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 -^.