc structures.

An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on M lead to results about whether M is an Einstein manifold or conformally equivalent to one. An appendix contains a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds.

This book is suitable as a text for courses in advanced differential geometry and global analysis, and can serve as an introduction for further study in these areas.

Library of Congress Cataloging-in-Publication Data

Priedrich, Thomas, 1949-[Dirac-Operatoren in der Riemannschen Geometrie. English] Dirac operators in Riemannian geometry / Thomas Priedrich ; translated by Andreas Nestke.

p. cm. — (Graduate studies in mathematics, ISSN 1065-7339; v. 25) Includes bibliographical references and index. ISBN 0-8218-2055-9 (alk. paper) 1. Geometry, Riemannian. 2. Dirac equation. I. Title. II. Series.

QA649.F68513 2000 516.373—dc21 00-038614

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to rspr int-psnnissionOams.org.

© 2000 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at URL: http:/ /www.ams.org/

10 9 8 7 6 5 4 3 2 1 05 04 03 02 01 00

Contents

Introduction xi

Chapter 1. Clifford Algebras and Spin Representation 1

1.1. Linear algebra of quadratic forms 1

1.2. The Clifford algebra of a quadratic form 4

1.3. Clifford algebras of real negative definite quadratic forms 10

1.4. The pin and the spin group 14

1.5. The spin representation 20

1.6. The group Spirf 25

1.7. Real and quaternionic structures in the space of n-spinors 29

1.8. References and exercises 32

Chapter 2. Spin Structures 35

2.1. Spin structures on £0(n)-principal bundles 35

2.2. Spin structures in covering spaces 42

2.3. Spin structures on G-principal bundles 45

2.4. Existence of spinc structures 47

2.5. Associated spinor bundles 53

2.6. References and exercises 56

VII

Vl l l Contents

Chapter 3. Dirac Operators 57

3.1. Connections in spinor bundles 57

3.2. The Dirac and the Laplace operator in the spinor bundle 67

3.3. The Schrodinger-Lichnerowicz formula 71

3.4. Hermitian manifolds and spinors 73

3.5. The Dirac operator of a Riemannian symmetric space 82

3.6. References and Exercises 88

Chapter 4. Analytical Properties of Dirac Operators 91

4.1. The essential self-adjointness of the Dirac operator in L2 91

4.2. The spectrum of Dirac operators over compact manifolds 98

4.3. Dirac operators are Fredholm operators 107

4.4. References and Exercises 111

Chapter 5. Eigenvalue Estimates for the Dirac Operator and Twistor

Spinors 113

5.1. Lower estimates for the eigenvalues of the Dirac operator 113

5.2. Riemannian manifolds with Killing spinors 116

5.3. The twistor equation 121

5.4. Upper estimates for the eigenvalues of the Dirac operator 125

5.5. References and Exercises 127

Appendix A. Seiberg-Witten Invariants 129

A.l. On the topology of 4-dimensional manifolds 129

A.2. The Seiberg-Witten equation 134

A.3. The Seiberg-Witten invariant 138

A.4. Vanishing theorems 144

A.5. The case dimWlL(g) = 0 146

A.6. The Kahler case 147

A.7. References 153

Appendix B. Principal Bundles and Connections 155

B.l. Principal fibre bundles 155

Contents IX

B.2. The classification of principal bundles 162

B.3. Connections in principal bundles 163

B.4. Absolute differential and curvature 166

B.5. Connections in Z7(l)-principal bundles and the Weyl theorem 169

B.6. Reductions of connections 173

B.7. Frobenius' theorem 174

B.8. The Freudenthal-Yamabe theorem 177

B.9. Holonomy theory 177

B.10. References 178

Bibliography 179

Index 193

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Introduction

It is well-known that a smooth complex-valued function / : O —> C defined on an open subset O C M2 is holomorphic if and only if it satisfies the Cauchy-Riemann equation

^ - 0 with —--(— i — dz dz 2\dx dy

Geometrically, we consider M2 here as flat Euclidean space with fixed orien­tation. Changing this orientation results in replacing the operator J | by the

differential operator ^ — \ ( J^ — i-^-). Taking both operators together we

obtain a differential operator P : C°°(IR2; C2) -> C°°(M2; C2) acting via

dg< f\ I dz

= 2i 9J \dl

\dz>

on pairs of complex-valued functions. An easy calculation leads to the fol­lowing alternative formula for P:

V* o) dx \-i oy dy

Denoting the matrices occurring in this formula by ^x and j y ,

yields

\z uy • 0 1

- 1 0

9 d

XI

X l l Introduction

as well as

7x = -E = 7?, Ixly + 7y7x = 0.

The square of the operator P coincides with the Laplacian A on

dx2 dy2

Thus we have found a square root P = v A of the Laplacian within the class of first order differential operators, and its kernel is, moreover, the space of holomorphic (anti-holomorphic) functions.

In higher-dimensional Euclidean spaces the question whether there exists a square root \/A of the Laplacian was raised in the following discussion by P.A.M. Dirac (1928). Let T be a free classical particle in R3 with spin \ whose motion is to be studied in special relativity. Denoting its mass by m, its energy by E and its momentum by p = , v™ =, we have

y l - r / c 2

E = \/c2p2 + ra2c4.

In quantum mechanics T is described by a state function ip(t,x) defined on IR1 x M3, and energy as well as momentum are to be replaced by the differential operators

E i—> i/i— and p \—> —i/igrad,

respectively. The state function ip then becomes a solution of the equation

ih^- = Vc2h2A + m2c4 ^ at

involving the 3-dimensional Laplacian A = — -^ — -^ — J^. Mathemati­cally speaking we now move to an n-dimensional Euclidean space und look

n 2

for a square root P = \fK of the Laplacian A — — ]T -^. The obvious as-2 = 1 l

sumption that P should be a first order differential operator with constant coefficients leads to the ansatz

d

2 = 1

p'^>ir* n

Now the equation P2 = A = — ]T -^ holds if and only if the coefficients 7Z 2 = 1 l

of P satisfy the conditions

1i=-E, i = l , . . . , n ; 7 i 7 j + 7 j 7 i = 0, i^j.

Introduction xin

For n = 3, there is an obvious solution to these equations. The vector space

c 2 can be identified with the set of quaternions via ( I = z\ + JZ2, \Z2 )

and 71,72,73 : C2 = H —> H = C2 then correspond to multiplication by the quaternions i, j , fc G H , respectively. Writing these as complex (2 x 2)-matrices, we obtain

i 0 \ / 0 - l \ / 0 i 71 = U - i , / ' ^ = V l 0 ^ ' 7 3 = V * °

The algebra multiplicatively generated by n elements 7 1 , . . . , 7n satisfying the relations

is called the Clifford algebra Cn (W.K. Clifford, 1845-1879) of the negative definite quadratic form (Rn, — x\ — . . . — xn). Thus, the question whether there is a square root yfK of the Laplacian leads to the study of complex rep­resentations K : Cn —> End (V) of the Clifford algebra. It turns out that Cn

has a smallest representation of dimension dime V = 2^1. The correspond­ing vector space is denoted by An and its elements are the Dirac spinors. Moreover, y/A is a constant coefficient first order differential operator acting on the space C°°(IRn; An) of smooth An-valued functions on Rn.

Spinors can be multiplied by vectors from Euclidean space. In order to define this product we represent a vector x G W1 as a linear combination with respect to an orthonormal basis e i , . . . , en,

n ,1,

x — y x Ci,

and then define its product x • i\) by a spinor i\) G A n as n

From the defining relations of the Clifford algebra one immediately deduces the formula

x • (x - ip) = —||x||2^.

In particular, the product X-I/J vanishes if and only if either the vector x G Kn

or the spinor ij) G An is equal to zero. There is no non-trivial representation e of the linear or the orthogonal group in the space An of spinors that is compatible with Clifford multiplication, i.e. which satisfies the relation

A(x).e(A)(iP) = e(A)(x-iP)

XIV Introduction

for every A E SO(n; M), x G K n and i\) € An . Hence spinors on Riemannian manifolds cannot be defined as sections of a vector bundle that is associated with the frame bundle of the manifold. It is for this reason that in differ­ential geometry the question to what extent the concept of spinors could be transferred from flat space to general Riemannian manifolds remained open for decades. In 1938 Elie Cart an expressed this difficulty in his book "Legons sur la theorie des spineurs" with the following words:

"With the geometric sense we have given to the word 'spinor' it is impossible to introduce fields of spinors into the classical Riemannian technique."

Only the development of the framework of principal fibre bundles and their associated bundles as well as the general theory of connections within dif­ferential geometry at the end of the forties made it possible to overcome this difficulty. The group SO(n;R) is not simply connected. For n > 3 its universal covering, the group denoted by Spin(n), is compact and cov­ers SO(n;M,) twice. On the other hand, there exists a representation e : Spin(n) —> GL(An) of the spin group which is compatible with Clifford multiplication. Considering now those special Riemannian manifolds M n , today called spin manifolds, the frame bundle of which allows a reduction to the double cover Spin(n) of the structure group SO(n] R), we can define the vector bundle S associated with this reduction via the representation e : Spin(n) —» GL(An), the so-called spinor bundle of Mn. Then spinor fields over Mn are sections of the bundle S and, as in the Euclidean case, the Dirac operator D can be introduced by the formula

n

Here V denotes the covariant derivative corresponding to the Levi-Civita connection of the Riemannian manifold.

Therefore, spinor fields and Dirac operators cannot be introduced on every Riemannian space; but, nevertheless, they can be introduced for a large class. The existence of a Spin(n)-reduction of the frame bundle of Mn

translates into a topological condition on the manifold, i.e. the first two Stiefel-Whitney classes have to vanish:

Wl(Mn) = 0 = w2(M

n).

In dimension n = 4, for a compact simply connected manifold M4 , this topological condition is equivalent to the condition that the intersection form on H2(M4; Z), considered as a quadratic form over the ring Z, is even and unimodular. The algebraic theory of quadratic Z-forms then implies that the signature a is divisible by 8. Surprisingly, in 1952 Rokhlin proved

Introduction xv

a further divisibility by 2: the signature <J(M 4 ) of a smooth compact 4-dimensional spin manifold M 4 is divisible by 16:

< J ( M 4 ) / 1 6 G Z.

This additional divisibility of the signature of a 4-dimensional spin manifold, which does not result from purely algebraic considerations, was an essential aspect for the introduction of spinor fields and Dirac operators into mathe­matics. The consideration behind that may be outlined as follows. Could it be possible that there exists an elliptic operator P on every compact smooth 4-dimensional manifold with even intersection form on H2(M4; Z), the index of which coincides with a/16? Today we know the answer to that question: it is essentially given by the Dirac operator on a spin manifold, eventually introduced for Riemannian manifolds by M.F. Atiyah in 1962 in connection with his elaboration of the index theory for elliptic operators. Since then it has occured in many branches of mathematics and has become one of the basic elliptic differential operators in analysis and geometry.

This book was written after a one-semester course held at Humboldt-Uni­versity in Berlin during 1996/97. It contains an introduction into the theory of spinors and Dirac operators on Riemannian manifolds. The reader is as­sumed to have only basic knowledge of algebra and geometry, such as a two or three year study in mathematics or physics should provide. The pre­sentation starts with an algebraic part comprising Clifford algebras, spin groups and the spin representation. The topological aspects concerning the existence and classification of spin reductions of principal 50(n)-bundles are discussed in Chapter 2. Here the approach essentially requires only el­ementary covering theory of topological spaces. At the same time, each result will also be translated into the cohomological language of character­istic classes. The subsequent Chapter 3 deals with analysis in the spinor bundle, the twistor operator and the Dirac operator in detail. Here the general techniques of principal bundles and the theory of connections are applied systematically. To make the book more self-contained, these results of modern differential geometry are presented without proof in Appendix B. Chapter 4 contains special proofs for the analytic properties of Dirac op­erators (essential self-adjointness, Fredholm property) avoiding the general theory for elliptic pseudo-differential operators. Eigenvalue estimates and solution spaces of special spinorial field equations (Killing spinors, twistor spinors) are the topic of Chapter 5. We mainly discuss the general approach, referring to the literature for detailed investigations of these problems. The book is concluded in Appendix A with an extended version of a talk on

XVI Introduction

Seiberg-Witten theory given by the author in the seminar of the Sonder-forschungsbereich 288 "Differentialgeometrie und Quantenphysik" in Berlin on February 9, 1995.

Since the eighties a group of younger mathematicians at Humboldt-Univer-sity in Berlin has been working on spectral properties of Dirac operators and solution spaces of spinorial field equations. Many of the results from this period are collected in the references. On the other hand, the present book may serve as an introduction for a closer study. I would like to thank all those students and colleagues whose remarks and hints had an impact on the contents of this text in various ways.

I am particularly grateful to Dr. Ines Kath for her careful and detailed cor­rections of the text, and to Heike Pahlisch, whose typing of the manuscript took into account every single wish.

Thomas Friedrich

Berlin, March 1997

The English translation of this book has been prepared in the beginning of the year 2000. It does not differ essentially from the original text, although I made many changes in details which are not worth listing. During the last three years many new results have been published in this still dynamic area of mathematics. I included the corresponding references in the bibliography of the translation. During the academic year 1996/97 Dr. Andreas Nestke provided the exercises for students of my lectures at Humboldt University which furnished the starting point for this book. Two years later he had to leave the University. I thank him, as well as Dr. Ilka Agricola and Heike Pahlisch, for all the work and help related with the preparation of the Eng­lish edition of this book.

Thomas Friedrich

Berlin, March 2000

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Index

absolute differential, 58, 166, 167 adjoint operator, 92 algebra of complex numbers, 9 algebra of quaternions, 8 almost-complex manifold, 73, 74 almost-complex structure, 60, 80, 146, 147,

151 Ambrose-Singer theorem, 178 anti-canonical spin c structure, 79, 81 associated fibration of a principle bundle,

159 associated spinor bundle, 75 associated vector bundle, 161

Bianchi identity, 168 bilinear form, 1, 7

direct sum of, 7 index of, 2 nondegenerate, 1 rank of, 2 signature of, 2

canonical basis, 2 canonical connection, 83 canonical spin c structure, 79

of an Hermitian manifold, 77, 78 canonical spin c(4) structure, 147, 149 Casimir operator, 86, 87 Cauchy-Riemann equations, 71 center of an algebra, 9 characteristic class, 108 Chern class, 108, 141, 163, 171, 172 Clifford algebra, 4, 10, 11 Clifford multiplication, 21, 32, 53, 68, 70,

133 complete Riemannian manifold, 98 complex n-spinors, 14

complex projective space, 40, 42, 48, 161 complexification

of a real algebra, 11 of a real quadratic form, 11

conjecture, ^ , 131 connection, 163, 165, 169

holonomy group of, 177 locally flat, 168 reduction of, 174

continuous spectrum, 91 covariant derivative, 58, 60, 67, 68, 70, 166,

167 covering spaces, 40 curvature form, 62, 135, 167, 169 curvature tensor, 62

de Rham cohomology, 171 integral, 172

determinant bundle of spin c structure, 52-54, 108, 113

Dirac operator, 68, 69, 71, 93, 96, 101, 107, 127, 136, 148

eigenvalues of, 113, 116, 126, 128 G-function for, 103 index formula for, 110 index theorem for, 109 spectrum of, 99

Dirac spinors, 14, 69, 113, 115 direct sum of bilinear forms, 7 distribution, 175

integral, 175

eigenspinor, 102, 103, 112, 114, 126 eigenvalues, 91, 126

of the Dirac operator, 113, 116, 126, 128 Einstein space, 118 ^ conjecture, 131

193

194 Index

equivalent fibrations, 166 equivalent A-reductions, 158 equivalent spin structures, 35 equivalent spin c structures, 51 essentially self-adjoint operator, 92-94, 96 ?7-function, 105, 112

for the Dirac operator, 103 exponential map, 18 exterior differential, 73

fibration, 156, 159 equivalence of, 156 locally trivial, 155

first integral, 124 frame bundle, 158 Fredholm operator, 107 Freudenthal-Yamabe theorem, 177 Probenius theorem, 174

global, 176 local, 174, 175

fundamental group, 26

Gaschiitz proposition, 39 gauge field theory, 130, 131 gauge group, 135 gauge transformation, 135, 169 Ginzburg-Landau model, 131 G-principal bundle, 156 Grafimannian manifold, 56, 88

index, 108 of a form, 2

index formula for Dirac operators, 110 index theorem for Dirac operators, 109 integrable distribution, 175 integral de Rham cohomology, 172 integral manifold, 176 intersection form, 109, 130 isomorphic principal bundles, 157 isotropic subspace, 3

Kahler manifold, 61, 81, 82, 89, 116, 147, 150, 151, 153

Killing number, 116, 118, 119 Killing spinor, 116, 118, 119, 121, 124, 125,

128 imaginary, 120 real, 120

Lagrange theorem, 2 A-reduction, 158, 173

equivalent, 158 Laplace operator, 71

on spinors, 68 Levi-Civita connection, 57, 61, 81, 84, 113,

125, 135, 148, 164 Lie algebra

of Spin{n), 17, 18 of Spirf(n), 29

linear operator, 91 locally flat connection, 168 locally trivial fibration, 155

manifold without boundary, 175 Maurer-Cartan form, 83, 164 moduli space, 140, 147

for Seiberg-Witten theory, 136

nondegenerate bilinear form, 1 null subspace, 3

orientation, 159

parallel spin c spinor, 67 parallel spinor, 67, 89 parallel spinor field, 67 parallel transport, 166, 167 Picard manifold, 172 Pin(n), 15 point spectrum, 91 Pontrjagin class, 108 principal bundle, 157

associated fibration of, 159 G-, 156 isomorphic, 157 S1- , 163 Z2- , 163

projective space complex, 40, 42, 48, 161 real, 55, 56

q-foim of type p, tensorial, 165 quadratic form, 1 quaternionic structure, 29, 30, 110

in A n , 32, 54

rank of a bilinear form, 2 real projective space, 55, 56 real structure, 29, 30

in A n , 32, 54

harmonic spinors, 81, 82 heat equation, 112 Hermitian manifold, 75, 78, 147

canonical spin c structure, 77, 78 spinor bundle of, 79

Hermitian metric, 53, 68, 74, 80 Hermitian scalar product, 24 Hilbert-Schmidt operator, 103, 104 Hirzebruch-Hopf proposition, 133, 146 Hirzebruch signature theorem, 109 holonomy group of a connection, 177 homogeneous spin structure, 85, 87 homotopy classification theorem, 162 homotopy theory, reduction theorem of, 178 Hopf bundle, 161 Hopf fibration, 157, 161, 172, 173 horizontal lift, 165

Index 195

reducible solution of the Seiberg-Witten equation, 140, 142

reduction of a connection, 174 A-, 158, 173

equivalent, 158 £/(&)-, 48, 60, 61, 81

reduction theorem of homotopy theory, 178 Rellich lemma, 100 residual spectrum, 91 resolvent set, 92 Ricci tensor, 64, 118 Riemannian manifold, complete, 98 Riemannian metric, 141, 159 Riemannian symmetric space, 82, 87 Rokhlin's theorem, 110

S1 -principal bundle, 163 scalar curvature, 111, 113, 118, 135, 144,

145, 148, 149, 151 Schrodinger operator, 127 Schrodinger-Lichnerowicz formula, 73, 100,

110, 113, 134, 145 Schur-Zassenhaus proposition, 39 second Stiefel-Whitney class, 40 section, 156 Seiberg-Witten equation, 131, 134, 136, 138,

140, 153 reducible solution of, 140, 142

Seiberg-Witten invariant, 144-146, 149, 151 Seiberg-Witten theory, moduli space for, 136 self-adjoint operator, 92

essentially, 92-94, 96 spectral theorem for, 93

signature, 109 of a form, 2

spectral measure, 92, 93 spectral theorem for self-adjoint operators,

93 spectrum

of a Dirac operator, 99 of an operator, 91, 92

sphere, 43, 88, 116, 125, 128 spin bundle, 54 spin representation, 14, 23, 25, 54, 58, 75,

133 of Spin(n), 20

spin structure, 35, 36, 38-40, 42-45, 47, 50, 53-55, 60, 79, 113

equivalence of, 35 homogeneous, 85, 87

spin c structure, 47, 48, 50, 51, 53, 57, 60, 78, 93, 96, 111, 131, 153

canonical, 79 of an Hermitian manifold, 77, 78

determinant bundle of, 52-54, 108, 113 equivalence of, 51

Spirf{4) structure, 134, 141, 146 canonical, 147, 149

Spirf(n), 25, 26 Spinc(n) representation, 28 Spin(n), 15

spin representation of, 20 spinor bundle, 53, 78

of an Hermit ian manifold, 77 spinor derivative, 59 spinor field, 67

parallel, 67 Stiefel-Whitney class, 163

second, 40 structure identity, 168 submanifold, 175

weak, 176 Sylow subgroup, 2-, 39, 44 Sylvester's theorem, 2 symmetric operator, 69, 92, 93 symmetric space, Riemannian, 82, 87 symplectic manifold, 158 symplectic structure, 151, 159

tangent bundle, 155 tautological bundle over C P \ 161 tensor product of Z2-graded algebras, 7 tensorial 1-form, 62 tensorial g-form of type p, 165 twist or equation, 128 twist or operator, 69, 70, 121 twistor spinor, 121, 123 2-Sylow subgroup, 39, 44

E/(fc)-reduction, 48, 60, 61, 81 unitary group, 27 universal covering, 19

of 5 0 ( n ) , 16 universal G-bundle, 162

vanishing theorem, 140 vector bundle, 161

associated, 161 von Neumann theorem, 92

weak submanifold, 176 Weyl spinors, 22, 32 Weyl tensor, 118, 121 Weyl theorem, 138, 172 Wit t decomposition theorem, 3 Wu's proposition, 133

Yang-Mills equation, 130

Z2-principal bundle, 163 C-function for the Dirac operator, 103