http://dx.doi.org/10.1090/gsm/025 Selected Titles in This Series

25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto 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 Martin 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 Elliott 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 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin 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 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994 2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993 1 Ethan Akin, The general topology of dynamical systems, 1993 This page intentionally left blank Dira c Operator s in Riemannia n Geometr y This page intentionally left blank Dira c Operator s in Riemannia n Geometr y

Thomas Friedrich

Translated by Andreas Nestke

Graduate Studies in Mathematics Volume 25

^™ ^ America n Mathematica l Societ y IK Providence , Rhod e Islan d Editorial Board James Humphreys (Chair) David Saltman David Sattinger Ronald Stern

2000 Mathematics Subject Classification. Primary 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 Geometrie. 1. Auflage (1st edition)" © by Priedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1997

Translated from the German by Andreas Nestke

ABSTRACT. This text examines the Dirac operator on Riemannian manifolds, especially its con• nection 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 rsprint-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 Vlll 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 This page intentionally left blank 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 orien• tation. 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 fol• lowing alternative formula for P:

V* o) dx \-i oy dy

Denoting the matrices occurring in this formula by ^x and jy, 0 1 \z uy • -1 0 yields

9 d

XI Xll 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 yl-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^. Mathemati• cally 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 p'^>ir* 2=1

n 2 Now the equation P = A = — ]T -^ holds if and only if the coefficients 7Z 2=1 l of P satisfy the conditions

1i=-E, i = l,...,n; 7i7j + 7j7i = 0, i^j. Introduction xin

For n = 3, there is an obvious solution to these equations. The vector space c2 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,/' ^=Vl 0^' 73=V* °

The algebra multiplicatively generated by n elements 71,... , 7n satisfying the relations

is called the Clifford algebra Cn (W.K. Clifford, 1845-1879) of the negative n definite quadratic form (R , — x\ — ... — xn). Thus, the question whether there is a square root yfK of the Laplacian leads to the study of complex rep• resentations K : Cn —> End (V) of the Clifford algebra. It turns out that Cn has a smallest representation of dimension dime V = 2^1. The correspond• ing 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 n n on the space C°°(IR ; An) of smooth An-valued functions on R .

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 ei,... , en,

n ,1, x — y x Ci,

and then define its product x • i\) by a spinor i\) G An 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

n for every A E SO(n; M), xGK 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 differ• ential 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 dif• ferential 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 cov• ers 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 Mn, 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 n e : Spin(n) —» GL(An), the so-called spinor bundle of M . 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: n n Wl(M ) = 0 = w2(M ). 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

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 mathe• matics. 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-Uni• versity in Berlin during 1996/97. It contains an introduction into the theory of spinors and Dirac operators on Riemannian manifolds. The reader is as• sumed to have only basic knowledge of algebra and geometry, such as a two or three year study in mathematics or physics should provide. The pre• sentation 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 el• ementary covering theory of topological spaces. At the same time, each result will also be translated into the cohomological language of character• istic 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 op• erators (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 cor• rections 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 Eng• lish edition of this book.

Thomas Friedrich Berlin, March 2000 Bibliography

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absolute differential, 58, 166, 167 complex projective space, 40, 42, 48, 161 adjoint operator, 92 complexification algebra of complex numbers, 9 of a real algebra, 11 algebra of quaternions, 8 of a real quadratic form, 11 almost-complex manifold, 73, 74 conjecture, ^, 131 almost-complex structure, 60, 80, 146, 147, connection, 163, 165, 169 151 holonomy group of, 177 Ambrose-Singer theorem, 178 locally flat, 168 anti-canonical spinc structure, 79, 81 reduction of, 174 associated fibration of a principle bundle, continuous spectrum, 91 159 covariant derivative, 58, 60, 67, 68, 70, 166, associated spinor bundle, 75 167 associated vector bundle, 161 covering spaces, 40 curvature form, 62, 135, 167, 169 Bianchi identity, 168 curvature tensor, 62 bilinear form, 1, 7 direct sum of, 7 de Rham cohomology, 171 index of, 2 integral, 172 nondegenerate, 1 determinant bundle of spinc structure, 52- rank of, 2 54, 108, 113 signature of, 2 Dirac operator, 68, 69, 71, 93, 96, 101, 107, 127, 136, 148 canonical basis, 2 eigenvalues of, 113, 116, 126, 128 canonical connection, 83 G-function for, 103 c canonical spin structure, 79 index formula for, 110 of an Hermitian manifold, 77, 78 index theorem for, 109 c canonical spin (4) structure, 147, 149 spectrum of, 99 Casimir operator, 86, 87 Dirac spinors, 14, 69, 113, 115 Cauchy-Riemann equations, 71 direct sum of bilinear forms, 7 center of an algebra, 9 distribution, 175 characteristic class, 108 integral, 175 Chern class, 108, 141, 163, 171, 172 Clifford algebra, 4, 10, 11 eigenspinor, 102, 103, 112, 114, 126 Clifford multiplication, 21, 32, 53, 68, 70, eigenvalues, 91, 126 133 of the Dirac operator, 113, 116, 126, 128 complete Riemannian manifold, 98 Einstein space, 118 complex n-spinors, 14 ^ conjecture, 131

193 194 Index

equivalent fibrations, 166 Kahler manifold, 61, 81, 82, 89, 116, 147, equivalent A-reductions, 158 150, 151, 153 equivalent spin structures, 35 Killing number, 116, 118, 119 equivalent spinc structures, 51 Killing spinor, 116, 118, 119, 121, 124, 125, essentially self-adjoint operator, 92-94, 96 128 ?7-function, 105, 112 imaginary, 120 for the Dirac operator, 103 real, 120 exponential map, 18 exterior differential, 73 Lagrange theorem, 2 A-reduction, 158, 173 fibration, 156, 159 equivalent, 158 equivalence of, 156 Laplace operator, 71 locally trivial, 155 on spinors, 68 first integral, 124 Levi-Civita connection, 57, 61, 81, 84, 113, frame bundle, 158 125, 135, 148, 164 Fredholm operator, 107 Lie algebra Freudenthal-Yamabe theorem, 177 of Spin{n), 17, 18 Probenius theorem, 174 of Spirf(n), 29 global, 176 linear operator, 91 local, 174, 175 locally flat connection, 168 fundamental group, 26 locally trivial fibration, 155

Gaschiitz proposition, 39 manifold without boundary, 175 gauge field theory, 130, 131 Maurer-Cartan form, 83, 164 gauge group, 135 moduli space, 140, 147 gauge transformation, 135, 169 for Seiberg-Witten theory, 136 Ginzburg-Landau model, 131 G-principal bundle, 156 nondegenerate bilinear form, 1 Grafimannian manifold, 56, 88 null subspace, 3 orientation, 159 harmonic spinors, 81, 82 heat equation, 112 parallel spinc spinor, 67 Hermitian manifold, 75, 78, 147 parallel spinor, 67, 89 c canonical spin structure, 77, 78 parallel spinor field, 67 spinor bundle of, 79 parallel transport, 166, 167 Hermitian metric, 53, 68, 74, 80 Picard manifold, 172 Hermitian scalar product, 24 Pin(n), 15 Hilbert-Schmidt operator, 103, 104 point spectrum, 91 Hirzebruch-Hopf proposition, 133, 146 Pontrjagin class, 108 Hirzebruch signature theorem, 109 principal bundle, 157 holonomy group of a connection, 177 associated fibration of, 159 homogeneous spin structure, 85, 87 G-, 156 homotopy classification theorem, 162 isomorphic, 157 homotopy theory, reduction theorem of, 178 S1-, 163 Hopf bundle, 161 Z2-, 163 Hopf fibration, 157, 161, 172, 173 projective space horizontal lift, 165 complex, 40, 42, 48, 161 index, 108 real, 55, 56 of a form, 2 q-foim of type p, tensorial, 165 index formula for Dirac operators, 110 quadratic form, 1 index theorem for Dirac operators, 109 quaternionic structure, 29, 30, 110 integrable distribution, 175 in An, 32, 54 integral de Rham cohomology, 172 integral manifold, 176 rank of a bilinear form, 2 intersection form, 109, 130 real projective space, 55, 56 isomorphic principal bundles, 157 real structure, 29, 30 isotropic subspace, 3 in An, 32, 54 Index 195

reducible solution of the Seiberg-Witten Spirf{4) structure, 134, 141, 146 equation, 140, 142 canonical, 147, 149 reduction Spirf(n), 25, 26 of a connection, 174 Spinc(n) representation, 28 A-, 158, 173 Spin(n), 15 equivalent, 158 spin representation of, 20 £/(&)-, 48, 60, 61, 81 spinor bundle, 53, 78 reduction theorem of homotopy theory, 178 of an Hermit ian manifold, 77 Rellich lemma, 100 spinor derivative, 59 residual spectrum, 91 spinor field, 67 resolvent set, 92 parallel, 67 Ricci tensor, 64, 118 Stiefel-Whitney class, 163 Riemannian manifold, complete, 98 second, 40 Riemannian metric, 141, 159 structure identity, 168 Riemannian symmetric space, 82, 87 submanifold, 175 Rokhlin's theorem, 110 weak, 176 Sylow subgroup, 2-, 39, 44 S1 -principal bundle, 163 Sylvester's theorem, 2 scalar curvature, 111, 113, 118, 135, 144, symmetric operator, 69, 92, 93 145, 148, 149, 151 symmetric space, Riemannian, 82, 87 Schrodinger operator, 127 symplectic manifold, 158 Schrodinger-Lichnerowicz formula, 73, 100, symplectic structure, 151, 159 110, 113, 134, 145 tangent bundle, 155 Schur-Zassenhaus proposition, 39 tautological bundle over CP\ 161 second Stiefel-Whitney class, 40 tensor product of Z2-graded algebras, 7 section, 156 tensorial 1-form, 62 Seiberg-Witten equation, 131, 134, 136, 138, tensorial g-form of type p, 165 140, 153 twist or equation, 128 reducible solution of, 140, 142 twist or operator, 69, 70, 121 Seiberg-Witten invariant, 144-146, 149, 151 twistor spinor, 121, 123 Seiberg-Witten theory, moduli space for, 136 2-Sylow subgroup, 39, 44 self-adjoint operator, 92 essentially, 92-94, 96 E/(fc)-reduction, 48, 60, 61, 81 spectral theorem for, 93 unitary group, 27 signature, 109 universal covering, 19 of a form, 2 of 50(n), 16 spectral measure, 92, 93 universal G-bundle, 162 spectral theorem for self-adjoint operators, vanishing theorem, 140 93 vector bundle, 161 spectrum associated, 161 of a Dirac operator, 99 von Neumann theorem, 92 of an operator, 91, 92 sphere, 43, 88, 116, 125, 128 weak submanifold, 176 spin bundle, 54 Weyl spinors, 22, 32 spin representation, 14, 23, 25, 54, 58, 75, Weyl tensor, 118, 121 133 Weyl theorem, 138, 172 of Spin(n), 20 Witt decomposition theorem, 3 spin structure, 35, 36, 38-40, 42-45, 47, 50, Wu's proposition, 133 53-55, 60, 79, 113 Yang-Mills equation, 130 equivalence of, 35 homogeneous, 85, 87 Z2-principal bundle, 163 spinc structure, 47, 48, 50, 51, 53, 57, 60, C-function for the Dirac operator, 103 78, 93, 96, 111, 131, 153 canonical, 79 of an Hermitian manifold, 77, 78 determinant bundle of, 52-54, 108, 113 equivalence of, 51