Progress in Mathematical Physics

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2 Progress in Mathematical Physics Volume 24 Editors-in-Chiej Anne Boutet de Monvel, Universite Paris VII Denis Diderot Gerald Kaiser, The Virginia Center for Signals and Waves Editorial Board D. Bao, University of Houston C. Berenstein, University of Maryland, College Park P. Blanchard, Universitiit Bielefeld A.S. Fokas, Imperial College of Science, Technology and Medicine C. Tracy, University of California, Davis H. van den Berg, Wageningen University

3 Jan Cnops An Introduction to Dirac Operators on Manifolds Springer Science+Business Media, LLC

4 Jan Cnops Department of Computer Sciences Gent Polytechnic Schoonmeersstraat 52 B-9000 Gent Belgium Library of Congress Cataloging-in-Publication Data Cnops,Jan. An lntroduction to Dirac operators on manifolds / Jan Cnops. p. cm.- (Progress in mathematical physics; v. 24) Inc1udes bibliographical references and index. ISBN ISBN (ebook) DOI / Clifford algebras. 2. Quantum theory. 3. Manifolds (Mathematics). 4. Mathematical physics. 1. Title. II. Series. QC20.7.C '.57-dc CIP AMS Subject Classifications: 15A66, 81RXX, 22E70, 53AXX Printed on acid-free paper Springer Science+Business Media New York Originally published by Birkhlluser Boston in 2002 Softcover reprint of the hardcover 1 st edition 2002 All rights reserved. This work may not be translated. or copied in whole or in part without the written permis sion of the publisher, Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN SPIN Reformatted from the author's files by 1EXniques, Inc., Cambridge, MA l

5 Contents Preface vii 1 Clifford Algebras 1 1 Definition and basic properties 2 2 Dot and wedge products Examples of Clifford algebras 13 4 Modules over Clifford algebras Subgroups Manifolds 25 1 Manifolds Derivatives and differentials 28 3 The Spin group as a Lie group 32 4 Exterior derivatives and curvature 38 5 Spinors Spinor fields Dirac Operators 61 1 The vector derivative 67 2 The spinor Dirac operator The Hodge-Dirac operator 79 4 Gradient, divergence and Laplace operators 81 4 Conformal Maps 91 1 Mobius transformations Liouville's Theorem Conformal embeddings Maps between manifolds 115

6 Vi Contents 5 Unique Continuation and the Cauchy Kernel 1 The unique continuation property. 2 Sobolev spaces The Cauchy kernel The case of Euclidean space 6 Boundary Values 1 The Cauchy transform Boundary values and boundary spinors. 3 Boundary spinors and integral operators Appendix. General manifolds 1 Vector bundles Connections Connections on SO(M). 4 Spinor bundles... Bibliography List of Symbols Index

7 Preface Dirac operators play an important part in several domains of mathematics and mathematical physics. Index theory, theory of elliptic pseudodifferential operators, theory of electromagnetism, particle physics, representation theory of Lie groups: all are touched by notions related to Dirac operators. The innocent reader in one of these-and several other-topics might be baffled by the technical complexity of the material involved, and wish to understand more of the basic ideas underlying the theory of Dirac operators. This book is not meant to be a complete overview of the study of the subject. We have aimed at an exposition of the basic theory of the Dirac operator, and the analysis of its properties, which is clear rather than complete, understandable rather than fully general. We do not assume the reader is well-versed in differential geometry or operator theory. If he is, he might well find the treatment of some topics less general than what he is used to, but on the other hand, he might get to see some things in a different light, and we hope he will find that the main ideas necessary to understand the importance of the Dirac operator are all there. In the first chapter we introduce Clifford algebras. These algebras, also called geometric algebras, are the basic tool to describe geometric objects in finite-dimensional metric spaces and, by extension, on metric manifolds. Clifford numbers are used in much the same way as complex numbers are used in the plane. A vector is defined as an abstract entity having direction and size. But if the dimension is greater than one, we also have k-vectors: k-dimensional entities the direction of which consists of k-vectors, and a k-dimensional volume. The Clifford algebra unites all these k-vectors and is an efficient tool for manipulating them. Moreover, since it incorporates the metric, it allows for an efficient description of orthogonal transformations. In the second chapter manifolds are studied. Fundamental ideas are introduced here. We describe the two main kinds of 'functions' on a manifold: Clifford fields (which can be identified with differential forms) and spinor fields.

8 viii Preface Related to these, the generalisation of derivation to connections on the manifold is given. We have opted for a description using the embedding of a manifold in a metric space. This makes it easier to grasp the essential concepts involved than does the abstract approach; for a general description we have included an appendix which describes everything without resorting to embeddings. To the reader who is not familiar with manifolds, but wants to get acquainted with the theory in its generality we suggest reading the appendix after having worked through the second chapter. To the reader already acquainted with the theory of bundles on manifolds, the second chapter may give a fresh view of the theory, as the framework used allows for a quite visual approach. In the third chapter we define Dirac operators. The fundamental concept here is Stokes' equation, relating integrals of functions on the boundary of a domain to the Dirac operator on the interior of this domain. It is possible to put the notion of a Dirac operator in a general framework, which encompasses not only Clifford and spinor fields, but more general objects. But here, again, we have opted for a more particular approach, which will make it easier for the reader to understand the fundamental ideas. Some related operators, such as gradient, divergence and Laplace operators, are compared with the Dirac operator. The second part of this book, chapters four to six, is dedicated to the study of the properties of Dirac operators, and more in particular to the spinor Dirac operator. In the fourth chapter we consider the relation between Dirac operators and conformal maps. The geometrical nature of Clifford algebras is of great help here, and we describe the conformal maps on a (pseudo)-euclidean space in terms of orthogonal transformations in a higher-dimensional space. Then we consider the transformation of a Dirac operator under a conformal map, showing that it remains essentially unchanged. The last two chapters deal with the invertibility of Dirac operators. In the previous chapters there was no big difference between the Euclidean case, with its definite metric, and the pseudo-euclidean one. But from this point on, the differences are great. On Euclidean manifolds, the so-called elliptical case, Dirac operators have quite strong invertibility and unique continuation properties not shared by pseudo-euclidean Dirac operators. Apart from an example in Chapter 5, where we show that, in ~ 1, there l, exists a limited unique continuation property, pseudo-euclidean manifolds will be abandoned altogether, although it is still possible to use Euclidean manifolds embedded in pseudo-euclidean spaces.

9 Preface IX In the fifth chapter the problem of inverting the Dirac operator is described. A first topic here is the unique continuation property of monogenic functions, the null solutions of the Dirac operator. To what extent do the values of a locally monogenic function on some set determine the values on a larger set? We give some configurations where they do, and show that on some manifolds every globally monogenic function must be identically zero because of this property. Also we show the existence of a function, the Cauchy kernel, which is central in finding the inverse of the Dirac operator. Classically, the Cauchy kernel is set in the framework of distributions, but we have opted for a new approach using Sobolev spaces. This is a more natural environment for the Cauchy kernel, as it immediately links the Cauchy kernel as a means of inverting the Dirac operator with the boundary value theory as given in the next chapter, which also uses the Sobolev space setting. In the sixth chapter we tum our attention to some topics in relation to boundary values. If, for a certain domain, a monogenic function which is zero on the boundary must be identically zero on the domain, then it must be possible to reconstruct an arbitrary monogenic function from its boundary values. It turns out that again the Cauchy kernel plays a central part in this problem. Finally we introduce spinors on the boundary, and show the relation to boundary values of monogenic functions. As already stated, we avoided using technical results from different areas of mathematics. Some properties of Dirac operators mentioned in Chapters 5 and 6 are usually proved using theories of elliptic operators and pseudodifferential operators. To keep the text essentially self-contained, we have avoided referring to these theories. Therefore the required know ledge needed for reading this book is limited. The reader is supposed to be familiar with the basic notions of real analysis, but the specific function spaces needed for the text, such as Sobolev spaces, are explained in full. Of course knowledge of complex analysis will be helpful, since the Cauchy-Riemann operator is the classical example of a Dirac operator, but little if any direct reference is made to complex function theory. Differentiable manifolds as well as Clifford algebras are explained starting from scratch, so no previous familiarity with the notions of this part of mathematics is needed. The book was written while the author was working as a postdoctoral researcher for the Flemish Science Foundation, F.W.O. He also gratefully acknowledges the Department of Mathematical Analysis of the University of Gent, which acted as host institution during that time, and provided him with stimulating discussions and coffee. Members of the Clifford Analysis Group have

10 x Preface all helped tremendously in forming and writing out the ideas contained in this volume, especially R. Delanghe, its director, who encouraged the project and who proofread an early draft. JAN CNOPS Gent, Belgium

11 An Introduction to Dirac Operators on Manifolds