MATHEMATICS: THEORY & APPLICATIONS
ELLIPTIC BOUNDARY PROBLEMS FOR DIRAC OPERATORS BERNHELM Booji-BA VNBEK KRZYSZTOF P. WoJcIEcHowsKl
BIRKHA USER
Mathematics: Theory & Applications Editors
Richard V. Kadison Isidore M. Singer
Bernhelm BooB-Bavnbek Krzysztof P. Wojciechowski
Elliptic Boundary Problems
for Dirac Operators
Birkhäuser Boston • Basel • Berlin
1993
Bernhelm BooB-Bavnbek IMFUFA Roskilde University 4000 Roskilde Denmark
Krzysztof P. Wojciechowski Department of Mathematics IUPUI Indianapolis, IN 46202 USA
Library of Congress Cataloging In-Publication Data Booss-Bavnbek, Bernhelm, 1941Elliptic boundary problems for Dirac operators I Bernhelm Booss -Bavnbek, Krzysztof P. Wojciechowski cm. -• (Mathematics) p. Includes bibliographical references and index. ISBN 0-8176-368l-l (H : acid-free). -- ISBN 3-7643-3681-3 (H acid-free) I. Differential equations, Elliptic. 2. Boundary value problems. 3. Dirac equation. I. II. Title. III. Series:
QA377.B686 1993 515'.353--dc2O
Wojciechowski, Krzysztof p., 3953. Mathematics (Boston. Mass.) 93-22006
Birkhäuser Boston 1993
Printed on acid-free paper
CIP
Birkhä user
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Contents Preface List of Notations
vii
xv
I. CLIFFORD ALGEBRAS AND Diit&c OPERATORS
1. Clifford Algebras and Clifford Modules 2. Clifford Bundles and Compatible Connections 3. Dirac Operators 4. Dirac Laplacian and Connection Laplacian 5. Eudidean Examples 6. The Classical Dirac (Atiyah-Singer) Operators on Spin Manifolds 7. Dirac Operators and Chirality 8. Unique Continuation Property for Dirac Operators 9. Invertible Doubles 10. Clueing Constructions. Relative Index Theorem
3 10 19
26 29 36 40 43 50 59
PART II. ANALYTICAL AND TOPOLOGICAL TOOLS
11. Sobolev Spaces on Manifolds with Boundary 12. Calderón Projector for Dirac Operators 13. Existence of of Null Space Elements 14. Spectral Projections of Dirac Operators 15. Pseudo-Differential Grassmannians 16. The Homotopy Groups of the Space of Seif-Adjoint Fredhohn Operators A. Elementary Decompositions and Deformations B. The Homotopy Groups of C. 17. The Spectral Flow of Families of Seif-Adjoint Operators A. Continuity of Eigenvalues B. The Spectral Flow on Loops in F. C. Spectral Flow and Index D. Non-Vanishing Spectral Flow
67 75 95 105 111
127 133
138 140 145 157
III. APPLICATIONS
18. Elliptic Boundary Problems and Pseudo-Differential Projections 19. Regularity of Solutions of Elliptic Boundary Problems
163 180
vi
Contents
20. Fredhoim Property of the Operator AR 21. Exchanges on the Boundary: Type Formulas and the Cobordism Theorem for Dirac Operators 22. The Index Theorem for Atiyah-Patodi-Singer Problems A. Preliminary Remarks B. Heat Kernels on the Cylinder C. Duhamel's principle. Heat Kernels on Manifolds with Boundary D. Proof of the Index Formula E. L2-Reformulation F. The Odd-Dimensional Case. A Three-Dimensional Example 23. Some R2marks on the Index of Generalized Atiyah-Patodi-Singer Problems 24. Bojarski's Theorem. General Linear Conjugation Problems 25. Cutting and Pasting of Elliptic Operators 26. Dirac Operators on the Two-Sphere
188 .
205
211 214 231 239
242 248 253 262 276 282
Bibliography
289
Index
303
Preface
Elliptic boundary problems have enjoyed interest recently, especially among C-algebraists and mathematical physicists who want to understand single aspects of the theory, such as the behaviour of Dirac operators and their solution spaces in the case of a non-trivial boundary. However, the theory of elliptic boundary problems by far has not achieved the same status as the theory of elliptic operators on closed (compact, without boundary) manifolds. The latter is nowadays rec-
ognized by many as a mathematical work of art and a very useful technical tool with applications to a multitude of mathematical contexts. Therefore, the theory of elliptic operators on closed manifolds is well-known not only to a small group of specialists in partial differential equations, but also to a broad range of researchers who have specialized in other mathematical topics. Why is the theory of elliptic boundary problems, compared to that on closed manifolds, still lagging behind in popularity? Admittedly, from an analytical point of view, it is a jigsaw puzzle which has more
pieces than does the elliptic theory on closed manifolds. But that is not the only reason. Having worked with the topology, geometry, and analysis of elliptic boundary problems for many years, the authors feel that the existing books on elliptic boundary problems, excellent as they are, present the theory in too great a generality and by means of too intricate a machinery for our intended readership: graduate students in, for example, geometry, topology or mathematical physics, and researchers from fields other than partial differential equations. They will need insight into specific situations, which involve manifolds with boundary and their Dirac operators. Our aim is to give these readers a book that presents the absolute minimum of the necessary machinery and illustrates how the methods work with interesting examples.
To follow our presentation, the reader should be familiar with the basic calculus of pseudo-differential operators on closed manifolds. Knowledge of the first pages of P. Cilkey's book [1984; pp. 1—36] (new edition in preparation) is quite satisfactory for our purpose.
viii
Preface
Our Chapters 17 and 22—26 furthermore presuppose knowledge of the
index theory of elliptic operators on closed manifolds. Once again Gilkey's book provides the reader with all needed material. As an auxiliary reference, we also refer to BooB & Bleecker [1985].
Analytically, the heart of our book consists of deep results proved long ago by A. Calderón and R. Seeley: in our book Theorem 12.4, Theorem 13.1, Corollary 13.7, and Theorem 13.8. Our presentation closely follows, with some simplifications from elementary constructions developed in our previous work, the original work of Seeley (see Seeley [1966], [1969] and Palais [1965a; Seeley's contribution to Chapter 17]). Inevitably, some proofs are relatively long and complicated,
but still we hope that the aforementioned simplifications make the theory of elliptic boundary problems for Dirac operators accessible to a larger audience. In fact, even the theory seems to benefit from those simplifications. They lead, for example, to recognizing that the Cauchy data spaces of a Dirac operator and its adjoint are orthogonal after Clifford multiplication. This is crucial to almost all index calculations presented in the text.
The text itself consists of 26 chapters, divided into 3 parts. The first part — Chapters 1-10 — gives a detailed exposition of Dirac operators acting on sections of bundles of Clifford modules. In the first 7 chapters, we closely follow standard references like Atiyah, Bott & Shapiro (1964], Lawson & Michelsohn [1989], and especially Gillcey [1989] and Branson & Gilkey [1992a]. It is worth mentioning that Chapter 3 distinguishes three different classes of operators. We say that A is an operator of Dirac type, if the principal symbol of A2 is defined by the metric tensor; A is a generalized Dirac operator, if A is the composition of a connection operator and Clifford multiplication; and we call A a compatible Dirac operator, or in short, a Dirac operator, if the connection used to define A is compatible with Clifford multiplication. For simplicity, all results in the book are proved only for Dirac operators, but they also hold for operators of Dirac type. In fact, except for certain index formulas, they hold even for arbitrary linear elliptic operators of first order. The reason why some of our index formulas do not hold for all first-order elliptic operators is that they are computed by reducing the index problem to a problem on the boundary. That works when the unique continuation property (or uniqueness of the Cauchy problem—UCP) for the solutions holds. For general elliptic operators, the lack of UCP implies that not all solu-
Preface
ix
tions have non-trivial traces on the boundary. That would lead to an additional term in our index formulas.
Chapter 8 proves the simplest among different variants of the unique continuation principle for Dirac operators. Let v be a solution of the equation Av = 0, where A is a Dirac operator on a connected
manifold. Assume that v =
0
on some open set; then v =
0 every-
where. The proof closely follows the beautiful exposition in F. Treves [1980], which he based on ideas due to Calderón. Typically, we apply the UCP to a solution v of Ày = 0 with v = 0 on a closed submanifold of codimension 1, so that v vanishes everywhere. The result allows the reducing of index computations for elliptic boundary problems or elliptic transmission problems (general linear conjugation problems) to calculations on the boundary. Chapter 9 shows that any Dirac operator on a compact manifold
with boundary extends to a Dirac operator on a closed manifold. Moreover, the constructed extension is an invertible operator. That greatly simplifies the proof of the Calderón-Seeley theorem, the crucial analytical result of our book (Theorem 12.4).
Chapter 10 applies the glueing construction of Chapter 9 to the relative index theorem and to the excision property for Dirac operators.
The second part of the book — Chapters 11-17 — presents some basic analytical and topological tools that enable us to analyse the nature of the index of elliptic boundary problems. In Chapter 11, we investigate Sobolev spaces on manifolds with boundary. Let v be a section on a compact manifold X with boundary Y and let v belong to the Sobolev space H1. Then the main result is that v, provided
8>
has
a well-defined trace v y, which is an element of the Sobolev
space H8 The crucial analytical result is contained in Chapter 12: Let A be
a Dirac operator on a compact manifold X with boundary Y. First we construct a Poisson type operator K+, mapping sections on Y to (smooth) solutions of A in X \ Y. Then we obtain the Calderón operator by restricting to the boundary Y, where h is a section on Y. We show that is a projection onto the space H(A) := {vly Av = 0 in X \ Y} of Cauchy data of the operator A. Moreover, is a pseudo-differential operator and K+ provides an isomorphism of H(A) with the space of solutions. In the modern literature, the operators and K+ are often constructed modulo smoothing operators. This leads only to approximate projections and
x
Preface
approximate isomorphisms, but it simplifies the construction and is still sufficient to prove the regularity theorems and develop the index theory. However, this is the point where usually extra machinery appears, and generality is achieved at the expense of precision and detail. Therefore we chose a different path. The most important properties of and are collected in Theorem 12.4. The proof of Theorem 12.4 is rounded off in Chapter 13, which, among other results, proves the existence of a well-defined trace on the boundary for any solution of Av = 0. Chapter 11 showed that v has a well-defined trace, if v belongs to the Sobolev space H with s> Here we show that for any s, an element v of H3 with Av = 0 in X\Y has a well-defined trace in on Y. Chapter 14 investigates another operator, which lives on the boundary and is induced by the tangential part B of A. B is a total Dirac operator, and in particular it is seif-adjoint. Then, for any real number a, the 3pectral projection P>a of B onto the interval [a, oo) is a pseudo-differential operator which differs from P+ by a compact operator. For the operator A, these spectral projections define specific boundary conditions first considered by Atiyah, Patodi & Singer in [1975]. We discuss generalized Atiyah-Patodi-Singer problems in the third part of the book. In Chapter 15, we switch to topological tools and introduce the Grassmannian Cr, which is the natural object for studying the index theory of elliptic boundary problems. The Crassmannian is the space of all pseudo-differential projections, which differ from by a compact operator. The idea of studying the infinite-dimensional Grassmannian is due to B. Bojarski [1979). We investigated the topological structure of Cr and found applications to the cutting and pasting of the index of elliptic operators in a series of papers (see Wojciechowski [1981], [1985a], [19861, and BooB & Wojciechowski [1985b], [1986];
more recently, in the context of spectral estimates, BooB & Wojciechowski [1989] and Douglas & Wojciechowski [1991; Appendix B]). The Grassmannian appeared in other contexts in the works of other authors (see, for example, Sato, Miwa & Jimbo [1979], Sato [1981], Segal & Wilson [1985), and Pressley & Segal [1986]). Chapter 15 stud-
ies the topological structure of Gr and shows that, for P E Cr, the operator PP÷ (acting from the range of P÷ to the range of P) is a Fredholm operator and that its index i(P, distinguishes the connected components of Cr. For P, orthogonal, the integer i(P, P÷) can also be interpreted as the spectral flow of a corresponding family {(2P — Id) + — P)} of seif-adjoint operators.
Preface
xi
That is why Chapter 16 studies the topology of the non-trivial component F. of the space of self-adjoint &edholm operators. It is well known that F. is a classifying space for the functor K' (see Atiyah & Singer 119691). To avoid the original and quite complicated proof, we present instead a computation of the homotopy groups of
that space. In particular, the first homotopy group is equal to the integers. This is all we need to know in this book. Chapter 17 is devoted to the study of the spectral flow of a family in F.. It is a nice feature in {Bt}tEsl which is just the class of that it has a straightforward analytical interpretation: the number of the elgenvalues of which change their sign from — to + as t goes along S', minus the number of eigenvalues which change their sign from + to —. We discuss various topological formulas, giving the specis a family of elliptic operators of non-negative tral flow in case order. Such families arise naturally, when elliptic boundary problems or elliptic transmission problems induce specific parametrizations of the tangential parts of Dirac operators in a collar of the boundary, reap. the partitioning submanifold.
That is explained in the third part of the book — Chapters 18—26 — which presents the basic theory of elliptic boundary problems and other applications of the machinery that we have developed. Chapter 18, following the beautiful lectures of Seeley [1969], introduces the concept of an elliptic boundary condition for a Dirac operator A. It is a pseudo-differential operator R (of order 0), which satisfies a specific compatibility condition with the operator A. Following Seeley, we furthermore assume a technical condition, namely that the range of the operator R is closed in suitable Sobolev spaces. This last condition implies, by an argument from M. Birman's and A. Solomyak's work [1982] on elliptic fans, that the orthogonal projections onto the range and the kernel of the operator R are pseudodifferential operators. One consequence is that the boundary integral RP÷ : H(A) range(R) is a Fredhoim operator (Corollary 18.15). In that way it brings the index theory of elliptic boundary problems back to the Grassmannian Cr. Chapter 19 investigates the unbounded operator AR : dom AR —, L2 — the L2-realization of A under the boundary condition R(vly) = is finite-dimensional 0— and shows that the space of solutions of (Corollary 19.2). Moreover, Chapter 19 proves that any solution of AR is smooth up to the boundary and shows the range of the operator AR to be closed.
xii
Preface
The structure of the index of the operator AR is discussed in Chapter 20. First we show that AR is a closed operator in L2 and then we construct the adjoint operator. A nice consequence of Seeley's aforementioned technical condition is the explicit form obtained for the adjoint elliptic boundary condition Q which gives (AR)* = (A*)Q. That way around we show that AR : dom AR L2 is a Fredhoim operator with kernel and cokernel consisting of smooth sections. We reduce the index computation of a global elliptic boundary problem to computing the index of the composition of two projections over the boundary which differ by a compact operator: (0.1)
index AR = dim kerAR — dim = index{RP÷ :
—* rangell}
=
Then we explain the relations between the indices of the realization AR and the full mapping pair (A, R) in the case of local elliptic bound-
ary conditions i.e. when the principal symbol of the boundary condition can be written as the lifting of an isomorphism of the original vector bundles to the cotangent sphere bundle. It is worth mentioning that the boundary conditions we call local elliptic are called elliptic or conditions in the classical literature on partial differential equations. Those boundary problems which we, in accordance with the terminology of global analysis, call global elliptic or simply elliptic were either not treated or are a subclass of what is called injectively elliptic in certain parts of the literature (see e.g. Grubb [1990]). Notice also that local elliptic boundary problems, for which the designation elliptic used to be reserved, play an important but secondary role for Dirac operators where they only appear in odd dimensions or for systems and transmission problems. Chapter 21 uses formula (0.1) to prove the Agranovië-Dynin formula index — index AR2 = index RiP+(R2)*. Next, the Agraformula is used to prove the cobordism theorem for Dirac
operators, which says that the index of a Dirac operator, which bounds, is equal to 0. Historically, the theorem was the basis of the first proof of the Atiyah-Singer index theorem (see Atiyah & Singer [1963) and also Palais [1965a]). That supports the perspective, elaborated in the rest of the book, of looking at the seemingly more elementary index theory of elliptic operators on closed manifolds through the glasses of the seemingly more intricate, but in reality more fundamental, index theory of elliptic boundary problems.
Preface
xiii
The Atiyah-Patodi-Singer index theorem, the index formula for the operator is investigated in Chapters 22 and 23. Recall that
:= Po denotes the spectral projection of the tangential part B of A onto the interval [0, +oo). The operator Ap> finds so many applications in geometry, topology, and number theory that it deserves a separate treatment. (For one approach, aimed specifically at manifolds with corners, see Melrose 11992]). We only prove the index formula for Ap>. In doing so, we follow the original paper by Atiyah, Patodi & Singer [1975], beginning with the Atiyah-Bott formula
= trexp(_t(Ap>)* Ap>) — trexp(—tAp> (Ap>)*). We obtain the corresponding heat kernels by glueing the heat kernels on the closed double X of X with the heat kernels of the corresponding operators on the cylinder Y x [0, +oo). Therefore, in the formula we have a contribution from the interior, a contribution from the cylinder, plus the error term. We take the interior contribution from the local index theorem for Dirac operators, but compute the cylinder contribution. The only problem left is to show that the error disappears, as t approaches 0. Here we use Duhamel's principle, which is explained in detail in Chapter 22C. We could have used the finite propagation speed property, but we chose the more classical approach still used in recent literature. Chapter 22 is rounded off with an L2-reformulation of the results and a discussion of a three-dimensional example treated before by Taubes [19901 and Yoshida [19911. Having proved the Atiyah-Patodi-Singcr formula, we use it together
with the Agranovië-Dynin formula in Chapter 23 to obtain an index formula for Ap, where P is an arbitrary element of the Grassmannian, and to discuss the additivity of the index of Dirac operators with spectral boundary conditions. This allows the writing of the signature of a 4k-dimensional compact oriented manifold with boundary as the true index of an elliptic boundary problem - without the error term coming from the kernel of the tangential part of the signature operator. In Chapter 24, we continue discussing additivity problems and the index of Dirac operators on partitioned manifolds. We prove a theorem conjectured by Bojarski (1979]: Let M be a closed manifold and Y a submanifold of codimension 1, which divides M into two parts M = U X_, where fl X_ = Y. Let denote the restriction of A to Then we have the formula index A = index{(Id = index
:
(A)
H_ (A)1 }
Preface
xiv
where 'P4. denotes the Calderón projector of A4. and C denotes Clifford multiplication by the unit normal vector. We also discuss a formulation of this theorem in the language of Fredholm pairs and then proceed to the general linear conjugation problem. We are looking for s....), which are solutions of the following problem on M couples
A+s+=0 (0.2)
and
As_ = 0 s_=Ts+
inX4. in X_
onY,
where T is a suitable bundle isomorphism. Bojarski's theorem constitutes the special case of T = C. We prove the formula
indexLcp(A,T) = indeXLcp(A,G) + indexl'÷TP4., under a certain consistency assumption on the isomorphism T, where indexLcp(A, T) denotes the index of the general linear conjugation problem (0.2).
Chapter 25 investigates the underlying geometrical problem of surgery — cutting and pasting — of Dirac operators, that is, we cut M along Y and glue it differently by using a diffeomorphism and a corresponding automorphism of the bundle of Clifford modules. Assuming consistency of the data on Y, we obtain a new Dirac operator and come up with a formula for the difference of the indices. The final Chapter 26 illustrates those results, using a simple, but multifaceted example. We investigate the case of M = S2, which is a perfect illustration of all the phenomena discussed in this book. It might be a good idea to start reading this book with a quick look at this chapter.
Many colleagues and students have helped to improve this book, and we heartily thank them all. Our emphasis on Dirac operators and the related geometrical concepts owes much to remarks by Peter Cilkey. Special thanks also to Ron Douglas, who suggested writing this book, to Ulrich Bunke, Jerry Kaminker, Slawomir Klimek, and Anders Madsen for continued and inspiring discussions of many details, to Ingrid Jensen for drawing the figures, to Birthe Wolter for typing, and to Krystyna and Wiadek Uscinowicz for providing us with housing in Szczecin for our monthly meetings some years ago when we began our joint work.
List of Notations a-index, analytical index, 146 A, (total) Dirac operator, 20 partial (chiral, split) Dirac operators, 41 invertible extension to closed double, 51 Aleft, A right, quaternion Dirac operators, 33 Amin, Amax, minimal and maximal extension, 196-198 realizations, 165, 180, 195 AR, A, elliptic operator after cutting and pasting, 277 glued operator, 60 A(N, As, Atiyah-Singer operator, 37 automorphism groups, 268 Aut, A, Atiyah-Patodi-Singer realization, 211 index density, 60 reduced operator of order 0, 139 B', suspension, 148 B, algebra of bounded operators, 129
ch, Chern character, 146 c, Clifford multiplication, 5 c0,. , c3, Clifford matrices, 34 •, P>), spaces of smooth sections C000, on semi-infinite cylinder, 216 C€m, Clifford algebra of Rm, 29 Ct(V, q), Clifford algebra of (V, q), 3 Ct(X), Clifford bundle over X, 10 chiral components, 4 C, Calkin algebra, 129 dA, covariant derivative, 250
Levi-Civita connection, 13 Ds, spin connection, 15 covariant derivative with respect to v, 13 D, 7)', normalized cylindrical Atiyah-Patodi-Singer problems, 215 classical Dirac operator (on spin manifolds), 37 'PA, twisted signature operator, 250 Laplace-Beltrami operator, 21 cylindrical Dirac Laplacians, 222 partial quaternion Dirac operators, 35 5, Dirac distribution, 80, 224
e+, extension by zero operator, 72
List of Notations
erfc, complementary error function, 227 E", bundle after cutting and pasting 277 Eli, EU0,, eUiptic seif-adjoint operators of positive order, 142, 153 ettx, special group of invertible elliptic operators, 112 6tiX invertible elliptic operators, 116 subgroup of (,continuous extension operator of codimension one, 72
£(t,x,y), heat kernel, 59
£, e.,
heat kernels on manifolds with boundary, 214, 237
, heat kernels on the semi-infinite cylinder, 223 ea, ed., heat kernels on the closed double, 231
77B, eta-function, 229
too, deformation retract of
140
Fred2(H), Fred2H÷ (H), spaces of Fredhoim pairs of subspaces, 265 F, space of Fredholin operators, 127 F., spaces of seif-adjoint Fredhoim operators, 128 if F, Thom isomorphism, 150 g, genus of Riemann surface, 249 C, Green's form, 24 c, group of units of Calkin algebra, 129 Q, seif-adjoint units of Calkin algebra, 129
C, C±, C, unitary retracts, 132 special groups of invertible operators, 113 bounded invertible operators, 266 Grassmannian of pseudo-differential projections, 111 connected component of the Grassmannian which contains P.,., 116 full Grassmannian, 133 Gr(H), standard trace operators, 68, 95, 180 r, Gamma-function, 229 r5, Gamma-five matrix, 8 Christoffel symbols, 15 kernel dimensions, 242, 245 Hk, higher Hopf bundle, 283 H, cohomology functors, 272 H'(R'), H8(X), H8(Y), Sobolev spaces, 67, 216 H±(A), H±(A, s), spaces of Cauchy data, 76 spaces of limiting values, 244 h,
index(., .), index of Fredhoim pair, 263 l(I'2, P1), virtual codimension, 119 orthogonal projection onto the range of R, 166
J, reflection operator, 72
List of Notation8
8), spaces of distributional solutions, 76
K(M), Crothendieck group of complex vector bundles, 128
K1(M), higher K-group, 146 Ko(X, Y), relative K-homology, 58 Poisson type operator, 78 K, ideal of compact operators, 129 A(t), symmetric integral, 223 u, y), symmetric heat kernel, 223 £+, continuous extension operators of codimension zero, 72 L2-ker, L2-solution spaces, 243, 244
L2-index, index of L2-solutions on non-compact manifold, 158, 243 LCP(A, 4), linear conjugation problem, 268 CR, linear span, 31 A(V), A(X) = A(TX), exterior algebra, 8, 41 A+, Riesz operator, 96
M', manifold after cutting and pasting, 277 range of the boundary projection symbol, 90, 164, 194 M(N, C), full matrix algebra, 29 NR, orthogonal projection onto the kernel of R, 166 V, covariant differentiation in exterior algebra, 14 connection 1-form, 15 w, standard orientation form, 40 spaces of differential forms, 41
p.4., boundary projection symbol, 79 p, canonical projection onto Calkin algebra, 129 Pa, P<, spectral projections, 105 f orthogonalized projection, 93 Proj°°(H), space of non-trivial projections, 266 Calderón projector, 79 p5, basic operators on A(V), 8 f, 31 f p,
restriction operator, 67, 71 curvature tensor, 26 space of realizations, 196 su(2), Lie algebra of SU(2), 250 sf, spectral flow, 141 f sign(M), signature of 4k-dimensional manifold, 258 spec, spectrum, 107 essential spectrum, 129
xvii
List of Notations
str, symmetric trace, 59 spinors of even/odd chirality, 40 S±, bundles of extended spinors, 53 associated bundle over spin manifold, 37 (73, Pauli matrices, 33 OB, principal symbol of operator B, ((7B], symbol class, 147 stable symbol class, 148
19
t-index, topological index, 150 tr, trace of operator, 60 correcting operator after cutting and pasting, 279 r, Todd class, 150 U, Up÷, spaces of unitary operators, 133
X, closed double, 51 non-compact Riemannian elongation, 243 Euler characteristic, 280
PAirrI
Clifford Algebras
and Dirac Operators
1. Clifford Algebras and Clifford Modules
We study bundles over a point, recalling the definition of the Clifford algebra C £( V, q) of a real vector space V of dimension m equipped with a positive definite inner product q; the Z2-grading of Clifford algebras is shown, followed by an introduction of complex representations of Clifford algebras and the concept of complex V, q)-modules and of Clifford multiplication; the isomorphism classes of irreducible Ct( V, q)-modules are studied.
We begin by fixing the notation: Let V be a real vector space of finite dimension m equipped with a symmetric positive definite bilinear form q. Think of V as the tangent space of a Riemannian manifold in a given point endowed with the corresponding Euclidean metric q. For most of the following results the positivity is not needed and we may take V = R4, e.g. the Minkowski space quadratic form instead of the Euclidean. Denote the Clifford algebra of (V, q) by
Ce(v, q) or in short, C€(V). It is the associative algebra with unit generated by the elements of V (and the identity 1) under the relation
v•w+w.v=—2q(v,w)1, v,WEV.
(1.1)
Assume that Ct(V, q) is not generated by any proper subspace of V.
Remarks
1.1. (a) We have the following reformulation of our defi-
nition. Let (1.2)
T(V) := R
V
(V ® V)
...
denote the complete tensor algebra of V, and define Iq(V) to be the two-sided ideal in T(V) generated by all elements of the form v v runs through V (or — equivalently — generated by all elements of the form v 0w + w 0 v + 2q(v, w), where v and w run through V). Then we have (1.3)
C€(V,q) = T(V)/Iq(V).
(b) The canonical map V —+ Ci(V, q), given by the composition
I. Clifford Algebras and Dirac Operators
4
is an injection. Moreover, if {v1,.. . , } is a basis for V, then 1 together with the products ... = 0 or = 1) } (with form a basis for Ce(V, q); hence in particular dim C€(V, q) = For •
further relations between the Clifford algebra C€(V, q) and the exterior algebra AV of V, see Example 1.4 below. (c) The Clifford algebra C€(V, q) is up to isomorphisms characterized by the following properties: (1) Ci(V, q) is a real algebra with unit; (2) V is contained in Ct(V, q) and generates C1(V, q); (3) v v3 + v3 for a chosen q-orthonormal basis for V; = (4) C€(V, q) is not generated by any proper subspace of V. (d) An equivalent characterization of Ct(V, q) is obtained by the following universal property C€(V, q) is the unique associative (real)
algebra with the property that any linear map f of V into an associative (real) algebra A with the unit extends uniquely to an algebra A, if f satisfies f(v). f(v) = —q(v,v)1 for all v E V.
homomorphism f : C€(V, q) (1.4)
V '—
Ct(V,q)
Fig. 1.1 The universal property of C€(V, q)
(e) Following (b) we see that Ce(V, q) decomposes into the direct sum (1.5)
Ce (V, q) are the subspaces spanned by those elements of the basis which have an even, resp. odd number of exponents equal to 1. The Z2-grading can also be obtained from (d), namely, by extending the linear map := —v from V into Ct(V, q) to an automorphism a: C€(V,q) —' C€(V,q) which in fact is an involution (&2 = Id). Then q) are the ±1 eigenspaces of a. Clearly (V, q) is a subalgebra, and we have that Ct C and cr = or c or. (f) Let V' be a subspace of V of codimension 1, let q' := qlv'xv' donote its inner product, and choose I V'. Then the correspondence
or
.
v++v—I_pv++v—.vn,
v±Ece±(vI,qF)
is a Clifford algebra isomorphism of Ce(V', q') with CtF(V, q).
1. Clifford Algebras and Clifford ModuLes
5
Recall the concept of Clifford modules and Clifford multiplication.
Definition 1.2. (a) A C-representation of the Clifford algebra C€(V, q) is an R-algebra homomorphism
c : Ct(V,q)
Homc(W,W)
into the algebra of C-linear transformations of a finite-dimensional complex vector space W. Any vector space W together with a C-representation of C€(V, q) in Homc(W, W) is called a (complex) ce(v, q)-module and we shall use the terms C-representation of Ct(V, q) and C€(V, q)-module synonymously. We shall refer to the corresponding action
ço.w:=c(ço)(w) for çoEC€(V,q)andwEW as Clifford multiplication of w by p. (b) A C-representation c : q) —' Homc(W, W) is called reducible, if the vector space W can be written as a non-trivial direct sum W=W1EBW2
such that C W3 for j = 1, 2 and for all ço E Ct(V, q). := c(p)1w1 for In this case one can write c = c1 where
j
= 1, 2. A representation will be called irreducible, if there are no proper invariant subspaces. (c) Two representations:
j = 1, 2,
c3 : C€(V, q) —, Homc(W3, W,),
are said to be equivalent, if there exists a C-linear isomorphism F: W1
W2
such that Fci(ço)F' =
for all
E C1(V,q).
The following result about the isomorphism classes of irreducible C1(V, q)-modules is usually derived from general properties of simple (matrix) algebras. Here is a direct proof.
Lemma 1.3. Let V be a real vector space of dimension m equipped with a positive definite inner product. Then every (complex) C€(V)module decomposes uniquely as a sum of irreducibles. More precisely:
(a) If m =
2n, then there is (up to isomorphisms) exactly one irre-
ducible Ct(V)-module
and we have dim
=
1. Clifford Algebras and Dirac Operators
6
(b) If m =
2n + 1, then there are (up to isomorphisrns) exactly two
irreducible Ci(V)-modules and we have = Moreover, c(r) = ±1 on for any normalized orientation r of V. Note.
of V defines a normalized
Any orthonormal basis
orientation
=
(1.8)
v1 ....
.
of V. Here [-] denotes the greatest integer function; the normalization
is chosen so that r r =
moreover, r commutes with V in Ct(V) if m is odd, and anti-commutes if m is even. 1;
Proof of Lemma 1.3. Begin with the case m =
2n and fix an for V. Consider a (complex) C1(V)orthonormal basis {vi,... , module W and the related representation c : Ct(V) Homc(W, W).
Let
and (1.9)
a,, := i c2,._1 c2,, for 1
One sees that the {a,,} are a commuting family of involutions in Hom(W, W) inducing a decomposition of W into complementary ±1 eigenspaces of a,, (1.10)
If e = (es, .. . ,e,) is a choice of signs, we have (1.11)
where
W(e):={wEWIa,,w=e,wforl
are the simultaneous eigenspaces and the sum in (1.11) is taken over all sign combinations. Let := (+,... , +). Then multiplication by c(vfr)) defines isomorphisms between W(e0) and W(e), where
vfr) :=
fl
v2,,.
1. Clifford Algebras and Clifford Modules
7
if e, = +, and anti-
The reason is that c(v(e)) commutes with
commutes with a1, if e1, = —. Consequently we get from (1.11)
dimW =
(1.13)
Let
2';
.dimW(eo).
,ç,z} be a basis for W(eo). For any a' e {1,... ,a} we
shall define an irreducible C€(V)-module W0 such that W = and all W, are mutually isomorphic. Set
Wa
W, := £C{C(v(E))çoa}c, where {c(v(e)),
denotes the linear span. This space is invariant under Since these elements generate Ct(V) as an algebra,
the complex vector space Wa is a C€(V)-module and is actually irreducible by construction. This yields the decomposition W = Furthermore, the map
extends to a Ce(V)-module isomorphism between W1 and Wg for any a; so := W1 and dims = 2'; because of (1.13). This ends the proof
of(a). Now let m = 2n + 1. Recall that the normalized orientation r is W" decomcentral in C€(V) and that r2 = 1; hence W = poses into the ±1 eigenspaces of c(r). Then are C1(V')-modules, where V' := vm_i). Now use (a) to decompose W± into . . , the direct sum of Since resp. a irreducible Ce(V')-modules {V', r} generates C€(V) as an algebra, we obtain W = as
. ..
. ..
a Ct(V)-module, where the first sum consists of a+ copies of
(where the representation is provided by c) and the second of a copies (where the representation is provided by — c). Then (b) follows.
Notice that, for m even, there is only one irreducible C€(V)module and that c) and — c) are isomorphic. Since c(r) anticommutes with c(V), c(i-) provides the isomorphism. [Fm is odd, then the two irreducible V)-modules := (s', c) and &. (s', — c) are not isomorphic. fl
1. Clifford Algebras and Dirac Operators
8
Examples 1.4. (a) Our first example of a Clifford module is the itself. (complexified) Clifford algebra (b) As a second example, recognize also that the exterior algebra AV (omit the complexification) is a Ct(V)-module. Begin by specifying how V acts on AV. Let {v1,... , Vm} be an orthonormal basis for V and let denote its dual basis, i.e. = Consider the corresponding elements of Hom(AV, AV), namely,
ext,.1 = mt,,1 =
int(v) :=
exterior multiplication by the vector interior multiplication (contraction) by the covector v.
which implies that c14 := ext,,1 — Then ext,,, int,, + ext,,1 = satisfies the defining relation (1.1) for the generators of Ce(V). Hence c of V into the action a '—b Hom(AV, AV) which satisfies the Clifford condition (1.4) and uniquely extends across the whole of C€(V) by the universal property (see Remark 1.1.d above).
(c) The two representations id : C4V) —, Hom(Ct(V), C€(V)) and
C: C€(V)
Hom(AV, AV)
are equivalent under the isomorphism
c'
'—'
extj(1),
where I denotes a multiindex. Notice that F maps the unit 1 E Ct(V) onto the identity 1 A°V and that
when I'5 :=
denotes the Gamma-five matrix, as in physics terminology. Considered as an element of (the complexifled) Clifford algebra C€(V), it is equal to the normalized orientation introduced in (1.8). (d) If m = 2n is even, the representation of C€(V) in Hom(AV, AV) provides an alternative construction of the irreducible Ct(V)-module introduced in Lemma 1.3. Let c1 . . Cm
and
1. Clifford Algebras and Clifford Modules
for ,u = 1,.. . ,n and define
:=
9
with
Observe that
a basis of the (complexified) Ct(V)
and that so dimc
= £c{p'ps}. = 2's, and C€(V)
irreducible.
Hence
,
i.e.
p's) as vector space,
is simple and
2. Clifford Bundles and Compatible Connections
We show how the natural operations for vector spaces with quadratic forms carry over to vector bundles with metrics. For a Riemannian manifold X (with or without boundary), we obtain the Clifford bundle Ct(X) := Ce(TX,9). We show that there exists a connection D for any bundle S of complex left modules over which is compatible with Clifford multiplication and extends the Riemannian connection on X to S.
There is a well-known principle by which the natural operations on vector spaces carry over canonically to vector bundles. In the same way the natural operations for vector spaces with quadratic forms carry over to vector bundles with metrics.
Definition 2.1. Let X be a compact Riemannian manifold (with or without a smooth boundary). (a) The Clifford bundle C€(X) of X is a bundle over X, where the fibre over a point x X consists of the Clifford algebra denotes the Riemannian metric for the real tangent vector here space TX1. The fibrewise multiplication in Ct(X) gives an algebra structure to the space of sections of Ce(X); the decomposition (2.1)
C€(X) =
®
is inherited from the plus-minus decomposition of the algebras. (b) A representation of Ct(X) (or synonymously a Ce(X)-modeae or a left Clifford multiplication of module sections by sections in the Clifford bundle) is a fibre preserving algebra morphism c:
Hom(S,S),
where S is a smooth complex vector bundle over X. Without loss of generality one can always assume that S is endowed with a Hermitian structure ( ) which makes Clifford multipli-
cation by tangent vectors skew-adjoint. This follows from the next lemma.
2. Clifford Bundles and Compatible Connections
Lemma 2.2.
11
Let X be a compact Riemannian manifold (with or
without a smooth boundary) of dimension m, S a complex vector bundle over X, and c : TX —' Hom(S, 5) a bundle map with c(v)2 = —IvI2Id
:
for v E
Then there exists a Hermitian Clifford-skew-adjoint metric on S so that at each x E X we have (c(v)s, s') =
for all s, s' E
— (s, c(v)s')
and all vectors v E
i.e. c(v)* = —c(v).
Lemma 2.2 provides the nice property that Clifford multiplication in a Ct(X)-module by unit vectors of TX is unitary, i.e. at each x E X Note.
(2.2)
(v
for all s, 8'
E
s, v s') =
(s, —v2s') = (s, s')
Sr and all unit vectors v E
Proof. One proves the lemma in two steps: first locally, then globally using a partition of unity. Since step 2 is canonical, we shall concen-
trate on step 1. Assume that the dimension of X is even, m = 2n. (Then the case m = 2n+ 1 is also solved by the structural splitting, as in the second part of the proof of Lemma 1.3 above.) For fixed x E X choose an orthonormal basis v1,. . . , V2n of With the arguments and notations of the proof of Lemma 1.3 we obtain a decomposition of the fibre into 'eigenspaces' and canonical = isomorphisms c(v(e)) :
where e = (es,.. , is a choice of signs, Co = (+,. .. , +), and v(e) E C€(TXZ) the product of those basis vectors vp.,, for which the .
sign
= —. any Hermitian metric on and expand the metric on by the isomorphism c(v(e)). This gives a Hermitian metric satisfying the following two conditions:
Choose
all on (1) (2)
..L
for all sign choices e is a unitary isomorphism.
I. Clifford Algebras and Dirac Operators
12
We write c3 := c(v3)
and
check that c =
— c,. Indeed,
let s, s'
E
sx,
Then one has
(c3s,s')s =
= c(v(S))'
c3
= =
Here
c3 : for
all j =
1,.
.. , 2n and sign choices e, where
:= (El,... with •
elU+1)/2)
f+
if eIu+1)/21
—
else.
—
It is easiest to check the transformations for the component of
in
s since
here by definition —c3 :
—÷
is the inverse of c3 - both for j even and odd. From c =
c=—c.
— c3
follows Li
The non-trivial aspects of Clifford modules are related to the choice
of a suitable connection which provides the geometry of the Dirac operator.
Definition 2.3. Let X
be a compact Riemannian manifold (with or without a smooth boundary). (a) Let k denote the field of real or complex numbers and let E be a smooth k-vector bundle of fibre dimension N over X. A connection
2. Clifford Bundles and Compatible Connections
13
('covariant derivative') on E is a k-linear first order partial differential operator
D : C°°(X;E) —i s '—i
C°°(X;rXeE)
Ds
such that
(2.3) D(fs) = df®s+fDs
for f E C°°(X) and s E C°°(X;E).
Given a smooth vector field v: X —. TX on X, we thus obtain a map
: C°°(X;E) -+ C°°(X;E)
called the covariant derivative with respect to v. At a given poin := (v, only depends on vt,, and on the values of s x E X, in a neighbourhood of s. (b) A connection D on a Riemannian (or Hermitian) bundle E will be called Leibnizian (sometimes also Riemannian), if it satisfies the inner product rule (2.4)
(8, s')
=
s') + (s, Des')
for all v E TX and 8, 8' E C°°(X; E), where (.,.) denotes the inner product in the fibres of E and 'v•' the derivation of functions in v direction. (c) A connection D on a C€(X)-module S is defined to be compatible with the C€(X)-module structure, if it is Leibnizian with respect to a suitable Hermitian structure on S, and D c = 0. Here c is considered to be the composition
C°°(X;TX) —+ C°°(X;Ct(X)) -* Hom(S,S) = C°°(X;SØS)
and D is lifted to C°°-sections in the bundle TX 0 (S 0 S*) by (Dc)(v)(s) := D(c(v)s) — c(D9v)s
—
c(v)(Ds),
denotes the Leviwhere v E C°°(X; TX), s C°°(X; S), and Civita connection on TX for the Riemannian metric g. (When we extend the Levi-Civita connection on the whole Clifford bundle C1(X) in the standard way, Dc is defined in the same way for v C°°(X; Ct(X)).) Hence our requirement Dc = 0 means that the
I. Clifford Algebras and Dirac Operators
14
connection D on S is a module derivation which extends the LeviCivita connection on X to S:
D(c(v)s) =
+ c(v)(Ds).
It is quite natural to demand a connection to be compatible, since Dc = 0 means that the Clifford multiplication acts parallel in all The most elementary example of a compatible connection fibres is the natural covariant differentiation in the exterior algebra
V: C°°(X;
C°°(X;
® A(TX))
of a Riemannian manifold X which by definition is compatible with the Clifford multiplication introduced in Example 1.4 and which extends the Levi-Civita connection of X. The next lemma is the first ingredient in establishing the existence of a compatible connection D for any C€(X)-module S.
Lemma 2.4.
Let X be a Riemannian manifold of dimension m; let a local orthononnal frame for TX over a contractible open set U C X, and let S be a (complex) Ct(X)-module. Then we can choose are constant. a local frame for S over U so that the matrices } be
Proof. The orthonormal framing of
TXIu=RmxU
provides the trivializations
and
Let the elgenspaces S(e) be defined pointwise as in (1.12) above in the proof of Lemma 1.3. Because of (1.13), these spaces have constant fibre dimension a := 2 tm/2) dimS and define smooth vector subbundles of S. Since U is contractible, the bundle S(eo) (with the choices of signs named as in the proof of Lemma 1.3) is trivial over U; thus we can find a smooth frame for Sfro) over U. This . . permits us to repeat the construction of Lemma 1.3 to show: I (2.5) DIu =
I
I I.
a times .
times
xU
if m is even a_ times
x U if m is odd.
This shows that the Clifford module structure locally is a product.
0
Now we shall prove the existence of compatible connections. Let
S be a C€(X)-module with Clifford multiplication c
:
Ct(X)
2. Clifford Bundles and Compatible Connections
Hom(S, 5). Fix a local orthonormal frame {v1,... , a contractible open set U. We denote the matrices
for TX over
: U — Hom(SIu,SIu)
by (2.6)
Then we obtain C,4;j,
=
:=
+
Here denotes the matrix of 's ordinary partial derivatives in for S the v,, direction with respect to a chosen local frame . . . over U; w denotes the connection 1-form of D which is an N x N matrix
of 1-forms and defined by W := (w,,), when Ds, = >kWjk 0 the usual Lie bracket denotes the connection 1-form evaluated at — c1 is denoted by [wi,, c,1]; and denote the Christoffel symbols of the Levi-Civita connection for TX relative to the frame {vi,...,vm} defined by (2.7)
= As a first consequence of (2.6) we see that the restriction Du of is compatible, if and only if all Cp;p = the connection D on vanish. + Next use (2.6) to show that there always exist compatible connections for a Ct(X)-module S. It suffices to prove this locally, since we can patch together compatible connections using a partition of unity, and since a convex combination of compatible connections is compatible. This follows from (2.6) which shows that scalar coefficients of a connection would enter linearly in the expression for Cp;y. We fix an arbitrary orthonormal frame {v1,... , } for TX over a contractible open subset U CX and choose a local frame {Si,. .. ,SN} for S over 0. We define the spin connection Ds U by Lemma 2.4 so that for S as an extension of the Levi-Civita connection for TX by explicit determination of its connection 1-form :=
(2.8)
Recall that Ds is then defined by (2.9)
Dss =
f
(dfk 0 Sk +
0
I. Clifford Algebras and Dirac Operators
16
for any section s E C°°(X; S) which locally is decomposed in the form
s(x) =
f,(x)8,(x) with f, E C°°(X).
We prove Ds c = 0. It remains to check that [c$, vanishes for all p, ii. In fact,
+
= =
(2.10)
by the Clifford relations. Since by definition of the Christoffel symbols
in (2.7)
we obtain
=
= Now,
in the present situation of an orthonormal frame {v,j for TX, := as defined in (2.7) with
identify the Christoffel symbols Then [we,
+
= 0.
+
=
This proves that the spin connection Ds is locally well-defined and is also a Riemannian connection since compatible; = —we. Hence, for the whole of S, it provides a compatible connection. Therefore one has the following
Proposition 2.5. Let
S be a complex Ct(X)-module over a compact Riemannian manifold X. (a) A connection D for S over a contractible open set U c X is comHom(S, S)), if patible (with the Ce(X)-module structure C: Ct(X) and only if
=0
(2.11)
for a suitable choice of a local orthonormal frame for TX over U. (b) There exist compatible connections on S which extend the Riemannian connection on X to S. By definition, a compatible connection provides a module derivation, i.e. Note.
(2.12)
s) =
.
s
+
(Ds)
2. Clifford Bundles and Compatible Connections
17
C°°(X; C€(X)) and all s E C°°(X; S), where denotes Clifford multiplication and the canonical extension of the Levi-Civita connection on Moreover, to any compatible connection on S, which is built on local spin connections, there belongs a Hermitian metric on S for which Clifford multiplication becomes skew-adjoint and hence satisfies condition (2.2) above. for all
Remark 2.6.
The delicacy of the preceding proposition is due to three circumstances: (1) The decomposition of a given Ct(X)-module S in a direct sum of irreducible Ct(X)-modules as in Lemma 1.3 is only locally possible and in general not globally possible. Actually, these irreducible bundles exist globally, if and only if X admits a spin structure. But there are famous topological obstructions (the non-vanishing of the first two Stiefel-Whitney classes) which prevent some X from admitting a spin structure. In general the spin connections Ds, defined above in (2.8) and (2.9), are not globally defined either. (2) Nevertheless Proposition 2.5.b ascertains the global existence of a compatible connection for any Ct(X)-module S, whether X is spin or not. (3) Locally this compatible connection has a standard form as the following proposition shows. The global uniqueness is lost by patching the compatible local connections together.
Proposition 2.7. Let D be a compatible connection on S and let U C X be a contractible open set. Choose a local orthonormal frame for TXIu and use Lemma 2.4 to decompose —5 U1
if m is even if m is odd
&(U)®V..
with suitable trivial bundles V0, V÷, V_. Then there exist unique con-
nections D°, D+, D on these bundles so that
f
Ds 0 1
10 D°
if m is even
1ØD} ifm is odd.
Proof. We suppose m even; the case m odd is analogous. Let w be the connection 1-form of the given connection D and let
® 1 be the
18
1. Clifford Algebras and Dirac Operators
connection 1-form of the spin connection Ds defined in (2.8), (2.9). Since both D and Ds are compatible, w — (wS ® 1) commutes with the Clifford module structure, hence
1®O forsomeOeEndV0. Let 9 define a connection D°, then
D Ds ® I
1 ® D°.
9
3. Dirac Operators
We define a canonical first order differential operator A C°°(X;S), called the Dirac operator of S. Next we find the principal symbols of A and A2 and show that A is formally seif-adjoint with an explicit Green's formula.
There are many different concepts of a Dirac operator: AtiyahSinger (or classical Dirac) operators and twisted Atiyah-Singer (or twisted Dirac) operators on spin manifolds; operators of Dirac type,
generalized Dirac operators, and compatible Dirac operators on arbitrary Riemannian manifolds; full and split (odd-parity) Dirac operators, boundary Dirac operators, and so on. As we shall see, each definition has its own merits and range of application. Let X be a smooth manifold (with or without boundary) of di-
mension m and let S be a complex vector bundle over X. Recall the definition of the principal symbol of a differential operator P: C°°(X; S) C°°(X; S) of order k 1. If, in local coordinates, we have
P= >
and
=
E
T*XX,
then
(3.1)
>
IaI=k
Now we can formulate:
Definition 3.1. Let X be a Riemannian manifold (with or without boundary) of dimension m and let S be a complex vector bundle over
x.
(a) Any first order differential operator A : C°°(X; S) C°°(X; S) with principal symbol of A2 defining the Riemannian metric, i.e. aA2(x,e) =
,
I. Clifford A)gebras and Dirac Operators
20
shall be called an operator of Dirac type. Then the operator A2 is called the Dirac Laplacian. (b) Let S carry the structure of a (complex) Ci(X)-module bundle
over X. We write the left Clifford multiplication as a mapping C: C°°(X;TXØS)
Let D: C°°(X;S)
C°°(X;S).
(J°°(X;T'XøS) be a connection on Sand let
J: Coo(X;T*XØS)
C°°(X;TX®S)
denote the isomorphism of vector and covector fields. The composition
A := coJoD defines a canonical first order differential operator A: S) C°°(X; S) called the (generalized) Dirac operator of S. Thus, in terms we have of an orthonormal base {ev},,i,...,m of (3.2)
As(x) =
.
denotes the covariant derivative of the section s E C°°(X; S), taken at the point x E X in the direction and denotes where
the Clifford multiplication. (c) If the connection D is compatible with the C€(X)-module structure of S and the extended Levi-Civita connection, we call A a compatible Dirac operator or, in short, a Dirac operator.
Remarks 3.2.
(a) The original Dirac operator is defined for the indefinite relativistic metric and is not elliptic. Historically the search for a square root of the Laplacian (for Dirac the Klein-Gordon operator) with its scalar principal symbol was the beginning, see Chapter 5 below. Systematically we have a correspondence between operators of Dirac type and Clifford multiplication, since the principal symbol OA defines a Ce(X)-module structure on S, if the principal symbol 0A2 has the form of a Riemannian metric. Hence any operator of Dirac type belongs to the subclass of generalized Dirac operators modulo operators of lower order. (b) An example: We illustrate the construction of the Dirac operator for the exterior algebra A(TX) of a Riemannian manifold X; see also
3. Dirac Operators
21
Example 1.4 and our comment before Lemma 2.4 above. We identify
TX with TX and obtain an invariantly defined operator
A : C°°(A(TX))
COO
(T*X 0 A(TX))
In terms of an orthonormal basis A(Q)(x) =
C°° (A(TX)).
we get
for —
since c = ext — mt. It turns out that A actually coincides with the de Rham complex
(d + d) : C°°(A(TX)) —' COO(A(T*X)), see e.g. Gilkey [1984; Section 3.1]. The Dirac Laplacian (d + d)2 and one derives the coincides with the Laplace-Beltrami operator Weiizenböck formula
(d+d*)2
->V,V3
where 1%.— 1,—Em
I
p
i'
+1Z,
r
a
p110-p
with c1 = ext1 — mtt denoting left Clifford multiplication, C" = ext" — int" denoting the corresponding right one, and sc denoting the scalar curvature of X. For details see Gilkey [1991; Lemma 8.1]. In Theorem 4.4 below we shall prove the general Bochner identity as a generalization of the Weitzenböck formula. (c) We postpone the discussion of the various other classical (natural
and geometric) Dirac operators in the Euclidean case and on spin manifolds (Sections 5 and 6 below). Here we shall only notice that some of them, like the operator d + d of the de Rhain complex over a Riemannian manifold and the Atiyah-Singer operator of the spin complex over a spin manifold, are defined by compatible connections. Others are not, like the operator of the Dolbeault complex —f : C°°(X;S)
—'
C°°(X;S)
for S :=
I. Clifford Algebras and Dirac Operators
22
over an almost complex manifold X, namely when the error term r becomes a non-trivial differential operator of order 0, i.e. a nontrivial local section of Hom(S, S), cf. Gilkey [1984, Sections 3.5-3.6]. More generally, we have proper inclusions of the three classes of Dirac operators of Definition 3.1: compatible Dirac
generalized Dirac
Dirac type.
We shall prove the last inclusion in the following lemma. (d) Most of the results below for manifolds with boundary hold for arbitrary first order elliptic differential operators. However, the exposition is much simpler for operators of Dirac type because the symbolic calculus can be reinterpreted geometrically. That is also the reason
why the unique continuation property holds for operators of Dirac type, see Theorem 8.2 below.
Lemma 3.3. The principal symbol of a generalized Dirac operator A of a C€(X)-module S is left Clifford multiplication with E TX TX times the imaginary unit: (3.3)
For the Dirac Laplacian we obtain the scatar symbol (3.4)
crA2(x,e) =
:
—+
In particular, both A and A2 are elliptic operators. Proof. Determine the principal symbols of A and A2: Fix x E X and an orthonormal basis {e1,. . . , em } of Choose local coordinates (x1,... ,Xm) on X at x such that x corresponds to 0 and e,L correT*XX, sponds to for each Under the identification the tangent vector e,L also corresponds to the differential lo. We see that (3.5)
A=
lo
+
zero-order terms
since the covariant derivative on S takes the form De0 = (O/Ox,t)ü
+
zero-order terms
3. Dirac Operators
23
for any local trivialization of S near x. Hence, for any cotangent
=
'SO.
•SO =
For the principal symbol of the Dirac Laplacian we find
=
= is invertible for all e fore A is elliptic. So
=
=
and there-
0, hence also
9
As permitted by Lemma 2.2, assume that the Ce(X)-module S is
endowed with a Hermitian metric which makes Clifford multiplication by unit tangent vectors skew-adjoint. We see at once that the principal symbol of any operator of Dirac type is then self-adjoint: (3.6)
=
=
denotes the Hermitian metric in the fibre where (.; If we assume that A is defined by a compatible connection, i.e. A is a Dirac operator, we have much more: not only is the symbol self-
adjoint in the fibres, but the operator itself is formally self-adjoint with an explicit boundary integral in Green's formula. That is very important and fundamental to (infinitely) many calculations on manifolds with boundary.
Proposition 3.4. Let X be a compact Riemannian manifold with or without boundary; let S be a C€(X)-module, and let A denote its Dirac operator (i.e. built on a compatible connection for S). (a) Then A is formally seif-adjoint, i.e. (3.7)
(Asj;s2) = (si;As2)
for all
E
where the inner product on C°°(X; S) is induced from the pointwise inner product (..; by setting (3.8)
and
(sl;s2) := S) denotes the subspace of sections with support in X\OX.
I. Clifford Algebras and Dirac Operators
24
(b) More generally, the following equality holds for all smooth sections
(3.9) (As1;s2) — (si;As2)
=
where Y denotes the boundary of X and G(y) denotes Clifford multiplication by the inward unit normal vector n E for
yEY.
Remark 3.5. We shall refer to (3.9) as a special case of Green's formula (for the general case see e.g. Palais [1965a; Theorem XVII.1j),
and to C as
Green's form of the Dirac operator. From the rule (2.2) for Clifford multiplication by unit vectors in TX we get the
G=
(3.10)
—c = C—'
Proof of Proposition 3.4. First fix an x
X and choose an or-
thonormal tangent frame field (v1,.. . , in a neighbourhood of x so that = 0 for all t'. This can be achieved by extending a frame at x by parallel translation along geodesic rays emanating from x. Then we have at x: (3.11)
=
>(v,i. DVMS1;s2)
(2.4)
—
=
V,4
.
(2.12)
+ (si;v,4.
— V,4
(Si; V,4
= To evaluate the second summand, use
the following trick: Let r the vector field defined by the condition that (3.12)
=
—
be
3. Dirac Operators
25
and x' E X. Then
for all w E
=
(r;v,j}12
=>J{v,,
(3.13)
—
=
= From (3.11) and (3.13) we obtain not only (3.14)
=
+div(r)1
at the chosen point x, but in fact (Asj;
=
+ (Si;
for all x' E X, since all expressions in (3.14) are independent of the chosen frame
Stokes' formula (see e.g. Dubrovin, Fomenko & Novikov [1984; 4.26]) gives
IM
div(r)dvol(x)
=
(r; —n) dvolQj) JOM
which proves (a) and (b).
dvol(y) JOM
[I
4. Dirac Laplacian and Connection Laplacian
We discuss the general Bochner identity which gives an expression of the Dirac Laplacian A2 in terms of the connection Laplaci.an D*D and certain bundle endomorphisms.
Now we shall express the Dirac Laplacian A2 in terms of the connection Laplacian D*D and certain bundle endomorphisms. We begin
with the definition of the operator DD. Definition 4.1. Let X be a compact Riemannian manifold and let S be any Ce(X)-module with compatible connection D. The connection Laplacian
D*D:
C°°(X;S)
is defined by taking the trace
DDs := — tr(D2s),
(4.1)
where s E C°°(X; 5) and D2
Coo(X;S* ®S)
C°°(X;TX EITX)
denotes the invariant second derivative given by
:=
(4.2)
—
DDVWS
for any pair of tangent vector fields v and w.
Remarks 4.2. (a) Clearly the operator Dr,.
only
depends on the
value since this is a general property of the covariant derivative. This follows from Also, moreover, only depends on the value
the identity fl2 where
—
fl2
__D —
R denotes the curvature tensor of S.
4. Dirac Laplacian and Connection Laplacian
27
(b) In terms of a local orthonormal frame {v,.1 } for TX we have
DDs =
(4.4)
(c) The concept of the connection Laplacian is of great generality in
differential geometry and not restricted to the case of solely discussed here.
Proposition 4.3. The operator DD : C°°(X; S)
S) has
the principal symbol OrD.D(x,e) =
(4.5)
fore
TXX,
and so DD is elliptic. It is also non-negative and formally selfadjoint. In particular, (DDs1;s2)
(4.6)
JM
(Dsi;Ds2)
for all
E C°°(X; S), provided that either or has support in the interior of X. (As always we assume that X is compact.). Here
(Dsi; Ds2) is defined in terms of local frames {v,j for TX by the expression
(Dsi;Ds2) =
(DV,sl;DVM52)
Proof. We reproduce the main arguments from the proof of Lemma 3.3 and Proposition 3.4 above, replacing left Clifford multiplication by
covariant derivation. Fix an x X and choose a local orthonormal frame {v1,. . . , } for TX with the property that = 0 for all p, &'. Then, at the point x, one has that
DDs =
(4.7) and so as
=
—
since
=
zero-order terms
observed earlier.
From (4.7) we obtain (all inner products are taken at x) that
(D*Dsi;s2) = (DV,Asl;s2) —
= —div(r)+ (Dsi;Ds2)
I. Clifford Algebras and Dirac Operators
28
where r now is the tangent vector field on X defined by the condition (r; w) = 82) for all w E TX. Then
Now the proposition follows by integration.
Theorem 4.4. (The general Bochner identity). Let X be a compact Riemannian manifold (with or without boundary), and let S be a Ct(X)-module with compatible connection. Let A2 denote the Dirac Laplacian and let D*D denote the connection Laplacian. Then
A2 =D*D+1Z. Here IZ is a canonical section of Hom(S, S) defined by the formula
jz,v=1 } is any orthonormal tangent frame at the point in question, is the curvature transformation of S (cf. (4.3) above), and the dot "•" denotes left Clifford multiplication.
where
Proof. We make the same assumptions about x and preceding proof and find at x that A2 =
=
v1,1
= by (4.3).
=
in the
v1,
using (2.12) and
= >v,.,
as
}
+
= —
0
at x
5. Euclidean Examples
Some important examples are discussed: Dirac's search for a square root of the Klein-Gordon operator; the Dirac operator on 11.2 = C considered as mapping even to odd spinors which is exactly the Cauchy-Riemann operator; the Dirac operator on H-valued functions; and the quaternion analogue for the Cauchy-Riemann operator and its expression by Pauli matrices.
The first class of examples we want to consider are Dirac operators on or on a compact connected submanifold X C of codimension 0 with smooth boundary. Let Ctm denote the Clifford algebra of I1.m. It has real dimension Let W be a (complex, N-dimensional) C€m-module, and let c : Ctm Hom(W, W) denote the related representation, i.e. left Clifford multiplication in W with elements of C€m. We study the Euclidean Dirtzc operator A : C°°(X; S) C°°(X; S), where S := X x W is endowed with the trivial (compatible) connection. it is a constant coefficient operator of the form
where all are linear maps : W —. W and from their definition = c(e,1) inherit the relations as (5.1)
for all
ii.
el,... ,em denote the standard basis vectors of If we choose a basis for W, we can identify Hom(W, W) with the full matrix algebra Here
M(N, C), and the c,4's will be represented by elements of the group GL(N, C) because of (5.1) for = ii. Equation (5.1) also implies that A2 = In the coordinates for W it is an N x N diagonal matrix with the (positive) Laplacian = — repeated in the diagonal.
Remarks 5.1. (a) The physics literature distinguishes carefully between Euclidean (elliptic) Dirac operators, which are operators of the type discussed here, and Minkowskian (hyperbolic) Dirac operators,
30
I. Chiford Algebras and Dirac Operators
which are the operators originally investigated by P.A.M. Dirac in the 1920s. Dirac posed the following question: Can the Klein-Gordon operator + m2, where
0 := with gIW denoting the coefficients of the Minkowski metric, be expressed as a product of two first-order differential operators as follows:
[I +m 2?=
1
0
.
\1
.
0
Dirac realized that the coefficients must be matrices which generate the algebra M(4, C) and satisfy the relation (5.1), where the right
side is replaced by with S = R4 x W and W =
In physics, the space C°°(R4; S) is called the space of (4-component) spinors. See e.g. Dubrovin, Fomenko & Novikov [1984; Section 40]. (b) If we only consider irreducible Cim-modules we can determine the following data for each integer m 1: First, find C€m. All C€m are explicitly known, they have real dimension and there is actually a kind of periodicity with period 8; see Lawson & Michelsohn [1989; Section 1.4]. Then determine the irreducible representations C: C€m —, Hom(W, W) along the lines of Lemma 1.3. They are uniquely determined (up to isomorphisms) if m is even, i.e. m = 2n; then the complex dimension of W is N = 2's; if m is odd, m = 2n +1, we have two non-equivalent irreducible representations, each module again of complex dimension N = r. Each vector of the canonical basis {e,j of am provides an endomorphism cM = of W which satisfies the relations in (5.1) (as, in the opposite direction, each system of m endomorphisms of W defines a representation of Ctm, if it satisfies C4
(5.1)). Now choose a basis for W and calculate c,, which is an N x N matrix, the entries of which are linear combinations of the partial derivatives 0/Oxi,... , 0/t9Xm. This matrix, which is enormous for higher m, has the property that its square is where E denotes the N x N identity matrix. Finally, the principal symbol of the Euclidean Dirac operator restricted to unit cotangent vectors provides a generator aA(x,.) : 5m_I GL(N, C) of the homotopy group 7rm_i(GL(N,C)) Z form even and 2N m. As Lawson & Michelsohn [1989, p. 120] put it: "This explicit form is seldom, if ever, useful. It is always simpler to use the structure of
5. Euclidean Examples
31
the Clifford module." Nevertheless, below we shall derive the explicit forms in the cases m = 1, .. . , 4 for three reasons: (1) to illustrate and exercise the basic concepts and constructions of Clifford algebras, Clifford modules, and Dirac operators;
(2) to provide the basic bricks for all geometrically defined operators. This may sound strange, since Euclidean Dirac operators do not immediately extend to elliptic operators over closed manifolds — that is inhibited by the characteristic form of the principal symbols which have non-vanishing local index
for even m. However, later in this book, we shall show how to build up arbitrary classical operators by pasting Euclidean Dirac operators together, see Chapters 24 — 26; (3) and to ease reading the literature, where some calculations are carried out through ingenious manipulations with these matrices, see e.g. Goldstone & Jaffe [1983], Schmidt & Binzer [1987], Schmidt [1987], or Kori [1993].
Letm= 1,sothatC€1 =Cn{1,ei}= C. ThenW=Candc(1)= 1. There are up to equivalence two irreducible representations, depending on whether c maps e1 into i or —i, and these two representations
are not equivalent, since F: z operator is just
is not C-linear. Then the Dirac
(5.2)
The Z2-grading Ct1 = Cit Ctj gives the splitting in real and imaginary parts which the Dirac operator interchanges. Let m = 2, so that dim Ct2 = = 4 and in fact Ct2 = £R{1,el,e2,ele2} = H. The decomposition of H into cEB C is natural and corresponds to the Z2-grading with = CR{1, eje2} and C4 = Ca{ei, e2}. We are looking for a complex vector space W such that Cl2 Homc(W, W). Then the complex dimension of W has to be 24m = 2. Actually W := C C is an irreducible C€2-module with Clifford multiplication given by
c(1) :=
/1
o\ i)'
c(ei)
:=
(0
—i\ 0
c(e2) :=
/0
i
0
With the notation of Example 1.4 we can write down the two basic operators on AC2 Cl2:
p=c(ei)+ic(e2)
and
I. Clifford Algebras and Dirac Operators
32
with the relations
p• p = (c(ei)
+ i
c(e2))(c(ei) + i c(e2))
= —c(1)
+ic(ei)c(e2) +ic(e2)c(ei) +i2c(e2)2
=0
and
= —2(c(1)+ic(ei)c(e2)) so
-4 as required in Example 1.4d. Then
p•
=
—2(c(1)
+ ic(ei) c(e2))
and
1.
=
= (c(ei)
—
ic(e2))
form a basis for W, and Clifford multiplication with p and does not lead out of the subspace W of C€2. Indeed, we have p• W C W, since
p' (p
=
0
and trivially p. (1
= (p
and an invariant description of
and W, determine the ±1-
elgenspaces of the orientation form
= i c(ei) c(e2). For the ease of presentation we omit the c in c(e,) and write just e3 for Clifford multiplication with e3. Then we get
= hence
—2iele2(1
= —2(1 +iele2)
generates W÷; and
= iele2(eI hence
+ieie2) =
j3 generates W.
—
ie2)
=
—(e1 —
ie2)
=
=
5. Euclidean Examp'es
Correspondingly the Dirac operator A = has the form (5.3)
33
(8/Ox,) + e2 (8/Ox2)
e1
—i), := 0/Oxi + jO/Ox2
where 8/Oz := 8/Ox, — jO/Ox2 = 28, and
28 is twice the standard Cauchy-Riemann operator. Thus the Dirac operator on R2 = C, considered as mapping even to odd spinors, is exactly the (double) Cauchy-Riemann operator.
Our choice of c(ei) and c(e2) is by no means canonical. There are actually three matrices, the Euclidean analogues of the classical Pauli matrices Note.
(i o\
)'
c2:=c(ei),
ando3:=c(e2),
which generate the full matrix algebra M(2, C) and satisfy the relations (5.1). Every two of them together generate an irreducible in complex form. By Lemma 1.3 these reprerepresentation of sentations are equivalent. This is easily checked, e.g. for the two representations c3 : Ct2 M(2, C) defined by
c1:=c =
Then
trary a
F'
forjt=1,2.
and for
= 1,2 if F =
(a
for arbi-
0.
Letm=3. W=
Clearly we have two non-equivalent representations of H on H, namely, left quaternion multiplication and right quaternion multiplication. We begin with left quaternion multiplication which gives the following expression for the Dirac operator on H-valued functions on R3: (5.4)
.8 =1— Ox,
.0 0x2
0 Ox3
If we write q H = R4 in the form q = a + bi + ci + dk with real coefficients a, b, c, d and the canonical basis {1, i,j, k}, we can write left quaternion multiplication as 4 x 4 real matrices (corresponding to
1. Clifford Algebras and Dirac Operators
34
L(i)q = i q =
—b
+ ai — dj + ck, L(j)q =
...,
and
L(k)q =...) and
obtain
/0 I
Aleft
101
=1
0
—02 03
03
0
03 02 —01
0
—02
:= If we choose right quaternion multiplication, we obtain a different Dirac operator where
(5.6)
Aright
=
+
+
which can be written in real coordinates (corresponding to R(i)q =
q•i=—b+ai—dj+ck,R(j)q=...,andR(k)q=...)as
/0
—03
—03
0
1
0
Now decompose H = C ® C and rewrite
q= a+ bi+ cj + dk = ((a+ bi), (c+ di)). Then right quaternion multiplication may be described by complex
2 x 2 matrices. Check that R(i) = a1, R(j) = a2, and R(k) = O'3, where
are
the Pauli matrices defined above.
Letm=4. ThenCt4 The representation C: Ct4 Homc( W, W) is uniquely determined up to isomorphisms. So we can just play around with the Pauli matrices until we get complex 4 x 4 matrices c4 = c(e,4) for /L = 0,1,2,3 which satisfy the Clifford relations (5.1). Clearly the following matrices will cO= (?
and
(°
forp=1,2,3.
Then the Dirac operator can be expressed with respect to the splitting
(5.9)
5. Eudlidean Examples
35
where
o
— 0
0
.0 0
.0 .
0
k
Note the analogy with dimension two. For a thorough exploitation of the analogy, see e.g. Delanghe, Sommen & Souëek [1992], where quaternionic analysis and other higher dimensional analogies of complex analysis are described. An analogy of holomorphic differential forms in higher dimensions builds on formal similarities between the Cauchy-Riemann operator and the other Euclidean Dirac operators.
6. The Classical Dirac (Atiyah-Singer) Operators on Spin Manifolds
We consider a spin manifold with a spin structure on its tangent bundle, and a spinor bundle endowed with its canonical connection. We formulate the Lichnerowicz vanishing theorem.
One might argue that this chapter is merely of historical interest now that the fact is established that, for any bundle S of complex there always exists a connection which is left modules over compatible with Clifford multiplication and extends the Levi-Civita connection on X to S; see Chapter 2. Since that is true for any bundle of Clifford modules over a Riemannian manifold X, this somewhat removes the prominence of spin manifolds in earlier work in index theory and the exclusiveness of spin manifolds in admitting non-Euclidean Dirac operators. Nevertheless, for various calculations, one can take great advantage of the symmetry of spin manifolds, of the associated spinor bundles, and of the related Dirac (Atiyah-Singer) operators. Introductions to
the subject can nowadays be found in many places. We especially refer to Gilkey (1984; 3.2 and 3.3] or Lawson & Michelsohn [1989; 11.2, 11.5, and 11.8]. Here it suffices to explain the relation to our previous introduction of isomorphism classes of irreducible Clifford modules. We shall begin with an intuitive argument. Let X be a compact oriented Riemannian manifold of even dimensionm = 2n. For every point x E X, we have three different objects: of real dimension m, (1) the tangent space (2) the Clifford algebra C1(TXZ) of real dimension which contains
(3) and, by Lemma 1.3 (up to isomorphism), the uniquely determined irreducible complex of complex dimension 2", which we shall denote by Whereas the families {TXZ}XEX and {Ce(TXX)}XEX carry natural topological and differential structures which make them bundles, namely, TX and Ct(X), the parametrized family {SX}XEX admits the structure of a continuous and smooth bundle only if X is a spin man-
6. The Classical Dirac (Atiyah-Singer) Operator
37
ifold. In other words, a principal bundle Pspin(TX) over X is given which has Spin(m) as structural group and is (roughly speaking) a double covering of the principal bundle Ps0(TX) of orthogonal oriented frames, or (more precisely) accompanied by an oriented vector bundle isomorphism
TM
Xp Rm,
denotes the double cover. where p: Given such a spin structure we make, of the parametrized family {SX}XEM, the associated bundle as by regarding modules and identifying t)
(x,
(x, s, ic), x E U4 fl U,ç,
are suitable coordinate transformations for the spin manifold X. Then the associated bundle carries a natural Hermitian structure satisfying (2.2) and is endowed with a natural compatible connection induced by the Levi-Civita connection for TX. where
:
fl
Definition 6.1. (a) For spin manifolds of even dimension, one has a (unique) irreducible complex spinor bundle for any choice of a spin structure. We shall denote this bundle by The Dirac operator in this case will be written (b) More generally, a (complex) spinor bundle, associated with TX, is a bundle S of modules over C€(X) of the form
S=
Psp1,,(TX)
M,
where M is a complex left module for the complexified Clifford algebra C€(Rm) 0 C. Then, once again, the Hermitian structure of S is compatible with Clifford multiplication (in the sense of equation (2.2)) and S carries a natural compatible connection as explained, for example, in Lawson & Michelsohn [1989; 11.4]. The Dirac operator will in this case be called (following Lawson & Michelsohn) the Atiyah-Singer operator. We write A or As. The classical necessary and sufficient topological conditions for a Riemannian manifold to admit a spin structure is derived and discussed, for example, in Lawson & Michelsohn, namely, the vanishing
I. Clifford Algebras and Dirac Operators
38
of the second Stiefel-Whitney class. The most elementary examples are:
• All 2-connected manifolds (like homotopy spheres and simplyconnected Lie groups) carry a unique spin structure. • Any manifold, whose tangent bundle is stably parallelizable
trivial after adding a suitable trivial bundle), is spin. This includes any Lie group and any orientable manifold of dimension <3. • A complex manifold is spin, if and only if its first Chern class (i.e.
is even. We have:
Theorem 6.2. (A. Lichnerowicz). Let X be a spin manifold and let S be a spinor bundle associated with TX with spin connection D. Then (6.1)
denotes the Dirac Laplacian, D*D denotes the connection Laplacian (see Definition 4.1), and ,c denotes the sco.lar curvature, A2
expressed by the formula
=
(6.2)
—
>
any local orthonormal tangent frame at the point in the curvature transformation of S; see equation question and
where
is
(4.3).
Corollary 6.3. We have (6.3)
kerAS=O
for any spinor bundle associated with TX, if the underlying manifold is spin and closed (compact and without boundary) and if the scalar
curvature is 0 and > 0 at some point. Theorem 6.2 is an immediate consequence of the general Bochner identity (Theorem 4.4), and the corollary follows by integration of (6.1). For details, cf. Lawson & Michelsohn [1989; p. 161].
6. The Classical Dirac (Atiyah-Singer) Operator
39
Examples 6.4. (a) We can see C€(X) as a bundle of left modules over itself by left Clifford multiplication and endow S := Ct(X) with its canonical Riemannian connection. The Dirac operator in this case is a square root of the classical Hodge Laplacian. (b) By tensoring we can generate ever new examples from the basic examples. More precisely: Let S be a (complex) C€(X)-module over X (not necessarily spin) with compatible connection DS, and let E be an arbitrary Hermitian vector bundle with connection DE over X. Then the tensor product S 0 E is again a C€(X)-module and is equipped with a compatible connection.
7. Dirac Operators and Chirality
We emphasize the decomposition of a Ct(X)-bundle S = S and the related splitting of Dirac operators. It is illuminating to treat the signature operator and other geometrically defined operators in this context.
Now we are back to our general situation, where X is a Riemannian oriented (not necessarily spin) manifold of dimension m and S a C€(X)-module with compatible connection. We want to globalize the Z2-grading explained in Remark 1.le and decompose the Dirac operator. We do that in a series of lemmata. Let w be a section of C€(X) given at any point x E X by (7.1)
is a positively oriented orthonornial basis of where This provides a well-defined global section of Ce(X). Since w is independent of the choice of basis, for any x , we may choose a local frame {eM} such that Depeyiz = 0 for all v. Hence we have the following lemma:
Lemma 7.1. On any oriented Riemannian X of dimension m, we have a canonically defined section w of Ct(X) which satisfies the following relations:
=
(7.2) (7.3)
(7.4)
wv =
for any section v of TX,
Dw=0.
Now define a bundle automorphism multiplication (7.5)
:=
: S —' S by left Clifford s.
It follows from Lemma 7.1 that, for even-dimensional X, the bundle S splits: (7.6)
with v 5+ C S and v• S C St where spanned by the eigensections of ±i, if m 2 mod 4, or ±1 if m
are the subbundles corresponding to the eigenvalue
0 mod 4.
7. Dirac Operators and Chirality
Lemma 7.2. For the (total) Dirac operator A
41
:
C°°(X; 5)
C°°(X; 5) we have
= From this lemma it follows that the Dirac operator splits in even dimensions. The components
(7.7)
:
AICOO(x;S±)
:
C°°(X;
C°°(X;
are well-defined elliptic operators. That construction provides interesting indices, because in difference to A the operators A± are not formally seif-adjoint. We collect our results:
Theorem 7.3. Let X be a Riemannian, oriented manifold with or without boundary and let S be a Ct(X)-module with compatible connection. If dim X is even, the bundle S and the total Dirac operator A decompose to
1)
and
and the operator A— is formally adjoint to the operator At
Examples 7.4. (a) One of the most prominent Dirac operators is the signature operator. Let X be a compact oriented, Riemannian manifold (with or without boundary) of dimension 4k. Usually one defines the signature operator via differential forms: Recall that the operator A := d + d* acts on the space := C°°(X; A(TX)) of all differential forms. Denoting by the ±1-eigenspaces of the involu-
tion r:
—'
11, defined by
3s
it follows that A interchanges 11' and defines an operator
*
E
and hence by restriction
which is called the signature operator. This corresponds to the preceding definition given in (7.7) when we considered the basic case of ce(x)-module bundles, namely,
S := C€(X) =
=:
S.
Here is identified with A(T*X). Moreover, the Z2-grading is given by the volume element once again.
42
1. Clifford Algebras and Dirac Operators
(b) To get the required form
close to the boundary Y of X, assume that X is isometric to a product
near the boundary. Identifying the restriction of to Y with the space 12(Y) = COO(A(T*Y)) of all differential forms on Y, one finds that
Bs=
where the sign in the formula depends on the parity p of the differential
form; see Atiyah, Patodi & Singer [1975; (4.6)] and Gilkey [1984; pp. 261 if J. One sees that B is a self-adjoint operator on Y which * preserves the parity of forms on Y and commutes with 8 so that B splits into B = B°" and Be%c is isomorphic to Bo&I. We get
:=
where s =
—
E C°°(X; V) and V :=
see also
Branson & Gilkey [1992a, Example 2].
(c) For Dirac operators coupled to vector potentials equal to Dirac operators with coefficients in an auxiliary bundle, we refer to Example 6.4b and Atiyah & Singer [1984]. (d) A simple construction of Dirac operators on S2 with coefficients in line bundles is given in Chapter 26.
Remarks 7.5. (a) Instead of our choice of the orientation section w in (7.1) we could have taken a normalized orientation 'r like in our note to Lemma 1.3. This makes only one difference, namely, r2 = 1 instead of the more complicated formula (7.2) for Then our subbundles become the ±1 eigenspaces of r. (b) From Lemma 1.3 we obtain that the smallest fibre dimension for any total Dirac operator over an even-dimensional manifold X is the
dimension of the irreducible module, i.e. N = 2m/2 for dim X = m. Therefore we obtain
as minimal fibre dimension for the split operators. In fact there is no first-order elliptic differential operator over an open subset of defined on a smaller number of complex valued functions; see also Atiyah [1970].
8. Unique Continuation Property for Dirac Operators
We give a direct proof of the unique continuation property of a Dirac operator by exploiting its simple product decomposition.
Definition 8.1. An operator A over a smooth manifold X (with or without boundary) has the unique continuation property, if any solution s of As = 0, which vanishes on an open subset of X also vanishes on the whole connected component of the manifold.
All classical (Euclidean) Dirac operators eo ipso satisfy the unique continuation property when they are not coupled to a vector potential (that is, not tensored with an auxiliary coefficient bundle; see Chapter 5 and Example 6.4b above). This follows immediately from the Cauchy-Kovalevskaya theory of elliptic equations with analytic coefficients, since the Eudidean Dirac operators are analytic and can even be written as operators with constant coefficients.
It is a remarkable property of all other operators of Dirac type that they also obey the unique continuation principle. In recent years several authors have exploited this property; see, for example, Roze [1970], Jaffe & Taubes [1980], KaIf [1981], Simon [1982], Freed & Uhlenbeck [1984], Donaldson and Kronheimer [1990], and Taubes [19901.
Usually reference is given to the Aronszajn-Cordes uniqueness theorem for second order differential operators A = EIQI<2 with elliptic scalar real valued principal symbol. We want to keep our treatment of Dirac operators self-contained and show that the unique continuation principle for Dirac operators is a consequence of their simple product form. For unique continuation, the decisive property of Dirac operators is that even in the interior of the manifold does the locally well-defined product A = G(y, + (resp. to a chosen Riemannian metric) have locally a tangential with elliptic self-adjoint part part + Ba). Theorem 8.2. Let X be a compact Riemannian manifold (with or without boundary) and S a C€(X)-module with Ct(X)-compatible con-
nection D. Then the unique continuation principle is valid for the corresponding Dirac operator A : C°°(X; S) -+ C°°(X; S).
44
I. Clifford Algebras and Dirac Operators
We shall mostly apply the unique continuation principle in the following form which, via Green's formula, is an easy consequence of the preceding theorem (for details, cf. Lemma 9.2).
Corollary 8.3. Let X = ifold with X÷ flX... =
U
X_
be
a connected partitioned man-
= Y. Lets E C°°(X;S) satisfy As =
0
and sly=0. Thens=OonX. Remark 8.4. In most parts of this book we assume that the Riemannian structure of X and the Hermitian structure of S are products + B) with unitary near Y. This implies a product form A = C and seif-adjoint B close to Y, where C and B do not depend on the normal variable ti. Then, close to Y, the unique continuation property of A follows immediately from elementary harmonic analysis when we expand a solution of As = 0 near the boundary in the form
s(u, y) = where is a spectral resolution of L2(Y; Sly) generated by B (cf. the discussions in Chapter 22; e.g. the proof of Proposition 22.20).
The crucial point of Theorem 8.2 is that one obtains the unique continuation property also in the interior, where no product structure of the metrics can be assumed a priori. In the interior we may of course also introduce a product structure locally for the ease of calculation. But then we can no longer assume that a given Dirac operator will split there in a product with the parts C and B independent of a local "normal" parameter u.
Proof of Theorem 8.2. Let s E C°°(X; S) be a solution of As = 0, andassumethat s=0 inanopenset LetxoE OV. Weshall work locally in the neighbourhood of x0. By fixing a local chart for X we get spherical coordinates locally. The situation is described in Figure 8.1. First choose a positive real r sufficiently small and a point p V at a distance r from SO such that the ball with center at p and radius r is contained in V. We are going to show that for some real T> 0, the section s is equal 0 in the ball with center at p and radius r + This proves Theorem 8.2 since X is connected.
8. Unique Continuation Property for Dirac Operators
45
X\V ov
Fig. 8.1 Local specification for the Carleman estimate
The result is a consequence of the following estimate of Carleman type:
(8.1) RI: uo I
112 dy
IIv(u,
-
du OAv(u, y)112 dy du,
s;.;'
where R is a sufficiently large positive number and
v(u,y) := w(u)s(u,y) with a smooth cut-off function chosen in such a way that = 1 for u and so(u) = 0 for u> It follows (see Figure 8.1) that
(8.2) suppv C [0,
x
and
suppAv C
x
The proof of Theorem 8.2 is split into two lemmas. First we show that (8.1) ensures that s also vanishes in the annular region u < Next the inequality (8.1) is proven. U Lemma 8.5. If (8.1) holds for any R> 0 sufficiently large, then s
is equal to Oforu <
I. Clifford Algebraa and Dirac Operators
46
Proof. We have
j2
IIs(u,y)Il2dydu
=
Ils(u, y) 112 dy du
cf
T
C
Dv(u,y)Il2dydu
jTj. IIAv(ii,y)Il2dydu
1T
j
IIAv(u, y)112 dydu;
hence we have (8.3)
j2
Js;.;1 IIs(u,y)Il2dydu
f which gives the result as R —'
00.
T
Js;,;1
IIAv(u, y)112 dy dtz,
[1
Now (8.1) must be established.
Lemma 8.6. For T sufficiently small there exists a constant C such that the inequality (8.1) holds for any positive R.
Proof. First consider a few technical points. The Dirac operator A has the form on the annular region [0, TI x and + it is obvious that we may consider the operator instead of + A. The operator is a seif-adjoint elliptic differential operator on Note that the metric structures depend on the normal variable U.
To simplify the computations, deform the Riemannian and the Her-
mitian structure in a neighbourhood of such that they do not depend on u; but keep the operator A fixed. Still it is enough to prove (8.1) for the new metric. Then, however, the tangential part is
8. Unique Continuation Property for Dirac Operators
47
not a seif-adjoint operator with respect to the new structures, but its self-adjoint parts := are elliptic differential operators (at least when T is sufficiently small). Now make the substitution
v =:
(8.4)
which replaces (8.1) by (8.5)
Rf
T
Is;,;' IIvo(:,y)Il2dydu
Cf
+Bvo+R(T—u)voII2 dydu.
Now we prove (8.5). Decompose + B + R(T — tL) into its symmetric part + R(T — u) and its anti-symmetric part + B_, where
B This gives
ff (8.6)
+Bvo + R(T — u)votll dydu
=11
+ ff
dydu
VU
+ R(T — u))v0112 dydu
dydu. Integrate by parts and use the identity for the real part
=
I. Clifford Algebras and Dirac Operators
48
in order to investigate the last term on the right side of (8.6). This yields
+ B_v0; B÷v0 + R(T — u)vo) dy du + R(T — u)vo) dydu
= +
+R(T_u))}vo) dydu
= (8.7)
(vo; B_B÷vo) dydu
—
=
ffJf(vo; (vo; —
V0
+ Rvo) dy du
+
=RJ
+1! (vo; where ii'
urn denotes
+ [B+; B_]vo)
the m-th Sobolev norm on Sir_i. It follows
from (8.6) and (8.7) that the proof of (8.5) will be completed when it is shown that
+ [B+;B_]vo)
(8.8) k
(RI
+
j
ii(B + R(T — u))voiI2 du)
for some constant 0 < k < 1. The operators of first order, hence 11111'
=
are elliptic
8. Unique Continuation Property for Dirac Operators
for any section
49
f of S on S;' (and 0 u T). Then
(8.9)
JJ(vo;
+ [B+;B_Jvo)
cicf
ca jT
+
iivoiio
iivoiio Ilvolli du
IIvoIIo(IIBvoIIo + Ilvollo) du
0
cicJ
IIvoIIo 0
{II(B + R(T — tL))voIIo + (R(T — u) + l)IlvoIlo} du
fT
cic(RT+1) / Jo
+cjc / II(B+ +R(T—u))voIIo Ilvollodu. Jo
The integrand of the second summand is equal to (8.10)
II(B+
IIvoIIo)
+
II(B+ + R(T
—
with the inequality due to the estimate ab < (8.10) in (8.9) we obtain
f f(vo;. So
.
.)
II(B+
the desired result holds for T and
+ + b2). By inserting
+ R(T —
sufficiently small.
[]
Remark 8.7. Clearly the split Dirac operators introduced in Chapter 7 inherit the unique continuation property of the full Dirac operator, since, by decomposition of A, any solution u of A
=
0.
9. Invertible Doubles
We discuss an important extension result for Dirac operators on manifolds with boundary which provides an invertible differential operator on the closed double.
Next we derive an important extension result for operators of Dirac type on manifolds with boundary. Once again it holds for any elliptic operator of first order, but the restriction simplifies the notation. We recall: Our main object on even-dimensional manifolds is the operator :
In
C°°(X;S).
a collar neighbourhood of the boundary Y
of
X it takes the product
form
a family of Dirac S and where G is a unitary morphism G: operators over Y. When investigating the ti-invariant, the focus is on odd dimensions and on B = B0. (Here the it-function of B will appear only a few times.) If the odd-dimensional compact manifold Y is not closed, but has a boundary Z, then, in a collar neighbourhood of Z, the odd-dimensional Dirac operator B takes the form
C over Z. They play the role of A, A+, A. This time, however, we have r2 = — Id. with even-dimensional Dirac operators C,
Since we are mainly interested in the even-dimensional case, we concentrate on the extension of the operator A+. Also the treatment becomes more transparent due to the chiral notation. However, all arguments can easily be transferred to the odd-dimensional case. Transcriptions are left to the reader as an exercise.
Theorem 9.1. Let
: C°°(X; —b C°°(X; S-) be a Dirac operator on an even-dimensional manifold X with boundary Y. There
9. Invertible Doubles
51
exists a closed manifold X, which contains X, and a Dirac operator on X such that
= and
is an invertible operator. More precisely, ker
(9.2)
= {0} =
coker A+
is an elliptic pseudo-differential operator of order —1.
and
The existence of an invertible extension of A+ (and of A as well) is a nice property of Dirac operators which is not found in the literature. We shall use it in Chapter 12 in order to construct the Calderón projection onto the Cauchy data of Note.
Proof. It is standard that
takes the form
(9.3)
= G(u, y)(ôu +
in a collar neighbourhood N of Y
X.
Here u denotes the inward oriented normal coordinate which gives a parametrization of N as I x Y; C(u) is Clifford multiplication by the unit inward vector, hence G(u, y) is a unitary bundle isomorphism; denotes the corresponding (total, self-adjoint) Dirac operator on the manifold Y x {u} which may be identified with Y. In general, C and B depend on the u-coordinate. Only in the cylindrical case can we equal a fixed Dirac operator B: C°°(Y; obtain that the —p C°°(Y; independent of u, and that Clifford multiplication with the unit inward tangent vector in a distance u from a boundary point y yields the same as Clifford multiplication at the boundary point under by local orthonormal and = identification of the fibres frames.
To eliminate the u-dependence and obtain a true product form of we therefore follow the usual procedure and attach a cylinder (—1,01 x Y to X and extend the operator A+ in such a way that, on a small collar neighbourhood of { — } x Y, it takes the form (9.4)
=
+ B)
close to {u =
I. Clifford Algebras and Dirac Operators
52
where C and B do not depend on the normal coordinate u. This can always be done and corresponds to a deformation of the Riemannian structure on X and of the Hermitian structure on S into a product in extends to an a collar neighbourhood of the boundary. Therefore, operator on the manifold X1 := X U ((—1,0] x Y) such that x Y is of the restricted to (—1, is still of Dirac type and that form (9.4).
+B)
here
here A+ = A+ Fig. 9.1 The extension of
into product form
To simplify the notation we change the parametrization in the normal direction in such a way that we obtain a collar neighbourhood N1 := (—1,3) x Y with the following properties: (9.5)
X=X1\((—1,2]xY), and
on(—1,1)xY,
and
of Dirac type,
(see Figure 9.1). Of course the point is that the extensions
of are again spinor bundles of positive/negative chirality, namely, as splittings of the extension of the C€(X)-module S with compatible connection D to a Ct(X1 )-module S1 with compatible connection D1. That means that is still a true Dirac operator and does not lose essential properties, like the unique continuation property, under the deformations and extensions. To avoid excessive use of subscripts, denote the extended bundles and the extended Dirac again by operator again by
9. Invertible Doubles
53
du
Fig. 9.2 The making of X
Now take X2 := (—X1), a copy of X1 with reversed orientation. Hence its corresponding collar neighbourhood of Y is of the form N2 = [—3, 1) x Y. Clue these two copies together to obtain the (extended) doubleX with U :=X1flX2= (—1, 1)xY in X; see Figure over and S, take the bundles 9.2. To introduce the bundles with the element X1 and S over X2 and identify any element s E for (u, y) E U. Notice that Clifford multiplication with G(y)s E S; see (1.5) a tangent vector by definition toggles between and (7.5)/(7.6). However, the orientation on X2 has been reversed so that the negative spinors become the positive ones. Therefore denote the bundle obtained in this way by (9.6)
:=
S.
In the same way define the bundle of spinors of negative chirality on X: (9.7)
S := S Us-i St
I. Clifford Algebras and Dirac Operators
54
A smooth section of is a couple (si, 82) such that is a smooth section of S÷ over X1, is a smooth section of S over X2, and on U the following coupling condition is satisfied (see Figure 9.3):
s2(u,y) = G(y)si(u,y)
(9.8)
for yE Y
and tiE (—1,1).
SI
S2
Fig. 9.3 Cut through
Now we shall show that
a global section
defined by the formula
A+(si,s2) := (A+si,As2)
(notice A = (Ak) by Theorem 7.3) provides a well-defined elliptic S+) —* C°°(X; S—). : differential operator of first order In fact, by (9.6) and (9.7), the bundles can be considered as the positive and negative chirality subbundles of a well-defined Ct(X)module S with compatible connection; appears just as the Dirac operator associated with the bundle To prove this we only have to check the pasting condition on U for the couple As2). We need to obtain a section of S—. In order to check the compatibility of with As2 in the sense of (9.7), recall that and satisfy (9.8) and that the operator A is formally adjoint to This implies that A is of the following form on U: (9.9)
A
=
+ B)C' =
+ GBG')
for u E
(—1,1).
9. Invertible Doubles
55
The choice of signs in (9.9) needs an explanation: Formally, the formal adjoint to an operator of the form (9.4) is given by But
this is the formula on Xi, where u is the inward normal coordinate. On X2 the coordinate u is the outward normal coordinate. Therefore we have to change the sign in front of Now it is easy to check the pasting condition (9.7). We have the following equalities on U: 9 10)
+ B)G's2 =
As2 =
+ B)G'(Gsi) =
+ B)si
= G1G(ÔU + B)s1 =
From the construction it follows that
=
(9.11)
hence index is a truly in= 0. Now we want to show that vertible operator. We postpone the proof that A+is invertible until after the next lemma, and the proof that hence (A+ ) 1 is an elliptic pseudo-differential operator of order —1 to Proposition 9.5. []
Lemma 9.2. Let X be a closed partitioned Riemannian manifold of even dimension, X = X1UX2 with X1flX2 = Y = OX1 = OX2, : operator, and let C°°(X; S-) be a Dirac where are the ±-chirality components of a C€(X)-module S with with compatible connection. Then any 81 e C°° (X1; =0 and SiIy = 0 can be continued to a smooth solution for the operator over the whole manifold X by setting
onX1
onX\X1. Proof. We show that the extension of si by zero is a weak solution for over the whole of X. Apply Green's formula (3.9):
Av)
= IM =
Av(x)) dvol(x)
(si(x);Av(x))dvol(x)
+f0
=
+ j (G(0,y)si(y);v(y))dvol(y) = 0
I. Clifford Algebras and Dirac Operators
56
for any v E C°°(X; S—). By the regularity of the solutions of elliptic equations over closed manifolds it follows that C°°(X; and
0 Proposition 9.3. The operator defined above is an invertible operator (i.e. has trivial kernel and cokernel).
Proof. For any 5€
ker A+
we show that
=
0.
Recall from
the definition of that s can be written as a couple (81,s2)E C0o(i; S+) subject to the coupling condition (9.8). Then s E
means A+si =0 and A82 =0; hence 0=
—
(s1;As2) Green's formula
=
1
2 — i (G(siIy);s2ly)dy= 11821v11
Jy
It follows from Lemma 9.2 and the unique continuation property (Theorem 8.2 and Remark 8.7) that 0 and 0, hence ker(A+) = {O}. We apply the same argument to the operator A- = A— U to show that ker(A+) = kerA
{0}.
To complete the proof of Theorem 9.1, we must show that the is an elliptic pseudo-differential operator of order —1. This follows from standard pseudo-differential theory. We begin with a proposition from elementary functional analysis which we shall also apply later on. inverse
Proposition 9.4. Let T : H1
H2 be a bounded operator acting between (complex separable) Hubert spaces H1, H2. Then range(T) is closed, if and only if the inverse operator of
T denote the operator Tl(kerT)i. Assume that range(T) = is closed. Then the operator (T)' is bounded by the closed graph theorem. For the opposite direction we give an elementary proof. Assume
that there exists c> 0 such that (9.12)
ll(TY1vII
for any v E range(T)
9. Invertible Doubles
57
and that range(T) is not closed. Hence we have a sequence with
C
y
—,
range(T).
< k for Next notice that {z,j is an unbounded sequence. If would be a Cauchy sequence some k > 0, then = (because (T)' is bounded) and we would have
x := fl—.oo lim
and
E
y = Tx.
Finally, by eventually passing to a subsequence, we may assume that
> n and This yields
< hull + 1.
—
=
> n,
hence (T)1 is not bounded.
Proposition 9.5. Let A : C°°(M; E)
C°°(M : F) be a bijective elliptic pseudo-differentzal operator of order m over a closed manifold M. Then A—' is a pseudo-differential operator of order —m. Proof. Since A' = (A*A)_1A*, we can assume that A is a selfadjoint and non-negative operator on C°°(M; E). Now A is a Fredholm operator; hence for any k the extension
A has a closed range. By the preceding Proposition 9.4, this means that the inverse operator of A(k) is a bounded operator
E)
:
Hk(M; E).
Let Q denote a parametrix of A (9.13)
QA=Id+K,
where K is a smoothing operator. We multiply (9.13) by A' from the right which yields the following formula for A1: (9.14)
A' = Q — KA'.
The operator KA' is a smoothing operator, hence it has a smooth kernel and A' is a pseudo-differential operator of order —rn []
58
I. Clifford Algebras and Dirac Operators
Remarks 9.6. (a) The operator is not a seif-adjoint operator though it is isomorphic to A = (A+) by (9.11). To be precise, let J : X X denote the isometry which interchanges the two factors X1 and X2 of X by mapping xof the one copy into J(x) := x of the other copy. The pullback J (S+)is equal to S and we can extend which on sections is given by the J to a bundle automorphism of formula
J(s1,82) := (82,81). We have the following formula:
= A-J which gives the required isomorphism.
(b) The construction of operators of the form A U A* was introduced in Wojciechowski [1985b], where the analytical realization of the relative cycles for the K-homology group Ko(X, Y) is discussed. See Douglas & Wojciechowski [1989] for a detailed exposition of this aspect of the theory. There the odd-dimensional case is also treated and operators of the form B U (—B) are constructed, where B is a formally seif-adjoint Dirac operator. Our construction is also applied in some further work, where we discuss spectral invariants of seif-adjoint operators and boundary problems; see Boos & Wojciechowski [1989], Douglas & Wojciechowski [1991], Klimek & Wojciechowski [1992], [1993], and Wojciechowski [1993].
We can apply the glueing construction in a more general context as well. We only need (1) a partitioning of a closed manifold M = U U V into two submanifolds with common boundary V , and (2) Dirac operators A1 on U and A2 on V which are equal on i.e. with common Green form C and boundary Dirac operator B. on M. The details Then we can always construct the operator A1 are left to the reader; see also Wojciechowski [1985b].
10. Glueing Constructions. Relative Index Theorem
We explain the relative index theorem of Cheeger, Gromov & Lawson and offer an analytical interpretation which also works in the odd-dimensional case. We also give a simple proof of a special case of the excision principle for indices.
Here we want to give an analytical interpretation of the relative index theorem of Cheeger-Cromov-Lawson (see Gromov & Lawson [19831 and Lawson & Michelsohn (1987}). We set aside some definitions and technical details, since there are quite a few excellent expositions of the theorem including also various K-theoretical interpretations (see also Borisov, Muller & Schrader [1988], Julg [1988), and Bunke [1992aJ, (1992b]). We only want to offer a simple analytical interpretation that works in the odd-dimensional case as well. The situation is as follows: We have E1 and E2 bundles of Clifford modules over X1 and X1 respectively, where X1 and X2 are two complete Riemannian manifolds (not necessarily compact). Assume
that there exist K, compact subsets of X,, such that there exists E1 IXI\Ki —' E21x2\K2 a unitary isomorphism of Clifford bundles.
Let / denote an isometry of X1 \ K1 onto X2 \ K2 covered by 4' This assumption implies that the corresponding Dirac operators I), are equal outside of K,, j = 1,2. Now assume that dim X, is even. Then the following fact is straightforward.
Proposition 10.1.
Let 5,(t; x, y) denote the kernel of the operator
Then
(10.1)
index(D1,D2) := urn
t—.o
[J-'1strEi(t;x,x)dvoli— J
is a well-defined integer which we call the relative index of the opera-
tors D1 and 1)2. Here the symmetric trace is defined by str e -ti)2I = tr e
as usual.
I
I
—
tr e
i
i
I. Clifford Algebras and Dirac Operators
62
We can also apply our glueing construction to prove the excision principle for indices. We obtain an explicit operator expression for the differences. Following Donaldson & Kronheimer [1990; Section 7.11 (see also Seeley [1965, Appendixj) we consider two sets of data
(M; U, V, D1, D2)
and
(N; W, Z, A1, A2),
where M is a closed Riemannian manifold (compact and without boundary) which is the union of two open submanifolds M = U U V. Let D : C°°(M;E,) —. C°°(M;F2), i =1,2 denote a pair of elliptic differential operators over M. Assume that there exist bundle isomorphisms and
such that
= over We make corresponding assumptions about the second set of data, the second quintuple, i.e. roughly speaking: N = W U Z and A1 is D2
isomorphic to A2 over Z. (See Figure 10.3.) DiIv = D21v
4" U
w z
Fig. 10.3 The excision data: two quintuples
10. Clueing Constructions. Relative Index Theorem
In addition to Seeley and Donaldson & Kronheimer, we make two further assumptions. First consider only (compatible) Dirac operators. Second, assume that U \ V has a smooth boundary Y and that all structures are products in a bicollar neighbourhood of Y. This gives the following variant of the relative index theorem:
Proposition 10.4.
Let a quintuple (M; U, V, D1, D2) be given, i.e. a
closed Riemannian manifold M = U U V given as the union of two open 8ubmanifolds and two Dirac operators D1, D2 over M which coincide under suitable bundle isomorphisms over V. We assume that U \ V has a smooth boundary Y and that all stn&ctures are products in a bicollar neighbourhood of Y. Then we have
U D,u,
index D1 — index 1.32 = index
denotes the restriction
where
j = 1,2,
:= DjIu\v, and double
denotes
U of U \
V
by glueing
Dirac operator obtained over the closed the pieces together along Y.
the
Now assume that we have two quintuples as above:
(M; U, V, D1, D2)
(N; W, Z, A1, A2).
and
Assume also that there exists a diffeomorphism r : U —. V by suitable bundle isomorphisms such that .1
Theorem 10.5. Under the property for
r..#A
7Th
preceding
covered
assumptions we get the excision
indices
index A1 — index A2 = index D1 — index D2. Proof.
indexAi — indexA2
= index 7.
•
Prop. 10.4.
=
indexAi,w UA2,w
* U Dv.,
Prop. 10.4.
=
index D1 — index D2.
D
Remark 10.6. The explicit description of the index differences is due to two circumstances, namely that we deal with Dirac operators and not arbitrary elliptic differential operators, and that we restrict ourselves to the smooth, relatively compact case.
PART II
Analytical and Topological Tools
11. Sobolev Spaces on Manifolds with Boundary
We investigate the continuity properties of taking traces in Sobolev spaces over manifolds with boundary.
Let X be a smooth compact n-dimensional manifold with boundary Y and fix a Riemannian structure on X. Use the Riemannian structure to construct a collar neighbourhood N = Y x I of the boundary Y and denote the (inward) normal coordinate by t E I = [0,1). Actually, we may assume that X is a submanifold of a closed (= compact, without boundary) smooth manifold M, e.g. M = X Uy X, the closed double of X. We extend the identification Y x [0, 1) = N to an imbedding '1' : Y x [—1,1] '—i M.
Definition 11.1. The following chains of Sobolev spaces are naturally associated with our data (s R, s 0): (a) The space H8(M) consists of all complex valued L2-functions u over M which yield elements of in local coordinates. More precisely: Let with open in M and : —p R'2 be a smooth atlas for M. Then, for any smooth function over M with compact support contained completely in there exists an element such that = o E (b) The space H8(Y) is defined in the same way. (c) The space H(X) consists of the restrictions u e H8(M)}, where r+ : L2(M) L2(X) denotes the restriction operator u i—. ulx.
The preceding concepts extend to s <0 by duality. For convenience we recall the definition of Note.
:= {v I v E
and e
'—' (1
+
where the Fourier transform is defined by
:= f Then the Fourier inversion formula yields (11.2)
v(x) =
f
E
II. Analytical and Topological Tools
Note that Dvii.
:=
(27r)_hh/2 11(1 +
ii
which can be regarded as L2(R'; (1 + provides a norm for From the L2-space with respect to the measure (1 + which depend this we get norms for the spaces H3(M) and and a corresponding on the choice of a smooth atlas {UL, = 1: and 0, supp4 C partition of the unity with iiuii. :=
(11.4)
u) 0
cii.
for u E H5(M) and similarly for u
It is remarkable that all these norms for H'(M) are equivalent and define the same topology:
Proposition 11.2. Let
be a diffeomorphism of an open set Ui C be an open, relatively Let onto another open set U2 C compact subset of U2. Then we have for all s iiv 0
Citvii.
for oil v E
C is independent of V. Proof. Accounted for e.g. in Gilkey 11984; p. 281 or Boof3 & Bleecker 11985; p. 177].
0
Corollary 11.3. Let U1 be an open subset of a closed manifold M1 and a diffeomorphism of U1 onto an open 8ubset U2 of (or of another closed manifold M2 (or of RTh). Let K be a compact subset of U1 and let H8(K) denote the space of elements of H8(M1) with : v i-÷ v o continuously support in K. Then maps
into H'(4'(K)). The aim of this chapter is to prove the following trace theorem.
Theorem 11.4. For u E C°°(M) we define ut(y) := : Y x [—1,1] where M is a fixed parametrization of a collar neighbourhood of Y. Then for t E (—1,1), the mapping u Ut from C°°(M) to C°°(Y) can be extended to the whole space H8(M)
for s > 1/2 by continuity. It provides a continuous linear map which is uniformly bounded for t belonging to any compact subinterval of (—1, 1). For given u E H8(M), the
'Vt
H8(M) —+
11. Sobolev Spaces on Manifolds with Boundary
=
mapping t i—' (—1, 1) into
Ut
69
is a continuous mapping from the interval
The delicacy of the trace theorem is due to the fact that we cannot in general apply the bounded Sobolev embedding C which is valid only for s > n/2. A special feature of Theorem 11.4 is the emphasis on the dependence and independence of all constants of the normal parameter t. The main ingredient of the proof is the following lemma:
Lemma 11.5. ForeachtER,sER, ands>1/2 we have (11.5) IIftIIa_1/2 where
J_oo(1+T2)_SdT
is the restriction off on
Proof. First assume t = all y E (11.6)
because
0.
for f E
x {t}.
Note that fo(y) =
f°°00 J(y, t)dt for
In fact /0(y)
= =
J
J
2ir Rn—I
JJ
e_itTf(,7, t) dt dr dire
—00 —00
of the Fourier inversion formula for
f
2ir
r) drj dr dt.
It"
(Since f has compact support, the multiple integrals converge absolutely and the order of integration can be inverted.) Therefore, 2
00
IJo(y)12 = (27r)_2
J(y,
(J
If(y,t)R1 + v2 + t2)8/2(l+ y2 + t2)_8/2dt) 2 IJ(y, t)12(1 + y2 +
j(i
+ y2 +
Since 3> 1/2, the second integral exists and we can apply the HolderSchwarz inequality. Now if we make the substitution t = (1+ y2)'/2r we get
f(1 + y2 +
=
(1
+ y2)8+l/2 J_oo(1 + r2 )_8dr.
II. Analytical and Thpological Tools
70
Therefore,
(1 +
(2ir)_2J
(1
+T2)_8dTj
IJ(y,t)12(1
+y2 + t2)dt,
and integrating with respect to y gives the desired result for the restriction f '-+ ft for t = 0. For general t we make the dilatation g(y, s) := f(y, s+t) and receive (1 +
lift 118—1/2 = 11901Is—l/2
hf 118(j (1 +
=
the dilatation of the variable yields only a phase shift of the Fourier transform: because
= = = for
=
E
f(y, s + t) ds dy
J
f(y,
J
dy
eteTJ(e)
hence
=
and lIgD.
= 111118.
Proof of Theorem 11.4. We suppose that H3(M) and We choose are equipped with fixed norms and an open covering {Uj of Y, an atlas of charts of onto open of unity. Let and a corresponding C°° partition subsets of t belong to a compact subinterval J of (—1,1). We choose a real C°° function x which is equal 1 on J and has support in (—1,1).
Then for any s> 1/2 and uE C°°(M) we have lI14tIIHa_.1/2(Y) < C1
where Cl does not depend on it. Also
= c2IX(t)'I
11. Sobolev Spaces on Manifolds with Boundary
71
where t) := Now, using Lemma 11.5, note that (11.9)
Corollary 11.3 here says (11.10)
IIV&IIH.(Rfl) C4IIUIIH.(M).
with All the inequalities together imply IIUtIIHa_1/2(y) c independent of t and u. Therefore the mapping u Ut can be extended to all of H(M), s> by continuity, and yields, for each
t, a continuous map of H(M) into H''/2(Y) which is uniformly bounded for t varying in any compact subinterval of (—1,1). To finish our proof we shall show that the parametrization t '—' Ut for any given is a continuous mapping from (—1,1) into U E H'(M). This follows from the fact that the function (y, t) ut(y) is an infinitely differentiable function in Y x (—1,1) for u E and from the earlier derived continuity properties. fl
shall now give various reformulations and extensions of the trace theorem. We
Corollary 11.6. Theorem 11.4 remains true, if we replace the closed manifold M by the manifold X with boundary. More precisely: For X of the s > 1/2 and a given parametrization 'I' : Y x [0, 1] collar of Y in X there exists a natural separately continuous mapping which is uniformly bounded on any H°(X) x 10,1) compact subinterval of [0, 1).
For the proof of this corollary and for later use, we recall various restrictions, extensions, and reflections in function spaces over manifolds with boundary. Not all of the mappings are natural; some depend on the choice of smoothing functions and local coordinates. Definition 11.7. We distinguish the following natural mappings (for simplicity s 0): (a) The restriction operator : H8(M)
(11.11)
is given by u
uIx.
H3(X)
II. AnaJytical and Topological Tools
72
(b) The extension by zero operator
L2(M) is defined
L2(X)
by
Iu
(11.12)
onX
onM\X.
Then the composition e+r+ yields the cut-off operator. (c) The reflection operator J : u(y,t) .— u(y, —t)
(11.13)
is only well-defined for functions with support close to Y. It can be defined for the whole function spaces by multiplication with a smooth function x which is equal to 1 near Y and vanishes outside a neighbourhood of Y. (d) A continuous extension operator of codimension 0
forkENLJ{O} is given by graded globalized reflection t)
(11.14)
ift>O
( u(y,t)
if t <0, a,u(y, —jt) x(t) whore x is a smoothing function equal to 1 close to Y and vanishing outside a neighbourhood of Y and the a, will be picked to yield k+1
form=0,1,...,k. j=1
extends to continuous operators The operator H3(M) for s k with the property = idH.(x).
: H8(X)
(e) A continuous extension operator of codimension 1
H8(M)
for s>
is given close to Y in local coordinates by (11.15)
t) := X(t)
J
—
tip)
where x is a smoothing function equal to 1 close to Y and vanishing outside a neighbourhood of Y, and is C°° with compact support on a coordinate patch of Y. Note that 0. —, (f q5)g as The claimed properties of the natural maps follow immediately from the definition. Then Theorem 11.4 implies Corollary 11.6 by the con: H8(X) H8(M). tinuous extension
11. Sobolev Spaces on Manifolds with Boundary
73
Note that the definitions and statements of this section remain valid, if we replace the function spaces by spaces of sections in arbitrary smooth Hermitian vector bundles. By local trivializations we obtain:
Corollary 11.8. Given a bundle E over X we can extend E over M. Near Y we can represent it as x (—1,1) so that sections u of E can be written in the form ut(y) = u(y, t) with y e Y, t E (—1,1) and ut(y) E EJy = EIyx{o} EIyx{t}. Then, for s> 1/2, the trace map y: H'(E) x [0,1) —*
H'"t2(EIy)
is well-defined, separately continuous and uniformly bounded on any compact subinterval of [0,1). We end this chapter with a brief discussion of a few examples.
Examples 11.9. Let
denote the n-dimensional torus is generated by the complete orthonormal sys:= tem of C°° functions {eV}VEZn with (b) For arbitrary s, the space Ha(Tvz) consists of the completion of under the norm (a) The space
:=
Iü(v)12(1
+
11,12)8,
where the zi-th Fourier coefficient is given by
:=
J
In other words, any u belongs to exactly when the converges absolutely to u in the II• norm. sequence E (c) Let y i-4 (y, 0) be the natural embedding of x T' 118
S'. Then for any u E H8(T") and arbitrary s, r, the restriction is well-defined and belongs to
:=
1
exactly when
(1 +
A€Z"-1 pEZ
Comments on the example. (a) is well-known from the theory of Fourier series. (b) may be deduced from our Definition 11.1 by
II. Analytical and Topological Tools
74
straightforward calculations or, more smoothly, by a characterization of H'(M) as the domain of the differential operator in L2(M). This characterization is valid for arbitrary Riemannian M. We obtain (c) from noticing u(y, 0) = ü(Â, where the sum is over all (A, t) E Z = Z', and the convergence on T' is uniform. = (2ir)1/2, it follows that Since
u(y,O)=
eA(y)>u(A,p),
where the series converges uniformly on
Thus we have
)(A) = (27r)_h/2
9
Remarks 11.10. (a) From Example 11.9c we gain insight into the possible loss of summability under restrictions of Sobolev spaces (trace theorem) in contrast to the gain of differentiability in C' theory. Take e.g. the Fourier series over T' :=
Then the series
,t)12(1 + A2 +
converges
for s = 0
an element of H°(T2) = L2(T2), whereas the series (1
+
not converge for r = 0 and J3 1. Hence uIsi is not well-defined in H°(S') = L2(S'). (b) One can also easily construct examples of functions which lie in but with trace not belonging to For example does the function u(x) := lxI'x(x), where x is a C°° function on does
R with compact support and equal to 1 on the unit ball, belong to exactly when 2a> 2 — n. So u(x) 1x1"4x(x) is an element of H'(R3), but neither does the first trace 4, ., 0) belong to H' (R2), nor the second trace u(., 0,0) to H'(R). The example furthermore shows that there is no continuous linear map H8(R') —p which extends the trace map —+
12. Calderón Projector for Dirac Operators
For any Dirac operator over a compact manifold with smooth boundary, the orthogonal projection of sections over the boundary onto the space of Cauchy data is a pseudo-differential operator of order zero with principal symbol given by the related spectral projection.
We need to fix and explain the notation. As in the preceding chapter we assume that M is a closed smooth n-dimensional manifold which is partitioned into two manifolds and X_ with = t9X_ = Y by a smooth closed submanifold Y of codimension 1. A Riemannian structure is fixed on M which defines a parametrization Y x [—1,1] of a bicollar neighbourhood of Y in M.
Fig. 12.1 The partition of X = X_
U
Following the fundamental tenets of numerical analysis, we build our analysis on the concept of the spaces of Cauchy data. The point is that the following concepts and results are well-defined and true for the total Dirac operator A and the component A+ (and for even and odd dimensions); hence in the following definition A may denote any operator of Dirac type. Definition 12.1. For any Dirac operator
A: C°°(M;E) -. C°°(M;F)
II. Analytical and Thpological Tools
76
over
a partitioned manifold M = X_
U
with X_ 11
= Y we
define
(a) the spaces of Cauchy data (along Y) by
:=
(12.1)
I
u
E C°°(E) and Au =0 in
and we denote the closure of these spaces in (Ely) by H±(A, s) for any real s; (b) for any real s we define the solution spaces (null spaces) by
ker±(A,s) := {u E H3(EIx±) Au = 0 in
(12.2)
\ Y}.
Remarks 12.2. (a) Since Dirac operators satisfy the unique continuation property (see Chapter 8), there are no non-trivial sections in ker A or in ker+(A, s) with support contained in \ Y. Notice that the unique continuation property also excludes non-trivial solutions which vanish identically on the boundary. This is standard: Let u E C°°(EIx+) and Au =0 in and =0. Then the extension by zero e+u is a weak solution for A over the whole of M because
= fM
= =0
A*v)
=
(u; A*vIx÷)
+J0
(Au; vlx+) -I-f (G(y, O)u(y); v(y))
by decomposition of Green's formula (Proposition 3.4) for any v E C°°(F).
By the regularity of the solutions of elliptic equations over closed manifolds, it follows that w := C°°(M; E) as desired; hence 0 by unique continuation. (b) As in Chapter 9, we shall in the present chapter restrict our presentation to the case of even-dimensional manifolds. The only reason is to take advantage of the easy to follow chiral notation in the even-dimensional split case. However, the results remain true and the arguments remain applicable also in the odd-dimensional case. We leave the necessary transcriptions to the reader. To avoid an inflation of subscripts and superscripts, we shall use the notations X and and S, and A interchangeably throughout this chapter, namely, for the object8 on the primary (plus) side of the closed double X. This ambiguity seems tolerable, since w
12. Calderón Projector for Dirac Operators
77
all results are true for the total Dirac operator and the chiral Dirac operator as well. Note that, corresponding to the ambiguity in the notation of S and A (whether they refer to the formally seif-adjoint total Dirac operator or its non-seif-adjoint chiral component), we have
an ambiguity in the notation S and A-. The ambiguity is whether they designate the same bundle and operator as on the plus side, but copied to the secondary (minus) side of X, or the image bundle and the formally adjoint operator, which are the same for the total Dirac operator, but different bundles and different operators in the chiral split case. Both of these ambiguities disappear once it is specified whether one argues about the total or the chiral split Dirac operator.
Let A (instead of A+) denote the invertible extension A of on the closed double constructed in Chapter 9. We begin our analysis of the behaviour of Dirac operators close to the boundary by presenting the following fact.
Lemma 12.3. Let
C°°(X;Sj
be a Dirac operator over an even-dimensional manifold X with boundary Y and let A denote its extension over the closed double X constructed in Chapter 9. Then the spaces H÷, H_ of Cauchy data of A intersect only in the o section.
Proof. Let u
—'
:
H÷(A)n H_(A). Then there exists s+ E C°°(X; such that =0 and As_ =0 and
S)
=U=
(12.3)
Define a continuous section s of S+ on X by
(12.4)
=
( s-i-
t.s-
on
=
x
onX_.
We show that s is a weak solution of A8 = we have over the closed manifold X
0.
Let w E C°°(S—). Then
(12.5)
(As;w) =
(s, (A)*w)
= (st; ((A)*w)Ix+) + (s_; ((A)w)Ix_)
Recall from Chapter 9 that A takes the form + B) on X÷ close to Y, and the form + — B) on X_ close to Y, with the normal parameter B) =
II. Analytical and Thpological Tools
78
t inward (and therefore opposite) oriented in both cases. Hence, by Green's formula (Proposition 3.4),
(s+;((A)w)lx+) = and
(s_; ((A)w)Ix_) = (As_;wlx_) + ((—C). Since
(12.6)
and As_ vanish, we finally obtain (As;w) = (C.
= 0.
—
By the regularity theorem for elliptic operators over closed manifolds this implies that s E C°°, hence s = 0 for the injectivity of A (see Chapter 9) and therefore u = Sly = 0. 0
The main result of this chapter is the following construction of a precise and pseudo-differential projection onto the space of Cauchy data of a Dirac operator.
Theorem 12.4. (A. P. Calderón, R. T. Seeley). Let X be a smooth compact even-dimensional Riemannian manifold with boundary Y; let S be a bundle of C€(X)-modules with compatible metric, and let the corresponding Dirac operator be denoted by : C°°(X; C°°(X; S-). We assume that the Riemannian structure of X is a product near Y. (Hence the operator A extends to an invertible operator A over the closed double X, and near Y it takes the form + B) as explained above in Chapter 9. In the following, the tangential part B does not have to be self-adjoint although our models,
the Dirac operators, have self-adjoint B). Let denote the restriction operator from sections over X to sections over X and let — be the trace map which was introduced in Chapter 11 and is well-defined and continuous for s > Then:
(a) For any real s, the mapping (12.7)
:=
—p
extends to a continuous map K÷ : ker÷(A, s) which is surjective, and is bijective if restricted to the Cauchy data space s). Here denotes the adjoint of (12.8)
: H1(X;S) —*
12. Calderón Projector for Dirac Operators
S1y) —'
It is continuous from
79
5) for r <
converge uni the sections (b) For all g E formly to limits and to limits —P_g as e —+ O_; the as e —, operators Pj, are projections onto the spaces of Cauchy data H± along and are pseudo-differential operators of order 0. Let b denote the principal symbol of the tangential operator B. Then the principal sym-
ofl'÷ at
yE Y, and 77 E TY\0 is the projection onto
the subspace of spanned by the eigenvectors corresponding to the eigenvalues of b(y, : with positive real part.
Note. The operator
is called the Calderón projector of A. By
a projection here, we mean an operator P with P2 = P, but not = P (see also Lemma 12.8 below). In other words, we show that 7'+ is idempotent. necessarily
The operator (A)'
LJA—)—' is sometimes called the volume
potential of A; it plays the role of the Newtonian potential in classical potential theory. Recall that the Newtonian potential V(x) := clog Ixl provides the first stage in finding the solution of the Dirichiet problem and
(12.9)
uIy=g
for a given compact domain X C R2 with smooth boundary Y and given functions v on X and g on Y: Let v be extended in any conis a venient way to the whole R2. Then v'(x) := f V(Ix — solution of Lxv' = v, but in general with 9' := v'Iy 5é g. To get the boundary values right, the next step is to solve the problem
This
is done for the
wly=g—gl=:g2.
and
(12.10)
disc
[2w
— 2irRj0 1
:=
Ixl
g2(Rcos9, Rsin9)(R2 — r2)
dO
v'R2+r2—2RrcosO
for general, not necessarily circular, Y by the classical method of double layer potentials by writing to in the form and
w(x)
OV(x—
f
f(y)dy,
II. Analytical and Topological Tools
80
where n denotes the normal outward derivative and f has to be determined by the boundary integral equation
cf(y)
+
j
= 92(Y)
for y E Y
with a certain pseudo-differential kernel L. Then u := v' + w solves the original problem. may be called the surface potential or the Poisson The operator operator of A; it plays the role of the mentioned double layer potential. The use of these three operators and in solving elliptic boundary value problems for Dirac operators is explained below; see especially Chapters 18 and 20.
Proof. The essential part of the statement and of the proof of all parts of this theorem is the uniform convergence in part (b) with its delicate estimates. However, for a better demonstration of what is going on, we prefer to begin with the proof of part (a) (assuming that some of the results of (b) have already been proved). (a) Let g belong to
The distribution
(12.11)
:= [(vly;g)
is defined by
JY
is nothing but the distribution öy ® g. v E C°°(X; S). So When we apply the Green form C we obtain a distribution w = G(t5y ® g) on the whole of X, which in fact belongs to H 4, and for
A'w is C°° out of Y simply because of the fundamental property of pseudo-differential operators, namely, the pseudo-locality sing supp A 1w C sing supp w.
We shall in (b) below show that and all its derivatives converge uniformly as This means that Kg := E L2(X; S). Then AK9 = is supported (as a distribution) in Y; hence AK9 = 0 in X \ Y and K÷g = E ker÷(A,0). Now we show that for any g E C°° I y) and any real s (12.12)
IIK÷g118
=
C independent of g. We shall do this for s equal (For s < the estimate follows immediately
to an integer k 1.
12. Calderón Projector for Dirac Operators
from bounds on A' and
Set u := K÷g =
and By part (b) of this theorem, the limit exists uniNow choose a continuous extension (Y) (X) as in Definition 11 .7e and obtain a Hk(X) with v smooth over X \ = go, and Next assemble a section to L2(X) by
go := formly and operator section v = e90 Ilvilk
(12.13)
to =
In r(v)
fort>O fort —<0,
I.
where r denotes the restriction to X_ \ Y = {t <0}. Note that u and v coincide on V = {t = 0}. It must now be shown that to and hence u belong to Hk. To do
that we show that Aw has support in X_: In fact for any smooth section 4) we have (Aw; 4))
J(w; (A)*4)) = c—eO+ lim
=
x
= lim
J
J
(to; (A)*q5)
(Aw; 4)) + urn J (C(y, thew; 'y€4)) Y
ItI>c
—
urn
Here the two integrals over V cancel as
and rw =
j
—
(G(y, —€)'y_f to; 0,
since r+w = ii =
rv have the same trace over Y; recall also that Au =
vanishes in the interior of X. This yields (Aw; 4))
= j(Aw;
=
(r (Av); r (4))).
But Ais an operator of first order, so HIc_l(X) with IIAvIIk_l and also Aw with the same norm because the supC3IIvIIk port of Aw is completely contained in X_. Since A' is an operator of order —1, we finally obtain w E and IIUIIk IIWIIk
= 11A'AwIIk = C2lIegoIIk
which proves (12.12).
C,IIAwIIk_l = C,IIAvIIk_,
82
IL Analytical and Topological Tools
Thus the potential operator K+ extends to a continuous map H8_5(S+Iy) ker(A,s) for real s < and for integer s 1 and by interpolation for all real s. Also K+ is surjective: Let u belong to ker(A, s), i.e. u E H3(X; and Au = 0 in also We set g := and v := Then X \ Y; s > belongs to ker(A, s) and it follows that A(u — v) = 0 in X \ Y and (u — v)Iy = 0, hence u — v = 0 because the Dirac operators fulfill the unique continuation property. This ends the proof of statement (a) of Theorem 12.4 for s> The general case needs additional arguments and will be discussed in the next chapter.
= (b.1) First we want to show that the limits exist, are uniformly approached, and in fact provide a pseudodifferential operator P± of order 0 with the claimed principal symbol. It suffices to consider the local situation. Since the Green form C is invertible, it may be neglected. Thus we consider operators of the are form with := where the locally supported real valued C°°-functions on X of a C°°-partition of unity suited to a finite covering of X with open sets. Distinguish three situations. (i) When the support of does not intersect Y, then all is trivial since = 0. (ii) When and has a smooth kernel as integral op: H8(Y) erator and C°°(M) is continuous for any 8 < 0 and therefore for all s. From the well-known Sobolev lemma (or from the stronger Theorem 11.4) we then have that converges uniformly as e 0+ and the limit yields an operator of orhave disjoint support, M#1 A—'
der —00.
have support in an open set U which is contained in the strip {ItI < 1}. We choose coordinates is trivial. Unfortunately, no diand assume that x: U (iii) Suppose now that
and
rect approach is available for investigating will lead to an operator T But a slight modification of with well-defined traces Then the investigation of the correcting terms will yield the claimed convergence, pseudo-differential and symbol properties. More precisely, choose a C°°-function ço with supp C U and 1 =: V. Then is a pseudo-differential on supp(l&I +
operator with the (total) symbol given explicitly by C_k with c_1= (a1)' on V. Here a1 = u1(A) denotes the principal symbol of A. We denote by ck(x, D) the homogeneous pseudo-differential
12. Calderón Projector for Dirac Operators
83
operator of order k over U given there in the local coordinates x by
ck(x, D)u(x) =
(12.14) Set
fork =
Ck :=
J
T :=
(12.15)
Then for any k0 EN
—l,—2
C_k
—
k
an operator of order —k0. By Theorem 11.4 we obtain a continuous mapping is
(12.16)
Hk0_l(S)
:
which, for sufficiently high k0, by the Sobolev embedding theorem gives a continuous map
—p
C°(S). Hence
has a well-
defined trace on Y and it remains to prove that the errors converge uniformly as —. 0+. This is not a trivial result, since which, for even for g E C°° we have, a priori, only example, for k = 1 does not necessarily have a well-defined trace. To overcome that difficulty it is crucial to know certain details about the symbol of and fa = 1. We set C (—1, am(t) := ma(mt). Then the sequence {am 0 9}m=I,2,... converges weakly to in H8(X) for each 8> In fact (am ®g; v)
=
Jf
(am(t)g(y); v(y, t)) dydt
0)dy
for v E H8(X) as m — oo. Now recall
á1(r, ij)
=
f
f =f =
Then we obtain (12.17)
= lim = lim
m—'oo
f =
t)
J J
t;
e*t1â(r/m)c_k(y, t;
r1)drdi1
T)dT)
II. Analytical and Topological Tools
84
Now we replace the integral
for any E by an integral over a finite path r(q) in 0). In a certain sense this replacement is the decisive step in establishing the Calderón projection, since it provides the uniform boundedness of the integrands which permits passing inside the integrals. Then for (12.17), this yields the expression (12.18)
t)
=
eitrc_k(y, t;
J
r)dr1 )
(
= as m —i f Before drawing conclusions from (12.18), return to the replacement
since
of
â(r/m) converges boundedly to â(0) =
Let q(i-), r E R denote the integrand of
q(r) :=
t;
00.
1
in (12.17)
r).
We are going to show that q
• is continuous on R (for t 0), • extends to a holomorphic function in IZR := {i- E C R} for some R, 0, In • and suffices q(r) = for some N, when r —' oo in 11R• Then one obtains I
p00
J
—oo
pR
q(r)dr
=
j
—R
pO
q(r)dr —
J
q(Re8)RiezOdO
which is sometimes emphasized by denoting both sides of the equation by the expression q(r)dr. We shall do likewise. First find the necessary holomorphic extension of c_k(y, t; 'r) in an appropriate region of 0). All the ck are rational functions of r, hence meromorphic. In fact, their poles are the roots of a1. More generally we can argue as follows: From the definition of (c_k) k=l,... and the symbol composition formula for pseudo-differential operators, we obtain the following re-
12. Calderón Projector for Dirac Operators
lations for (y,t) c_1a1 c_ 1a0 +
c_2a1 —z
= Id
0a1
Oc_1 0a1
.
=0
—
3=2
c_kal+c_k÷iao+...c_la_k+2—z
h—
Oc_k+i 0a1
.02c_k÷202a1 +*
i.e. C_k can
be written as
(12.19) aj1 x (sums and
products
of derivatives of a1, a0,... ,a_k+2 and of c_l,c_2,... ,c_k+1).
Here
ak denotes the asymptotic expansion of the (total) symbol
of A. Now ai(y, t;z7, ,-) is linear in (ij, r) and non-singular for 1zi12+r2 > 0,
and r E R. Allowing r to be complex, there is a compact set Z in the complex plane C, not intersecting the real axis, such that for I'iI = 1 and (y, t) E supp the complex values of r, for which a1 is has a holomorphic extension not invertible, lie in Z. Thus = to C \ Z for such and (y, t). Then by induction it follows from (12.19) that all C_k and their derivatives have such extensions, each of which is bounded by some polynomial in In. See also Figure 12.2. Let denote the part of Z which belongs to > 0}. We choose a positive real R such that C {InI ( R}. For any 77 E define a closed path '1 E
r(77) := 0 ({r I ml max(1; R1771)}
0})
with counter clockwise orientation and a closed path r0(77) as indicated in Figure 12.3. Our choice of R ensures that the integrand ettT&(r/m)c_k(y, t;77, r) of our real integral in (12.17) is holomorphic for r in the region which is enclosed by r0(77) for all natural m and k. Let us denote
II. Analytical and Topological Tools
86
Fig. 12.2 The complex values of r for which al (y, t;
r) is not invertible
•. .
.
S S
S
S
S
S
.
S T
Fig. 12.3 The closed paths
and
the joint arc of the contours and above by r1, and the segment of close to 00 by r2. Note that is bounded for
t 0 and
0 and that
decays exponentially as In
—+ co
in {E(r) O}. From all this we get
12. Caiderón Projector for Dirac Operators
(00
87
fRIII
j.oO
J...=J...+J...=J ."+J".+ ...
...
r(,7)
since we can move the segment r2 arbitrarily close to oo and thereby let the suppressed integrals go to zero without changing R(v) and the contour by the contour integral over the finite path Having replaced obtain a dominated convergence for the integrands (bounded we for all m by a common polynomial in ftI). We may therefore pass inside the integrals. This proves (12.18). t) in (12.18) is C°° for t > 0, and all its It is clear that derivatives in y and t extend continuously to {t 0}. This gives the in L2(Y;S+Iy) fore 0+. uniform convergence of From (12.18) it also follows that the hereby well-defined operator is a pseudo-differential operator of order 1— k, since its symbol P—k(Y,'7) :=2ir-1-J
c_k(y,O;?1,T)dr
in the integral representation (
f
=
urn
P—k(Y,
and hence belongs
is homogeneous of degree 1 — k in i7 for large
to the standard symbol class Si_k. More precisely, we have for real
8>
1,
=
f
=
r(817)
f
since C...k remains homogeneous in its holomorphic extension,
= 3—k+I
f
c_k(O,y;r/s,r1)s'dr = 3—k+i
r(a17)
=
3—k+1
f
c_k(0, y; r', ii)dr',
fl {IrI = 81—k 2irp..k(y, ii).
since
f
c_k(0,y;T',71)dT'
r=
is the boundary of
0) and therefore equal
r
II. Analytical and Topological Tools
88
(It is not crucial to the argument that s—'r(sij) = but only that the two paths are 0-homologic — form a boundary — in the region where the integrand is analytic.) In our investigation of let us see how far we have gone. First we saw that the corrected operator T = —
C....k of (12.15) is an operator of order —k0 and
that
is well-defined for g E C°°(Y; E C°(Y; Of course also extends on all Sobolev spaces and is pseudodifferential. This follows by the same argument, as in deriving (12.16),
replacing T by Dl', where D is any differential operator of order k, Ic E N. Then for sufficiently large Ic0, namely, Ic0> 1 + k + n/2, once again Sobolev's theorem and the continuity of —'
G is continuous from H y) to G°(S). converge uniformly as e —' 0. Hence is welldefined on all Sobolev spaces. Then we proved that for any Ic show that also
I
Hence
P-k =
urn
is a well-defined pseudo-differential operator over Y of order —k + 1. This shows that the sections converge uniformly to limits
+>
urn
k
Repeating the argument we get (12.20)
11911a+k
for g in C°°, with c independent of E. This establishes the existence
of 2+, which by definition is a pseudo-differential operator of order 0. We determine the principal symbol of Without further sup-
12. Calderón Projector for Dirac Operators
89
Sb' we compute
pression of the Green form C: S+ Iv
=
= = -1-.J 27r
+
=
G(y)dr
= 2ir
= =
.LJ
{r—ib(y,ij)}'dr
2irt
4J 2irz
which is the desired spectral projection. (b.2) Now define the operator 1'_ in an analogous way. Then only two
steps remain for the proof of (b), namely, to show (i) + P_ = Id, Since it was already shown in Lemma 12.3 and (ii) C that fl H_ = 0, it follows that and P.-.. are complementary projections onto Hj along First we show P-f + P_ = Id. Let g C°°(Y; Then for any V E C°°(X;S) one has (12.21)
j(C.
= (C
g;'yov)L3(Y) =
where the symbol [vJ means that the distribution is evaluated Then (12.21) may be continued: on v. Set Kg
=
= (AK9)[v] = = f(Kg; (A)v) = lim e—.O+ x
f
(Kg; (A)v).
ItI>e
Here we used the observation from the beginning of our proof of part
(a), namely, that Kg belongs to L2(X; S). This permits the introduction of by bounded convergence. From Green's formula (Proposition 3.4), we obtain
J = JItI>c
(Kg;(A)v) (AK9; v)+J
)—J
;
II. Analytical and Thpological Tools
= {t = ±e) denotes the two submanifolds of X of codimension 1 defined by the normal coordinate t = ±e. Since AK9 vanishes outside of Y, the two integrals and ft<e vanish and we obtain where
(12.22)
J(G •g; vly) = urn J e—.O+
= j(C.
(C. + P_g); vly).
inside the integrals because all sections are C°°, the domain Y is compact, and all Y±e may be identified with Y. Since C is invertible and {vIy I v E C°°(X)} = C°°(Y), equation (12.22) proves P+ + 1'— = Id. No new arguments are needed to show range C H+. We proved already that and that AK,.g = is supE in fact, in the interior of ported in Y and therefore =0 One could pass
E ker(A,s) for all s 0 if9 E
0
Remarks 12.5. (a) One observes that the principal symbol of the Calderón projector P÷ has locally (over the connected components of Y) constant rank for all non-zero covectors (y, Consider the subspace Ms,>,, of spanned by the eigenvectors corresponding to with positive real part. Since A the eigenvalues of b(y, : —' takes the form + B) close to Y, we get a(t, y; r,
= C(y)(ir + b(y, ii))
for the principal symbol of A close to Y as noticed above. The el-
lipticity of A therefore implies that b(y, has no purely imaginary eigenvalues. Hence no eigenvalues can pass from the one complex halfplane into the other, and the dimension of cannot change locally.
(b) Now consider the space (12.23)
12. Ca3derón Projector for Dirac Operators
91
The space consists of those solutions of the given system of ordinary If differential equations which are exponentially decreasing on we identify the solutions with their initial values, we obtain the space M> vu,.
Applying the preceding theorem, we shall give a simple geometric proof of the following result, observed in BooB & Wojciechowski [1985]. It answers the question of how to describe the kernel of the Poisson operator (12.24)
is bijective, By Theorem 12.4a, K+IH+(A,s) : H÷(A,s) —i and by Theorem 12.4b the closures of H÷ (A) and H_ (A) are complementary. This complementarity relies on our construction of the invertible double A and remains valid for all invertible Dirac oper-
ators over a partitioned manifold, but clearly breaks down when a given Dirac operator over a partitioned manifold is not invertible. On the contrary, the following corollary expresses the kernel of (12.24) intrinsically in terms of the given manifold with boundary and provides a short exact sequence for elliptic equations on manifolds with boundary, namely, (12.25)
0—' G_I(H+(A*,s)) '—. H84
ker+(A,s) —÷0.
Corollary 12.6. Under the assumptions of the preceding theorem, the subspaces and onal and complementary.
of C°°
are orthog-
Proof. First extend the Dirac operator A = : C°°(X; S) to the operator A = A over the closed double. Denote by P(A) and 2(A) the Calderón projections of L2(Y; ontotheclosureofH÷(A) = H+(A)andofH_(A) = G'(H+(A)) = Then and and the corollary follows from Theorem 12.4b. Another interesting consequence of Theorem 12.4 is that it provides
with the operator
a left-paraxnetrix for A. The point is
II. Analytical and Topological Tools
92
I
Id —K+-)o
Fig. 12.4 Sketch of
A' Af = I
—
of
that of course A'A =
Id over the closed manifold X, but there appears a correction term K+1o coming from the boundary data when
we restrict the operation to the manifold X with boundary Y. For a rough visualization see Figure 12.4. —, C°°(X; S-) Lemma 12.7. For any Dirac operator A : C°°(X; over a compact manifold X with smooth boundary Y, the operator
—.
(12.26) 3 well-defined and we have
(12.27)
r+A_1e+A = Id—K+'yo
Proof. Consider the operator of (12.26) on the distributional level.
12. Calderón Projector for Dirac Operators
Let f E
and
hE
93
We have
(12.28)
= f(Af;
=
)*e+h)dX
= =
ty)dy
— (G70f;
= = ((Id
—
It was pointed out that P(A) is just a projection, i.e. a not necessarily orthogonal idempotent. Actually one can always replace P(A) by the corresponding orthogonal projection in the case of product structure near the boundary and when A is an operator of Dirac type (namely with unitary G and seif-adjoint tangential part B). The orthogonal projection obtained is a pseudo-differential operator with principal symbol equal to the principal symbol of 1'(A). Lemma 12.8. (a) Let P be a projection in a separable Hubert space. Then Port
PP
+ (Id —P)(Id —P))1
is an orthogonal projection onto the range of P. (b) If P is a pseudo-differential projection with orthogonal principal symbol p, we have (TO(Port) = hence Port — P
is
= I',
a compact operator (operator of order —1).
Note. In the terminology to be introduced in Chapter 15, this means that P and Port belong to the same Grassmannian Grp. They
also belong to the same connected component of that PortlrangeP = Plrangep = Id, so index{PPort : range Port
cf. Theorem 15.12 below.
range P} = 0,
The reason is
II. Analytical and Topological Tools
94
Proof. (a): First {PP* + (Id _P*)(Id —P)} is invertible. In fact, pp. +(Id—P)(Id—P) >0, since
({PP + (Id—P)(IcJ—P)}u;u) =
+ II(Id—P)u112 0 with equality, if and only if Pu = ts and = 0, hence 0 (Pwu) = (PPu;u) = IIPuII2 = Hull2 and so u = 0. Next observe the vanishing of the commutator [PP; —P)} + (Id
and hence of the commutator [PP; {PP + (Id —P)(Id —P)}']. This implies
= {(pp') + ((Id_P*)(Id_P))*}_l(PP*)* =
= + (Id —P)(Id Then we find (keep track of the brackets {... }): (Port)2 =
{PP + (Id
=
—P)}' P)J — (Id
+ (Id = {[...J —
—P)} {... }
P)(Id P)} {... = {[Id —(Id —P)(Id —P)] [PP + (Id (Id
+ (Id
= {Id _(Id
Port
Port —P)]}
—P)}' Port
—P)} Port
= Port — (Id = Port — 0
Port.
..
{(Id P)P} P* (PPm + (Id —Pt)(Id
=
Now apparently PPort =
so we only have to show that Port P =
P. Let v = Pu. Then {PPt + (Id Port(v) =
(Pu)
+ (Id —P)(Id —P))' (Pu) = Pu— (Id—P)(Id—P)P{PP' —Pu. = Pu — (Id —Pt)(Id —P)
(b): If P is a pseudo-differential operator, then Port, too; and if P has an orthogonal symbol, then our construction does not change the principal symbol. U
13. Existence of Traces of Null Space Elements
The establishing of the Calderón projector for Dirac operators over compact manifolds with boundary is rounded off by showing that all elements of ker+(A, s), even the distributional solutions, have well-defined traces (boundary values).
We still have not completed the proof of Theorem 12.4a. The argument presented in Chapter 12 works only in case s > To apply the closing argument of the proof of Theorem 12.4a for s we have to show the existence of the trace in for v E (A, 8). This is the subject of the next theorem which is crucial to the whole theory of elliptic boundary problems. We adopt the notation of the preceding chapters, especially • the construction of the invertible double A of chapter 9, • the parametrization {Ye}tEJ of a collar neighbourhood of the boundary Y, the restriction operator r+, and for s > the trace operator-y : xI of Chapter 11, • and for arbitrary real s the null space ker(A, s) and the Poisson type operator : —. ker÷(A, s) of Chapter 12.
Theorem 13.1. Let v E ker÷(A, s) (then in particular v is smooth in X \ Y, Av = 0 in X \ Y, and for e > 0, the trace is a welldefined element of converge to an element 9 E
Then, as e —' 0÷, the sections
to a section 'i) E H3(; S+) v (see Definition 11.7d). Then w := Ai3 is an element
Proof. First we extend v E H8(X;
such that of H8_1(i;S_) which vanishes on X \ Y. Therefore it is enough to prove that, for any distribution w E Ht(X; Sj with w = 0 in X \ Y, lim
exists in
II. Analytical and Topological Tools
Use a partition of unity to localize the problem. One has to investigate the limit lim 'Ye(McpA1M,1,W),
and are smooth functions with compact support contained in a neighbourhood of a given point y E Y. Hence consider the folwhere lowing problem: Split = R. x where
n-
— 1/
— U—
It is a pseudo-differential Let Q denote the operator operator of order —1 in R'. Then, to prove Theorem 13.1, one must only show:
(with values in CN) vanishes for is a well-defined element of
(13.2) Assume that w
U > 0.
Then
To prove (13.2) we use the operator
:= in denotes the Laplacian — and i/i + is a positive_pseudo-differential operator of order 1 with total symbol a(y, = We have the following classical + lemma.
Here
Lemma 13.2. For any s E R, the operator A+ gives an i.somorphism of Banach spaces
—' Proof. To prove that A+ extends to a bounded operator from H3(RTh) it is enough to show that i/i + extends to such a mapping. Let / E H8(R'). We may assume s > 0 since the general result then follows by duality. We wnte (x, = (u, y; r, x TB?' with y = (xi,... We have and (= (Ci,...
f
+
13. Existence of Traces of Null Space Elements
97
hence
IN/i +
= 1k/i +
+
+ J lf(e)l2(1 + =J
+
=
11111.,
thus proving the continuity. Also the proof that A.,. is an isomorphism
is trivial. Notice that (A+)* = Ou+V11 + =: A_. Then the operator (A+)A+ _(O)2 + + 1 = 1+ denotes the is an isomorphism between H8 and H8_2; here Laplacian in R'1. El The reason for using the At-operator to prove (13.2) is the following statement. Proposition 13.3. Let w be a distribution in with support in the half-space {u O}. Then for any t E R, the distribution also has support in {u ( O}. Proof. Let (w, f) denote the duality pairing between elements of 8' and S. Then for A...., defined as above, we have = (w,M..f). It suffices to show that At..f vanishes for u 0, if f E S vanishes for
u e for some e> 0. The point is that for such an f
J(r,
f(u, y) dydu
= .f has a holomorphic extension to the half plane {c(r)
—
ib,
=
() =
which tends to 0 in S as b —+ +oo. Moreover, (13.4)
(M..f)(u,y) = t
=
JRn_1
=J
dC j
etL1 [(1
+ iT] J('r,
+
d(R—.oo urn J ei" —R
+ iT
(1 + 3=1
1(r, ()dr.
II. Analytical and Topological Tools
98
R
Fig. 13.1 Integration path for Cauchy's theorem
Now apply Cauchy's theorem. Since / is an element of 8, the integrals along the sides (I) and (II) disappear as R —+ 00. Using (13.3) and (13.4) yields (13.5)
f
= urn
I
/
a=—R
R_•OOJa=R
da
(
—1
t
n—i
I
1
+ ia + b)
+
t
n—i
=1
—
j=i ib, ()d(da.
I j=1
= 0.
Taking u < 0 and letting b —, +oo yields
Unfortunately A+ is not quite a pseudo-differential operator, since its symbol A4. (x; = (ti, y; r, () = —ir + a(y, violates the clas-. sical estimate
+ oo. However, as noticed = (0,...) with > 1 and In in Lemma 13.2, still A+ is an operator of order 1. We furthermore also in other have the following obvious result which shows that
for any
aspects behaves like a pseudo-differential operator:
13. Existence of 'fraces of Null Space Elements
99
Lemma 13.4. Let m E Z. For any pseudo-differential operator Q of order m, the commutator QA÷ — A÷Q
[Q,
(and any iterated commutator erator of order < m.
Ak], A÷J,.. . A÷J, Ak]) is an op-
Proof. We are content to prove the lemma in our case of m =
—1
and
Q := where all is easy modulo small technical details. Notice, for example,
= +
+
Clearly for a pseudo-differential operator S of order k 0, the commutator IS1 is an operator of order k, hence [A, A+J an operator of order 1 and [My,, A+] an operator of order 0. 0
Resume the proof of Theorem 13.1. The proof goes via induction. Begin with the special, but most interesting case s = —1. Let wE
(136)
vanish for u > 0. We have
Qw =
+
—
= [Q,
[Q,
(w) E
and by the trace theorem 11.4
urn i'e[Q, A÷JA'(w) exists in trace
The same argument gives the existence of the urn
e
Now
= (—& + =
+
+ v'l +
II. Analytical and Topological Tools
100
Since the operator l'o
(v'i +
+
acts in tangential direction we have
+
=
=
+
('yeQA'(w)) =
We are left with the term
= =
vu
+
(7OQA;'(w)).
We have
+
The second sununand is smooth (for a suitable choice of and The first summand actually belongs to the null space of A on X÷ \ Y, since
= Here, let Xj, denote the two halves of the closed double X of X = Xi..
It follows from Proposition 13.3 that A' (to) has support in u 0, hence h := has support in X_. Then A'h belongs to H1(X; S+) and has a well-defined trace in Y. We also have
0= urn
= lim
+
= c ( urn 'ye(Ou + B)(A'h) =
G
( urn
+ B-yo(A'h)
which implies (13.7)
lim
= —B'yo(A'h),
and the theorem is proved for s = —1. The argument works for any 8> — To prove statement (13.2) for arbitrary s we argue by induction, using step by step the lifting argument Qw QA' (to) of (13.6). To explain the induction, let us repeat the arguments: Let v be a distribution which is a solution of Av = 0 in X \ Y. As
in the last part of the proof for s =
—1, we obtain that the traces 10((OuYcv) exist for any k 1, if the trace yov exists. The point is that we have
(13.8)
= 0,
13. Existence of 'fraces of Null Space Elements
and that we can solve that equation to obtain a formula for 70((Ou)Icv)
in terms of
I 0 j < k} as in (13.7) for k = 1. In particular, let w E H8(R") with supp w C {u 0} and let k be sufficiently large, namely k> —s — Then it follows from the preceding argument that the distribution E
has a well-defined trace
E
H8+4(RTh1).
Now use that result for the induction. To explain the induction, let us begin once again with a special case. Let w E HS(Rfl) with suppw C {u 0}, but now 8> Then
+
Qw = [ci,
= =
A÷]A',
+
+
By Lemma 13.4, the first summand belongs to hence it has a trace by the general trace theorem 11.4. By the same argument QA2w has a trace. The third summand is equal to —
+
—
The traces are well defined, since gentially, it follows that
2(9utJl
and
+ i) QA2w.
+
as explained above, exists. And, since acts tanis well defined, and we are left
with the trace of All we need to show is that QA'w has a well-defined trace in since then our method of solving (13.8) for the normal derivatives yields that The same is true for has a well-defined trace in %/i +
and hence for A÷QA'. Now consider
QA'w = QA'w —
+
= [Q, A+JA2w + The first summand belongs to and hence has a well-defined trace in With regard to the second summand, we nohence also tice that QA2w has a well-defined trace in
11. Analytical and Topological Tools
102
by our (13.8) argument. This shows the existence of the
trace of in For general s, use the formula k
Qw =
The first summand is in hence it has for k> —s— a has trace, and we argue as before to show that then each a well-defined trace in [J
show that the traces established in the preceding theorem are and the reasonable with respect to the Poisson type operator Calderón projector 'P+, we prove Proposition 13.5. Let v E ker÷(A,s) and r := 'yo(v). Then To
v=K÷(r)
(13.9)
and
Now let s be arbitrary Proof. The proposition is proved for s> s). Let x E X \ Y and choose e so that and v E
:= X \ {[0,2ej x Y}.
E
The distribution v is C°° in Choose
hence v =
in
and t/' in C°° (X) so that
and
(13.10)
Then
v(x) =
(13.11)
The operator
is a smoothing operator, and we have a conver-
gence in
—,
S—)
where t is a negative number s. This shows that for any e and any xE
=
v(x) = Therefore we have
v = K+r
and
P+r =
urn
= lim
'YcV =
r.
0
To complete the proof of Theorem 12.4a, we need two more results.
13. Existence of
of Null Space Elements
Lemma 13.6. Let r E H8_4(Y;S+Iy) and P÷r = r. Then there exists v e H8(X;
such that Av =0 in X \ Y and
'yov= urn In other words, range(P÷) C
Proof. Let =
denote a sequence of smooth sections such that S+Iy). Then the previous estimate r in is a Cauchy sequence in H8(X; (12.12) implies that hence it has a limit and of course
urn K+ (ru) = K+r = v,
v—,00
and once again we have
r = 2+r = urn '7eV. e—O+
Recall
that
(A, s) denotes the closure of
Corollary 13.7. The range of and the operator injective, when restricted to
(A) in H8
Then
is the space Ha_i
in :
s).
Proof. Assume that we are given r1 = 1'+ri and r2 = P+r2 in =v=
such that
Then
urn
This corollary ends the proof of Theorem 12.4. Next we use Theorem 12.4 and Lemma 12.7 to show the existence of traces of other distributional sections not necessarily belonging to the null space.
Theorem 13.8. Let v E Ht(X; t, s real, and s > — in
Then
S) with
the trace of v on Y is well-defined
Proof. If t > s + 1, we have t> from Theorem 11.4. Assume t < s
and the assertion simply follows is + 1. Then w
well-defined in and we have Aw = Av in H8(X;S); hence — w E ker+ (A, t). Theorem 12.4 implies that there exists a
II. Analytical and Topological Tools
104
unique
r=
E
Ht_4 (Y;
such
a well-defined trace in we have
that = v — w and w has C Ht_4(Y;S+ty). Therefore
'yov= urn We have one important case which has already been established in the second part of the proof of Theorem 13.1.
Corollary 13.9. Let v E
and Ày
L2(X;S). Then v
has a well-defined trace in H4(Y; Remark 13.10. With Theorem 13.8 and its corollaries we sharpen the trace theorem of Chapter 11, which gave a trace in Ht 4(Y) for u Ht only if t> Theorem 13.8 shows that the assumption t> is dispensible, if Au E H8 for some 3> — In Chapters 19 and 20 below, it turns out that the theorem and its corollaries are decisive for regularity and closedness of (global) elliptic boundary value problems.
14. Spectral Projections of Dirac Operators
We account for the construction and the basic properties of the spectral projections associated with the tangential part of a Dirac operator.
In this chapter we discuss one more pseudo-differential projection which plays an important role in the theory of elliptic boundary problems for Dirac operators. Unlike the Caiderón projection, which is defined by global data of a Dirac operator A over a manifold X with boundary Y, we shall employ only the tangential part B for the construction of the spectral projection. In this chapter we may therefore forget about X and A and just consider arbitrary pseudo-differential elliptic self-adjoint operators over a closed manifold Y. Let B : C°°(Y; V)
C°°(Y; V)
be such an operator acting on sections of a Hermitian vector bundle V over Y. Assume (though it is not necessary) that B is of first order. it is well-known that B has a discrete spectrum contained in R numbered like (14.1)
and one can find an orthonormal basis {CJ}JEZ of L2(Y; V) consisting of (smooth) eigenfunctions of B (i.e. Be3 = j E Z), see called e.g. Gilkey [1984; Lemma 1.6.3]. Such a system e,}3€z a spectral decomposition of L2(Y; V) generated by B, or, in short, a spectral resolution of B.
Definition 14.1. For a given self-adjoint elliptic operator B and for any real a we shall denote by L2(Y; V) —' L2(Y; V) the spectral projection, that is, the orthogonal projection of L2(Y; V) onto the subspace spanned by {e1 I A, a}, where {A3; e3 },€z is a spectral resolution of B.
II. Analytical and Topological Tools
106
Note. We shall write P>a(B), when we want to stress the dependence on B; we shall omit the a when a = 0; and we shall adopt the notation P< := Id — Hence
Po(B)
P< :=
and
= Id —
It is well-known that the spectral projection P>a is a pseudodifferential operator of order zero (see e.g. Atiyah, Patodi & Singer [1976, p. 48], Baum & Douglas [1981, Proposition 2.4], or BooI3 & Wojciechowski [1985, Lemma 2.2]). We shall repeat the arguments as this proposition is basic for the following:
Proposition 14.2. The spectral projection P>a is a pseudo-differential operator of order zero for any real number a. The principal symbol of P>o does not depend on a, and for any T*Y\{0} it is equal to the orthogonal projection :
—i
of V1, onto the direct sum of the eigenspaces of the automorphism :
corresponding to the positive eigenvalues. Here b denotes the principal
symbol of the operator B.
Proof. For any pair of real numbers a < b, the operator P>a — Pb is the projection onto the direct sum of eigenspaces corresponding to the eigenvalues from the interval [a, b). This is a finite-dimensional space with a basis consisting of smooth sections; hence is an operator with smooth kernel. Thus it suffices to — prove the proposition for a = 0. Let B' denote the bounded operator (in L2(Y; V), i.e. operator of order zero) (14.2)
B' := (Id+B2)112B.
It has a spectral decomposition (14.3)
;e3
I.. V/i +
3EZ
14. Spectral Projections of Dirac Operators
of instead of II is Auo to two facts, The namely that — the new spectrum is contained in the interval (—1,1), — and there still exists e > 0 such that specB'rl(—2e,2€) C {0}.
(14.4)
It is well-known from spectral theory, cf. e.g. Kato [1976, Section 111.6.4], that the operators and P< = = Po(B) = P<0 = Id — P> have the integral representation (14.5)
(B'—A)'dA,
2irz
and
(B'—A)1dA,
27r2
where (14.6)
:= {(1 + e)ett + 1},
and
r< := {(1 — e)eit
and e is chosen so small that [—e,0) fl spec(B') =
0, see
—
1}
Figure 14.1.
C
Fig. 14.1 Separation of the spectrum of B'
It is easy to determine the full symbol of the operator (the resolvent) (B'—A)' for A E C\spec(B') in any local chart. Its asymptotic expansion has the form P—, (y, A), where the Pj are rational with poles at A = ±1 (see e.g. Gilkey functions of A of order —j [1984, p. 51f]). In particular we find for the principal symbol of B' b'(y,
= p÷(y, ,j) — (id
(y, 7) = p+
—
p_.
II. Analytical and Topological Tools
This yields (b'
—
A)'(y,ii) = [(b' —
= 1
=
1
1 1
1
+
This shows that the principal symbol of P> is equal to 2irt
(14.8)
=
:J{
dA — 1
1
= p+(y,tñ. In the same way we compute (14.9)
Ck(y,fl) =
_L 2irz
Jr> pk(y,n;A)dA.
Let Ck denote an operator with symbol standard argument, one obtains that P> —Ck
cL(y, 77). Then, by the is an operator of order
—k — 1.
Corollary 14.3. Let us assume that the operator B is the tangential a, Pa(B) — P÷(A) is part of a Dirac operator A. Then, for any an operator of order —1.
Proof. The operators Pa(B) and symbol.
have the same principal []
Remarks 14.4. (a) To prove that P(B) is a pseudo-differential operator, an alternative argument may be summarized as follows:
Let B = UBIBI be the (unique) polar decomposition of B, see e.g. Kato [1976, Section VI.2.7]. Here IBI denotes the non-negative square root of B2 = B*B. It is an elliptic pseudo-differential operator of the same order as B by Seeley's theorem on (complex) powers of pseudo-differential operators. The operator UB is a partial isometry
14. Spectral Projections of Dirac Operators
from L2(Y; V) in itself with a decomposition of L2(Y; V) into the three eigenspaces m+, m_, and mo with eigenvalues equal 1, —1, and 0. These eigenspaces are generated by the eigenfunctions of B corresponding to positive, negative, and zero elgenvalues (the last being the kernel of B). The space mo = ker B is finite-dimensional and consists of C°°sections. Let k denote the L2(Y; V) orthogonal projection onto ker B,
kg :=
(14.10)
where e1,. . . , ej is an orthonormal basis of ker B. This shows that k can be defined by a C°°-kernel, so k is a pseudo-differential operator of order —oo, i.e. a smoothing operator. We observe that
P(B)=
(14.11)
Therefore, in order to show that P (B) is a pseudo-differential operator of order zero, it suffices to show that UB is a pseudo-differential operator of order zero. have the same kernel. Hence Notice that the operators B and IBI+k is a pseudo-differential operator of the same order as B, namely of order 1, elliptic and invertible, and (IBI + is an (elliptic) pseudo-differential operator of order —1. Then we can write UB = Un(IBI + k)(IBI + k)' = UBIBI(IBI
since UBk = of order 0.
0.
+ k)' = B(IBI +
This shows that UB is a pseudo-differential operator
(b) Notice that the Calderón projector of a Dirac operator over a manifold with boundary and the spectral projection P> (B) of its tangential part coincide for the Cauchy-Riemann operator over the disc (see Example 21.1 below). One might ask whether that coincidence is a peculiarity of dimension two. For higher dimensions and certain symmetric spaces, harmonic analysis e.g. establishes related 1-1 correspondences, namely between the space of harmonic forms over a symmetric n-dimensional domain with boundary and the Hardy space of harmonic eigenforms to non-negative eigenvalues over the i-dimensional Silov boundary.
110
IL Analytical and Topological Thols
More specifically, we always have, by definition, the inclusion
kerB C rangeP>(B), whereas the question
kerB
range 1'÷(A)
is non-trivial: Roughly speaking, a harmonic form over the 1-codimensional boundary might have too many conditions to satisfy to be extendable over the whole manifold. Nevertheless, Kori [1993; Theorem 5.10] obtains = P(B) for the Dirac operator over the 4dimensional ball with boundary S3 in a suitable metric. (c) A more general variant of spectral projections of elliptic operators is discussed in Wojciechowski [1985a; Sections 2—4], where the index
theory for families of operators with two rays of minimal growth is discussed.
15. Pseudo-Differential Grassmannians
The homotopy groups of the space of pseudo-differential projections with given principal symbol are computed. Criteria are given for two projections belonging to the same connected component.
The next tool needed to explain what the index of global elliptic boundary problems determines is the Grasamannian of pseudodifferential projections. We shall consider the set Gr9,. of all pseudoand with differential projections P with principal symbol equal to the topology given by the standard norm topology in the space of all bounded operators acting on the space L2(Y; Say). (In this chapter we only assume that is a Herinitian bundle and do not care whether originates from a full or split bundle of Clifford modules.) 0 and We assume that is an orthogonal projection and that Id or, equivalently, that dim ker P = oo = dim range P. We caland it turns out that the closure culate the homotopy groups of of this space is a classifying space for the K° functor; but we shall only exploit the fact that this space has countably many components. Remarks 15.1. (a) The homotopy groups of the corresponding total (3rassmannian in a separable Hilbert space (the closure of are well-known (see e.g. Wojciechowski [1981], BooB & Wojciechowski [1982/1985], Wojciechowski [1985a], and Pressley & Segal [1986]); they only differ by the choice of equivalence classes, namely, whether
one takes classes modulo operators of finite rank, modulo compact operators, or modulo operators of Hilbert-Schmidt type. (This makes no difference to the respective homotopy groups, as shown by Palais [1965bJ). Fuglede [1976] observed earlier that the total Crassmannian consists of infinitely many connected components. The point of our
presentation is that we have to restrict ourselves to the pseudodifferential Grassmannian. (b) We assume here that the term projection denotes an idempotent (P2 = P). However, as base point for our homotopy groups, we choose an orthogonal projection = = Pg). We can do that without loss of generality, as explained in Lemma 15.11 below. More specifically, we let P.÷ be an orthogonal projection onto a subspace of
II. Analytical and Topological Tools
112
is the L2(Y; Sly). In the context of this book, a natural choice of orthogonal projection onto the space H+ (A) of Cauchy data of a given Dirac operator A. As explained in Lemma 12.8, the projection P÷ is a pseudo-differential operator and has the same principal symbol as the Calderón
projector constructed in that chapter. The reason is that
in our case the principal symbol orthogonal.
C) of the Calderôn projector is
We introduce a group of invertible elliptic operators on Sly: (15.1)
:= {g g is an elliptic invertible operator of order 0 on Sly with principal symbol equal to I
The key to the whole problem is the following elementary lemma:
Lemma 15.2. Let P0 liP'
—
and P1
Poll < 1. Then the
such that
operator
T = Id+(P1 — Po)(2P0 — Id)
(15.2) belongs
denote two elements of
to EIIX and satisfies:
TP0=P1T.
(15.3) Proof. We have liT
— Id
ii
= il(P' — Po)(2P0 — Id)il
lIP1 — Poll ll2Po —
Id
Ii
ilPi—Poll<1, since (2P0—Id)2 = Id for any projection P0 and hence ll2Po—Id Ii = 1. So T is invertible. For the principal symbol of T we obtain ci(T) = Id +0 = Id. Finally, an elementary check yields TP0 = P,Po = P,T which proves (15.3). 9
Corollary 15.3. The space
locally contractible and, in particular, the connected components are path-connected. is
Proof. A deformation retraction of {P1 I liP1 given by the homotopy :=
(T(P,,t))'P0T(P,,t),
—
for
Poll
<
0 t 1,
onto P0
is
15. Pseudo-Differential Grassmanians
where
T(Pi,t) := Id +t(P1 —P0)(2P0—Id). Clearly IJo(Pi) =
Po,
Hi(P1) =
P1,
and
remains in
Later on we shall use this lemma to show that any single connected component of our Grassmannian is a base space of a certain principal fibre bundle with the total space (Theorem 15.5). Hence, in for k > 0, we have order to compute the homotopy groups Irk to compute the homotopy groups of Wx. They are well-known, but there are no computations published to establish them.
Proposition 15.4. The homotopy groups of
(with base point
Id) are given by ir
(0 forkeven tZ forkodd.
Proof. First recall that these are exactly the homotopy groups of the groups and given by Bott periodicity (see Karoubi 11978; 1.3.14 and 11.3.191) and Palais' results (see Palais [1965b]). Here denotes the group of invertible operators acting on L2(Y; Sly), which differ from the identity by a compact operator, and CL°° denotes the group of invertible operators acting on L2(Y; Sly), which differ from the identity by an operator of finite rank. Now it is enough to show that any family of operators of the form to a family of operators can be deformed within f: from This will be done in four steps: EftX be a continuous family. Deform fo(x) := unitary operator fi(x) := f(x)If(x)1', where If(x)I
1st step. Let f:
f(s) into the denotes the
—'
unique positive square root of the operator f(x)f(x), x E The deformation is provided by combination with the standard retraction gg(x) := tld+(1 — t)lf(x)I, 0 t 1 of the space of positive operators onto the identity which rests within e.e€x.
step. The unitary operator This means belongs to that its spectrum is contained in the circle S' and that 1 is the only point of the essential spectrum of fi (z). Thus we have a (possibly 2nd
II. Analytical and Topological Tools
114
spec 111(f1(x))
spec Ii (x)
H1(specfi(s))
Fig. 15.1 Spectral deformation of a unitary fi(s) into H1 (f1(x))E
infinite) set {Am} of eigenvalues of finite multiplicity contained in 5',
and 1 is the only limiting point of this set (it may be an eigenvalue of infinite multiplicity). Then for any small e > 0 we can find a 1 t 2 with /2(5) E CL°° and deformation ft : Sk and 1 t 2. In fact, we can lift(s) — fi(x)Il < for all x E
choose a small 5 > 0 and a related "deformation retraction" of 51 onto 5! in such a way that
H0=Id H,(etO)=1 for—S<9<S Hg(e$O)=eW for3S
forO
{f2(x) := Hj(fl(s))}XESk of operators belonging to CL°°; for S sufficiently small, we get
iIHt(fi(x))
—
f'(x)ll <
as required. (See also Figure 15.1).
for 0< t < 1
15. Pseudo-Differential Gras8rnanians
The technical difficulty here is that CL°° is not contained in eeeX. An element g of GL°° is a pseudo-differential operator, if and only if g — Id is an operator with a smooth kernel. However, it is obvious that we can approximate any family h: Sk —' CL°° by a family of operators h1, where h1 — Id is a family of operators of finite rank with smooth kernels. We shall sketch the proof. For any x E SC, the image of the operator Siyi step.
Id —h(s) : L2(Y; Sly) —, L2(Y; Sly)
a finite-dimensional vector space V(x). Since is compact, one can find a finite-dimensional subspace V of L2(Y; Sly) such that for any x, the image of h(x) — Id is contained in V. By a small deformation, one V for can change h to_h and V to V such that Id —h(s) : L2(Y; Sty) N = dim V all x E 5k and V consists of smooth sections. Let denote an orthonormal basis of V consisting of smooth spinors which is
may be completed to an orthonormal (smooth) basis of V. Then for s
L2:
(Id —h(s)) s =
(s; i=1 2=1
it is obvious that the continuous functions can be approximated by smooth functions up to an arbitrary order. This implies that to determine the set CL°°] = lrk(CL°°), we
may use a family h :
CL°° of maps, where the range of h
consists of operators belonging to e€ex Thus we finally have a family 13 :
—'
fl
with
l1f3(x) — 12(x) II <
4th step. The family f3 was obtained by continuous deformations
within the space But only the first homotopy and the results of the second and third homotopy are confined to However, since f3(x) belongs to and 1113(x) — fi(x)II <e for sufficiently small e, we can choose the linear path
tf3(x) +
(1 —
t)fi(s)
is totally contained in as a new homotopy. Thus we have obtained a deformation of I = fo within (t€X into a family /3 which consists of operators belonging to CL°° (see Figure 15.2). Therefore which
15. Pseudo-Differential Grassmanians
In fact, provides a homeomorphism and the base space between the homogeneous space / eeex To show the injectivity of the mapping, check that for any g, belongs to the operator h = go E £UX with = for 9 E
and h E
i.e.
=
=
=
=
To show that the map is surjective, take an arbitrary P E is path-connected (Corollary 15.3), we have a finite seSince P, and such that P1 = quence {P1}11,... ,N E From Lemma 15.2 11P1+1 — P,fl < 1 for all i = 1,... , N — 1. with we then obtain operators T1 E T := TN_i . ... . T1 E EWE. P= To prove that
is
a (principal) fibre bundle with fibre and structure group
has a local cross-section it only remains to assure us that Tp mapping a in i.e. the existence of a function T : P into 6€iX and neighbourhood U of = {P÷ continuously "+) } such that = P for each P E U. Choose
U:=Up+={PIPEGrp+ andIlP—P+II<1}. Tp := Id +(P — — Id), the operator from Lemma 15.2. The function T is continuous in the operator norm and we have = = P. Then our theorem follows from the Set
bundle structure theorem; see Steenrod [1965, p. 30fJ, see also Figure 15.3.
[1
Remark 15.6. Notice that a local trivialization of the total space e,ex over the neighbourhood Up÷ is given by the formula (15.4) g
where T:
'—'
—'
= gP+g';(T,(9))_19) E Up1. x
eiex is the local section from the preceding proof.
In fact
= . = . = (T,(9))4i(g)T,(9) = P+,
IL Analytical and Topological Tools
118
= P for all P E
since This proves
and so also for P = 4)(g).
as claimed in (15.4). Actually (15.4) E defines a connection on the bundle A parallel transport of an element 9 E 4v '(P1) to a fibre over P2 (where P1, P2 E is given
by Tp2 . (Tp, )' g, and there is an obvious construction of a global
connection on
We shall not discuss this topic further here.
x
TIN Cr
P Fig. 15.3 Local section and parallel transport
Recall the assumption that neither the symbol i+ nor Id —p÷ van= dim range(Id ish identically. Then we have dim range oo, and the operator P+ decomposes the space L2(Y; Sly) into the and range(Id —P4. With respect to this direct sum of '÷ decomposes into two factors such that splitting, the group each of them has the homotopy groups of GLC(L2) as in the proof are of Proposition 15.4. Therefore, the homotopy groups of given by
('0
(15.5)
forkeven forkodd.
Now compute the homotopy groups of Grp4. Apply the exact homo—p Grp÷ with fibre and topy sequence of the fibre bundle 4 : see Steenrod [1965, p. 90f]: structure group based at lrk+1(Crp+)
lrk(Crp+) —4
lrk....l(&tiXP+)
—4
15. Pseudo-Differential
Here j
:
denotes the natural inclusion. The homotopy
groups are relative to a fixed base point, say P.1.. The induced map i on the homotopy groups is 0 for even k and j. (m, n) = m + n for odd k. This gives the short exact sequence Z
0—'
Z
Z —'
0
which proves the following theorem:
Theorem 15.7. For a fixed projection P.1., the homotopy groups of the connected component are given by of lrk(Grp÷,P÷)
I Z for k even and k> 0 =
fork odd.
The remaining part of this chapter is devoted to showing that has countably many components classified by the index of certain natural Fredhoim operators (Theorem 15.12 below). Following Brown, Douglas & Fillmore [1973] we define:
Definition 15.8. Let P1, P2 be pseudo-differential projections with the same principal symbol. The integer l(P2,P1) := index {P2P1 : rangePi —' rangeP2}
(15.6)
is called the virtual codimension of P2 in P1. Note. since
Clearly P2 P1 : range P1
range P2 is a Fredhoim operator,
ker{P2P1 : range P1 —, range P2 } = ker T 11 range P1
and range{P2P1 : range P1 —' range P2} = range T fl range P2,
where the 0-order pseudo-differential operator (15.7)
T := P2P1 + (Id —P2)(Id —P1)
has principal symbol equal Id, hence is elliptic. The following lemma is an obvious corollary from Lemma 15.2:
11. Analytical and Topological Tools
120
Lemma 15.9. Let P, and P2 belong to the same connected component of Then the virtual codimension i(P2, P1) vanishes.
Proof. Lemma 15.2 implies the existence of g E
gP1g' = P2, and
such
that
it is obvious that:
(15.8)
index P2P1 = index gP,g'P,
= index{g : range P1 range P2} + index P1g =0 + index P, (Id +compact)P1 = 0,
since, for g E
'P1
belongs to S€IX and can be written as
Id + compact.
Note. The operators g and can be continuously deformed into the identity within the path-connected space Silx. This follows from formula (15.2). Now we want to show the opposite direction, namely, that P1 and P2 belong to the same connected component, if i(P2, P1) is equal 0. First we prove this for orthogonal projections and afterward we will explain why it works in the general case. We need a stronger variant of Lemma 15.9.
Lemma 15.10. Two orthogonal projections P1, P2 E belong to the same connected component of the Grassmannian, if and only if
i(P2,Pi) = 0. Proof. The only if was shown in Lemma 15.9. We show the if. Consider the operator
T = Id-i-(P2 — P,)(2P, — Id) = P2P1 + (Id—P2)(Id—P,) +Tbefore in Lemma 15.2 for the special situation of liP2 — Pill < 1. Now investigate the general situation. Then T is not necessarily invertible, but still elliptic as observed earlier after investigated
equation (15.7). Clearly ker T
ker T+
ker T_
and
coker T
coker T+
coker T_
15. Pseudo-Differential Grassmanians
We
find
(15.9)
kerT+
= ker{P2P1 : range P1
range P2}
—.
= {s (Id —P2)s = s A (Id —P1)s = =ker{(Id—Pj)(Id—P2)
O} —'
:
=coker{(Id—P2)(Id—Pi):
—t
= coker T_. Of course here we exploit that P1 and P2
are
orthogonal projections,
so that e.g. (15.10)
ker{P1P2
: range
= coker{ (P1
P2) *
One can check (15.10)
P2 —, range P1}
=
=
P2P1 : range P1 —* range P2 }.
directly:
coker{P2P1 : range P1
—,
range P2 }
= {s P2s = s and s I range P2P1} = {s P2s= sand (s;P2P1I) =0 for all f} = {8 I P2s = sand (PiP2s;f) = 0 for all f} I
I
= {s
ker P1P2}
Similar to the derivation of (15.9), by interchanging P1
and P2, one
gets ker T_ = coker T+
Now i(P2, P1) = 0. It suffices to construct an operator T' and satisfies T'P1 = P2T'. Then the lemma is which belongs to proved, since is connected. First construct T': Since i(P2,P1) = 0, we have
dim
= dim (kerT.
cokerT+))
Hence one can fix a unitary morphism V : ker T+ —. ker T_ which defines an involution W : ker T ker T by the formula (15.12)
(0
v
0
II. Analytical and Thpological Tools
with respect to the decomposition (15.13)
Extend
ker T 'I' (by
= ker
ker T_ = ker P2P1
ker P1 P2.
0) to the whole Hubert space L2(Y; Sly) and define T'
:=T+'I'
to be the perturbation of T by the finite rank operator 'I'. Then also the operator T' is a pseudo-differential elliptic operator of order 0 with principal symbol idg. Moreover ker T' = {0} = coker T'; hence An elementary computation shows 7" E
T'P1=P2T' which implies component.
that P1
and P2 are elements of the same connected U
To extend this result to the general case of not necessarily orthogonal projections, we need the following lemma.
Lemma 15.11. For any P E let P denote the orthogonal projection onto the range of P. Then P and P belong to the same connected component of Grp+. We have It follows from Lemma 12.8 that P E that P and P belong to the same connected component of more we argue with the elliptic operator construction
Proof.
T := Id-F(P — P)(2P
—
to show
Once
Id) = PP + (Id —P)(Id —P).
Again, T is a pseudo-differential operator of order 0 which differs from
the identity only by a compact operator, so it has principal symbol equal and is elliptic. Therefore the index of T vanishes. We show that T is invertible and hence belongs to the connected space £W<: Let s E ker T. Then
PPs + (Id —P)(Id —P)s = 0, hence
(15.14)
PPs = 0 and (Id—P)(Id—P)s = 0.
Since Pp = P and (Id —P)(Id —P) = Id —P, we get from (15.14):
s=Ps+(Id-P)s=0+0=0. Therefore kerT {0}, hence also cokerT = {O}, and so T E eeex. Our assertion follows from T = PP = TP. U
15. Pseudo-Differential Grassmanians
Theorem 15.12. Two projections, P1, P2 E
belong
to the
same connected component of Grp+, if and only if
i(P2,P,) = 0. Proof. All we must
know is
i(P2, P1) = index{P2P1 : 15.10
range P2}
range P1
P2T2 TI' P,T1 : range P1
=
: range P,
range P2 }
range P2}
= index{P2(Id+ compact)P1 : range Pi
rangeP2}
= i(P2,P,). Then the theorem follows from Lemma 15.10.
Corollary 15.13. The Grassmannian
has
enlLmera bEe many
connected components:
Z. Proof. Fix an orthogonal projection P.,. = P0 which belongs to
Then the proof has two parts. First (1) construct a projection P with non-trivial virtual codimension I(P, Fo) of P in P0 and get, more precisely, i(.,Po)
z
surjective.
Then (2) show that this codimension is sufficient for distinguishing the connected components, so
z
injective.
(1) Let {e,},€z be an orthonormal basis of L2(Y; Sly) which consists of smooth sections of Sly, such that (15.15)
H0 := range P0 = £C{e,}j>o
k E Z, and let Ph denote the orthogonal projection onto H,. Then Ph e for any k and More generally, let Hk := £c{e3}3>k for
l(Pk,PO) = k.
II. Analytical and Topological Tools
124
(2) Now let P1, P2 E
be orthogonal projections such that
i(Pj,Po)=k=i(P2,Po)
(15.16)
We shall show that then P1 and P2 belong to the same connected component of We shall reduce the various cases k 0 to the case k = 0 which was already explicitly solved by the preceding theorem. Actually, we only have to distinguish two situations, k < 0 and k > 0. Case k < 0: This implies that dim coker{P1P0 range P0 —i range Pi} > —k,
and the same holds for the operator P2 P0. We have cokerP1P0 = {s E L2(Y;S(y) P1s = sand P0s = O};
notice that it consists of smooth sections since it is a subspace of the cokernel of the elliptic operator P1P0 + (Id —P1)(Id —Po). Let , si} (1 —k) be an orthonormal basis of coker P1P0. Introduce P13 := P1s —
The operator P1 is a projection onto the orthogonal complement of the space W1 := Cc {si,... , s—k} in range P1. In the same way remove
the space W2 from range P2 and obtain P2. We write P1 = P2 + Q where Q denotes the orthogonal projection onto W1, i = 1,2. and it is obvious that Then F1, P2
= 0= Hence P1 and P2 belong to the same connected component of as P0, therefore by the preceding theorem i(P2, P1) = 0. So we also have
i(P2,P1) = index{P2P1 : rangePi —i rangeP2}
= index{P2P1 + P2Q1 + Q2P1 + Q2Q1 : range P1 —. range P2} = index{P2P1 : range P1 —' range P2} = 0, = index{P2P1 : range P1 —* range P2} =
15. Pseudo-Differentia' Crassmanians
125
which shows by Theorem 15.12 that P1 and P2 belong to the same connected component of Case k > 0: Now remove some subspaces V1, i = 1,2 from ker{P1P0 : range P0 —+ range P1}
which are found in quite the same way as the spaces W1. One obtains two perturbated projections Po,t := P0 —
denotes the orthogonal projection of range Po onto
where
i(P1,Po,1) =
(15.17)
0
Hence
=
Moreover (15.18) P0,1)
= index{Po,2P0,i : range = index{Po,2P0,1 : range Po,1
range Po,2} V2
range P0,2
V1}
= : range Po —. range Fo) = index{(Po — R2)(Po — R1) : range P0 —p range P0} = i((P0 —R2)+R2,(Po —R1)+R1)
=i(Po,Po)=O. Because of (15.17) and (15.18), P1 and P2 belong to the same connected component. [J Note. For later use we repeat the construction of an operator with nontrivial index for the special case k = 1 as an exercise: Let f E We define L2(Y; Sly) be a unit section belonging to rangeP÷ =:
P1 by the formula: (15.19)
Pj(s) :=
—
(s;f)f.
It is immediate that ker P1 P÷ is spanned by f and that range P1 = the orthogonal complement of {cf}CEC in H+. Since the orthogonal complement is the whole range of P1, we get for the operator P1 : range P÷ —' range P1 a vanishing cokernel and hence i(P1,P+) = index = 1.
II. Analytical and Topological Tools
126
Remark 15.14.
As shown in Theorem 15.12, the obstruction for Po, Pi E belonging to the same connected component is the integer i(P1, P0), and as shown by Corollary 15.13, this integer is in fact crucial for distinguishing the connected components of the Grassmannian Based on the notion of the spectral flow, which is explained in Section 17, we get an alternative expression which is sometimes easier to calculate. Assume that P0, P1 are orthogonal.
Then := t(2P1 —Id) + (1 — t)(2P0 —Id)}tEJ
is a family of elliptic seif-adjoint operators of order 0 parametrized over the unit interval I = 10,11. Compare also the different families constructed for the proof of Corollary 15.3. The family {Bt } joins the operators B = (Id —P2) — (i = 0, 1) with spectrum equal to {—1, +1}, and an easy computation shows that
= i(P1,Po)
(15.20)
that the
From Corollary 15.13 follows the basic property of diagram of Figure 15.4 is index-commutative: range P1
P3P/ range
P2 —'
P3P2
range P3
Fig. 15.4 i(P3,Pi) = i(P3,P2) + Proposition 15.15. Let P1,
P2,
i(P2,Pj) Then
P3 E
i(P3,Pi) =i(P3,P2)+i(P2,Pi)
= index{P3P2P1
: range
Pi
—+
range P3}.
Proof. We may assume
(15.21)
P,
for suitablek, €Z; j= 1,2,3,
where denotes the orthogonal projection of L2 ( Y; Sly) onto the subspace Hk3 introduced in the proof of Corollary 15.13. We have to consider different possibilities
andsoon to analyse index{Pk3Pk2
: Hk1 —p range Hk3}. Elementary compu-
tation shows that the proposition is true in each case.
0
16. The Homotopy Groups of the Space of Seif-Adjoint Fredhoim Operators
We offer an elementary and relatively simple calculation of the homotopy groups of the space of seif-adjoint Fredhoim operators.
We determine the topoiogy of the space of seif-adjoint Fredhoim operators in Hubert space. This was done in Atiyah & Singer [19691, but though transparent, their computations are quite long and complicated. We make three simpilfications: We do not bother to classify properties in K-theory, but calculate only the homotopy groups; we do not consider skew-adjoint Fredhoim operators, but (complex) selfadjoint ones; and we do not repeat all arguments of Atiyah & Singer [19691 ab ovo, but use only a simple argument from the beginning
of their work which permits us to appiy the concepts and methods of infinite-dimensional Grassmannians provided in Chapter 15 of this book.
Actually the story is even more simple: In Chapter 15 we restricted ourselves to Grassmannians of pseudo-differential projections with given principal symbol. This was differential analysis. Now we need not be concerned with all the intricacies we had to surmount at that time, since we make our deformations in the full Grassmannian of all (orthogonal) projections which differ from a given non-trivial projection by a compact operator (and the corresponding Grassmannian of seif-adjoint involutions in the Calldn algebra). This is functional analysis. As such, this chapter solves a problem posed by Singer [1971; p. 951], namely to find an "ele,nentcry" or "direct" proof of the periodicity of the homotopy groups of F. (see Proposition 16.2 and Theorem 16.10 below).
1GA. Elementary Decompositions and Deformations First a few rather generally known observations about the spectrum of Fredhoim operators and seif-adjoint Fredholm operators.
Proposition 16.1. Let H be a separable infinite-dimensional complex Hilbert space and let F denote the space of all &edholm operators
II. Analytical and Thpological Tools
128
on
H. Then
(a)
small A.
(b) If T E F is seif-adjoint, then 0 belongs to the discrete spectrum of more precisely there exists an e > 0 such that (—e, e) fl spec(T) is empty or equal to {O} which is an eigenvalue of finite multiplicity.
Proof. (a) is obvious since F is open in the normed space B of all bounded operators. To prove (b) we recall that the range of a Fredholm operator is closed, hence by Proposition 9.4 the inverse operator (Tlker T)
'is bounded. Therefore there exists e > 0 so that IIT'pII
for 'p E (kerT)1.
The first remarkable fact about the topology of the space of selfadjoint Fredholm operators is that it decomposes into three connected components which are distinguished by the essential spectrum. This will be proved in the following proposition. For a better understanding of this result, recall from ordinary Fredhoim theory: The space F, equipped with the operator norm, decomposes into Z connected components which are distinguished by the index. More generally, for any continuous family {Am}mEM of Fredhoim operators
parametrized by a compact topological space M, we can define a homotopy invariant, the index bundle index{Am}m€M E K(M),
where K(M) is the Grothendieck group of complex vector bundles over M. The Atiyah-Jänich theorem states that the index bundle defines a bijection [M, F] —' K(M), where [.., ..] denotes the homotopy
classes of mappings. Hence the homotopy type of the space F is completely identified: it is a classifying space for the functor K. It turns out that one can develop an analogous theory for the space F of self-adjoint Fredholm operators, not based on the index which vanishes on F, but on spectral invariants, namely the essential spectrum and the spectral flow. The most profound result is the following:
Proposition 16.2. Let F denote the space of self-adjoint Fredholm operators on a fixed complex separable Hubert space H. It has three
16. The Homotopy Groups of F
connected components F÷, 1..., and F.,. characterized by
T
is a positive (negative) operator for some compact operator k T— k
C
T
F. :
T not in T has both positive and negative essential spectrum.
Moreover, the components F+ and F_ are contractible. Proof. First check the equivalence of the two alternative descriptions given in the statement of the proposition: of the sets F+, F....., and If T — k is a positive operator for some compact operator k (one also says T is essentially positive), we have k compact
=
—
k)
T—k positive
C
(0, +oo).
Vice versa, if C (0, +oo), it follows from Proposition 16.1 that there exists e > 0, such that there are only finitely many discrete eigenvalues in the interval (—oo, e); hence T is a positive operator on the complementary space of the finite direct sum of these finitedimensional eigenspaces. The same arguments apply for F_ and F. so that also the spectral characterization of these spaces are established. To proceed further it is suitable to turn to the images of F,
and F. in the Calldn algebra C := B/K of the Hubert space H. Let Q denote the group of units of C and the subspace of seif-adjoint invertible elements in C (with respect to the natural Ce-algebra structure for C). Let p : B —p C denote the canonical projection. A widely
exploited fact is that F = p'(G) which is obvious, since the Fredholm operators on H can be characterized as those which are invertible modulo X. We also see that p maps F into g, and the mapping —, is also surjective: for g E and T E F with p(T) g g and T+2T E F. g By definition
specp(T) := (Al p(T) — Therefore the subsets ing subsets of
A
and F. are the inverse images of correspond-
under the projection p. Of course, here the
130
II. Analytical and Thpological Thols
speces5
—1
0
Fig. 16.1 The decomposition of
1
into its three connected components
point is that the
are defined via the essential spectrum and the g-subspaces only via the spectrum. These subspaces of Q are both open and closed and hence also its inverse images F± and Therefore the space F is not connected and decomposes into the union of these three mutually disjoint subsets. It turns out that each of these subsets is connected. This is obvious for Then (16.2)
does not lead out of
since by (16.1) spec
= (1—
=
(1
—
t)(specp(T)) + t t
T c Hence (16.2) provides a contraction of F÷ to the operator Id, cf. also Figure 16.1. Similarly F_ contracts to the operator — Id. We postpone the proof that is also connected. fl
16. The Homotopy Group8 of F
c.
p(T) Fig. 16.2 The retraction of
onto
: F.
Lemma 16.3. The natural map
131
is a homotopy equiv-
alence.
Proof. Recall the theorem of Bartle and Graves (see e.g. Bessaga & Pelczynski [1975; p. 86]) which states that each surjective, continuous, linear operator from one Frechet space into another Frechet space possesses a right (not necessarily linear) continuous inverse. Since the projection p: B —+ C is surjective, continuous, and linear, we get from the theorem of Bartle and Graves a section s in the bundle p: B C over C = B/K with total space B and fibre K, namely a continuous (in our case definitely not linear) right inverse s of p. Then s(g) +
g
(s(g)) 2
is a section (continuous right inverse) for the mapping
Modulo homotopy, the section we can retract .?. back onto := (1 —
is
also a right inverse of linearly by
t)T + ts1(p(T)),
0 t 1
since
II. Analytical and Thpologicai Tools
132
that all and in fact to belong to the same fibre p'(p(T)) of .1,. This is clear since psI = id, so sip(T) — T E K and therefore also T E K. See also Figure 16.2. for arbitrary T E
Note
[1
Thus the topology of the F-spaces has been replaced by the topol-
ogy of the c-spaces. However, the c-spaces are still rather large and their spectra are not easily grasped. For further topological investigations, it is convenient to replace these spaces by suitable deformation retracts, which only consist of single points, or for which at least the spectrum only consists of single points. We apply the standard algebra retraction (16.3)
:=
g
((1
—
t) id
o
t
i.
As a special case, we have that the group C of unitary elements of the Calkin algebra C is a deformation retract of (and hence homotopy equivalent with) C. Fortunately the c-spaces are stable under the retraction (16.3) because (16.4)
g*=g
g=gt.
Therefore (16.5)
S fl C is a deformation retract of S
for SE Denote the corresponding unitary retracts by Roman letters. Then
=gandg*g=idhenceg2=id}, (16.6)
Ô+={id},
Ô_={—id},
and
From Lemma 16.3 we get with (16.5) the following theorem:
Theorem 16.4. The space
is homotopy equivalent with the space
G*. Now we shall determine the homotopy groups of Ô..
16. The Homotopy Groups of Q *
133
16B. The Homotopy Groups of C. Let H be a complex separable infinite-dimensional Hubert space. consists of the seif-adjoint We have seen in (16.6) that the space involutions in the Calkin algebra C(H) = 8(H)/X(H) of H which are not equal to the identity:
Ô. = {j E C(H)
If = j, j2 = id, and j
±id}.
Thus it can be identified with a suitable Grassmannian
Gr(C(H)) := {q
(16.7)
E
C(H)
I
q2
=q =
0, id).
The isomorphism is given by
j
(16.8)
E
Gr(C(H)).
Consider the case of the corresponding Grassmannian in 8(H): (16.9)
Gr(H) := {P
If we fix a Po neous space
8(H) I P = P2 = and dim range P =00= dim range(Id —P)}.
Gr(H), then Gr(H) is isomorphic with the homoge-
(16.10) U(H)/U(Po(H))
—Po)H)
U(H)/U(H)
where 8(X) denotes the C-algebra of bounded operators on X for XE {H, P0(H), (Id—Po)(H)} and U(X) its unitary retract. The isomorphism is given by the map (16.11)
Rp0 : U(H)
U
UP0U'
E
Gr(H).
Repeating an argument from Chapter 15 (cf. especially the proof of Theorem 15.5 and Remark 15.6, see also Remark 24.5b below) we obtain the following theorem:
II. Anaiytical and Topological Tools
134
Theorem 16.5. The mapping R,'0 : U(H) —p Cr(H) of (16.11) defines a principal fibre bundle with fibre and structure group U(H) Note. By Kuiper's theorem, it follows that Gr(H) is the image of a contractible space and therefore itself contractible.
Now repeat this argument in the case of G.. The only essential difference is that now neither the total space, nor the structure group of the bundle is contractible. As before, let C denote the unitary group of C(H). Similar to how proved Lemma 16.3, see that C has the homotopy type of the space of Fredhoim operators and in particular (16.12)
Irk (C,30)
forkeven =
forkodd
0
for any base point E C. We fix jo (3m. As already noticed (in the proof of Lemma 15.2), P0 —(Id —P0) is an involution for any projection P0. Now the next elementary observation is proved:
Lemma 16.6. There exists P0 E Gr(H) such that p(Po—(Id —P0)) = jo, where p : B(H) — C(H) = B(H)/K(H) denotes the natural projection. Proof. Consider the following diagram which is not necessarily com-
mutative and contains a semi-continuous mapping P> (because of jumps of dimensions):
(16.13)
Gr(H) Here
p
P(T) denotes the spectral projection ofT onto [0,oo) and
is defined by (16.14)
:= p(2P — Id).
16. The Homotopy Groups of : —, is Begin with the fixed class jo E C. C g.. Since surjective, there exists T E.?. such that p(T) = jo• The operator T operator with = {±1}, and T is is a seif-adjoint essentially an involution:
(16.15)
:=
E
p'{id} = {Id+K}.
On the other hand 2 p2 (T) — Id is a true involution and
(T)) =
The following corollary completes the proof of Proposition 16.2.
Corollary 16.7. The spaces C. and F. are connected. Proof. First we show that C. is connected. Fix a P0 E Gr(H) and consider the following composition of maps
U(H)
Gr(H) —f-' &.
The first map is clearly continuous and surjective by Theorem 16.5.
The second map is continuous by construction and surjective by the
preceding lemma. Hence C. is the image of the connected (even contractible) space U(H) under a continuous map and therefore connected. Since, by Lemma 16.3, F. is homotopy equivalent with C., it is connected as well. U As seen below, the continuous image C. does not inherit the con-
tractibility from U(H). Similar to (16.10) above and Theorem 16.5 we now define a map r,0 : C —, C. on the unitary group C of the Calkin algebra by the formula (for fixed io) (16.16)
r,0(u) :=
We want
(1) to show that
a principal fibre bundle, (2) to derive its long exact homotopy sequence, (3) and to determine the homotopy groups of C.. We begin with the following result: : C —+ Ô. defines
II. Analytical and Topological Tools
136
Lemma 16.8. The map rj0 : C —'
is surjective.
Proof. Let j E a..
follows from Lemma 16.6 that j = p(P — Cr(H), and it follows from Theorem 16.5 that there exists a unitary operator U U(H) such that (Id —P)) for an appropriate P
UP0U' =
P, where P0 E Gr(H) was fixed such that p(P0 —
(Id —Po)) = jo• Set u := p(U). Then
j = p(P
(Id —P)) = p (U(P0
—
—
(Id —Po))U')
=
Now investigate the fibre
= {u E C I ujou1 = jo}.
Go :=
it is a subgroup of C. Moreover:
Proposition 16.9. We have Go Note.
C.
C
This result can also be formulated as follows: Go = p(Up0(H)),
(16.17)
where (16.18)
Up0(H) :
{U eU(H) UP0 — P0U
E AC(H)}
and P0 E Gr(H) with p(Po — (Id —P0)) = Then Proposition 16.9 states that any class u C0 can be expressed (in the Calkin algebra) as a class of direct sums of unitary operators on P0(H) and on (Id—Po)(H).
Proof. Define := ±jo). Assume that a Po E Gr(H) is given with jo = p(Po). Then for any class u Go, we have a decomposition (16.19)
u=u+u.... =u_.u+,
where
(16.20)
:=
+
The map iz (u+, u_) is an isomorphism of the group Co of unitary elements of the Calkin algebra onto
16. The Homotopy Groups of •
137
denotes the unitary part of the group p(.:F(P0H)) (or
where
—P0)(H))) resp.) which is a deformation retract of it.
U
To follow the line of arguments in proving Theorem 15.7, we now construct a local section of the bundle r30 : C Let j E with Iii — iou < 1. Then, following Lemma 15.2, the
id+(j — jo)(2jo
t(j) :
(16.21)
id)
—
is invertible in C(H) and we have
t(j)jo = jt(j),
(16.22)
therefore (16.23)
j = t(j)jot(j)'
hence
,
r30(t(j)) =j.
So t defines a local section. To be more precise, we have to take the unitary part of t(j) in order to get an element of C, since a priori only t E p(F(H)). This can always be done. Now following the proof of Theorem 15.7 we get the short exact sequence (16.24)
Z —. lr2k(ô.)
exploiting that lr2k(G
C)
Z
Z
and
lr2k(G)
Z.
Here as usual the homotopy groups are defined relatively to fixed base points. From (16.24) the main result of this chapter is obtained:
Theorem 16.10. Let T0 =
Po — (Id —Po) E and jo E be fixed Then the homotopy groups of the non-trivial component of the space of seif-adjoint Fredhoim operators and of the corresponding space of 8elf-adjoint non-trivial involutions of the Colkin algebra are given by
elements.
—
-.
fori
.
ir,(F.,To) = 7r(G.,30)
=
Z fort
even odd.
17. The Spectral Flow of Families of Self—Adjoint Operators
The spectral flow is investigated for families of seif-adjoint elliptic operators with non-vanishing symbol class over an oriented closed smooth manifold.
Our main tool for the treatment of global elliptic boundary problems for operators of Dirac type is the concept of the spectral flow, this peculiar blend of classical analysis and differential topology, which will be explained in detail in the present chapter.
17A. Continuity of Eigenvalues From a topological point of view, the most fundamental property of the spectrum of seif-adjoint operators is its continuity under continuous changes of the operator:
Lemma 17.1.
Let A be a seif-adjoint bounded linear operator in a Hubert space H and let a be a positive real number such that the inter-
section spec A fl (—a, a) consists only of a finite system of eigenvalues
(all the eigenvalues repeated according to their multiplicity). Then, for all seif-adjoint operators A' sufficiently close to A, the intersection spec A' fl (—a, a) consists of the same number of eigenvalues
and one has —
)41 hA
—
A'II for each j
E
{k,... , m}.
Proof. First decompose A into the difference A+ — A_ of two nonnegative operators. The operator A+ has the eigenvalues , and A_ has the eigenvalues —Ak,... , Since IIA+ — A'÷hI < — A'hI, it suffices to consider the case when A is non-negative.
17. The Spectral Flow
139
Recall that then the j-th eigenvalue A, can be characterized by the minimum-maximum principle (see e.g. Reed & Simon [1972] or Kato [1976; Section 1.6.10]) through
Ajsup{UA(vI,..,vj)Ivl,...
,v,€H},
where UA(v1,... ,v,) =
inf{(Aw,w) liwli = 1 I
and
WE £C(vl,
Now for each choice of vi,... ,v3 and for each > 0 w with liwli = 1, orthogonal to v1,... ,v3 such that (Aw,w) — UA(V1,... It
choose
a vector
,v,) <e.
follows that
UA'(Vl,...
,v,)
(A'w,w) = (Aw,w) +
((A'
UA(V1,...
,vj)+IIA—A'II+e;
- A)w,w)
hence
UAS'(Vl,... ,v,) UA(V1,... ,v,)+IIA—A'II.
In the same way we get
UA(v1,... which
,v,) UA'(Vl,... ,v,)+IIA—A'II,
proves the lemma.
Remarks 17.2.
(a) Recall that seif-adjoint elliptic operators of positive order over a closed manifold have a discrete spectrum {Aj},Ez of finite multiplicity. Moreover, there exists no essential spectrum, and the eigenfunctions span the whole L2E. We get the continuity of the eigenvalues immediately from Lemma 17.1. As usual, operators of order r can be reduced to operators of order 0 by the substitution
be a spectral decomposition of the More precisely, let {A,, Hubert space of square Lebesgue-integrable sections in a Hermitian vector bundle E generated by B, i.e., the {Aj}jEz are the eigenvalues
140
II. Analytical and Topological Tools
of B : WE —' L2E, where H" is the r-th Sobolev space, with the corresponding eigenfunctions {e, }jEZ spanning the whole L2E. Then
+ a spectral decomposition of L2E generated by B with all eigenvalues bounded by +1. Hence the continuity of the eigenvalues of B carries over to the eigenvalues of B. (b) The continuity of a finite system of eigenvalues can be obtained for all closed, not necessarily bounded or seif-adjoint operators as shown in Kato [1976, IV.3.5]. is
Now we could provide the definition of the spectral flow directly by exploiting the perturbation theory of the spectrum of closed, not necessarily bounded operators, as mentioned in the preceding remark. However, the topological meaning of the spectral flow will be more transparent, if instead, we recall the homotopy investigations from Chapter 16 and first work with seif-adjoint (bounded) Fredholin operators in Hubert space. Later we shall carry the arguments over to elliptic operators of positive order.
17B. The Spectral Flow on Loops in Now we define the spectral flow of a family {Bt}tE[o,1] C F.. We begin with operator families belonging to the subspace
andBhas only finitely many eigenvalues}.
We can argue in a similar way as in the proof of Proposition 15.4 (second step) in order to show that F°° is a deformation retract of However, we do not need the full argument, since we only want
to explain the notion of the spectral flow. Let {Bt}tEsl be a periodic family of operators of F°°. Then the spectrum of can be described in the following way.
Lemma 17.3. Let B: S' —,
F°° be a continuous map. Then the
graph of the spectrum of B is given by a finite monotone sequence of continuous functions j : I [—1, 1],
i.e.,
= {jo(t),jj(t),... ,jm(t)},
tEl.
17. The Spectral Flow
Definition 17.4. Let £ be the integer such that jk(I) = jk+t(O) for all k, where we eventually define
i-ei-t+i
1 and
...
= 1.
The integer £ is called the spectral flow of the family B.
Remarks 17.5. (a) The spectral flow is therefore just the difference between the number of eigenvalues, which change the sign from — to + as t goes from 0 to 1, and the number of eigenvalues, which change the sign from + to —. In Definition 17.4, we rearranged the eigenvalues to get a monotone sequence.
(b) Consider a family B: to a family with values in F
in .?,. Now we might deform B
this is not needed since the discussion in Section 17A showed that for any family B with values in F,, the graph of the spectrum of B near 0 can be parametrized by a finite set of continuous functions j' : I R. This yields once again a well-defined integer £ such that jk(1) = jk÷j(0). .
However,
Proposition 17.6. The spectral flow defines an isomorphism sf : iri(F,) —' Z.
Since 5,
connected, the choice of a base 2oint in F, is irrelevant to the definition of the homotopy group ir1 (F,). Note.
is
Proof. First the spectral flow is a homotopy invariant of the families: Let A = and B = be two periodic families parametrized by t E I which can be deformed into each other, i.e., there exists a two-parameter family {Ft,U}g,UEI such that for = , = each t, and for each u. Let C = be the boundary = of i.e. A49
forO<sl/4
—
F1,49..1
for 1/4 < s
—
B3_4.,
for 1/2 < s <3/4 for 3/4 < s 1.
F0,4_49
1/2
Then sf C = sf A — sf B, and C can be deformed into a constant family C'. Let {Gt,U}t,UEJ be such a two-parameter family with = and = Let m be the spectral flow of the =
II. Analytical and Thpological Tools
142
family {Go,U}UEJ, and let {jk(t,u)}k be the sequence of functions
parametrizing the spectrum. Then =jk_,n(O,1) hence sf C = phism
0.
It is
clear
jk(1,O),
that the spectral flow defines a homomor-
In fact it is an isomorphism: We construct a periodic family with spectral flow equal to 1. Let {ek}kEz be an orthonormal basis of H. Take P—k)
+ (2t — 1)Po,
where P, is the orthogonal projection
onto Cc{ek};
:=
—
hence
t€I. The preceding proposition gives the maln result of this chapter.
Theorem 1'T.7. Let ifi(Y; E) be the space of elliptic seif-adjoint operators of positive order over a Riemannian manifold Y on sections of a Hermitian bundle E, and let B : I —. Ell(Y; E) be a family of operators, where the coefficients depend smoothly on the parameter t E I. (a) Then the graph of the spectrum of B can be parametrized near the 0-line through a finite set of continuous functions
(b) If B0 and B1 have the same spectrum, we get jk(l) = ik+z(O) for some integer which is the same for each k. That £ will be called the spectral flow sf{Bg} of the family (c) The spectral flow is a C°° homotopy invariant of periodic (i.e. B1 = Bo) families of elliptic seif-adjoint operators of positive order, which means that the spectral flow does not change under C°°deformations of the coefficients of the operators involved.
The situation may be illustrated by Figure 17.1, where e.g. jj(1) = j2(1) = j3(0), j3(1) = j4(0) etc., hence sf{Bt} = 1.
j2(0),
17. The Spectral Flow
143
1
t
Fig. 17.1 Spectrum of a family with spectral flow =
1
Examples 17.8. (a) Let H be a complex separable Hilbert space and let P_ be complementary projections, i.e. + P_ = Id, both having infinite-dimensional range. Then, for any v E range P_ and Here P, denotes the belongs to t E R, the operator — + projection of H onto £C{v}. The spectrum of P.,. — P_ + consists of the essential spectrum {1, —1} and the eigenvalue A = —1 + t of multiplicity 1. The spectrum of the family :=
—
P_ + 2tPv}tEI
is given by the left graph of Figure 17.2 and hence = 1. This family was already considered in the proof of Proposition 17.6. instead of Ps,, where V is a subspace of range P_ of finite dimension N, we get the same graph and a spectral flow equal to N by multiplicity argument. (b) Now consider the family Taking
I
d (IX
tES'
of ordinary differential operators over the circle S', parametrized by
t E S' = I/{O, 1). We have a spectral decomposition of H := L2(S')
144
II. Analytical and Topological Tools
by the system of eigenfunctions with corresponding eigeni.e. the spectrum of B is given by the right graph values {k + t}kEz, of Figure 17.2 and hence sf B = 1.
A
Fig. 17.2 The spectrum of
—
P_+
and of {—id/dx + t} (right)
B' := (Id + B2)'/2B leads to a family This family can be deformed further into the family with of (a), where P_ denote the projections onto k > 0 and k 0 respectively, and v denotes the constant function equal to 1 (corresponding to k = 0).
Note that the reduction B
The preceding Example 17.8b raises two natural questions:
(1) Instead of the family = —id/dx + t} of self-adjoint elliptic operators over parametrized by t S', we may consider its suspension, namely the (non-seif-adjoint) elliptic operator 0 .8 A:=—-z—+t Ox Ot
17. The Spectral Flow
over
145
the torus T = S' x S' acting on the space
C°°(T;E) := {f E C°°(I x S') I f(1,x) = for all x E S'}. How are the index of the operator A and the spectral flow of the interrelated? In Theorem 17.13 below we shall explain why these two integers must coincide. family
(2) We noticed that the preceding examples (a) and (b) are interrelated. More generally, instead of the operator A —8/Ot +
or (equivalently) the family we may consider the spectral projections and P< belonging to B0, and the gauge transformation B1 = for a suitable morphism g. But how are the spectral flow of the family (or the index of A) and the index of the desuspension P> —g P< related to each other? In Theorem 17.17 it will be explained why these two integers also must coincide. As we shall show in Chapter 24—26, both questions are fundamental
for the understanding of the cutting and pasting of elliptic operators. Especially the second question seems to be central to understanding the geometry of the Grassmannian of all elliptic boundary problems and to solving single boundary problems. See also Remark 15.14 above, which shows that two pseudo-differential projections with the same principal symbol naturally define a family of seif-adjoint elliptic operators, and that the spectral flow of that family is the obstruction for the projections belonging to the same connected component in the Grassmannian.
17C. Spectral Flow and Index Let S' —p be a family of self-adjoint Fredhoim operators, } where the circle S' is parametrized by u E 10,11. By Proposition 17.6, such a family defines an element of the homotopy group ir1
two families define the same element if and only if they have the same spectral flow. We shall relate the spectral invariant with topological invariants: the indices of related single operators (of the suspension, Theorem 17.13, and of the desuspension, Theorem 17.17). To that purpose it is suitable to introduce an intermediate, typically differential invariant, the Chern character of the analytical index, which is a certain vector bundle canonically associated with the family We shall make all constructions as explicit as possible. and
II. Analytical and Topological Toots
146
The basic notations of K-theory may be found e.g. in Karoubi [1978), Gilkey [1984), or BooB & Bleecker [1985].
Recall that the elements of
K (S' x S1 /({O} x S1 u S' x {i})) are determined by the homotopy classes of the clutching mappings g: S' U(N) of the bundles, and that the generator is the class of the map g(u) = et". In this case the Chern character is the isomorphism
H'(S';Z)
ch : K1(S')
Z)
given by the formula (17.1)
22rs
I Js'
trg'dg=degg,
where g is assumed to be differential. Consider the diagram a-index
K'(S1)
To describe the horizontal map a-index which gives the identification, we recall the construction of Example 17.8a: For fixed compleboth having infinite-dimensional range and a mentary projections non-negative integer k, choose a subspace V, of range(P_) of dimension k. (For negative k one may correspondingly choose a subspace of dimension —k). Then any element of ir1(F.) is V, of defined by a family + 2uPk}
—
= k. Definition 17.9. With the preceding notations the analytical index with
denotes the homomorphism a-index :
—
given by := [hJ, where tary matrices defined by (17.2)
=
—
:=
K'(S') denotes the family of uni-
(Id on(Vk)1 e2" on Vk.
The preceding definition of the analytical index gives at once the identification of and
17. The
Proposition 17.10.
Let
Flow :
147
S' —' F be a continuous family.
Then
=
= k and represent the class of
Proof. Assume by the standard family
= 2irs j0
in iri(F.)
Then by (17.1) and (17.2)
= tr
1
t'
—/
Idvb = dim Vk I du = k. Jo
The remaining part of this chapter is less elementary since we shall
exploit the full strength of the index theorem including the index theorem for families. We proceed with the topological study of families of self-adjoint
elliptic operators in the realm of symbolic calculus. Recall that the homotopy class of an elliptic operator B over a closed manifold Y only
depends on the homotopy class of its principal symbol 0B. For any such symbol, the usual theory of elliptic symbols constructs a symbol class [oBJ, which is an element of K(TY), and which leads to the computation of the index of B by the topological index homomorphism K(TY) —' Z. If B is a self-adjoint elliptic operator over Y, then again its homotopy class only depends on the homotopy class of its principal symbol,
that is, B can be deformed into another seif-adjoint elliptic operator C through a smooth 1-parameter family of self-adjoint elliptic operators if and only if can be deformed into cc in the class of self-adjoint and (for each non-zero cotangent vector) invertible symbols. This is a standard exercise. However, the usual symbol class LaB] E K(TY) is vanishing. In Atiyah, Patodi & Singer [1976, §3], it was found that the characteristic element of a self-adjoint elliptic
operator lies in the group K'(TY): Definition 17.11. (a) Let B be a seif-adjoint elliptic operator acting on the C°°-sections of a Hermitian vector bundle E over a closed Riemannian C°°-manifold Y, and let
yEY,
and I,iI=1,
be
and Thpological Tools
II.
148
the principal symbol of B. Consider the family
for ir
1
of elliptic symbols over Y parametrized by a point t on the circle S'. This yields an automorphism of the bundle irE, where ir: S' x SY —' Y is the projection (SY C TY is the sphere bundle of cotangent vectors) and hence defines an element K(S' x TY). Its restriction to {O} x TY is trivial and so can be regarded as
an element of K'(TY). It will be called the stable symbol class of the operator B. (b) Let Y and E be as in (a). Then, in the same way as in (a), we obtain, for a family B = {Bm}mEM of elliptic seif-adjoint operators on C°°(Y; E) with a compact parameter space M, a two-parameter V parametrized by m E M and t E S' and hence an automorphism of the bundle E lifted to 5' x M x SY. This yields an element in K(S' x M x TV), trivial on (0) x M x TV and hence an element in K'(M x TV), the stable symbol class of the family B. family of elliptic symbols
over
Lemma 17.12. Let B = {Bt}tEsl
be a family of elliptic seif-adjoint first-order operators acting on sections of a Hermitian bundle E over a closed manifold Y and let
B' :=
+ B&}tES1
be the associated first-order elliptic operator over the torus 5' x V acting on sections u(t, y) of the vector bundle p*E (= E lifted to S' x Y by the canonical projection p: S' x Y Y). Then we have
K'(S' x TY) 3
=
K(T(S' x Y)).
Proof. Clearly B' is an elliptic differential operator, but in general, B' is not self-adjoint since its formal adjoint is BI* = + Since the tangent space of 5' x Y is diffeomorphic to the product S'x RxTY bythemapping weget a natural isomorphism K(T(S1 x Y))
K(S1 x R x TY)
K'(S' x TY).
17. The Spectral Flow
149
Now consider 47B'(t,yT,71) =
E T(S' x Y) with 1T12 + 17112 = 1. Since is an operator of first order, the symbol OB is homogeneous in and we can rewrite
where t E S', y E Y,
aB(t, y; r, i,)
= —i
=
+ i1711
—i
r ir. As usual we join —i with the identity and so we get a homotopy between the clutching isomorphism by substituting r = cosr for 0
T ES', t
E SY
which defines the symbol class [ORB'] of the operator B' and the clutch-
ing isomorphism
for0r
f where r
S1, t E S1, (y,
E SY, which defines the stable symbol class of the family B. Since glueing by homotopic isomorphisms gives the same classes in K-theory, the lemma is proved. LI
Theorem 17.13. Let {Bt}tEsi be a family of self-adjoint elliptic operators of first order over a closed Riemannian manifold Y parametrized by t E S'. Then we have sf B = index B',
is the elliptic operator on Y x S' naturally where B' = {—& + associated with the family.
Proof. By Proposition 17.10 we have sf B = (cha-index B)[S'I, where [S'J is the fundamental cycle of S' in standard orientation, and a-index B E K' (S' x Y), the analytical index of B which equals its topological index; hence sf B =
II. Analytical and Topological Tools
150
Recall the double character of sider the corresponding diagram
K'(S' x TY)
K(S'
= [oB'J by Lemma 17.12 and con-
t-index
x TY)) K(T(S' x Y)) The diagram is commutative (check the commutativity for the periodic family + a}0es1 of operators on S' of Example 17.8b); x
hence
(ch t-index[o'BI) [5'] = t-index laB' I
which proves the theorem by the index theorem applied to the single operator B'. 0 As is usual in index theory, we want to relate the analytical index which is obtained globally, namely from the space of solutions, with the symbol class which is obtained locally, namely from the coefficients of the differential equations involved. In the case of one single elliptic
operator B over a closed manifold Y, this relation is given by the Atiyah-Singer index theorem:
a-index B = t-index[CB], where a-index B := [ker B] — [coker B]
K(point)
and
t-index: K(TY) K (point) Z denotes the topological index given by an embedding j Y —' by the excision of a tubular neighbourhood N of j(Y) and by the Bott periodicity K(TRk) K(point). In the language of :
characteristic classes, this becomes
U r(Y))[Y),
t-index[oBJ =
where it is supposed that Y is oriented, (Y] E
the fundamental
cycle of the orientation, n the dimension of Y, r(Y) E H(Y), the Todd class of Y,
ch: K(TY) the Chern character, and :
isomorphism.
Q) Q) —p H(Y) the Thom
17. The Spectral Flow
151
Corollary 17.14. Under the assumptions of the preceding theorem we get the following topological formula for the spectral flow:
sf B =
/
Ch[0B'] r(Y),
where n = dimY.
Next we consider a gauge invariant family of seif-adjoint elliptic operators, i.e. a family : C°°(Y; E) —+ C°°(Y; E)}jEJ with B1 = g1B0g for a suitable unitary automorphism g of the Hermitian bundle E. Then we shall express the spectral flow of the family {Bt}tEJr in terms of the gauge transformation g and the spectral projections P> and P< of the initial operator B0. (For the definition and the fundamental properties of the spectral projections we refer to Definition 14.1 and Proposition 14.2 above.)
Lemma 17.15. Let B be an elliptic seif-adjoint operator of nonnegative order acting on the smooth sections of a Herrnitian bundle E over a smooth closed Riemannian manifold Y. If the principal symbol
(or p4 of its spectral projection P> (or P<) vanishes, then B is half-bounded as an operator in L2(Y; E).
The conclusion is in fact that for an operator B of order zero, the essential spectrum of B lies on only one half line of R, i.e. B is essentially positive (or essentially negative). This is the case, if and only if the principal symbol of B is positive (or negative). Note.
of the pseudo-differential operaProof. Let the principal symbol tor P> vanish. Then P> is a pseudo-differential operator of order —1 and hence compact as an operator in L2. As a projection it has only eigenvalues {O, 1}. Since P> is compact, the multiplicity of the eigenvalue 1 has to be finite; hence it must have finite-dimensional range.
This proves that B only has a finite number of positive eigenvalues and hence
(Bu,u)<max{A,(u,u)IjEZ} forallueL2(Y;E). If P< vanishes, then similarly we only have a finite number of negative eigenvalues and we get
(Bu,u) min{A,(u,u) I
j
Z}.
provides a measure for the spectral asymmetry of B. More generally, let E± denote the bundles over Lemma 17.15 shows that
II. Analytical and Topological Tools
152
and p. What happens if the
SY which consist of the ranges of
bundles E÷ and E_ are pulibacks of vector bundles 4, F_ over Y by the projection 7r: SY —' Y, that is, if E1, = = are functions of y E Y alone. The situation is then more complicated than in Lemma 17.15. However, we find This is the case when
B=
+ p_Bp.. +
+
where the last term is an operator of order —1. + Hence, up to compact operators, the operator B is a direct sum of half-bounded (essentially positive or essentially negative) operators B
:
C°°(Y;E) —' C°°(Y;E)
with
point of view if p± defines, as its image, SY which is not a lifting from Y. Then B has infinitely many eigenvalues on both sides of 0, and it cannot be reduced to the sum of half-bounded (essentially positive or essentially negative) operators. We see that B is
non-trivial from our topological
a vector bundle over
The range bundle 4 of
is generally a vector bundle only over a connected component of SY, since the dimension of the range of may change when we pass to another component. Note.
Now we give a topological formulation of our results: The bundle
an element of the group K(SY). Neglecting the topo4logically trivial case discussed above, we actually get an element of defines
the group K(SY)/irK(Y)
K'(TY). This last isomorphism can
be derived, cf. Atiyah, Patodi & Singer [1976, §3] from the Cysin sequence
/
K(Y)
K(TY) K (SY)
K(BY)
and it follows that the element is equal to the stable symbol class of B introduced in Definition 17.11. This proves the following proposition.
17. The Spectral F'ow
Proposition 17.16. The stable symbol class
153
K1 (TY) of an
elliptic self-adjoint operator B of non-negative order vanishes, if and only if the operator decomposes into the sum of half-bounded (essentially positive or essentially negative) operators. This is the case if and P< of the principal symbols of the spectral projections B depend (modulo deformations within the class of idempotent symY. If on the contrary [os] 0, then we have bols) only on y
dimrangeP> = dimrangeP< = 00. We are now able to prove the following main theorem on spectral flow and gauge transformation (unitary equivalence). When it was first proved in BooB & Wojciechowski [19821 it was called the destzspension theorem.
Theorem 17.17. Let
be a smooth family of elliptic self-adjoint operators of non-negative order acting on the sections of a Hermitian
bundle E over a closed manifold Y and having the same principal Let g be an automorphism of E such that B1 = g'B0g.
symbol
Let {e,},€z be a spectral decomposition of the Hubert space L2(Y; E), i.e. an orthonorrnal basis of L2(Y; E) consisting of eigenfunctions
e3 of B0: B0e3 = Let P> and P< be the projections onto the
—gP<.
Remark 17.18. Recall that the space E110 consists of all elliptic self-adjoint operators of the same order acting on sections of a fixed Hermitian bundle and having the same principal symbol. Hence it is convex. This is an important difference from the case of the space F of all self-adjoint Fredhoim operators having the topological non-trivial component F., as seen above in Theorem 16.10 and Proposition 17.6. Hence every true periodic family in giL, could be deformed into the constant family. In order to get a topologically meaningful family, it is necessary to admit that B1 is different from B0. We assume that all have the same principal symbol. That permits us to reconnect B1 with B2 := B0 in the whole space F via
:=g'Bogg, tE [0,1], where {gt}t€I is a retraction of g to gi := Id in the contractible group GL(L2E) of all invertible operators on the Hubert space L2E.
17. The Spectral Flow
155
sphere bundle SY as However, the spectral flow is unchanged (and with it the class of the symbol in Assume that dim range P>
—
oo
= dim range P<,
since otherwise [ak] = 0 by Proposition 14.2; hence sf{Bt}tEJ = and obviously so does index — —g P< = 0. Now set S := We show that the family {Bt}tEJ can be deformed into the family
0
{Bt'}tEI := {(1 — t)S + t(9'Sg)}LEJ
without changing the spectral flow. Since Ella (the space of all elliptic self-adjoint operators of the same order — here of order zero — acting on sections of a fixed Hermitian bundle and having the same principal symbol a) is convex, we may assume that {Bt}tEJ already has the form
tEl. onto the straight Note that the retraction of the original path line connecting B0 and B1 does not change the spectral flow since it is a homotopy invariant by Proposition 17.6. By Proposition 14.2, we get for the principal symbol of S
as =
= a;
hence the operator S belongs to the same space E1L, as B. Let {C8}aEJ be the straight line connecting C0 := B0 with C1 Then the parallel move := (1— t)C3 +t(g—'C39),
S.
s,t El
defines a homotopy between the families {Bt}tEI and (and := its invisible closings and Thus we have = sf{Bt}tEI. have the same principal symbol, their difference Since and is an operator of order —1 and hence is compact. This proves that — B() = g'Sg — S = g'(Sg — gS). Sg — gS E K(L2(Y;E)), since Straightforward calculation gives, for H := L2(Y; E),
ker(P —gP<) = {u E H
I
u
= gP< u}
IL Analytical and Topological Tools
156
arid
coker(P —9P<) = {w E P H I gw E P< H}, hence
index(P> —9P<) = dim{w E P< H I 9W E
P H}
—dim{WEP>HI9WEP
=tr(g'P> determine the discrete spectrum of = (1 — t)S + tg'Sg by the point set {—1, 1, 1 — 2t, 2t — I} from the elementary calculation We
v
ifvEPHandgv€P>H (2t—1)v ifvEP>HandgvEP>H. —v
This gives a graph of the spectrum of the family
as shown in
Figure 17.4, where P0, /.L1 parametrize the eigenvalues 2t —1 and 1— 2t with multiplicity m0 and m1; hence = m0—m1. As seen above,
we have
m0 = dim{v E P< H I
E P> H) = dim ker(P> —gP<)
and
= dim{v E P> H I gv e P< H) = dimcoker(P —gP<) which proves the theorem.
[]
17. The Spectral Flow
157
t
Fig. 17.4 Spectrum parametrization of the family
17D. Non-Vanishing Spectral Flow At the end of this chapter about the spectral flow, we present an argument for non-vanishing spectral flow given by physicists. It shows that any solution s = s(u, y) of the equation on
)(
Y
gives rise to an eigenvalue A,(u) of which changes its sign as u goes from 0 to 1. Here is a family of seif-adjoint elliptic differential operators of first order acting on sections of a Hermitian bundle E with unitary equivalence B1 = gBog' for a suitable (nontrivial) choice of g. It is not difficult to fill the missing details into the chain of arguments below and to obtain the equality in this way. We leave that as an exercise for the reader, who is sufficiently familiar with the needed material involving the Atiyah-Patodi-Singer problem (see Chapter 22 below). such that The argument is adiabatic. Take a family B for u close to i = 0, 1. Assume that B, is invertible (we may replace by + e, where e > 0 is a small real number which
does not belong to the spectrum of B). Then extend this family to
II. Analytical and Thpological Tools
158
the whole real line
(Bo foruE(—oo,O) on[O,1]
(B1 foru1 on R x Y. A priori it is not and consider the operator A = + quite obvious that this operator is Fredhoim, but actually it is and its index is given by the L2-index formula of Corollary 23.9 below, which in our case gives: L2-index(A)
=
I
a(x) —
1
1
'7B_00(O) +
Rx Y
= IRxY
'1B+00(O)
a(x)
and Ba,. Here ü(x) debecause of the unitary equivalence of notes the index density which has finite integral. BR,0 I
B
Ru
U
i
O
U
Fig. 17.5 Adiabatic limit of the original family
Now we blow the normal variable u Ru and obtain a corresponding family := and an operator :
+
We are looking for solutions of AR(S) =
s(u,y) =
0
of the form
17. The Spectral Flow
159
corresponding to
where is a normalized eigensection of an elgenvalue of BU/R.
In reality such solutions do not exist; but when R
oo
we can
solve (17.3)
AR(S) = 0
modulo small errors. Equation (17.3) is equivalent to
= hence 1 ThPi,u/R U
+
=
The second summand on the left side is of size
We neglect this
term and are left with the equation
= or
= with a solution
= exp {—
hence
s(u,y) =
exp {—
/
j
dr},
AI,U,R(r)dr} coi,u/R(Y).
Now s must belong to V
+00>
= f:exP{_2
/
du.
For u negative we have and this = — fu° term must be positive for large lul. Hence the L2-norm of s is finite, if and only if S i.
positive foru>>0 negative for —u>> 0.
160
II. Analytical and Thpological Tools
This shows the following proposition.
Proposition 17.19. An L2-solution of the operator + (for large values of R) exists, if and only if there is a family of eigenvalues of which change the sign from — to + as u goes from —oo to +00. Similarly the solution of the adjoint operator + corresponds to a family of eigenvalues which change sign from + to —.
PART III
Applications
18. Elliptic Boundary Problems and Pseudo-Differential Projections
We explain what it means to pose global elliptic boundary problems and how pseudo-differential projections appear naturally in this context.
Now we can start the discussion of boundary problems for firstorder differential operators, say for a Dirac operator
A:
C°°(X;S)
a manifold X with boundary Y. (Here and in the following we A has a write A instead of closed range of finite codimension, but an infinite-dimensional kernel, which varies depending on the choice of the Sobolev space. Hence, over
to get regularity and a finite index, we have to restrict the class of admissible sections. The natural way of doing this is to just ignore almost all solutions of A.
Definition 18.1. Let R: C°°(Y;
—p C°°(Y; V) be a pseudodifferential operator of order 0. We call R a (global) elliptic boundary condition for A if the following conditions are satisfied: H8(Y; V) (1) For any real s, the extension H8(Y; of R has a closed range. :
(2) Let r denote the principal symbol of R. Then range(r) = range(p+). Hence the restriction range(r) is an isomorphism of vector bundles.
Recall that p.,.(y, () : —' for (y, () T*(Y) \ 0 denotes the principal symbol of the Calderón projector P+ (A) which is the orthogonal projection of onto the direct sum of the eigenspaces of the symbol b(y, corresponding to the eigenvalues with positive real part, see Chapter 12. As before, we assume that A is written in the form close to Y with unitary C and self-adjoint tangential operator B over Y with principal symbol b.
III. Applications
164
Remarks 18.2. (a) Definition 18.1 covers global Atiyah-PatodiSinger conditions as well as standard local elliptic conditions. We get the Atiyah-Patodi-Singer boundary condition when V := and R := P (B) the spectral projection of B for the interval [0, +oo). We
have seen (in Proposition 14.2) that P(B) is a pseudo-differential operator of order 0 like the Calderón projector and with the same principal symbol. The range of the extension
is the closure (B)8 of the space spanned by the eigensections of B corresponding to the non-negative eigenvalues, and the principal symbol of P> (B) restricted to range(p÷) is the identity. (b) For mathematical reasons and for the sake of applications, we define a subclass of the class of all elliptic boundary conditions which contains the original Atiyah-Patodi-Singer condition as well as compact perturbations of this condition. This subclass is distinguished We shall call such by requiring that R is a projection with r = boundary conditions generalized Atiyah-Patodi-Singer boundary conditions.
(c) We get a local elliptic condition when the range of r can be written as the lifting of the vector bundle V under the natural projection TY \ 0 —' Y. Then condition (1) of Definition 18.1 is automatically fulfilled. We shall discuss this in greater detail below (cf. Theorem 20.13 and Theorem 21.5). Note that on odd-dimensional manifolds, the decomposition of the spinors over the (even-dimensional) boundary defines a local elliptic boundary condition. We shall exploit this below in Chapter 21 to prove the cobordism theorem for Dirac operators. On even-dimensional manifolds we have a different situation, since for all Dirac operators there are topological obstructions which prevent the admission of local elliptic boundary conditions (see e.g. Boofi & l3leecker [1985, Section II.7.B] or Lawson & Michelsohn [1989, Section 11.6]). Nevertheless, systems of Dirac operators may admit lo-
cal elliptic boundary problems. This is always the case for systems associated with transmission problems, see especially Chapter 25 below.
(d) If, as in Remark 12.5b, we identify the space of initial values with the function space
= {w : R
(0, y;
=0
=
()
18. EBP's and Pseudo-Differential Projections
165
we can reformulate condition (2) of Definition 18.1 in the following way:
(2a) The equation r(y;
=
0
has no non-trivial solutions in
Thismeans
that
injective, and boundary conditions satisfying this condition are sometimes called injectively elliptic. is
(2b) For any 8
r(y, c(S), the equation r(y;
E
This means that
a solution in
=
8 has
:
ranger(y, () is surjective.
moreover, ranger(y, () = V,, then the pair (A, R) is sometimes called swjectively elliptic. Boundary conditions, which are both injectively and surjectively elliptic, we call local elliptic condiIf,
tions, whereas we require less for (global) elliptic boundary conditions, namely, that they only be injectively elliptic (2a) and that they satisfy (2b) and the technical condition (1).
What then is the meaning and strength of the concept of global elliptic boundary conditions? Such boundary problems present no serious analytical problems, but its index theory requires a different emphasis, namely somewhere between the broader class of injectively elliptic boundary problems (leading only to semi-Fredholm operators) and the smaller class of local elliptic conditions, where it suffices to
concentrate on the symbolic calculus with its approximative statements; in our situation of global elliptic boundary problems, as defined above, small changes of the boundary operator R without change
of its principal symbol r can lead to jumps of the index (see below Remark 22.25). Our main interests are the relations between the various expressions of a boundary problem, namely
• the full mapping pair (18.1)
(A,R):
given by the formula
(A,R)(u) := (Au,R(uly)); • the realization of A (18.2)
AR: domAR —4
as an unbounded operator in L2(X; domAR := {u
L2(X;S) with I
R(uIy)
IlL Applications
166
here R(uIy) =
0
means that u belongs to the kernel of the composite
continuous map
H1(X;S1)
L2(Y;V),
where W denotes the first Sobolev space of sections of the spinor bundle over X, '10 denotes the restriction to the boundary, and R0 is the well-defined continuous extension of R to H°(Y; = L2(Y; Note that '10 is even continuous H'(X) H'/2(Y) (Theorem 11.4 and Corollary 11.8); • and the boundary integral (18.3)
: H÷ (A) = range P÷
rangeR.
We have several questions:
(1) Under which restrictions on the operator R does the realization AR become a Fredhoim operator with smooth solutions and smooth solutions of the adjoint operator (AR)*? (2) Under which additional conditions for the operator R does the full mapping pair (A, R) become a Fredhoim operator with smooth solutions and smooth solutions of the corresponding adjoint problem? The question is whether the boundary conditions R, which provide a nice operator AR, are the same which give a nice operator (A, R). (3) Under which conditions does the boundary integral become a Fredhoim operator, and what about the stability of its index (which was called the virtual codimension i(R, P÷) of R in and discussed in Chapter 15 above)? (4) How can the general (global) and the conventional (local type) elliptic boundary problems be reformulated in the language of projections (pseudo-differential Grassmannians)? One of the reasons why we can answer these questions is that the projections, induced from the boundary condition R, are pseudodifferential operators, as we shall see now.
Definition 18.3. Let R: C°°(Y;
—, C°°(Y; V) be a pseudodifferential operator of order 0 such that the extension R(8)
: H8(Y; S1y)
H8(Y; V)
of R has a closed range for any real s. Then NR, resp. 'R shall denote the L2-orthogonal projections onto the kernel (Null space), resp. the range (Image) of R defined on resp. L2(Y; V).
18. EBP's and Pseudo-Differential Projections
167
When appropriate, we shall denote the restrictions of the resp. C°°(Y; V) by the same symbol. projections onto C°°(Y; Note.
Clearly we have
Lemma 18.4. With the preceding notation, the projection onto cokerR kerR can be written as HR• = Id 'R• The main purpose of this chapter is to prove the following result of Seeley [1968; Theorem IV.7 and Theorem VL6]:
Theorem 18.5. Let R be a (global) elliptic boundary condition in the sense of Definition 18.1. Then the L2-orthogonal projections HR, and HR. onto the kernel, the range, and the cokernel of R are pseudodifferential operators. The proof of Theorem 18.5 is divided into two parts. First we use condition (1) of Definition 18.1 to show that 0 is an isolated point of the spectrum of the seif-adjoint operator R*R (and of the principal
symbol of this operator as well). Then we apply the path integral representation from the proof of Proposition 14.2 to show that the orthogonal projection HR onto kerR is a pseudo-differential operator. We begin with some elementary functional analysis which is part of operator algebra folklore, but hard to find in written form in one place.
PropositIon 18.6. Let T: H1 —'
be a bounded operator acting between two Hubert spaces H1, H2. The following conditions are H2
equivalent:
is closed
(1)
range(T)
(2)
range(T) = (kerT*)i
(3) (4)
range(Tr) is closed
(5)
range(T*) is closed range(T*) =
(6)
range(T*T) is closed.
Proof. Since rangeT is dense in (1)
(2)
(3)
and
we (4)
clearly
(5)
have (6).
III. Applications
168
Now it only remains to prove the equivalence (3) (4). It follows from Proposition 9.4 that range TT is closed if and only if the self* adjoint operator I (ker T=ker
TT
has a bounded inverse, i.e. if and only if there exists a positive real number c1 such that I'r'T's V,V
(v,v)
for all v E (kerT*)
c1
or equivalently, (18.4)
ciIIvjI2 < (TTSV v) = IITvII2.
has a bounded inverse
The estimate (18.4) means that
which, once again by Proposition 9.4, is equivalent to closed range T*. U
be a bounded operator of closed Corollary 18.7. Let T: H1 range. Then 0 is an isolated point of the spectrum of the operator
T'T. Proof. It follows from Proposition 18.6.(6) that TT has a closed range and from Proposition 9.4 that
>6 for some e > 0.
U
Now we can prove Theorem 18.5: The results proved up to now
show that there exists 7> 0 such that the operator R*R — 'y is an invertible (elliptic) operator with kerR*R = and spec(R*R —'y) fl (—oo, 27) C {—'y}, where Va denotes the eigenspace of RR —7 corresponding to the real eigenvalue a.
Next we essentially repeat the argument given in the proof of Proposition 14.2. The projection HR onto kerR = ker RR is given by the path integral (18.5)
HR =
2irz
JA=ir
—
A)' dA.
To show that HR is a pseudo-differential operator we must compute the full symbol of HR. The only difficulty is to show that the principal This follows symbol of RR has no eigenvalues on the circle IAI = from the next proposition.
18. EBP's and Pseudo-Differentia' Projections
Proposition 18.8. Let T: C°°(Y; V)
169
C°°(Y; V) be a pseudo-
differential operator of order 0, acting on sections of a Hennitian vector bundle over a closed Riemannian manifold Y, and let 0T denote the principal symbol of T. Then the union (18.6)
spec(o'T)
spec ('T(y,()
U
(v,C)ET Y\0
of the symbol spectra, taken for all non-vanishing cotangent vectors of Y, is contained in the spectrum spec(T) of T.
Proof. Let Yo
Y, (e
Hence there is VO E
0, and let A E spec(or(yo, ()).
with (vo; vo) =
1
and Ày0 — UT(yo, C)vo = 0.
Extend v0 to a section v with (v; v) = 1 in a neighbourhood of Yo 0 for all y. and extend ( to a vector field e E C°°(Y; TY) with Let 0e,yo denote an approximation of the by smooth functions. From the expansion of the total symbol of T we obtain (18.7)
IIT(9e,yov)
0
—
for e
0.
By continuity we get A E spec(T).
The picture of the spectral correspondence between operator and principal symbol can be made even more clear.
Theorem 18.9. Let T be a self-adjoint pseudo-differential operator of order 0. Then spec(T) \spec(clT) consists of isolated points of finite multiplicity.
As for elliptic T, i.e. 0 spec(cYT), this generalizes the fundamental observation that its spectrum consists of isolated eigenNote.
values of finite multiplicity.
Proof. Let A0 E spec(T) \ spec(CT). We can assume that (18.8)
IAoI
>
I
yE Y and (E
Then the symbol A0 id —o-T(y, () is invertible and A0 Id —T is an elliptic pseudo-differential operator of order 0. Now either the Fredhoim operator Ao Id —T is invertible (which is impossible since A0 E spec(T)) or it has a non-trivial null space ker(Ao Id —T) of finite dimension, i.e. A0 is elgenvalue of T of finite multiplicity. We show that it is isolated. Let .M) denote the orthogonal projection onto
III. Applications
170
ker Ao Id —T. Then the operator A0 Id —T — M
is a seif-adjoint Fred-
hoim operator which is injective, hence surjective; and there must exist a whole real neighbourhood IA — Aol <6 with
0<6<
EY
and (E T%}
is invertible. If 0 < IA — Aol <6, it follows such that AId—T — that A spec(T). For if Ày — Tv = 0 for such a A A0, then the section v is orthogonal to the range of .Mj and so Av — Tv — N0v =0; hence v = 0 and A Id —T is in fact invertible. I]
Corollary 18.10. Let R be a pseudo-differential operator of order 0 over a closed Riemannian manifold Y defining an elliptic boundary condition. Let r denote the principal symbol of R. Then there exists T*Y, the spectrum of an interval [—Co, eoj such that for any (y, (r*r)(y,C) intersects [—eO,e0] only at 0.
Proof. Choose (18.9)
> 0 so that spec(RR) fl [—eo,eo] c {0}.
Then apply Proposition 18.8. Now it is obvious that formula (18.10)
NR:=_!_f 27r2
A=EO
the orthogonal projection onto the kernel of R. The argument from the proof of Proposition 14.2 furthermore shows that HR is a pseudo-differential operator of order 0 with principal symbol p(y, () which is the orthogonal projection onto the kernel of the principal symbol r(y, C) of R. That proves the first part of the theorem. To prove the rest of the theorem, recall from Lemma 18.4 that defines
= Id —HR defines the orthogonal projection onto the range of the Now we can repeat our considerations with the operator operator R replaced by R. This ends the proof of Theorem 18.5. Note. To prove the statement of Theorem 18.5 we only used assumption (1) from Definition 18.1. Assumption (2) has various specific
additional effects. It guarantees the existence of a suitable elliptic fan which implies the solutions of the boundary problem to be regular and brings about the finite dimension of the kernel. It also brings about
18. EBP's and Pseudo-Differential Projections
171
the finite codimension of the range of AR. This will be explained below.
The next proposition is a consequence of Theorem 18.5. It shows that each realization of a Dirac operator with arbitrary global elliptic boundary condition can be written as a realization with boundary condition defined by a projection. So, at least from the point of view of abstract theory, we can restrict ourselves to boundary conditions given by pseudo-differential projections with symbols satisfying condition (2) of Definition 18.1:
Proposition 18.11.
Let R be an elliptic boundary condition for a Dirac operator A and let 'R denote the orthogonal projection onto
the range of R, i.e. Id
equals the orthogonal projection APR onto the kernel of R. Then 'R• defines an elliptic boundary condition for A and we have (18.11)
AR=AIR.
Proof. It is easy to check conditions (1) and (2) of Definition 18.1 for the new boundary condition 'R•• Actually the range of a projection in a Hilbert space is always closed, hence also the range of the extensions on the Hilbertizable spaces H8(Y; V). Assumption (2) on the symbol follows directly from the definition of the projection IR' Then the equality (18.11) of the realizations is a straightforward corollary of Theorem 18.5. 0
Remarks 18.12. (a) Note that the principal symbol of the projection of the 'R' does not necessarily coincide with the principal symbol Calderón projector However, for any arbitrary elliptic boundary condition R, the principal symbol of the orthogonal projection 1p4. R' is equal to p+. Before proving that, we sketch an example (to be further elaborated in Chapter 21): Consider a Dirac operator which decomposes near the boundary into the form
'k)) with regard to a splitting g.f S y of the given Clifford bundle S over the boundary Y, and assume that the eigenvalues of the principal symbol
c(y,()
=
(0
C—
0
III. Applications
172
restricted to the cotangent sphere bundle are all contained in the set (1, —1} (e.g. for c_
= (ci) = (c4'). Then we get c.... 1
Consider the elliptic boundary condition defined by the projection onto the first factor
g) which clearly has a principal symbol r
of symbols we have
On the level
=
0)
and
hence the range of the principal symbol of RP÷ is the whole first factor
and so coincides with ranger. Therefore the orthogonal projections have the same principal onto the range of R and onto the range of symbol; and we shall see in Proposition 18.14a below that this always holds. Moreover
=
(
\\_i( —
E .. } .
s++s+ +
Hence the principal symbol of the projection R• onto the range of is in our example equal to and, not in that example alone, but always, as we shall show in Proposition 18.14b below. (b) So far we distinguished four ways of posing elliptic boundary problems for Dirac operators. We conclude:
(i) Global elliptic boundary conditions are defined by pseudodifferential operators of order 0 satisfying the assumptions of Definition 18.1. From the point of view of operator theory, they are the natural extensions of the restricted classical concept of local elliptic boundary conditions.
18. EBP's and Pseudo.Differential Projections
173
(ii) Elliptic boundary conditions defined via pseudo-differential orthogonal projections generate the same subset of elliptic boundary problem realizations in the space of unbounded operators in L2(Y; V) as in the first case; but calculations are much easier to carry out. (iii) Our main interest is devoted to a more narrow class of elliptic boundary conditions, namely those defined by pseudodifferential orthogonal projections with the same principal symbol as the Calderón projector. (In the terminology explained in Chapter 15, that means one chooses a boundary
condition in the Grassmannian of P4. We shall call this type of boundary conditions generalized Atiyah-Patodi-Singer conditions. Most of our geometric motivations and applications are related to this set of boundary conditions which is never empty itself always defines a suitable global elliptic boundary condition) and in fact of a remarkably rich structure as seen in Chapter 15 above and to be exploited later on.
(iv) Finally we have the (usually much smaller or even empty) space of local elliptic boundarp conditions in the classical setting. Indeed, when the boundary of the underlying manifold is odd-dimensional, then that space is empty for Dirac operators, but not necessarily for elliptic complexes, systems, and coupling problems. It has some special features which we shall show in Chapter 20 and apply in Chapters 21, 24, and 25 below.
For the boundary integrals and for the realization of boundary value problems, we shall establish the Fredhoim properties. To do that we need a version of Theorem 18.5 for the operator RP+. It is crucial to show that the range of that operator is closed. This can be derived from the existence of a left-parametrix for the boundary problem and will be discussed in Chapter 19. Here we present another proof based on a construction (the elliptic fan) of Birman & Solomyak [1982].
Lemma 18.13. For any elliptic boundary condition R, the operator (18.12)
:= (Id —P4 +
is an elliptic seif-adjoint operator of order 0.
Proof. Since (id —p+) is a seif-adjoint transformation, its surjectivity follows from injectivity, and hence it suffices to show that
III. Applications
174
its kernel is Oat any (y,() TY \ (id —p÷ (y, ()) v
{O}. Let v e
(y, ()r* (y, c)r(y,
+
Then (y,
=
0
implies (id
v
(y,
=0
and
p÷ (y,
(y,
C)p+ (y,
)v = 0;
hence
v =p+(y,C)v
(18.13)
= 0.
and
It follows from assumption (2) of Definition 18.1 that v = 0.
)v = [1
iY is an elliptic operator over a closed (= compact and withconsists of finitely many more precisely linearly independent smooth sections of Since
out boundary) manifold, the kernel of (18.14)
= {s since
Rs =
0.
IRs = 0 and P÷s = s}
E
C
= 0 implies (R'Rs; s) = 0 for s e hence (In our notation we do not always distinguish between
pseudo-differential operators of order zero defined on C°°-spaces and their unique bounded extensions to L2.) Let K denote the orthogonal projection onto the kernel of the operator is'. As an operator of finite-dimensional range, K is a smoothing operator (i.e. a pseudo-differential operator of order -.-oo); hence the operator (18.15)
is seif-adjoint) also vanishing
has vanishing kernel and (because
cokernel. Thus it is an invertible elliptic operator of order 0. Its inverse exists and is an elliptic pseudo-differential operator of
order 0.
Proposition 18.14. (a) For any elliptic boundary condition
R:
—p
C°°(Y;V),
onto the range of the operator RP÷ is the orthogonal projection a pseudo-differential operator and is given by the formula (18.16)
18. EBP's and Pseudo-Differential Projections
175
=
+ K as in (18.15). In particular, the principal symbol of 'RV+ is equal to the principal symbol of the projection 'R onto the range of R. onto the range ofP+R* (b) Similarly, the orthogonal projection where
is a pseudo-differential operator given by
:=
(18.17)
In particular, the principal symbol of 1p. R• is equal to
Proof. First we refine the arguments of the preceding proof. With and A2 := RP+ we clearly have, not only on the symbol A1 := Id level, but also on the operator level:
A1A=0
(18.18)
and
hence
rangeA.
= kerA1 Therefore the spaces range = 0 implies and +
A
=
andalsoforw2 :=As,i=1,
=
for all w1 and to2 hence 0
AA1K=0, i=1, 2,
(18.19)
Since where K again denotes the orthogonal projection onto ker K is seif-adjoint, so is AA1K and we also obtain = 0 from (18.19). This provides the following equalities
=
+ K)
= P+R*RP+R*RP+ = M'÷RRP÷, which imply (18.20)
P+R*RP+tl_l =
defined in (18.16), is idenNow we can show that the operator tical to the orthogonal projection onto the range of the operator The oppoBy definition (18.16), its range is contained in range site inclusion comes from the following identities. Use (18.20):
=
=
+
+
(18.21)
=
=
= Im,+RP+.
III.
176
Thus
is a self-adjoint pseudo-differential operator of order 0 with
range IRP+ = range RP+, and
it must only be shown that TRP÷ is a projection. Use equation
(18.21):
= (18.22)
=
=
=
It follows that
the principal symbol of the projection 'RI,.1., is the orthogonal projection onto the range of r(y, where r denotes the principal symbol of the operator R. Hence q÷ = q, where q denotes the principal symbol of the orthogonal projection 'R onto the range
of R. This ends the proof of (a). A calculation similar to the one presented above leads to the corresponding statement (b), namely (18.23)
=
P+R* =
which shows
range
R C range 'P1. R
(the opposite inclusion is again obvious from rewriting with (18.20)) and -r
.LP÷R.
2
'TI
=
T)* \
A
R=
='-P1.R'
From the definition and from (18.23), we finally get for the principal symbol of 'P-1. R•
(18.24)
+
=
id on S
onkerp+.
Whereas the space range P+R* may be difficult to understand, it is obvious that range p+r is the complement of cokerp+r = ker r is injective on This proves that the projection onto rangep+r*, is in fact equal to U The following corollary follows immediately from the preceding proposition. It establishes the Fredholm properties of the boundary integral and constitutes one of the main results of this chapter and of the elementary analysis of (global) elliptic boundary problems for Dirac operators on the whole:
18. EBP's and Pseudo-Differential Projections
Corollary 18.15. Let
denote the Calderón projector of a Dirac operator A over a manifold X with boundary Y and let R be an elliptic boundary condition. Then the operator
RP+ : H+ —' range R is a
denotes the space of
operator, where H+ = Cauchy data of A along Y.
is Proof. It follows from (18.14) that the kernel of the operator equal to the kernel of the operator which is an elliptic operator over a closed manifold. Hence ker RP÷ is finite-dimensional and contains only smooth sections. Moreover, by Proposition 18.14a, the principal symbols of TRP+ is compact. Therefore and of 'R are equal, hence — is a subspace of finite codimension in range R and closed since rangeR is closed. [I
The linkage between the index theory of elliptic boundary prob-
lems and the theory of pseudo-differential Grassmannians is provided by the following proposition, a reformulation of the results of the proceding Proposition 18.14 and Corollary 18.15.
Proposition 18.16. Let R:
Say) —, C°°(Y; V) be an elliptic
boundary condition for a (total or chiral) Dirac operator A over a denote the principal compact manifold X with boundary Y. Let symbol of the Calderón projector P÷ onto the space H+(A) of the Cauchy data of A. (a) We have Grq,
and
E
where q denotes the principal symbol of the orthogonal projection IR onto the range of R. (b) The operator
T :=
: H+(A)
range(R)
is a &edholm operator and (18.25)
indexT= i(R,P÷) dim kerT
—
dim coker T
III. Applications
178
Proof. All results, except the formula (18.25), were obtained in Proposition 18.14 and Corollary 18.15. The index formula is a consequence of the following decomposition of the operator T:
= index of this is
R
range(R). D
Remarks 18.17. (a) It follows from (18.14) that the kernel of which is an elliptic operator is equal to the kernel of the operator is finite-dimensional over the closed manifold Y. Hence and contains only smooth sections of the bundle 5+ ly. Later on (from Proposition 20.3 and Theorem 20.8) we shall deduce that also is of finite codimension in range(R) and even of finite codimension in C°°(Y; V) in the local elliptic case (see Lemma 20.11). (b) Proposition 18.11 implies that there are four different boundary integrals
1R P+, fl
and
'R P
naturally associated with every global elliptic boundary condition R. depends on the Dirac operaNote that the Calderón projector tor A over the whole manifold with boundary, whereas the spectral projection P depends solely on the tangential part B of A over the boundary. From Corollary 18.15, one can derive that all these boundary integrals are Fredholm operators in the respective spaces. We shall see later on that some of their indices coincide and others may by differ. More precisely, the index may change when replacing but not when replacing R by the projection 'R' onto the range of R*.
(c) It is worth noticing that there exists a left inverse of the operator RP÷ modulo operators of lower order, or, put differently: for any global elliptic boundary condition
R: C00(Y;S+Iy) —' there exists a pseudo-differential operator
T: C°°(Y; V) —i C°°(Y;
18. EBP's and Pseudo-Differential Projections
179
is a smoothing operator. This is a direct consesuch that TRP+ — with quence of Proposition 18.14. Take for T the operator as defined in (18.15). Then the operator TRP÷ equals the orthogonal projection 'P+R' onto the range of P+R* which has principal symbol i'+. Moreover, if the boundary condition R is a projection, then we obtain that also R is a smoothing operator for T, defined as above. This also follows from Proposition 18.14, since
and the principal symbols of 'RP+ and IR = R coincide.
As in the theory of elliptic operators over closed manifolds, the existence of a left-parainetrix provides a recursive argument for the regularity of solutions also in the case of elliptic boundary conditions. Alternatively, the regularity of the solutions of elliptic boundary problems can be derived from the regularity of elliptic problems over closed manifolds by reduction to the boundary. Both approaches will be explained in the following chapter.
19. Regularity of Solutions of Elliptic Boundary Problems
We derive norm estimates for boundary integrals and show the solutions of elliptic boundary problems for Dirac operators to have global regularity and all realizations to have a closed range.
The preceding chapter established some of the true meaning of the ellipticity condition of boundary problems in Definition 18.1, namely that given a Dirac operator A : C°°(X; —' C°°(X; S—) over a compact manifold X with boundary Y and given an elliptic boundary condition R: C°°(Y; —, C°°(Y; V), then —* range(R) is a Fredhoim operator, (1) : (2) RP+ has a left inverse T modulo smoothing operators, (3) and any realization AR can be rewritten as AIR., where Id — 'R• denotes the orthogonal projection HR onto the kernel of R.
As before we suppress for typographical reasons the superscript + the naming of the Dirac operator.
Before exploiting these results for the index theory of elliptic boundary problems, we shall discuss the other part of the meaning of the ellipticity condition: the regularity of the solutions of elliptic boundary problems. One major result of the theory of elliptic operators over closed manifolds is that the kernel only contains smooth solutions. However, the situation is radically different on manifolds with boundary, where the solution spaces ker÷(A, s) C H'(X; may depend on s, see Chapter 12 above. Nevertheless, the solutions of homogeneous elliptic boundary problems are still smooth.
Theorem 19.1. Let v E H8(X; problem (A, R), i.e. Av = section of S+ over X.
Proof. We have Av =
0
0
and
be a solution of the boundary = 0. Then v is a smooth
and R(-yov) = 0 which shows that h :=
is an element of ker
and
ker((Id —Pt) +
19. Regularity of Solutions
181
This last operator is an elliptic operator (see Lemma 18.13), hence h
is a smooth section of
Iv and
v=
(h) is a smooth section of
overX.
U
For later use notice:
Corollary 19.2.
The various null spaces of an elliptic boundary prob-
lem R for a Dirac operator A can be naturally identified:
kerAR = ler
Proof. Let
(j4)
—,
:
range(R)}.
denote the operator A with the domain
R(vly) = 0)
:= {v E
(19.1)
for any real 8 >
Here
R(vly) =
0
means that v belongs to the
kernel of the composite continuous map
the exwhere 'Yo denotes the restriction to the boundary and (We constantly write AR instead of tension of R to Clearly
\
/ ker
is independent of s and in particular
IA
(19.2)
ker
vE
the Cauchy data space 14(A) C = 0. This so 7'+(vly) = vly and RP÷(vly) = proves that the trace operator
III. Applications
182
maps ker AR into ker RP..,.IH+. Because the boundary data determine
the solution of Av = 0 uniquely (Theorem 12.4a which gives s)
s)), Yo
(19.3)
provides an isomorphism
ker(RP+IH+).
kerAR
It 25 a little more difficult to prove a general regularity theorem for inhomogeneous elliptic boundary problems. We choose the following V), and formulation: For given f E H'(X;S_), g E find a solution of
JAy
=1 1R(i'ov) =g.
Assume that v Ht is a solution of that problem. Notice that, by is well-defined. Then one wants Theorem 13.8, the trace y0v E Ht to show that actually v is an element of the space (X; Si). There are different ways of proving that result. We choose one which uses the existence of a left-parametrix for the operator
f
C°°(X;S)
A \
I
C°°(Y;V)
We begin with a lemma.
Lemma 19.3. Let R : C°°(Y;
C°°(Y; V) be an elliptic boundary condition for a Dirac operator A C°°(X; C°°(X; S—). Then the pseudo-differential operator :
(19.4)
f
R
\
—4
(
\Id—P÷J has
an injective principal symbol and
there exists an operator
C°°(Y; V)
(19.5)
—, C00(Y;S+)
Q=
C°°(Y; Sly) which is a left-parametrtx for the operator (Id (19.6)
IR\/
Qo1
\LU—r.f
j
=Id+k,
i.e.
19. Regularity of Solutionn
where k is a smoothing operator.
Proof. The principal symbol (id viously injective for any (y, C) E the fibre into ()
C)) of (Id
T'Y \ {O}, since one can decompose
range(id —p+(y, C)). Then r(y, () an injection on the first factor because of the regularity condition (2) of Definition 18.1, and id () is an injection on the second is
factor. Choose a pseudo-differential operator Q = (Qi, Q2) of order 0 with
I
principal symbol (qi, q2) providing a left inverse of (
j. This
V
implies (19.6) by the standard argument.
Corollary 19.4. For any real s, the extension (19.7)
\
/
H8(Y; V)
I
I
of the operator (Id
has
finite-dimensional kernel and closed
range as well as the operator (19.8)
:
—p C°°(Y; V)
and its extensions.
Proof. For convenience we write the operator in (19.7) as T: H1 —. "2 and its left-parainetrix T: H2 H1 so that TT — Id is compact. Then we show that T has closed range by applying the following standard chain of arguments. Clearly ker T C ker TT. Hence it follows from TT — Id compact that dim ker T dim ker TT < oo. Since (TT — Id) = T*TS — Id
is likewise compact, it follows from range(T) J range(T'T') that dimcokerTT < 00, so range(T) is closed, hence dimcokerT also range(T) is closed, e.g. by Proposition 18.6. Now it is immediate that
(19.9)
III. Applications
184
Moreover, also the range of the corresponding extension of RP+ is closed, since it can be written as (19.10)
=
range
E range
i
v
= o}.
Corollary 19.4 gives another proof of the existence of a pseudo-differential projection onto the range of the operator RP÷ (earlier established in Proposition 18.14a). Note.
it is a remarkable fact that any left-paraznetrix for the boundary integral
provides an explicit left-parametrix for the whole
problem
(To ease the reading we suppress the superscripts
(a)
of the various operator extensions when the context excludes ambiguity).
Theorem 19.5. LetQ = (Qi, Q2) be a left -paramet rix of the operator
I
Id
R\
Then the operator
S :=
(19.11)
K+Qi)
is a left-parametrtx for the operator
A ( R70
Proof. Check
SotIA R'yo
\
=
K+Q1) °
= (Id Lemma 12.7
—24'yo)
+ K÷Q1Ryo
+
Id
—K÷yo)
(19.12)
= Id (19.6) and
=
=
+ K÷ (Q2(Id —7'.,.) + Q1R) yo — K÷Q2(Id
Id
+ K+(Id +k)'yo — K+Q2(Id —'P÷)P÷'yo
19. Regularity of Solutions
185
eontrnuily for lb0 vai4ous maps
that for 3>
f
(19.13)
A \ I
(
\R'yoJ while
S)
8:
(19.14)
V)
Now we derive a lifting jack for regularity (see
Theorem 19.6. Let A be a Dirac operator over a compact (evendimensional) manifold X with boundary Y and let R be an elliptic be given, and assume boundary condition for A. Let v E Ht(X; for given s, t real, that Av and R(vly) E
and s> C=
Then v belongs to H3+l(X; S+) and we have a constant
C(s, t)
independent of v such that
(19.15)
IIt'IIs+i
Proof. Assume
+ IItIIt +
1144t'118)
s + 1. Then
(19.16)
The element
is a smooth section of S+; hence
v =8 (Rht)v —
(19.17)
is an element of Ha+l. V E
vly
Ht(X;S+)
"
E R4,
Av E H8(X;Sj Fig.
R(vly) E
(Y;
v E
19.1 Lifting jack by Theorem 19.6 (s > — 1/2 and t arbitrary)
III. Applications
186
Corollary 19.7. Let v E L2(X; St), Av E L2(X; S), and R7o(v) = 0. Then v belongs to H'(X; Si).
Remark 19.8. On the contrary, ( A does not in general have R'yo j \ a right-parametrix. Further details on left and right ellipticity and pararnetrices can be found in Grubb [1977], from which the proof of Theorem 19.5 is taken. Fortunately we only need Lemma 18.13 and Proposition 18.14 to show that the kernel and cokernel of the operators AR and
A
\ R'yoj
consist of smooth sections.
We shall close this sec)ion A
range of the operators
a discussion of the closedness of the
and AR =
That the respective
\ R'yo j ranges are closed is a consequence of the following elementary lemma. Lemma 19.9. Let T: H1
H2 denote a bounded operator between Hubert spaces H1 and Assume that T has a left-parametrix, i.e. there exists a bounded operator 5: H2 —, H1 such that
ST=IdH1-f-k, where k: H1 —p H1 is a compact operator. Then T has closed range.
Proof. According to Proposition 9.4, it must be shown that is invertible (i.e. the inverse is bounded). This is equivalent to the existence of a constant c> 0 such that (19.18)
IIfIIH,
cIITfIIH3
for any f E
Assume that such a constant does not exist. Then there exists a such that sequence {f, } c (ker (19.19)
= 1 and IITf,II <
k is compact, the sequence {kf3 } has a convergent subsequence and so also does the sequence {f, = STIJ — kf,}, which differs from { — kf, } only by the term STI, which converges to 0. Let fo denote the limit of this subsequence. Then fo is an element of (kerS)-'- with Il/oil = 1. On the other hand Tf0 = 0; hence fo has to be equal 0 which is a contradiction to (19.19). Hence the lemma is Since
proved.
Applying Lemma 19.3 and Theorem 19.5 yields at once:
[]
19. Regularity of Solutions
Corollary 19.10. For any s>
/
the
range of the operator
A \
I
\R70J is closed.
A direct consequence is the following proposition:
Proposition 19.11. The range of the operator AR Proof.
is closed.
be a sequence of elements of dom AR such that
Let
—* yE
L2(X;S).
= 0 for any n, hence
By definition
{(Axn)} is a sequence of elements belonging to the range of the operator
/ I
A
\
I:H'(X;Si—'
L2(X;S)
L2(Y;V)
\Ryo/
Then by Corollary 19.10 there exists w E H'(X;
(
Aw
'\_(y\ —
which ends the proof of the proposition.
such that
20. Fredhoim Property of the Operator AR
We
obtain the closedness of the realization AR, an explicit for-
mula for the adjoint boundary value problem, the standard Fredhoim properties of AR, and various relations between its index and the indices of the related mapping pair and boundary integral in the general (global elliptic) case as well as in the local elliptic case.
To keep the treatment transparent we use chiral notation and write
for A in this chapter, whether we deal with either the total or chiral Dirac operator. Recall the concept of the realization 4 of a A
Dirac operator C°°(X;S)
a smooth manifold X of even dimension with boundary Y under an elliptic boundary condition over
R:
—.C°°(Y;V)
and the major results obtained in Chapters 18 and 19: 1. By APR, we denote the operator with the domain (20.1)
:= {u e
IR(uly) = 0).
Here R(uly) = 0 means that u belongs to the kernel of the composite continuous map
where Y0 denotes the restriction to the boundary, and
is the well-
defined continuous extension of R to H°(Y; = L2(Y; Note that Yo is even continuous H'(X) H'/2(Y) (Theorem 11.4 and Corollary 11.8). Whether we call the operator unbounded in L2, the realization of A+ or of R, depends on our focus. 2. In the same way the operator is defined with the domain (20.2)
:= {u E
I
= 0}
20. Fredholm Property of the Operator AR
189
for any real 3> It maps dom into H8l (X; Sj and shares all important properties of the operator = 3. Proposition 18.11 showed that the orthogonal projection 'R• of C°°(Y; onto the range of R* is a pseudo-differential operator and defines an elliptic boundary condition for (20.3)
with
=
Note that Id — 'R' is the orthogonal projection of C°° (Y;
I
onto
the kernel of R. 4. In Corollary 18.15 we have seen that the operator : range
(20.4)
—'
is a Fredhoim operator. Here
tor of
rangeR
denotes the Calderón projec-
becomes the Cauchy data space so that range Roughly speaking, this is one half of the meaning of an
elliptic boundary condition. 5. In Theorem 19.1, resp. Corollary 19.2, we achieved that the kernel of the pair (20.5) (A+(t),R(t
))
:
Ht_l(X;S_)
Ht(X;S+)
is well-defined for any real t and consists of smooth sections. If t> it coincides with ker 4(t) as noticed in equation (19.2) above. Again, roughly speaking, this regularity property constitutes the second half of the meaning of an elliptic boundary condition.
However, these two aspects of elliptic boundary conditions, Fredhoim property and regularity, are intimately interrelated. We saw that in the preceding chapters and shall now elaborate further. We shall show that 4 dom —' V (X; S) is a Fredhoim operator. First we prove that is closed. (Proposition 19.11 established that the range of 4 is closed, which is quite a different statement).
4
Lemma 20.1. The operator
4: dom4
—*
L2(X;S) to L2(X; S).
is a closed operator from L2(X;
S), and let
Proof. Let u E L2(X; quence of elements of dom 4 (20.6)
u in L2(X;
such
be a se-
that
and
—'
w
in L2(X; S).
III. Applications
190
We have to show that u belongs to dom 4 and that = w. We use the left -paramet rix r+A1e+ for A+ introduced in Lemma 12.7:
=
(20.7)
—
where
—'
:
Then — Um)OHI =
— tim) —
—
Um)IIHI
— Um)IIL2,
shows that := the limit v := particular a limit in L2(X; which
is convergent in H' (X; S+)
—
= ti,1 — v,,} has in and, by Theorem 12.4, =
Therefore
to
is convergent in H4(Y; Now A+v = A+u in H1, hence it — v E ker+(A+,0) and the trace and Cauchy datum (it — v) E (Ak, 0) C H (Y; y) are well-defined. This shows that exists: I
= '1o(v) —7o(u — v)
(20.8)
E
and moreover,
/EH' Ryo(u)= R
= urn u E L2
by
assumption;
A+u= A+(v+u_v)=A+v+0= A+v = lim
—
n—.oo
=
urn
n—.oo
= w E L2;
and R('yo(u)) = 0. It follows from Corollary 19.7 that u E H'(X; and
so u belongs to the domain of AR.
IJ
20. Fredholm Property of the Operator AR
Remarks 20.2. (a) In Corollary 20.5 below, we shall show that (oo) the operator is actually the L2-closure of the operator C°°(X; R) —' C°°(X; Sj, where C°°(X; R) denotes the space of u which satisfy R(u(y) 0. (b) Instead of the preceding explicit sequence argument in the proof of Lemma 20.1, we could also argue in a distributional way to prove that extends to a bounded operator from the E L2: Since whole of L2 to the distributional Sobolev space H'(X; S-), we have = w in a distributional sense. More precisely, we obtain
=
=
=
(w;so)
e C°°(X;S) and hence for all E L2(X;S). belongs to L2(X;Sj and equals w. of (c) There are various other ways to prove that the realization for all test sections So
a boundary value problem is closed. The proof of Lemma 20.1 shows, however, that it is not necessary to apply the whole machinery of a priori inequalities and subtle analysis on the collar N of the boundary Y. Whether one uses our sequential or our distributional argument, the closedness of is a direct consequence of the simple lifting jack (Corollary 19.7) provided by the Calderón projector and the related explicit potentials and parametrices.
Now we can easily find the adjoint operator of a given realization It is an especially nice feature of the theory of (global) elliptic boundary value problems for Dirac operators that its class of realizations is closed under the taking of adjoints.
Propo8ition 20.3. Let R be an elliptic boundary condition for a given Dirac operator A+. We assume that A+ takes the form + B) close to Y with unitary C. Let 'R denote the L2orthogonal projection of C°°(Y; onto the range of R; hence = Then the L2-orthogonal projection kerR = kerlR. and (20.9)
: C°°(Y;S1y) -4 C°°(Y;S1y)
Q :=
is an elliptic boundary condition for the operator (Ak) = A—. Moreover (20.10)
=
=
III. Applications
192
Remarks 20.4. Before proving Proposition 20.3, we recall the three different meanings of taking adjoints: (a) For the pseudo-differential operator
R:
(20.11)
C°°(Y;V)
of order 0 acting between sections in Hermitian bundles over a compact Riemannian manifold without boundary, we denote the formal adjoint by R*; it is the uniquely defined pseudo-differential operator of order 0 C°°(Y;V) with (20.12)
(Rf; g)L2(y;v)
= (f; R*g)L2(y;S+ly)
for all f E C°°(Y; and g E C°°(Y; V). Moreover, the continuous extension of the formal adjoint R* to a bounded operator (R*)(o) : L2(Y; V)
L2(Y; is the adjoint operator of the extension in the true Hubert space meaning. (b) The Dirac operator (20.13)
C°°(X;S)
:
is a differential operator of order 1 acting between sections in Hermitian bundles over a compact Riemannian manifold with boundary. Here the formal adjoint is a differential operator of order 1 (20.14)
(Ak) : C°°(X;S)
uniquely defined by the condition (20.15)
V)L2(x;S—) = (u;
for all u E C°°(X; v E C°°(X; S-) with supptz, suppv contained in X \ Y. We have seen in Chapter 7 that (A+)* = A— is an operator with an immediate geometric meaning like A+. Moreover, we have shown Green's formula (20.16)
(A+u; v)L2(x;g-) = (u; AV)L2(x;S+)
—
VIy)L2(y;S_Iy)
for spinors with support not necessarily contained in the interior of X. So in contrast to the situation over closed manifolds (i.e. compact
20. Fredholm Property of the Operator AR
and without boundary), where formal adjoint operators become adjoint when extended to suitable Hubert spaces, we must here impose boundary conditions to make formal adjoint operators adjoint. (c) The adjoint operator of the realization (20.17)
—,
:
L2(X;S)
will be taken in the sense of the theory of unbounded linear operators in the Hilbert space, i.e. v E if and only if there exists a such that WE
= (u;w)L3(x;s+) for all u E
(20.18)
dom is dense in L2(X; Note that condition (20.18) is equivalent to the existence of a positive
real constant C depending on v such that (20.19)
S
for
all u E dom4.
In fact, C = IIAvII since
= I(u,Av)I
(20.20) see
IIAvII IItzII,
also e.g. Reed & Simon 11972; Definitions VIII. 1.4/5].
Proof of Proposition 20.3. Let us consider a fixed v E dom the form
A
(20.21)
(—Os + B)G(y)' =
—
C(y)B'C(y)).
Then Green's formula yields, for any u E dom (20.22)
(Av;u) =
+ j(CvIv;uIY) =
The integral over Y vanishes since 'R• (uIy) vanishes, so uly (rangelR.)1; and (Id — IR.)CVIY vanishes, so G*vly E
E
From (20.22), we get (20.23) which shows that v E
= I(ti;Av)I IIAvII hull cf. (20.19) in Remark 20.4c above.
III. Applications
194
For the opposite inclusion we consider a fixed v E dom(4), so v E L2(X; S—) and Av E L2(X; St); hence it follows from Corollary 13.7 that vly is a well-defined element of (X; S). We want to show that v belongs to dom It suffices to show that (Id — 'R• )Gvly vanishes as a distribution. So for any f E C°°(Y; we determine
((Id_TR.)G*(vly);f) = (C(vly);(Id—IR.)f) = (G(vly);uly) (20.24)
for any u E C°°(X; =
with uly = (Id — = (Av;çou) —
is any arbitrarily smooth function on X with ço 1 in an open neighbourhood of Y. By a suitable choice of we can make the V-norm of arbitrarily small, see Figure 20.1. This yields the Here
vanishing of (20.25)
I(Av;
IIAvII
.
Fig. 20.1 A suitable choice of
Since the Dirac operator, as a differential operator of positive order,
is not bounded in L2, we cannot apply the same argument to prove that also the second term in the final difference of (20.24) can be made arbitrarily small. Instead we exploit that v dom(4)' which yields (20.26)
I(v;Aiwu))I
20. Fredholm Property of the Operator AR
195
From (20.24), (20.25), and (20.26) we find Q(vly) = 0, as desired. It remains to prove that Q satisfies the conditions (1) and (2) of Definition 18.1. First (1): for any real s, the Green forms C and = G' extend to invertible bounded operators between the related Sobolev spaces and H8(Y;S1y). Since and Id — IRS are projections they have closed range, and hence also the extension of Q over Hs(Y; S has closed range. Now we must check the ellipticity condition (2), that is, we have to check the boundary condition Q on the symbol level. Therefore we have to show that the principal symbol of Q (20.27)
co(Q) =
(G(Id—IR.)C1) = GoO(Id—IR.)C'
provides an isomorphism (20.28)
range o'o(Q)(y,ij)
for any y E Y and E with 0. Here S'> denotes the vector bundle over TY \ 0 generated in (y, by the eigenvectors of the principal symbol of the tangential part _GB*G_l of (20.29) (A+)* =
= (—Of
+ B*)G* =
—
CB*G*)
close to Y,
namely
= _G(y)cri(B*)(y,,,)G_l(y)
:
—i
S;,
where we only take the eigenvectors which have eigenvalues with pos-
itive real part. (Since A± are elliptic, there are no purely imaginary eigenvalues and especially no vanishing eigenvalues neither of ui(B) (y, ti), nor of ai(—GB C') (y, 17) for 0). Clearly the vector space S;q> is generated by vectors of the form G(y)e, where e e is an eigenvector of 0i(B) (y, with negative real part eigenvalue. Recall from Corollary 12.6 that the closures of the infinite-dimensional spaces of Cauchy data and C'(H+(Aj) are orthogonal and complementary in L2(Y; Similarly, but more easily, we now obtain that the finite-dimensional vector spaces and are orthogonal and complementary in the space Moreover, range 00(IR. )(y, and range 0o(Id — TR• )(y, are orthogonal and complementary in by definition. Therefore, if 'R• is an elliptic boundary condition for its symbol provides an isomorphism from onto range oo(IR. )(y, and hence its complementary map (20.30)
C
—>
—i
III. Applications
196
must also be an isomorphism and so (20.28).
We can use the arguments of the preceding proposition to reprove the closedness of the operator 4 = which was obtained in Lemma 20.1. In fact, we have:
Corollary 20.5. Let 4 {u
(20.31)
Then 4
E
denote the operator
C°°(X;
I
the closure of 4(00)
with the domain
R(uly) = 0}.
as
an operator from a subset of
into L2(X;Sj.
Proof. A slight modification of the arguments, presented in the proof of Proposition 20.3, gives the following list of inclusions to be proved:
-
•
dom AQ c dom(AR
(1)
(ii)
dom 4 C C
where Q = G(Id —
dom4,
)G and 1R' an orthogonal projection with
—
— "IRS
Let T denote the operator 4
Then dom T C dom 4 which
implies (1), namely
domT j dom(4)* =
(20.32)
In order to prove (ii) we show for any v E domT that vly is welldefined in S1y) and Q(vly) = 0. We can repeat all the arguments of the second part of the proof of Proposition 20.3, where only C°°-test functions were involved. (i) and (ii) show that T* =
(20.33)
=
In order to prove (iii) we fix a u E dom 4 and apply once again Green's formula which yields the estimate
I(Av;u)I
(20.34)
for all v tion.
dom
since G*(vly) and uIy are orthogonal by defini-
Property of the Operator AR
20.
To prove (iv) we show that is well-defined in H4 (Y; and 'R (ulv) = 0 in a distributional sense for any u E We make the following modifications of the earlier proof: (20.35)
(IR.(ulv);f) = (uly;IR.f) = (CuIy;GIR.f) = (GuIy;wly) with WE C°°(X;S) and wIy = GIR.f =
=
(u;ATh,ow) —
shows the vanishing of IR' (uIy) for suitable C°°-functions with 1 close to Y and IIWIIL2 arbitrarily small. which in the (iii) and (iv) yield = so = +(oo) Hubert space means that AR is the closure of AR U which
.
The space of closed extensions of a Dirac operator A+
(or
of suitable
restrictions) does not only consist of the space 7Z. :=
I
R elliptic boundary problem)
of all realizations of elliptic boundary problems, but contains also two (see also Figure other prominent closed operators, and - --
20.2):
/
K. I
/
.0
..-1--
0
—— —
——
max
Fig. 20.2 The closed extensions of
III. Applications
198
Definition 20.6. Let tors in L2
which
the unbounded operaare determined by
and and
act like
denote
L2(X;S) and
:= {v E := {v E
E L2(X;S)}.
I
and
we define
Similarly
-y0v = 0},
are closed operaand Proposition 20.7. The operators into L2(X; Sj and in fact tors from subspaces of L2(X;
= {v E
I
= 0}.
Moreover
(A+ \*_A_ — max
an..d
(A+ \*_A_min max/ '
Note. We have C
C
for
all
E IZ
in the space of closed unbounded operators from L2(X;
into
L2(X; S—) (see Figure 20.2). is closed, let v, Proof. To prove that —i w in L2(X;S'), and l'ovn = 0. Then
(20.36)
v
in L2(X; St),
= (r+Ae+)A+vn;
= (Id hence, by Corollary 19.7,
—'
E H1 and
IIVnhIHl = II(Id —K+'yo)vflhIHl
is convergent in H' to a limit v' with 'y0v' = 0, but then it is convergent in V to the same = w. limit; hence v = v' and A+v = urn As in the proof of Lemma 20.1, we get that
To prove that is closed, we begin as above, but do not assume := (Id 7ovn = 0. Nevertheless, when we replace v,, by (20.36) remains valid and provides once again a limit v' E H' with A+vl
= n—.oo urn
—
= lirnA+vn = W.
Property of the Operator AR
20.
Moreover, K+ (i'OVn) = ker+(A+,O) and A+v = A+(v' + Similarly we prove =
v
v—
—
(v — v'))
199
v' E
= w.
Let v0, w0 E L2 and
w0) = (v; vo)
for any v E dom sequence w,
wO
We have to show v0 = Aw0. Choose a in L2, consisting of smooth sections, and get
= by Green's formula. C00° C dom hence dom is dense in L2. Therefore the sequence converges in L2. Let v' := urn := r+A_ hence = (Id is convergent in H' with := E H' and Aw' = = and we obtain w0) = lim(v; = (v; Aw'),
for any v E dom we get
hence v0 =
=
=
Aw' in L2. Since
is closed, and there is nothing left to prove.
Now we derive a formula for the index of the realization of an elliptic boundary value problem R: C°°(Y; C°°(Y; V) for a Dirac operator : C°°(X; —' C°°(X; S). Recall that the boundary integral —'
:
range(R)
is a Fredholm operator by Corollary 18.15. The following theorem provides a new proof of that fact and extends it to the realization.
Theorem 20.8. The operator =
:
dom4 —* L2(X;S)
is a &edholm operator with
= dim ker4 = =:
—
dim :
—'
ran ge(R))
III. Applications
200
Proof. Let T denote the operator
acting from
to
range(R). We have already shown that (1) is closed (by Proposition 19.11); (2) kerT (by Corollary 19.2); (3) and (At)' = (by Proposition 20.3); hence we only have to identify coker AR with coker T: (20.37)
= kerG(Id —IR.)G'P+(A)
cokerAR =
ker(Id 'R )(Id = =
: range(R) : range(R)
For the last equation we once more exploit the fact that the boundary
condition R (and hence R') has closed range by definition. That implies that R : range(R) —' is an isomorphism. Clearly the last space in (20.37) is isomorphic to coker T. Note.
U
In the following we shall return to the notation P÷ for
From Theorem 20.8 we get that the index of the Calderón boundary vanishes, since the corresponding boundary integral is problem just the identity. This can also be seen from Corollary 12.6 and Proposition 20.3 which give ) = A_. By definition of the Calderón projector, we have ker = ker = {0}. For the Atiyah-Patodi-
Singer boundary condition (= the spectral projection P> (At) onto correthe linear span of the eigensections of the tangential part of sponding to non-negative eigenvalues), the situation is quite different. First, we have
'
—
—A — P<(A—)
with an inevitable asymmetry in the boundary condition for a nontrivial kernel of the tangential component B. Moreover, the kernel of the realization does not necessarily vanish, and the indices of 4÷ and 4> coincide if and only if P+ and P belong to the same connected of the Grassmannian this more closely in Chapters 21, 25, and 26.
We shall investigate
20. Fredholm Property of the Operator AR
201
Next we explain special features of the index of elliptic boundary problems of local type. Let A be a Dirac operator or a system of Dirac operators which admit a local elliptic condition (cf. Remark 18.2c). Then R : C°°(Y; C°°(Y; W) is a pseudo-differential operator of order 0 such that
—'
rop+(y,C) :
is an isomorphism. In particular the principal symbol of R is surjective.
S: V) C°°(Y; W) be a pseudo-differential operator of order 0 such that the principal symbol s(y, () : of S is surjective for any (y, Then S has finite-dimensional cokernel and in particular the range of S is closed. Lemma 20.9. Let
Proof. The operator
: C°°(Y; W) —' C°°(Y; W) is an elliptic operator since its principal symbol is an isomorphism. Therefore ker SS = ker is finite-dimensional. 0 We reformulate Corollary 19.10:
Proposition 20.10. For any real s>
'A' "
the
operator
—,
/
has a closed range. It remains to compute the cokernel of
We have the following
elementary lemma which sharpens Corollary 18.15, but is valid only for local elliptic boundary conditions:
Lemma 20.11. For local elliptic boundary conditions the operator RP÷ : is a Fredholm operator.
Proof. Once again we may consider everything on the level of Sobolev spaces. It is enough to show that dim ker RP÷ < oo and dim coker RP÷ < oo. We have already proved the first estimate. Moreover
coker RP+ = ker(RP+ )8 = ker P+ R
= and the operator RP÷R* is elliptic.
=
9
III. Applications
202
Theorem 20.12. For local elliptic boundary conditions one has
=
index
: H÷(A) —' C°°(Y; W)}.
H(A)
Fig. 20.3 The parametrization of coker(A,R) by
Proof. With regard to the results obtained earlier, it only remains to identify the cokernel of (A,R) with cokerRP÷(A) = (see Figure 20.3):
coker(A,R) = {(v,h) E
(As;v)
+ j(R(slv); Ii) dy = 0 for any 8 E C°°(X; and by Green's formula
0 = (As;v) +
= (s; A*v) + J(sIv; Rh — G'(vly))dy. Once again this implies Av = 0 and coker(A,R)
{(q,h) E with
= qr and Rh = q}
{(r,h) E with
=
= 0 and r = R*h}
20.
Property of the Operator AR
203
For later use we give the following reformulation of Theorem 20.12 which determines the relation between index AR and index(A, R) (notice that we suppress the subscripts + of the Calderón projector and the Cauchy data space):
Theorem 20.13. Under the preceding condition8 38
index(A, R) = index{RP(A) : H(A) —+ C°°(Y; W)} = index AR — dim(C°°(Y; W)/range(R)).
Note. Decompose the kernel of the operator P(A)R* : C°°(Y; V)
C°°(Y; E'y)
H(A)
into two finite-dimensional subspaces kerP(A)R = Wi
W2
with W1 := ker R* and W2 the orthogonal complement of Wi in kerp(A)R* (see Figure 20.4). Since ker(A, R) = kerAR and coker(A, R) coker(RP) = W1 W2 are already established, Theorem 20.13 shows that (20.39)
cokerAR
W2.
ker
(coker AR)IY
kcrl?
Fig. 20.4 The decomposition of' kerP(A)R and the isomorphism (coker AR)IY
204
III. Applications
Corollary 20.14. For local elliptic boundand conditions we have index AR = index(A, R), if and only if R is a surf ective operator.
Remark 20.15. We have seen in Chapter 15 above that a compact perturbation of R can change the connected component of the corresponding projection 'R• and we shall see in Section 22F below that in general the index of elliptic boundary problems is not a topological invariant (see Theorem 15.12 and Remark 22.25). However, when we
restrict ourselves to the class of local elliptic conditions, the index becomes a topological invariant and we have a corresponding AtiyahBott index formula (see Atiyah & Bott [19641 and also Hôrmander [1985; Section 20]). We do not discuss the Atiyah-Bott index formula here and refer the reader to the works of Atiyah, Bott, and Hörmander mentioned before.
21. Exchanges on the Boundary: Agranovië-Dynin Type Formulas and the Cobordism Theorem for Dirac Operators
We prove two variants of the cobordism theorem.
formula and the
In the preceding chapter we obtained the following index formula for the realization AR of an elliptic boundary condition R = R2 (cf. Theorem 20.8): (21.1)
index AR = index {RP(A) : H(A) —' rangeR},
where P(A) denotes the Calderón projector onto the Cauchy data space H(A) corresponding to the Dirac operator A. The obvious question is whether this integer is a topological invariant. Contrary to the case of local elliptic boundary problems, the answer is negative. The index is not a homotopy invariant of the principal symbols of the operators A and R. it is not difficult to deliver a specific example.
Example 21.1. Let X := D2 and take the Cauchy-Riemann operator
0 .0 0— — +2— — 0x
0y
jO .10— + —— rOço
close to the where we take the standard polar coordinates (r, boundary. In this special case the choice of the orientation of the transversal coordinate makes no real difference, but only a sign change
for the whole expression, since r is in both components here. The C) Calderón projector P(s) is the orthogonal projection of (orthogonal with respect to L2(S'; C)) onto the space £c{ek(ço) =
of the Cauchy data of all holomorphic functions on the disk D2. According to the results of Chapter 12, the projection P(0) is a pseudo-differential operator of order 0 with the principal symbol
=
11
(0
III. Applications
206
Actually, in this particular situation the Calderón projector incides with the spectral projection P(iO/ rôp). Now let us consider projections Pk given by the formula P,
co-
for k E Z.
:=
It follows from Theorem 20.8, see equation (21.1), that:
index&b = index PkP(9) = index PkPO =
(21.2) since
=
and
m max{0,k}}.
Therefore
dimkerPkPo:
k0:
k<0:
dimcokerPkPO:
k
0
0
—k
proves (21.2). Nevertheless the principal symbol for any P, and hence is unchanged.
which
is
As an immediate corollary of Theorem 20.13, let us mention a simple variant of the Agranovië-Dynin formula for local elliptic boundary conditions:
Theorem 21.2. Let A : C°°(X;S)
C°°(X;S') be (possibly a sys-
tem of) Dirac operators over a compact manifold X with boundary Y, and for j = 1,2 let R, : C°°(Y; Sly) —. C°°(Y; V,) be a local elliptic condition. Then (21.3) Proof.
index(A,R2) — index(A,Ri) = indexR2P(A)Ri.
We apply Theorem 20.12:
index(A, R2) —index(A, R1)
C°°(Y; V2)} — index{R1P(A) H(A) C°°(Y; V1)} = index{R2P(A) : H(A) C°°(Y; V2)}
= index{R2P(A) : H(A)
:
+
: C°°(Y; V1) —.
H(A)}
= With regard to Theorem 20.13 and Corollary 20.14, the preceding theorem has the following direct consequence which is crucial for the proof of the cobordism theorem for Dirac operators.
21. Exchanges on the Boundary
207
Proposition 21.3. To the previous assumptions, add that, for j = 1,2, the local elliptic boundary operator R, 13 an orthogonal projection
onto a subbundle V3 of Sly, i.e. 11, =
= R,. Then
index AR2 — indexAR1
= index(A,R2) —index(A,Ri) = index{R2P(A)Ri : C°°(Y; Vi)
(21.4)
C°°(Y; V2)}.
Note. Elliptic boundary conditions of the type assumed in the preceding theorem exist for Dirac operators only in the odd-dimensional case, but for systems in the even-dimensional case as well.
Proof. Theorem 20.13 and Corollary 20.14 tell us that indexAR3 = index(A,R3) = Therefore we have index AR2 — index AR1
(21.5)
— i(Ri,P(A)) = = + = index{R2P(A)Ri : C°°(Y; Vi) —* C°°(Y; V2)).
Treating elliptic boundary problems as projections yields a correformula for global boundary sponding variant of the conditions of generalized Atiyah-Patodi-Singer type, i.e. belonging to the Grassmannian Grp+ (whereas in the preceding theorem the projections R1, R2 do not in general belong to Grp+):
Proposition 21.4. Let A be a Dirac operator over a compact manwhere denotes ifold X with boundary Y, and let R1, R2 E the principal symbol of the Calderón projector 1'(A) (= the principal symbol of the spectral projection P> of the tangential part B of A). Then (21.6)
index AR2 — index AR1 = i(R2, R1).
Proof. By Theorem 20.8 we have (21.7)
= i(R3,'P(A)).
Then formula (21.6) is a consequence of Proposition 15.15.
III. Applications
208
The main purpose of this chapter, however, is to give a proof of the famous cobordism theorem for Dirac operators based on Proposition 21.3 (applied to an odd-dimensional case). Let Y be an odd-dimensional compact Riemannian manifold with
boundary Z. We assume that the metric of Y is a product near the boundary. Let S be a bundle of Ct(Y)-modules over Y with compatible connection. Let
B: denote the corresponding Dirac operator. We have a decomposition of B close to the boundary Z of the form (21.8)
where r is a unitary bundle automorphism with r2 = hence r defines a decomposition of SIz into the direct sum
and
of the subbundles of the ±i-eigenvalues of {rZ}ZEZ.
Theorem 21.5. (a) The operator B takes the following form close to Z with respect to the decomposition SIz 5+ S :
IL.))
(21.9)
(b) Let : S —' Then the operators (21.10)
denote the orthogonal projections of S onto defined by
:= {s E H'(Y;S) I Pj(sjy) = 0)
LB±:=B
are well-defined local elliptic boundary problems, and we have the following equations: (I)
(II) index
=
(III) contain the famous cobordism theorem for the index of operators of Dirac type. The cobordism theorem provided the main step in the original proof of the index theorem (see Palais (1965; Chapter XVII]) and has various generalizations.
21. Exchanges on the Boundary
Corollary 21.6. (Cobordism theorem). Let A÷ :
C°°(Z; S) denote the restriction of a Dirac operator on a Closed evendimensional manifold Z to the spinors of positive chirality. Then the index of vanishes if the couple (Z, is a "boundary". This means that there exists a manifold X with boundary Z and a bundle S of Clifford modules over X such that S restricted to Z is equal to
SEEDS. Proof of Theorem 21.5. (a) The special form of the decomposition of B follows from Chapter 7. (b) As in Theorem 7.3, we have A_ = Az... This shows that B is formally seif-adjoint. That the unbounded operators and 8_ are mutually adjoint follows immediately from the definition, see Remark 20.4c.
Equation (III) is an elementary consequence of Green's formula. We show that
ker(8÷) =
(21.11)
{O}
= ker(8_).
Let s be an element of the kernel of where we write slz = (s+, s_), and
i.e. Bs =
0
and s+ =
0,
denotes the projection P±s.
Then
0 = (Bs; s) =
(s;
Bs) —
= —
=
f
f
(r(y)s(y); s(y)) dvol(y)
s(y)) dvol(y) =
if
(s_ (y); s (y)) dvol(y)
II8_IIL2(SIz)
It follows that (21.12)
ker(8±) = {s Bs = 0 and sIz =
= 0} =
{O}
= 0 because of (I). Now we prove (II). We recall the Agranovië-Dynin formula (Proposition 21.3): and then index
index BR1 — index BR2 = index R12(B)R2,
where the boundary conditions Rj are given by projections. Hence we have (21.13)
index8_ —
index P_P(B)P+.
III. Applications
210
We show that the principal symbol of the 0-order pseudo-differential is equal (up to a factor to the principal symoperator when restricted to the bol of the first-order differential operator cotangent sphere bundle 5(Z). Let us denote by a the principal symbol of the (total) tangential operator A and let a± denote the principal symbols of the operators They satisfy the following equalities on S(Z): (21.14)
a_at = Ids+
= Ids-.
and
Hence the eigenvalues of
a(z,() =
(0
a_ 0
are all contained in the set {l, —l}. We have shown in Proposition 14.2 that the principal symbol p..,. of the Calderón projection P(B) is equal to the principal symbol of the (positive) spectral projection of is the orthogonal projection —' the operator A. So p,. (z, C) spanned by the eigenvectors of a(z, C) correonto the subspace of sponding to the positive eigenvalues. Here z Z and ( E T*(Z)z IS of length 1. Then we have the equality :
1
/
2
and hence
= P_
1(0 o\fi a_\(1 o\ O)=2a+.
flo
1
22. The Index Theorem for Atiyah-Patodi-Singer Problems
We present a proof of the index theorem for Atiyah-PatodiSinger boundary problems.
22A. Preliminary Remarks Let : C°°(X; C°°(X;S) be a Dirac operator on an even-dimensional smooth compact Riemannian manifold X with
boundary Y. Actually the odd-dimensional case is not different and we shall make the necessary comments after the proof of the index theorem. We assume that the Riemannian structure on X and the
Hermitian structure on S are products in a collar neighbourhood N [0,1] x Y of the boundary Y in X, that is, the metrics do not depend on the normal coordinate u when restricted to so that can be identified with Sly. Recall that then has the following form on N
Y
= G(OU + B),
(22.1)
where C : S+ — is a unitary bundle isomorphism, namely the Clifford multiplication by the inward normal vector, and B C°°(Y;
denotes the corresponding Dirac op-
C°°(Y;
erator on Y. Note that C and B do not depend on the normal coordinate on N. Moreover, the tangential operator B is seif-adjoint and elliptic of first order. Let {Ak; çok}kEz be a spectral decomposition of L2(Y; generated by B with Ak 0 for k 0. We know from Proposition 14.2 that the orthogonal projection P> := Po is a pseudo-differential operator and we have a wellonto defined Fredholm operator (22.2)
A
:
—'
L2(X;S).
It follows from the elementary index theory presented in Chapter 20 (see Theorem 20.8) that (22.3)
indexA =
III. Applications
212
where 1'..,. denotes the Calderón projector of A+ and
= index{P P..f : rangeP+
rangeP>}
the virtual codimension of in P+. Formula (22.3) does not involve the geometric data suitable for applications to concrete problems. Deeper insight is provided by the geometric Atiyah-Patodi-Singer index theorem for the operator A (Theorem 22.18 below). Following Atiyah, Patodi & Singer [1975],
we shall present a proof of this theorem. As in the heat equation proof of the Atiyah-Singer index theorem (accounted for e.g. in Cilkey [1984] and Berline, Cetzler & Vergne [1992]), the base of the Atiyah-
Patodi-Singer index theorem is the study of suitable heat operators and the related heat kernels and traces. A novel feature is of course to keep control of the boundary conditions. This is possible due to our assumption of the product structure in the collar neighbourhood of the boundary providing for explicit computations of the traces of the heat kernels involved.
We begin with some formulas for the adjoint boundary problem arid the related natural problems of second order:
Lemma 22.1. Let A denote the closed operator from L2(X;
A=
to L2(X; S—) which acts like {s E
I
P(sb') =
O}. Then
(a) The adjoint operator A* acts like A— and is determined by
domA = {s E H'(X;S) I P< G(sly) = O}, where P< := P.<0 denotes the spectral projection of the tangential part corresponding to the eigcnvalues of B in the interval (—oo, 0). and is determined by (b) The operator A*A acts like
B of
domAA = {s E (c) The operator ..4A acts like
I
= 0 and = —B(P<(sly))}. and is determined by
domA4 = {s E H2(X;S) I P<(GsIy) = 0 and li_vS
Sy
(d) The null space ker A'A coincides with that of A, while the null space of .A..4 coincides with that of A.
22. The Index Theorem for Atiyah-Patodi-Singer Problems
213
Note. Here P>(sly) = 0 means that s belongs to the kernel of the
composite continuous map
where '10 denotes the restriction to the boundary. Note that '10 is even
continuous H'(X) — H'/2(Y) (Theorem 11.4).
Proof. Recall that P< := P<0 = Id — F> and = A = _G*(Ou — GBC)
(22.4)
on the collar. Then (a) is a reformulation of Proposition 20.3. As for (b) we have that
domAA = {s E
Is
E
E domA'};
domA and
and now (b) is an elementary consequence of the commutativity of the
tangential operator B with its spectral projections F> and P<. The computation of the domain in (c) follows in a similar way. Finally (d) is inherent in the concept of an adjoint operator, but can also be seen from the explicit form of the two boundary problems which actually fit appropriately. [] The preceding lemma allows us to use the traces of the kernels of the
corresponding heat operators to calculate the index of the boundary problem A. We prove an Atiyah-Bott trace formula for boundary problems in the same way as in the case of closed manifolds (see e.g. Cilkey [1984; Lemma 1.6.5] or Berline, Cetzler & Vergne [1992; Theorem 3.50]).
Let A E R
let nA and denote the dimensions of the Aeigenspaces VA of the operator AA and of the operator 4,4*• They are finite, since the null space of elliptic boundary problems is finite-dimensional (as shown in Corollary 19.4). Recall that the and
eigenspaces consist of smooth sections by Theorem 19.1. From Lemma 22.1 we get
A(AA) = (.AA')A and A*(AA*)
(A'A)A
so that A and A interchange the eigenspaces VA and over induce an isomorphism (22.5)
and more-
III. Applications
214
between them if A
= 0 for A
0. Thus nA —
0, and we compute
for t > 0
=
—
=
—
—
= dim kerAA - dim kerAA Lemma 22.Id
=
.
. dim ker A — dim ker A
Thm. 20.8.
=
index A.
This proves
Lemma 22.2. The following equality holds for any t > 0 index A = tr
(22.6)
—
tr
Formula (22.6) is the base of the proof of the Atiyah-Patodi-Singer index theorem. The proof splits into three parts:
(1) We construct the heat kernels tors
and
where
and t
•
of the opera-
denotes the corresponding
boundary problem on the infinite cylinder [0, oo) x Y. We have also given the corresponding heat kernels ed and ed * for the operators AA and AA* on the closed double X of X (see Gilkey [1984], Roe [1988], or Berline, Cetzler & Vergne [1992] for the construction). (2) We use Duhamel's principle to construct the kernels and for the heat operators and — (3) We study to obtain the final index formula.
22B. Heat Kernels on the Cylinder We construct the kernels and of the operators and on the semi-infinite cylinder [0, oo) x Y, where acts like A and A is given by formula (22.1), but with u E [0, oo). For simplicity consider the normalized Dirac operator (22.7)
on the cylinder acting on sections of p*(S+ly) lifted to xY under the natural projection p). In this section the assumption G = Id is no restriction, since
AA=
22. The Index Theorem for Atlyab-Patodi-Singer Problems
215
for arbitrary unitary G. Clearly D is elliptic and its formal adjoint is :=
(22.8)
We shall show
Proposition 22.3. Let V and ir denote the closure of the operators defined by D and D with domains given by in L2(R.,. x Y; smooth functions subject to the boundary conditions (22.9)
P(fI{o}xy) =
0
P<(fI{o}xY) =
0
and (22.10)
respectively. Then V and ir are adjoints of each other, and
(22.11) domD =
{f E
x
{f E
x Y;p*(S_Iy)) I
I
= 0)
and
(22.12) domV =
Here P(fI{O}xy) =
0
means for f E H'(R+ x
= 0).
that f
belongs to the kernel of the composite continuous map
H'(R÷ where 70 denotes the restriction to the boundary.
Note. The preceding proposition resembles Lemma 22.la but has a different message: Whereas Lemma 22.la and its originator, Proposition 20.3, are about elliptic boundary problems over a compact manifold (so that we can apply Fredholm operator theory), we are now dealing with open manifolds which require another set of arguments. We present these arguments in a series of lemmata about evolution equations which establish a parametrix for V (and similarly for and are of independent interest for the later computation of the heat kernels. We need some principal facts about the Laplace transformation. Nice proofs and additional information can be found in Carslaw & Jaeger [1906/1988] where one of the oldest, but most elementary
III. Applications
216
and satisfactory explanations, is given. For more advanced readers we suggest a modern presentation such as that of Doetsch [1970]. The following distinctions must be made: • Let xY; p*S+Iy) denote the C°°-sections with compact support (i.e. vanishing for IL C); P>) denote the space of all C°°x Y; • let sections satisfying the boundary condition (22.9) (for u = 0) x Y;p*S+Iy;P); with subspace
• and let C8°(R+ x Y;
denote
the space of C°°-
sections with (compact) support totally contained in the interior of the cylinder R÷ x Y (i.e. vanishing for u C and for
u c). For later use notice that the inclusions (22.13)
x
C
Y;p*(S+Iy))
C
C
are dense.
We shall show that any equation Df = g with g
f'VOO
has a unique solution f E
X
V.
IY
x Y;p*S+Iy;P). More precisely:
Proposition 22.4. For any normalized Dirac operator D = over the cylinder R+ x Y there exists a linear operator
Q:
x Y;p*S+Jy)
—' C°°(R.,. x Y;p*S+Iy;P>)
such that (i) DQ9 = g for aug E
(ii) QDf = f for alif E
+B
x Y;p*S+Iy).
x Y;p*S+Iy;P).
(iii) Q extends to a continuous map
1
for all integers
s 1. Here
denotes the space of sections which are locally in the Sobolev space H8.
(iv) The kernel Q(u, y; v, z) of is a section in a suitable homoand y, z in Y) morphism bundle (with u, v running in which is C°° for u v.
22. The Index Theorem for Atlyab-Patodi-Singer Problems
217
here lives
R4xY
here lives Fig. 22.1 Separation of variables over
xY
Proc4 To solve (22.14) with
Df = g
/E
xY;
for g E
we make the usual separation of variables (products of sections from span x Y; pS4iy), see Figure and from L2(Y; 22.1) and expand / and gin terms of the previously fixed orthonormal of elgenfunctions of B (here and in the following basis we repeat A E spec(B) corresponding to its multiplicity):
f(u, y) =
g(u,y) =
and
IA(U)(PA(Y)
Then the partial differential equation (22.14) reduces to a family of
ordinary differential equations
AEspec(B),
(22.15)
half of them with initial conditions
forAO.
(22.16)
The textbook solution of (22.15) is JA(u) = fA(O)e
AU
+
j
dv.
With (22.16) and an appropriate choice of (22.17)
fx(O) := —
j
eAt? 9A(v)dv
for A < 0,
III. Applications
218
we get explicit solutions
2/ \ =
JAtU)
( fue)t(t)_u)g(v)dv 0
for A >0
J
forA<0
so that formally the operator Q :=
with
QA
'—+
:
satisfies (1) and (ii).
To show that this formal solution converges, we estimate the Sobolev norms. This will prove (iii) as well. To do that, rewrite the equations (22.15) in terms of the Laplace transform
:= j and get
=
(22.19) We
compute the L2(R)-norms. Clearly A2 111 112
for
A 0.
For A <0 we need more subtlety. Using elementary arithmetic, we first deduce from (22.19) (22.20)
111i112
2
119A112
{
+ IfA(o)12j
}
From
1
( 2221 . )
and IleA
112. llgAIl2 Pa'aI
for A <0,
22. The Index Theorem for Atiyab-Patodi-Singer Problems
219
we get for
A2111 112<
A
<0.
Apply Parseval's formula once more and get (22.23)
IAI
Since the fA
satisfy
IlfAll
for all A E spec(B).
2
(22.15) this gives
(22.24) Recall Y;
that a suitable norm fl.
for the Sobolev space
may be defined by
:= hf 112 + since
x
+ hlBf 112
B is a first-order elliptic operator. So we set
{(i + A2)hhfAhI2 + 1101
=
112}
and deduce continuity of
Q: H°(R+
x
x
—*
by (22.23) and (22.24) which establishes (iii) for k = is actually a true continuous operator C)' := and for A =0, we use
fo(u)
=f
1.
What we have H° H',
go(v)dv
of (22.23). To establish (iii) for general k, multiply (22.23) by powers of A and differentiate (22.15) with respect to u. This leads to instead
II 3IAIP+ V
h19A11
+
11
V
III. Applications
220
We may repeat the arguments given above, but instead of the norm by fi. fli, we now introduce a norm II Ilk for the Sobolev space taking the square root of the sum of the L2-norms of all for
p+q k.
Proving (iv) is an exercise in heat equation arguments. From (22.18), one deduces that Q can be written as a convolution in the u-variable with the two-sided evolution operator
H(t) :=
(22.25)
(e_tIDIP> —
(
_gtlBI P<
t>O — t <0,
X where IBI := B P> —B P<. More explicitly, let g Y; y, z) denote the kernel of the operator H(t); lv)) and let
then Qg(u, y)
= JR+ x Y
fl(u — v, y, z)g(v, z)dvdz,
hence Q(u, y; v, z) = fl(u — v, y, z). So we verify (iv) by proving the smoothness of fl(t, y, z) for t 0. By (22.25), we only need to show that the kernel S(t, y, z) of the heat operator e_tIBI is C°° on R÷ x Y x Y, which actually is a well-known consequence of the spectral theorem and widely used in heat kernel expansions. Alternatively, the
smoothness of S can be deduced as follows: For 0 < t0
q
Jt0 =
J
YxY
L'
dt < (t1 — to)
—2toIAI
for arbitrary integers p, q. The last estimate follows from
which converges for large N. It follows that S belongs to all Sobolev spaces Hk for arbitrary k = p + q and so by the Sobolev lemma is coo. LI Remark 22.5. If we replace D by D and the boundary condition (22.9) by (22.10), the only difference in the proof above occurs for = 0, in which case we shift between (22.16) and (22.17) and set
fo(u) :=
_j9o(v)dv.
22. The Index Theorem for Atlyab-Patodi-Singer Problems
Therefore, D* also has a fundamental solution R iiit, iv') like Q in (i, jj, iii, iv).
with
221
properties
Proof of Proposition 22.3. As usual it needs some consideration to take the closure and the adjoint of an unbounded operator. Let us explain the problem by distinguishing certain levels: (I) As mentioned above, the operators D = and D = are for,nal adjoints of each other, namely (22.26)
(Df;g) = (/; (H)
for
all f, g
x
E
Next notice that (22.26) remains valid for all
/E
x Y;p*(S+I);P)
9E
x
and
f and g with respect to a complete orthonormal system of eigenfunctions of B. Note that = 0 and P<(gI{o}xY) = 0. Then Expand
(Df;g)
+ AIA(u))
çoA(y)
=
f.((u)OA(u)du+>2A1JA(u)9A(u)du
=
fA(u)Ag,1(tt)du
= + A
=0
= (f;Dg).
(III) Now we have to prove that the L2-closures V and V unbounded operators D
with
Dt
with
:= C°°(R+ x
and
domD. :=
x
of
the
III. Applications
222
are adjoints of each other.
Then, decompose the Hubert space
x Y; where H" involves the into two parts zero-eigenvalue of B and H' all the non-zero eigenvalues. Then V decomposes accordingly, since by definition D(H') C H', D(H") C H", and the projection of
on H' along H" does not lead out of dom V (cf. Kato 11976; Section 11.5.6]). Similarly V decomposes. We show that the parts 1)' and 1Y' of V and the parts D' and TY" of V in H', H" respectively, are mutually adjoint. On H" one has D = ô/t3u, D = —8/Ou and the adjointness of TY' and is clear. On H' the fundamental solution of Proposition 22.3 gives a bounded inverse Q' for TY, and similarly follows by we get a bounded inverse R' for D*I. Then R' = and continuity from the fact that (DI; g) = (f; D*g) for 1,9 E satisfying (22.9) and (22.10) respectively. Since adjoints commute with inverses (proved e.g. in Kate [1976; Theorem 111.5.30]), the follows. adjointness of D' and It remains to prove that
domD = ker(P Clearly we have
doml) C ker(P> 070) and domV* C ker(Pz Since P> and P< are complementary orthogonal projections and V = ad(V), we conclude dom V = dom ad(V*)
ker(P
u
Definition 22.6. With any (seif-adjoint) Dirac operator
B:
—'
over a closed manifold Y, we associate various operators over the cylinder R+ x Y and various kernels: (a) The seif-adjoint (closed) cylindrical Dirac Laplacians
VtV
and
* :=
22. The Index Theorem for Atiyah-Patodi-Singer Problems
both act like —02/0u2 + B2 and are distinguished by
{fE H2(R÷ xY;p*S+Iy)IP>(fJ{o}xy)=O and
1=o}
)tI=oJ
and
:=
=0
{i E H2(R+ x Y;p*S+Iy)
and
_Bf)} =o}. (b) The heat operators and are associated with and and by the spectral theorem, they are well-defined bounded (in fact smoothing) operators on x Y; p*(S+Iy)) for t > 0 (see also Figure 22.2).
the initial data live here
(t=o)
0
Ii
Fig. 22.2 The action of the cylindrical heat operator
(c) Forj E {c,c*}, let e,(t; is, y; v, z) denote the cylindrical heat kernel, i.e. the kernel of eta'; let AC(t; u, y) denote the symmetric (cijlindmcal) heat kernel, i.e. the kernel of at the point (u,y;u,y)
III. Applications
224
of the diagonal of metric integral
x Y) x
x Y); and let X(t) denote the sym-
j AC(t; u, y) dydu.
:= j
Remarks 22.7. (a) With regard to the polarization of the spinors x Y; p*(S+Iy)) introduced by the orthogonal spectral projections P and P<, the operator * respectively) is equal to the Dirac Laplacian —02/0u2 + B2 with Dirichiet boundary condition on the non-negative (negative) polarized spinors and a modified Neumann boundary condition on the negative (non-negative) polarized in
spinors.
(b) Recall that on
the scalar heat kernel is given by the exact
Poisson formula
e(t; x, x') =
(22.27)
where d(x, x') denotes the Euclidean distance. The scalar heat kernel is a smooth function on (0, oo) x RT' x which is a fundamental solution of the heat equation Ot + i.e. (1) it solves the equation
10
82\
02
under the initial condition that lime(t; x, x') is t—.o the Dirac distribution 8(x, x'); (2) alternatively stated, it defines an integral operator for all x' E
:=
J
which solves the heat equation for any given initial distribution 8;
(3) stated yet differently, it is the kernel of the heat evolution operator i.e. is approximated by (4) this yields a weak form of Weyl's theorem (22.28)
t —p0.
22. The Index Theorem for Atiyah-Patodi-Singer Problems
225
(c) Let P be any elliptic seif-adjoint differential operator of second order with a positive definite principal symbol acting on sections of a Hermitian vector bundle E over a closed rn-dimensional Riemannian manifold M. It is well-known that the existence of a heat kernel (homomorphism) e(t; x, x') of the operator can be established with properties corresponding to the properties (1, ... , 4) listed above; that it is usually impossible to find an exact expression for e(t; x, x'); that it can, however, be estimated for bounded positive t with certain constants c2 > 0 by (22.29)
d2(z x')
Ie(t;x,x')I
.
where d(x, x') denotes the geodesic distance. This inequality can be
derived from approximate solutions as explained e.g. in Gilkey [1984; Lemma 1.7.4], Roe [1988; Chapter 5], and Berline, Getzler & Vergne [1992; Theorem 2.301. Note that then the weak form of Weyl's theorem generalizes to (22.30)
tret' =
".'
ct_mi2
as t —'
0.
AEspecP
We note a minor generalization of (22.29) which will be useful in cutting and pasting heat kernels and discussing the 7?-invariant below. Let Q : C°°(M;E) C°°(M;E) be an auxiliary differential operator of order 1. Then Qe_tI' is a smoothing operator with kerQ acting on the xnel variable. Instead of (22.29), we get (22.31)
IQxe(t;x,x')I
.
for sufficiently small positive t and suitable positive constants c1, c2.
The following proposition extends the validity of (22.29) for M := Y, rn := n — 1, and P := B2 to an estimate concerning the cylindrical heat equation. (Below in Section 22C we shall apply (22.29) and (22.31) to the heat equation over the closed double with M := X, rn := n, and P := AA+). It asserts that after imposing bound= v) is still ary conditions, the contribution outside the diagonal asymptotically negligible.
Proposition 22.8. For any normalized Dirac operator the cylinder
x Y with dim Y = n—i, the kernels
+ B over
u, y; V7 z)
and
IlL Applications
226
u, y; v, z) of the corresponding heat operators and are exponentially small in t as t —' 0 for u v. More precisely they are bound by * (t;
C
(22.32)
e
for some constant C> 0 as t —+
0.
Proof. To solve the effiptic differential equation = 0 with / of B, as in dom one expands f in terms of the eigensections for the proof of Proposition 22.4. Then, if f(u,y) = each A one must study the ordinary differential equation
foruO
—74+AfA=O with the boundary conditions
ifAO,
fA(0)=0
(Dir?)
if A <0.
(modNeu<)
For A 0, consider the function (22.33) 2
eA(t;u,v) :=
2/fl1
(u+v)2
e
for u,v 0 and t > 0.
— e
)
It vanishes for u = 0; as t —* 0, it approaches 0 for u u = v; finally, for fixed t and v, apply the operator
0
(22.34)
+
—
v and oo for
A2,
and then all terms cancel as one can easily check. Hence it is a fundamental solution for the heat operator (22.34) with the boundary condition (Dir?). For the boundary condition (mod Neu<), standard Laplace transform methods (accounted for e.g. in Carsiaw & Jaeger [1959; Section 14.2]) give for A < (22.35)
eA(t; u, v)
(
0 (u—v)2
4t
te +
+e
(u-ft,)2
—
22. The Index Theorem for Atiyah-Patodi-Singer Problems
227
where u, v 0, t > 0 and erfc denotes the complementary error function defined by
j
erfc(s) :=
Note that erfc(O) = 1. The kernel u, y; v, z) of the heat operator is therefore explicitly given in terms of the eigenfunctions, and summing over
namely by multiplying eA(t;u,v) by
allAEspecB. for u, v 0.
Clearly (22.33) is bounded by Since
we have, for
<0, erfc
(u+v
<
—
—p—
21A1
e_A2t
e
—
e
so that the fundamental solutions (22.33) and (22.35) both are bounded by (22.36)
( e _xat
+
e
(u—u) 2
e
Using the inequality x ( x := [AR/i and taking into account the elementary bound it follows (z)I that is bounded by (22.37)
+
Since the kernel of on the diagonal of Y x Y is bounded by with dimY = n — 1 (see (22.29)), the proposition follows for
III. Applications
228
For the operator
the boundary conditions for each A are
(Dir<)
(dl,, i——+Af,,
(modNeu)
du
fA(0)=O
ifA<0,
=0
ifA0.
The fundamental solution e.,,(t; ti, v) of — + A2 for (Dir<) is again given by (22.33), while for (modNeu ) we must use (22.35) with A changed to —A. One sees that then the bounds (22.36) and (22.37) remain valid and the proposition follows also for 0 As announced in Definition 22.6c, we are primarily interested in the contribution from the diagonal and in the difference between and expressed by the symmetric (cylindrical) heat kernel K(t; u, y)
and the symmetric integral K(t) := f00°
K(t; u, y) dydu. For the cylindrical heat kernel, the following corollary provides the final formulas which we shall apply below in Section 22D.
Corollary 22.9. Let
+ B be a normalized Dirac operator over
the infinite cylinder R.4. x Y and let AC(t) denote the corresponding symmetric heat kernel integral. Then one has (a)
=
(22.38)
—
(b) For a(s) large, one has (22.39)
(c) If
jVC(t) +
dt =
has an asymptotic expansion
(22.40)
as t
'j-.
0,
k>-n then (22.41)
71B(25)
=
+ k=—n+1
+
22. The Index Theorem for Atiyah-Patodi-Singer Problems
229
for + 1). Recall that h := dim ker B is the multiplicity of the O-th eigen-
where ON(S) is holomorphic
and r is defined by r(s) :=
value, 7?B(S)
f°
for > 0 and extended to a meromorphic function on C with isolated simple poles at s = 0, —1, —2
Proof. (a) For t > 0, u 0, and y E Y one determines the kernel of x Y) x (R÷ x Y)
at the diagonal point (u, y; u, y) E by ftC(t;u, y)
{eA(t; u, u) —
= —
e,)(t; it, u)}
'ç-j( —e
e
®
—A2t
Id
+
}
+ Ae2"erfc
+
—
{ (22.42) e
—A2t
—A2t
—u2/tI
e —u2/t
+
IC(t)
I'PA(Y)12
=
j
(b) Note that, as t —'
j
}
Id
I'pA(y)12,
1. Integrating over R+ x Y, one
where, for convenience, sign(0)
(22.43)
}
+
=
obtains
'
u, y)dydu = >
Moreover, X(t) + in (22.43), )C(t) 0 exponentially as t —' oo. Also (22.43) shows that 1
1K(t)I
0
2
t
(22.30)
<
as
t
0.
III. Applications
230
Hence for
large
j(K(t) +
dt
converges. Integrating by parts, one gets
t'-1 dt =
sign(A)
—
=
—
by definition of ha(s). This follows directly from the definition of the
r(s + by setting 0 =
=
j
and then summing. Moreover, one uses
K'(t)
=
which is obtained by differentiating (22.43) with respect to t.
(c) This is an immediate consequence of (22.39).
D
Remarks 22.10. (a) Equation (22.41) gives an expression for the continuation of IB(28) to the whole s-plane. In particular, 'iB(s) is holomorphic near s = 0 and its value at s = 0 is given by
= —(2a0 + h).
(22.44)
(b) Finally, the asymptotic expansion (22.40) is unchanged if, instead x Y, we of defining K(t) by integrating over the infinite cylinder integrate only over a large finite cylinder [0, R] x Y for some R> 0. In fact, from (22.42) one gets that the difference to (22.43) is
+
sign(A) which is bounded by (22.30)
and hence is exponentially small.
Al
22. The Index Theorem for Atiyah-Patodi-Singer Problems
22C. Duhamel's Principle. Heat Kernels on Manifolds with Boundary This section investigates the kernels e and e. of the heat operators and Recall that A denotes the realization of the Dirac operator over the compact manifold X with boundary Y defined by the Atiyah-Patodi-Singer boundary problem. Following the presentation in McKean & Singer [1967] (see also Douglas & Wojciechowski [1991] and Klimek & Wojciechowski [1992], [1993]), we
prove the fundamental properties of the kernels which are needed in order to establish the index formula. As in the case of closed manifolds, referred to in Remark 22.7c above, it is usually impossible to find an exact expression for S(t, z, x'). The kernel can, however, be estimated for bounded positive t in a similar way as in the estimate (22.29); this inequality can be derived from approximate solutions obtained by patching the heat kernel Ed together. Here let &j denote the kernel of the heat operator of the invertible extension A of over the closed double X of X. Its restriction to X is well-defined and satisfies all of our demands for a fundamental solution for the operator Ot + AA, except for the boundary conditions of Lemma 22.1 and Definition 22.6a. Let denote the kernel of the heat operator of the normalized Dirac operator V over the cylinder x Y investigated above. Restricted to a collar neighbourhood N = I x Y of V in X, it satisfies the boundary conditions but does not extend automatically to the whole of X. (See Figure 22.3).
here Q = Ed here
Fig. 22.3 The making of a parametrw for
III. Applications
232
We construct an approximate fundamental solution for the operator Let sometimes called a parametrix of the operator p(a, b) be an increasing smooth function of the real variable u such
+ AA,
that (22.45)
p=
foru
(0 11
forub. —
of the normal variable We define four smooth functions u on N := 10,1) x Y which have obvious extensions to the whole of X, being extended xY to the whole of X, and to the whole of being extended by 1, see Figure 22.4): by 0 and
—
cP2
— 1
7
7
7
1
y
7
Fig. 22.4 Cut-off functions
56
:= 1 — p(—, —);
Then
} is
12
'P2 := p(—, —);
a smooth partition of
unity suitable to the open
covering {{u
> 3/7} and {u < 4/7)}
22. The Index Theorem for Atiyah—Patodi-Singer Problems 91
K with
(22.47)
1 on supp
233
and
dist
Fort>O, x,x'EXwedefine (22.48)
Q (t; x, x') :=
soj (x)E3
(t; x, x')iJ.', (x'),
:= E2 := 6d, and the ii,, are considered as multiplication operators. Clearly Q is an approximate fundamental solution of the operator +4' A over X which satisfies the initial and boundary conditions, but not the equation over the whole of X. We define a in an analogous way. parametrix Q, of the operator where
Lemma 22.11. Under the preceding conditions let C(t; x, x') :=
+ ..4'A)Q(;
•,
the error term induced by the approximate fundamental solution Q. It vanishes for all x E X \ ([1/7,6/7] x Y) and also for d(x, x') < 1/7. The following inequality holds: denote
(22.49)
IC(t;x,x')I
for some constants c1,c2 > 0 as t —' Note.
0.
The concept underlying the whole range of arguments
between the preceding Lemma 22.11 and Theorem 22.14 below is referred to as Duhamel's principle. It goes back to Duhamel [1833], where the heat equation with time dependent boundary conditions is solved by taking the derivative of a superposition (convolution) of the solutions at successive times for the corresponding boundary conditions with frozen coefficients, see also Carsiaw & Jaeger [1988; Section 1.141. Similarly one speaks about Duhainel's principle when solving the inhomogeneous wave equation by replacing the force term on the right side by a succession of suitable initial impulses applied at various times to the solution of the homogeneous equation. The underlying idea is always the same, namely • write the error term between the true solution and the approximate solution at time t as an integral over the derivative of a convolution of the true and the approximate solution,
III. Applications
234
• iterate and apply power series analysis to get Levi's sum for the fundamental solution, • derive the needed estimates which ensure convergence and the vanishing of the error term for t —' 0, • and estimate the pole of the heat kernel for t —' 0 up to terms of magnitude (See Levi [1909), Minakshisundaram & Pleijel (1949), and McKean & Singer (1967].)
Proof. Since both e Q —* Id for t — 0, —
and Q satisfy the initial condition E —, Id and we have on the operator level
Q(t) = 6(t)Q(0)
= /
—
e(o)Q(t)
d uS
=
(22.50)
=
=
+
—
jt (e(s){_(A*A+ d(t—
f
s)) ds
))}Q(t — s)) ds
i(s) C(t — s) ds.
Hence on the kernel level we have the equality (22.51)
,
e(t; x, z') = Q(t; x, x') + J
J ox
x, z)C(t — s; z, x') dzds.
Next we derive a formula for the kernel of C(t). We fix an x' E X. Then we have (22.52)
—C(t;x,x')
+A*A) Q(t;x,x')
=
+A*A)
= 0
=
{
(x)6,(t; x, x')i,L',(x') + x for d(x, x') <
(because of (22.47)).
22. The Index Theorem for Atiyah-Patodi-Singer Problems
235
To prove the estimate (22.49) we recall the corresponding bounds and Itc(t; u, y; v, z)I, see (22.31) and Proposition 22.8. for 16d(t; x, Then by (22.52)
IC(t;x,x')I
0 The preceding lemma shows that (0/Ot + ..4*A)Q is exponentially
small as t 0. Now let us find a bound for the true heat kernel S(t; x, x'). Using (22.51) we represent the heat kernel as a series which we can estimate. Let * fl(t; x, x') denote the convolution I.
* f3(t; x,
z') := j
a(s; x, z)f3(t — s; z, x') dzd8. J ox
We have the following sequence of equalities:
S(t; x, x') = Q(t; x, x') + (& * C)(t; x, x')
= Q(t;x,x') + >Q*Ck(t;x,x') k=1
(22.53)
= Q(t; x, x') + (Q *
Ck)) (t; x, x')
= Q(t; x, x') + (Q * C)(t; x, where C1 := C, Ck := C * Ck_1, and
C(t;x,x') := >2Ck(t;x,x'). To obtain the desired estimates of the heat kernels involved, we
need the following generalized triangle inequality.
Lemma 22.12. For any 0 < s
+
(22.54)
Proof. The inequality is obvious in the case
d(x,z) = d(z,y)
(
III. Applications
236
hence assume that a := d2(x,z)
and consider the function 1(8) :=
a
b
+
—
—
Then / is a smooth function on the interval (0, t) and
/ (s)
a
—
b
—
(t — s)2
f' is negative for 8
0
s2b
—
0.
a(t
s2(t
—
—
—
t, hence / has a minimum
and positive for 8
with f'(so) =
at a point
—
There is only one such point, namely
so—t
b—a
Hence the minimum of / is equal to b—a
a
'
b
t
t
and we have
d2(z,y) > (d(x,z)+d(z,y))2 > d2(x,y) t — t
+ t—s
S
0
Lemma 22.13. The following estimate holds for any z, x' E X and
0
IC(t;x,x')I
(22.55)
Proof. Begin with an estimate of C2(t; x, x'). (22.56)
1C2(t;x,x')I = IC * C(t;x,x')I
=
11 0
C(s; x, z) C(t — s; z, x')
SUppz C(ta;z,x')
22.11
fN e_ Lemma 22.12 S
c1 (c1 vol(Y) t)
2(z,/8e_c3d2,x')/(t_8) dzds 2
(z,y)/t
22. The Index Theorem for Atiyah-Patodi-Singer Problems
237
In the same way obtain ICk(t; x, x')I
9c_1
(ci
and finally
C(t; x, x')I c1 c1
vol Y)t vol Y)To e_c2d2(XPX')/t
c3e_c2d2(r,z)/t.
Now we can formulate the main result of this section:
Theorem 22.14.
Let A denote the Atiyah-Patodi-Singer realization of a Dirac operator A+ over a compact smooth Riemannian ndimensional manifold X with boundary Y. Assume that the Riemannian metric is a product close to Y. (a) Then the fundamental solution 6(t) = eLA'A is given by a convergent series of the form
(22.57)
6(t) = Q(t) +
Q(t) *
convolution in t and composition of operators. Here Q(t) denotes an approximate fundamental solution constructed by patching together fundamental solutions of the corresponding heat operators over the infinite cylinder R+ x Y and over the closed double X, C1 := (0/Ot + AA)Q, and Ck := C1 * Ck_1. where * denotes
In particular, for t > 0, 6 has a C°° -kernel which satisfies the
(b)
bound (22.58)
16(t; x, x')I
and differs from the kernel Q(t; x, x') of the approximate fundamental solution on the diagonal by the term t
—' 0
(22.59)
I6(t;x,x) — Q(t;x,x)I
Proof. (a) follows from (22.53) and (22.56) which state that the series defining 6 is absolutely convergent (for bounded t). We prove (b).
Ill. Applications
238
We begin by proving (22.58). According to (22.53) we may estimate Ie(t;x,x')l by the sum of IQ(t;x,x')I and I(Q *C)(t;x,a?)I. By definition, the first summand fits nicely with our claimed bound (22.58). Let us estimate the second summand: (22.60) I
(Q * C)(t; x,
dzds
c1c3
s_n/2e_c3d2,28)
C1C3 .L
dzds cic3e_c2d3(x,v)/(2t)
s_n/2e_c2d2(x,Z)/(28) dz
•
To verify that (22.61)
we use a partition charts such that
,z)/(2a) dz
Ix
of unity suitable to a covering of X with local
dz =
(z)
. . .
dz.
Focusing on one chart one gets (22.62)
Ix Xa(z)s
,z)/(2a) dz
dz
J
dz1
= Jftn
. . .
dz,1
by v :=
= Thus (22.61) is deduced, hence (22.60) and (22.58). It remains to show (22.59). It follows from (22.51) that I.
e(t;x,x)—Q(t;x,x) = / / Jo
e(s;x,z) C(t—s; z,x)dzds. C(t—s;z,x)
22. The Index Theorem for Atiyab-Patodi-Singer Problems
239
Thus we may assume The integrand vanishes whenever d(x, z) < that both e(t; x, z) and C(t; x, z) are bounded by for
x,z E X with d(x,z)
and 0< t
(22.63)
— Q(t;x,x)I
fJ
<
dzds C(t—s;z,x)
0
Pt
<
P
I I
e_cboh/(493)e_dboh/49(t_8) dzds
Jo
C(t—s;z,x) .
<
ds
vol(Y)
Remarks 22.15. (a) The result remains true for the heat kernel of the operator and the corresponding parainetrix (t; z, x') as well.
(b) The computations here are simplified by the fact that we restricted our considerations to the bounded time interval [0, To]. This does not work in more complicated cases when we must take larger time asymptotics into account (see Douglas & Wojciechowski [1991] and Klimek & Wojciechowski [1992), [1993]).
22D. Proof of the Index Formula Now we can prove the index formula for the operator A. The results of the previous two sections imply the following equalities: (22.64) urn
index Thm. 22.14
=
lirn {trQ(i)
—
—
t—4o+
{j 22.lOb
DeL 22.6c
urn
J F(t;x)iP2(x)dx} {j°° J K(t; (tz, y)) dydu +
{x(t)
F(i; x)1/,2(x)dx}
+ fx F(t;x)1P2(x)dx},
where ?C(t; u, y) and ftC(t) are defined as in Definition 22.6c and
F(t;x) :=
—
III.
240
Recall (see Gilkey [1984], Roe [1988], or Berline, Getzler & Vergne [1992]) that for small t, we have an asymptotic expansion k—n
where the coefficients are explicit local functions of the operaand and A"A over X. Since the operators tors are isomorphic in the collar N, via C, one has F(t; x) 0 for x in the F(t; x) and so (22.64) gives an asympcollar. Hence F(t; totic expansion for A(t):
A(t)riindexA — k_—n
(22.65) —
k—n
Applying Corollary 22.9, equation (22.41), we deduce that (22.66)
I
r(s+
+ indexA s
N
ak
+ON(8)
)
k=—n+1
where h := dim ker B and 0N (8) 15 holomorphic for n(s) > — This proves the following result: Proposition 22.16. Let Y be a closed Riemannian manifold and let B: C°°(Y;S) -' C°°(Y;S) be a Dirac operator which extends in the following sense: There exists a smooth compact oriented Riemannian manifold X with boundary Y S and and a bundle Sx over X of Clifford modules such that the operator B is the tangential part on Y of the corresponding Dirac Then the tj-function 'SB(s) of B extends meromorphically to the whole complex plane with simple poles at Sk = — k for k E N and :
Res$1k 'iB(s) = ak. In particular, 77B(5) is regular at s = 0, and we have (22.67)
riB(O) =
2J
ao(x)dx — (h + 2indexA),
defined by the AtiyahPat odi-Singer boundary condition and ao(x) denotes the index density for the operator A+ on X.
where A denotes the L2-realization of
22. The Index Theorem for
Problem8
241
Remark 22.17. Since that result was proved by Atiyah, Patodi & Singer, the structure of 'i is better known. We do not discuss that topic here, but simply refer the reader to Gilkey [1984], Bismut & Freed [1986], Bismut & Cheeger [1987], [1990aJ, [1990b], Branson & Gilkey [1992a], [1992b], Muller [1993], and the references given there.
Formula (22.67) is known as the Atiyah-Patodi-Singer index formula. We summarize the results of the whole chapter:
Theorem 22.18. (Atiyah, Patodi & Singer, 1975). Let X be a compact oriented Riemannian manifold with boundary Y and let
C°°(X;S)
:
that all the structures are products in decomposes over N into the special form
be a Dirac operator. Assume
N of Y. Then
collar
a
where u is the inward normal coordinate, C is a unitary bundle isomorphism S Ii', and B is the corresponding (seif-adjoint) Dirac operator on V. Moreover, (and A—) extend canonically to operators (and A—) over the closed double X of X. Let C°°(X; P>) denote the space of spinors s of positive chirality satisfying the boundary condition
P s(.,O)=O, where P> denotes the spectral projection of B corresponding to the eigenvalues 0. Then
C°°(X;S)
:
has
a finite
index given by
index
(22.68)
= where
h,
(i)
t —*
77B
are
J
h + 77B(0)
defined as follows:
is the constant term in 0) of the trace differences
e_t141(x)12 pEspec A+A
—
ao(x)
the asymptotic
—
AA+
expansion (as
242
III. Applications
denote the eigenvalues and eigenftinctions of where p, on the double of X, and p', are the corresponding objects for (ii) h := dim ker B = multiplicity of O-eigenvalue of B. (iii) 17a(8) sign AIAI8, where A runs over the eigenvalues of B. In (iii) the series converges absolutely for large, and then T7B(s) extends to a meromorphic function on the whole s-plane with a finite value at S = 0. Moreover, if the asymptotic expansion in (i) has no negative power oft, then 71B(s) is holomorphic for p(s) > — 22E. L2-Reformulatlon
To understand the Atiyah-Patodi-Singer index formula for a Dirac operator over a compact manifold X with boundary Y (and with + B) close to Y) in its correct analytical context, one has A= to reinterpret the action of the spectral projection P> defined by the tangential part B of which constitutes the Atiyah-Patodi-Singer boundary condition. Recall the analytical context of the Calderón projector namely its relation to the Poisson type integral which provides for all sections f over Y a global solution s of =0 with Roughly speaking, the action of the Calderón = projector is related to the inner extension problem of sections over the boundary to global solutions. The null space ker of the corresponding boundary problem, however, is uninteresting since by definition it consists only of the zero section. Now the remarkable fact is that the action of the spectral projection is opposite directed, namely toward the outer extension problem in such a way that can be identified with • the null space kerA (with A := the square-integrable solutions of = 0 on the complete non-compact elongation obtained from X by attaching the semi-infinite cylinder (—oo, 0] x Y to the boundary (see Figure 22.5); • the solution space kerA (with A* := a.) can be identified with the direct sum of all square-integrable solutions of A;s = 0 with a certain subspace of ker B which consists of the limiting values of all locally square-integrable solutions; • and that the dimensions of the two spaces of limiting values coming from = 0 and A,s = 0 add precisely to the
22. The Index Theorem for Atiyah-Patodi-Singer Problems
243
dimension of the null space of the tangential part B. (For the general case Muller [1988] and Melrose [1992, Chapter 7]).
xY
Fig. 22.5 The non-compact elongation We define:
Definition 22.19. Let X be a compact oriented Riemannian manifold with boundary Y, and let
C°°(X;S)
:
be a Dirac operator. Assume that the Riemannian structure on X and the Hermitian structure on S are products in a collar neighbourhood of Y. In particular, there we have A = with interior normal coordinate ti, unitary C, and self-adjoint B. (a) Since X is isometric to the cylinder Y x [0,1] near the boundary,
and since the Hermitian structure on S is product there, the noncompact Riemannian elongation
U (-oo,0]xY and related extensions of the bundles of Clifford modules to and of the Dirac operators to are well-defined and the unitary isomorphism C extends in the cylinder as well. (b) We define the space of L2-solutions of = 0 on := {s E
I
= 0}
and similarly (c) By an extended L2-solution of the equation = 0, we mean a solution which belongs to St,) such that, for large negative u, we have
s(u,y) =
8L2(U,y) + BIjm(y),
III. Applications
244
where
and 811m E ker B. Thus s has 8Iim as or limiting value as u —' —oo. We shall write
E
asymptotic
an
SE
A similar definition holds for extended L2-solutions $
+ 8ljm of
A;s = 0, but demanding G*SIim E ker B. (d) Let fl.1 C ker B denote the space of all limiting values of extended L2-solutions of =0, and define in a similar way C Gker B.
Proposition 22.20. As before, let A denote the closed operator from L2(X; to L2(X; Sj which acts like and is determined by domA = {s E H'(X : = 0}. Then we have natural I
isomorphisms ker A
(i) (ii)
and
L2
kerA*
fl.....
Due to the regularity of solutions of elliptic boundary
Note.
problems (Theorem 19.1) we have
kerA =
—'
:
G°°(X;S)}
and similarly kerA* = ker{A : C°°(X; S; P< G*)
Proof.
(i) Let
s
e ker A
C°°(X;
and expand it near the boundary in the
form
s(u,y) = is a spectral resolution of L2(Y; generated by B. Since = 0, we have ôs/Ou + Bs = 0, so that Since P>.(sly) = 0, we have fA(O) = 0 for A 0 fA(u) = and so where {A;
(22.69)
s(u, y) = >
fA(0)
A
This shows that s extends to a section on X, which satisfies = 0 and is exponentially decaying as u —co, hence certainly in L2. Conversely, a solution of
= 0 on
must obey
22. The Index Theorem for Atiyah-Patodi-Singer Problems
245
= 0 on the semi-infinite cylinder, so that it must be with of the form (22.69) if it is in L2, because terms involving gives the required isomorphism. A 0 are not in L2. Thus s s—' (ii) Let 8 E kerA*, then instead of (22.69) we get
+
(22.70)
s(u,y) = A>O
near the boundary. Here, for simplicity, we have suppressed the action
of C or, put otherwise, set C = Id. The natural extension of s to a is now an extended L2-solution (as defined above section on in Definition 22.19c) with limiting value Slim(Y) =
(Note that the summation is over the finite multiplicity of the 0is necessarily of the eigenvalue). Conversely, any E also gives an isomorphism. form (22.70) so that s i—' [J Since the cokernel of an elliptic boundary problem can be identified
with the null space of the adjoint boundary problem (Theorem 20.8), we deduce
Corollary 22.21. The index of the Atiyah-Patodi-Singer boundary A+ over a compact manifold problem A = X with boundary Y can be expressed by (22.71)
index A =
—
h_,
where h_ := dim 1L and fl.-. denotes the subspace of C(ker B) consisting of the limiting values of extended L2-solutions of = 0 (see Definition 22. 19d).
Let now h+ := dim 7t÷, where
denotes the subspace of ker B consisting of the limiting values of extended L2-solutions of = 0,
and recall that h := dim ker B. Then the non-negative integers h± are less than h and in fact add to h.
III. Applications
246
Proposition 22.22. For the three dimensions defined here, one has the formula
h = h.,. + h_.
(22.72)
Proof. First merge (22.68) and (22.71) yielding dx — h + 7)B(O)
(22.73)
=
h_.
—
Then deduce a formula similar to (22.73) with some but not all o the signs reversed. Consider the operator A := Aj<0 G• It has a slightly larger domain than A = c. and hence a different index. Applying (22.71), one gets
indexA= L2-index(A)
(22.74)
—
h÷.
Now A acts like A which near the boundary takes the form
From the construction of the index density it follows that ao(A) = hence applying (22.68) gives (22.75)
indexA= —
Note that h = 'lB(O)
dim ker
J
ao(x)dx — h + ?1...B(O)
B = dim ker(—B) is unaltered and that
= —71B(O). Thus (22.74) and (22.75) yield
(22.76)
Since by definition
dx — h
—
11B(O)
=
(22.72) from adding (22.76) and (22.73).
= L2-index(A)
—
we deduce
9
There is one more interesting observation regarding the L2-index
and the hi-dimensions:
22. The Index Theorem for Atiyah-Patodi-Singer Problems
247
Proposition 22.23. For any Dirac operator A+ on a compact manifold X with boundary Y, we have on the non-compact elongation
= =
(i) (ii)
The same also holds with
and
and A— interchanged.
Note. On the compact manifold X we have already seen in Lemma 22.ld that kerA coincides with kerAA taking regard of the respective boundary conditions. The preceding proposition claims that similar results hold for L2-sections on
Proof. We argue as in the proof of Proposition 22.20. Any solution of =0 is, expanded on the cylinder (—oo, 0] x Y, necessarily of the form
s(u,y) =
(22.77)
+
be rapidly decreasing as u —
—oo.
More precisely,
denote the compact manifold X U ([u, 0] x Y) C and let denote its boundary {u} x Y which is isometric to Y. Then we can estimate let
witha>0,
(22.78)
where &so
II denotes the L2-norm on Y
and C is independent of ti, and for all ti
(22.79)
0
for a suitable choice of a. Since A (and A;) are the formal adjoints - we suppress the subscript oo in the following), and A= we get B) also near = 0 and Green's formula (cf. Proposition 3.4b)
of (and since from
= (22.80)
=J
—
(G*(y)A+s(y);s(y))dvol(y) G(y)s(y))dvol(y)
= JY" / (22.78),(22.79)
0
asu—'—oo.
III. Applications
248
This proves (i). Now consider a section 8 E Then its expansion (22.77) can have non-zero terms a0 and bo; so instead of (22.78) we can only assert that —oo. HowIlls bounded as u ever, applying removes these terms and so (22.79) holds. This is enough to show that the integral in (22.80) tends to zero as u —' —00. As before we conclude that = 0, proving (ii). The corresponding statements with A interchanged are proved in exactly the same way.
22F.
[]
The Odd-Dimensional Case.
A Three-Dimensional
Example The proof given for the Atiyah-Patodi-Singer index theorem, Theorem 22.18, is valid in the odd-dimensional case as well, although in this case the index formula for A is simpler. Let X be an odd-dimensional compact smooth Riemannian manifold with boundary Y. Assume that, near Y, X is isometric to the product [0,1] x Y; here {0} x Y = OX. Let S be a bundle of Clifford modules over X. Let A : C°°(X; S) —, C°°(X; 5) denote the corresponding Dirac operator. Recall from Theorem 21.5a that then A has a decomposition near Y of the form
A=
+ B),
where r is a unitary bundle automorphism with r2 =
—
and
hence r defines a decomposition of Sly into the direct sum With respect to of the subbundles of the ±i-eigenvalues of this decomposition the operator A takes the following form near
(i o\(
(0 B 0
Observe that the formal self-adjointness of A = + B) implies —Be. That leads to the following simplification of Theorem
rB = 22.18:
Theorem 22.24. Let A := Ap>(B) denote the standard AtiyahPat odi-Singer realization of a Dirac operator over an odd-dimensional compact manifold X with boundary Y. Then (22.81)
index A = — dim ker
22. The Index Theorem for Atiyah-Patodi-Singer Problem8
where
249
comes from the splitting of A near Y into
A=r(OU+(B°+
a)).
Proof. If
is an eigenfunction of B, BcoA = with A 0, then rwA is an eigenfunction of B with eigenvalue —A. That means that the spectrum of B is symmetric with respect to 0, hence
AEapecB,
Moreover, we have proved above that index B+ = 21.5b). That means that dim ker B = dim ker dim ker
(Theorem B Then we get for the standard Atiyah-Patodi-Singer re0
alization of B, from the results of Atiyah, Patodi, and Singer mentioned above, that index Ap>(B) = Jao(x)dx —
+ dim ker B) =
since the index density a0 is equal 0, when dim X is odd (see e.g. Gilkey [1984; Theorem 1.7.6.a] and Berline, Getzler & Vergne [1992; Section 4.1]). [1
Remark 22.25. From our previous discussion (Theorem 20.8), we have
indexAp>(B) = i(P(B),P+(A))
(B) and (A) belong to and the index vanishes, if and only if Hitchin [1974] showed that the same connected component of dim ker B varies when we deform the metric (the most simple nontrivial examples are all Riemann surfaces Z with genus g > 3). This means that P(B) and P÷(A) change their connected components independently of each other and that under a smooth change of the metric the operator Ap(B) varies in a non-continuous way. So the homotopy properties of the Atiyah-Patodi-Singer boundary value problem differ completely from the familiar homotopy invariance
of the index of local elliptic boundary value problems. For them the homotopy invariance follows from Theorem 20.13.
250
111.
Applications
We illustrate the application of the Atiyah-Patodi-Singer index formula with an example which has gained some prominence in recent geometry and gauge theory, see e.g. Taubes [1990; Section 2, Proposition 4.9, and Lemma A.4] and Yoshida [1991; Sections 1-4J. Let X be a compact oriented Riemannian manifold of dimension dim X = 3 with connected boundary Y. Assume that, near Y, X is isometric to the product [0,11 x Y; here {0} x Y = OX. Assume also that the genus g of Y is 2. We define a twisted signature operator on the space (22.82)
0 su(2))
W :=
(Il' (X) ® su(2))
of differential 0- and 1-forms with coefficients in the bundle X x su(2). Here su(2) denotes the Lie algebra of SU(2) and IZP(X) = C°°(X; APTX), p = 1,2,3,4 denotes the space of smooth differential p-forms on X. Recall that every principal SU(2)-bundle over X is isomorphic to the trivial bundle X x SU(2), and that the space of smooth connections on Xx SU(2) can be identified with (11 (X)®su(2).
The Riemannian metric on X and the associated Hodge star operator * : AP(TX) —, ASP(T*X) define an V inner product (22.83)
(a; b) := —
f
tr(a A *b)
for a, b E
0 su(2).
Example 22.28. Let A be a smooth flat connection on X x SU(2) which restricts to a product connection B x Id in the collar neighbourhood [0, 1]xYxSU(2) ofYxSU(2) for an irreducible fiat connection B on Y x SU(2). We define the corresponding twisted signature operator 'PA : W W with coefficients in X x su(2) by (22.84)
'PA(a, b) :=
*dAa + dAb),
where a is a 0-form, b is a 1-form, and dA denotes the covariant derivative.
Here we follow the terminology of gauge theory and denote connections by the letters A and B which otherwise were reserved for elliptic operators in this book. The twisted signature operator is an operator of Dirac type (i.e. with principal symbol defining a Ct(X)module structure on the bundle A*X x su(2)) and hence elliptic: Its principal symbol is not inflicted by the choice of a flat connection and therefore coincides with the principal symbol of the (untwisted) signa-
ture operator. it is, however, not a true Dirac operator in our sense,
22. The Index Theorem for Atiyah-Patodi-Singer Problems
since it is not defined by a compatible connection. Actually, we have not used any compatibility of the connections with Clifford multiplication for the proof of the Atiyah-Patodi-Singer index theorem and may therefore now conclude:
Proposition 22.27. The index of the twisted signature operator 'PA with Atiyah-Patodi-Singer boundanj condition is well-defined, does not depend of the connection A, and can be expressed by
index('PA)p> =3—3g,
(22.85)
where g denotes the genus of Y.
Proof. In the collar [0, 1] x Y, any element (a(u), b(u)) decomposes as a triple (22.86)
(a, b) = (a(u), b(u) = p(u) + q(u)du) =: (p, q, a).
Then 'PA can be written in matrix form
fp\
(22.87)
'PA
(
fp\
qJ
(
q
where
(22.88)
1*0 r := ( 0 0
o\ Id I \o —Id 0)
and
'PB :=
f (
0
(dB)
\—*(dB)*
dB*dB 0 0
0 0
Here * denotes the Hodge star operator on (T Y) for j = 0, 1 and its section space and dB denotes the covariant derivative. Clearly we have (22.89)
r'PB+'PBr=o and I'2=—Id.
Hence we can apply Theorem 22.24 and get for the corresponding Atiyah-Patodi-Singer boundary problem (22.90)
index(4DA)p> =
It is well-known that ker'PB
dim ker'PB.
252
III. Applications
where
:= {w E 1l'(Y) ® su(2)
= 0=
(see Taubes [19901, Yoshida [19911 and the references given there).
Moreover, since B is irreducible, the covariant derivative dB has no is a (6g — 6)-dimensional real vector non-trivial kernel in 11°, and space isomorphic to the de Rharn cohomology groups HA (Y; su(2)) with su(2)-valued local coefficient system defined by the holonomy representation of B. This proves the formula (22.85). 9
23. Some Remarks on the Index of Generalized Atiyah-Patodi-Singer Problems
By combining the Atiyah-Patodi-Singer index formula with our theory of boundary integrals we obtain explicit formulas for all boundary problems in the infinite-dimensional Grassmannian
and a non-additivity theorem for the index of Atiyah-PatodiSinger boundary problems under pasting. We introduce an alternative adiabatic Atiyah-Patodi-Singer boundary problem which returns the signature as its true index.
We have the following immediate corollary of the formula (Proposition 21.4) and the index formula for Atiyah-PatodiSinger boundary problems (Theorem 22.18).
Theorem 23.1. Let P E
denotes the principal symwhere bol of the Calderón projector (and the spectral projection P> as well). Then the following formula holds: (23.1)
indexAp
+ dim ker B) + i(P,
=
—
Proof. Recall the definition of the virtual codimension (Definition 15.8)
i(P, P>) := index {P
: range P
formula (Proposition
Then (23.1) follows from the 21.4) which gives
indexAp
=
range P}.
index
+ i(P, P>),
and the Atiyah-Patodi-Singer formula for index Ap>.
U
Note. In Kori [1993; Theorem 6.4] the situation on S3 and S4 is discussed and the following formula is derived for X = D4: (23.2)
indexAp = i(P,P>).
III. Applications
254
This is in accordance with our result: The index formula depends on the metric on S4, and in Kori's metric, the boundary component operator B is invertible. Moreover, he proved directly that, in the situation he considered, the spectral projection P and the Calderón projector coincide. We refer to Kori [1993) for further details. We use Theorem 23.1 to discuss the non-additivity of the index. From a topological point of view, one special feature of the operator Ap> is that its index is non-additive, i.e. two conBistent AtiyahPatodi-Singer boundary problems do not in general sum up to the index of the corresponding Dirac operator over a closed manifold: Let X1, X2 be two compact Riemannian manifolds with boundary Y, and let A3, j = 1,2 be Dirac operators on X, such that, in a certain collar of Y, they are consistent in the following form: (23.3)
A1 = G(OU + B),
A2 =
—
where v is the normal inward coordinate on X2 which we can identify
with —u. We can apply the glueing procedure from Chapter 9. Notice on the collar. As a result of the pasting, that A2 is formally equal to we obtain an operator A = A1 U A2 on the manifold X1 U X2. One would like to have an equality (23.4)
Here
+
index(Ai U A2)
:= P(A,)>o, j = 1,2 denotes the Atiyah-Patodi-Singer
boundaiy operator corresponding to the operator A,. Recall that for any real a, P(Ai)0 denotes the orthogonal projection onto the subresp. onto the subspace of L2(Y; S space of L2(Y; by the eigenfunctions of B, resp. the eigenfunctions of
spanned
correbe a spectral sponding to eigenvalues greater or equal a. Let {A; := generated by B and {—)t; resolution of L2(Y; the corresponding resolution of L2(Y; S Iv) generated by —GBG'. in particular then denotes the orthogonal projection onto P0 := the a-eigenspace V0 := £c{cp,, I = a} of B. Then sb-A
otherwise
{ 0
and
° —
=
{
ifA=O
0 otherwise,
23. Some Remarks on the Index
255
hence (23.5)
+ Po)G".
= G(Id —
The conjecture of equation (23.4) does not hold in general, or, more precisely, it holds if and only if ker B = {O}.
Proposition 23.2. For Atiyah-Patodi-Singer boundary value problems, the following non-additivity holds under glueing: (23.6) index(A1)pi> + index(A2)p2> = index(Ai UA2) — dim kerB.
Proof. This follows directly from Theorem 22.18 which yields
=
aj(x) —
(dim kerB + 1713(0))
and
= =
J a2(x) —
/ a2(z) —
(dim ker GBG' + ll_GBG-1 (0))
2
(dimkerB — '7B(O)).
We give the most general result of this type. Much deeper nonadditivity results can be obtained for spectral invariants like the analytic torsion and the ti-invariant (see Klimek & Wojciechowski [1992], Matteo & Meirose [1992], BooB-Bavnbek & Wojciechowski [1993], Bunke [1993], Luck [19931, and Muller (19931). The following result also generalizes the calculations about exchanges on the boundary for the Cauchy-Riemann operator given in Example 21.1.
Theorem 23.3. Let X1, X2 be two compact Riemannian manifolds with common boundary Y and let 4,, j = 1,2 be Dirac operators on X3 satisfying the consistency condition (22.1). Let a E R and let (A3)0 denote the operator We have (23.7)
index(Ai)0 + index(A2)_0 = index(Ai U A2) — dim V0
and (23.8)
index(Aj
+ index(A2)0 = index(Ai U A2) +
P!,
— dim V_0.
III. Applications
256
Remark 23.4.
Recall
that
denotes the weighted spectral projec-
tiori of the operator B, namely the orthogonal projection onto the subspace Va spanned by the eigensections of B corresponding to eigendenotes the corresponding weighted values in [a, oo), and that
spectral projection of the operator —CBG'. Moreover, Va denotes the a-eigenspace of B (if a E spec(B); otherwise V0 := {O}). By Moreover, for a b c we have definition we have = and by Proposition 15.15 =
+
(23.9)
Ph).
=
Finally, for a> b, we have two short exact sequences p1
p'
range
range
—'
V1,
{O},
b
which yield
dim VA and
=
=—
dim VA.
bA
In particular we get that
is equal to the dimension of the direct sum of eigenspaces corresponding to the eigenvalues of B in the interval [—a,a).
Proof of Theorem 23.3. First we express the spectral projections of —GBG' in terms of the spectral projections of B, as in the calwe get culations for (23.5). From = (23.10)
= G(Id—PL0+P0)G'.
We have already proved (Proposition 23.2) that index(Ai)o + index(A2)a = index(Ai U A2) — dim kerB.
Let us choose e> 0 such that specBfl(—2e,2e) C {0}
and
specBfl(—a—2e,—a+2e) C {—a}.
23. Some Remarks on the Index
257
Then clearly
=dimkerB
and
=
We also notice
index(Ai)o = index(Ai)o +
= index(Ai)o
for —2e < S 0. With (23.10) we obtain (23.11)
= I(C(Id — PLa +
= i(Id
= •lThl =
—
P_0), Id
—
Po))
1T)l rO,r>_0 — F_a/\ = 1*1>0 — fl \ .ini ni — FO, —
flu
'>—o+e
—
P0,
—
dim V_0
PLa) — dim V_a.
=
formula (Proposition 21.4)
We use the generalized and obtain (23.12)
+ Po)G')
G(Id —
and
index(Au)0 = index(A1)o +
index(A2)>0 = index(A2)>o +
We have = {index(Ai U A2) — dimkerB}
lfldeX(Ai)a +
+
+ +
{... }
+ i(PL, PLC) +
—
dim V_a
PLa) — dim
= index(Ai U A2) +
+ = index(Ai U A2) +
—
dim V_0
PLa) — dim V_6,
258
III. Applications
which proves (23.8). In the same way we prove (23.7):
index(Ai)0+ index(A2)_a = {index(Aj U A2) — dim ker B}
+ (Dla' Dl—e/\ + •(fl2
+ i(PL, = {.. . } + — dim V0 = {. } + =index(A1UA2)—dimV0.
' —
dim
Va
El
Theorem 23.3 shows that we can not expect additivity of the index for Atiyah-Patodi-Singer problems. In the following we discuss the important case of the signature operator. For this operator we give a natural boundary condition which provides the desired additivity. First we notice that the signature of X is not the index of a true Atiyah-Patodi Singer problem From Atiyah, Patodi & Singer [1975, Section 4, Theorem 4.14 and Equation (17.9)J we extract the
two following formulas:
signX index
= =
£(x)
£(x)
—
h
—
—
and 7lBev(O),
and hence the signature deficiency fonnula:
signX = indexAp> + h.
(23.13)
Here X is an oriented Riemannian manifold of dimension 4k with smooth boundary Y (and X is isometric to a product near Y), A IL. denotes the signature operator introduced in Chapter 7 earlier. The signature operator takes the form A = + B) close to Y. The tangential component B splits into B = Be" + Bodd with isomorphic to (23.14)
(0)
=
Hence (0)
and
h := dim ker Be" =
dim ker B.
Moreover, C(x) := Lk(pl(x),. .. where Lk denotes the k-th Hirzebruch L-polynomial and the p, denote the Pontrjagin forms of the Riemannian metric. We introduce the notation V
and
23. Some Remarks on the Index
259
so orth.
V>.
There exists a natural projection PE : to the Grassmannian Crp+ such that (23.15)
—,
range PE belonging
signX = index ApE,
or equivalently, by (23.1), such that (23.16)
index{PEP : rangeP>
= indexApE
—
indexAp> = h.
Clearly the projection PE onto any subspace E of range P> of codimension h will provide (23.15), since then PEP : rangeP> —* E is surjective and rangeP>/E. We use results of Section 22E to describe a choice of E which appears in this context in a natural way. There is actually a very natural subspace of range = V = V> + ker B of codimension h, namely the space V> + (23.17)
V> :
where
£c{cOA A > O}
and 71÷ denotes the space of limiting values of the extended solutions, as introduced in Definition 22.19d. Then the projection onto V> +11÷ will do the job of our desired alternative signature-amended AtiyahPatodi-Singer boundary condition PE. We recall from Section 22E the concept of a non-compact RiemannIan elongation of related extensions, especially of the signature operator A to and the concept of an extended solution (extended square-integrable harmonic form of even parity). As seen above, these concepts are meaningful in much more general situations. The following lemma, however, depends heavily on the internal symmetry of the
tangential part of the signature operator. The lemma is proved in Atiyah, Patodi & Singer [1975; pp 64—66 (before Theorem 4.14)] but not explicitly stated there:
III. Applications
260
Lemma 23.5. Let fl÷ denote the subspace of ker(B) which consists of the limiting values of extended L2-harmonic forms of even parity. It has dimension h = dim ker Be" = dim ker B. We fix the orthogonal projection ê ker B —. ker B of ker B onto N÷ and introduce the projection P> of the L2 differential forms of even parity over Y onto the subspace V> defined in (23.17). Now we define (23.18)
PE:=P>+e.
is only a perturbation of the pseudo-differential projection P by an operator of finite range, we obtain from equation (23.1) with Lemma 23.5 and equation (23.16) the following theorem: Since
Theorem 23.6. The projection FE defined in equation (23.18) beand defines a generalized Atiyahlongs to the Grassmannian Pat odi-Singer boundor,j value problem with
index ApE = sign X.
Corollary 23.7. We have (23.19)
L2-index(A)
=
J a(x)
—
IlBav(O),
denotes the non-compact elongation of X, the L2-index is defined as the difference of the dimensions of the spaces of 'extended solutions' of even and odd parity over the elongation (see Definition 22. 19d above), A denotes the signature operator of X, and Cr(x) denotes the index density of A which is naturally extended over the whole of the elongation. where
Remarks 23.8. (a) The preceding theorem permits a reformulation of the cutting and pasting invariance of the signature which we shall discuss in Chapter 25. Moreover, it gives a nice frame for the analytical explanation of the Novikov additivity of the signature (see e.g. Jänich [1968], Karras et al. [1973], and BooB & Rempel [1982]) and, when treating Riemannian manifolds with corners, an analytical explanation of the Wall non-additivity, see Wall [1969], Rees [1983], and Melrose [1992].
23. Some Remarks on the Index
261
(b) Theorem 23.6 and Corollary 23.7 do not hold for arbitrary Dirac operators. The reason is that in general h = h(A) h(A). However, one has the following formula for the L2-index (Corollary 22.21): (23.20)
indexAp> = L2-index(A)
—
and more generally
(23.21)
L2-index(A)
=J
a(s) —
—
(h(A)
—
h(A)).
See also Meirose [1981] and Roe [1988].
Anyway, we have no problems calculating the L2-index in the case of vanishing ker B.
Corollary 23.9.
Let A be a Dirac operator with non-singular tan-
gential part B. Then index Ap> = L2-index(A)
=
J a(s) —
Notice that we used the preceding corollary in Section 17D above.
One more reason for the choice of our alternative boundary condition is the odd-dimensional case. It is well-known that for dim X odd, Ap> is not a seif-adjoint operator whenever ker B {0}. See BooB & Wojciechowski [1989; Proposition 5]. However, one obtains that ApE is an elliptic seif-adjoint problem in this case.
24. Bojarski's Theorem. General Linear Conjugation Problems
We recover the index of a Dirac operator A over a closed par= c9X_ = fl titioned manifold M U X_ with X_ = Y from the Fredholm pair of Cauchy data spaces along Y. Similarly, the index of the linear conjugation (or trnnsmis= (s.,.jy) is given sion) problem = 0 in by twisting the Cauchy data spaces with 4. Related local elliptic boundary conditions for systems of Dirac operators are considered.
In this chapter we present some applications of the theory developed in the previous chapters and explain the geometric meaning of the spaces of Cauchy data and of the Calderón projector, i.e. objects which live on the submanifold Y but originate from global data. We deal with a closed Riemannian manifold M which is partitioned into two manifolds with joint boundary: (24.1)
C°°(M; F) be a Dirac operator on M. We assume the splitting form (9.3) in a bicollar neighbourhood of V (see
Let A : C°°(M; E)
Figure 24.1). Then on V and close to V we have the objects and relations introduced earlier in a slightly more restrictive situation, namely for the closed double M := K,. and A :=
Theorem 24.1. Let to the manifolds
A.,. U (A+)*.
denote the restrictions of the operator A with M, A as above. Then the following
equality holds:
(24.2) index A =
:
The theorem was conjectured by Bojarski in the middle of the type proof seventies. He envisaged it as the first step toward a of the Atiyah-Singer index theorem via a double induction argument: with respect to the number of handles in a Morse decomposition of
24. Bojarski's Theorem. General Linear Conjugation Problems
263
M
Fig. 24.1 The partitioned manifold M with bicollar neighbourhood M of Y
the manifold, and with respect to the dimension of the manifold. This will be discussed (to some extent) in the next chapter. Bojarski formulated his conjecture as the equality of the index of A and the index of the Fredholm pair of Cauchy data of the operator A along the submanifold Y. Let us explain his formulation. We begin with the concept of a Fredholm pair of subspaces.
Definition 24.2. Let H1, H2 be closed infinite-dimensional subspaces of a separable Hubert space H. We call (H1, H2) a pair of subspaces, if the following conditions are fulfilled: (24.3) (24.4)
a = dim(Hi fl H2)
The integer a — /3 is called the index of the pair (H1, H2).
The notion was introduced by Kato [1976; IV.4.1J and used by Bojarski in his paper [1979] about the abstract linear conjugation problem for elliptic operators.
Clearly we have a natural embedding of the space of Fredhoim operators on H into the space of Fredhoim pairs of subspaces of H x H given by F s—' (H x 0, graph F) which preserves the index. More generally we have:
Lemma 24.3. A pair (H1,H2) of closed infinite-dimensional subspaces of a (separable) Hubert space H is a Predhoim pair, if and only
III. Applications
264
if the operator (Id—P2)P1 : H1
H21
is a &edholm operator. Here P, denotes the orthogonal projection onto H,, i = 1,2. In this case we have index(H1, H2) = index(Id —P2)P1 = i((Id —P2)P1).
(24.5)
Proof. We have ker(Id—P2)P1 ={f E H1 I(Id—P2)f=O}= H1 ni?2 and
coker(Id—P2)Pj ={fe
=(H1+H2)1,
which proves the lemma.
[I
Proposition 24.4. Let (Hi, H...) be a pair of orthogonally complementary subspaces of a fixed Hubert space H with orthogonal projections of H onto Then for any bounded invertible operator on H, which commutes with P+ modulo compact operators, the pair H_) is a Fredholm pair of subspaces and we have
index(4H+, H_) = index(P_
(24.6)
—
H_) is just a Fredholm pair of (not necessarily orthogonally complementary) subspaces of H, we also get that Note.
If
H...) is a Fredhoim pair by compact perturbation. However, formula (24.6) has to be replaced by (24.7)
= = index{(Id — P4c17'÷ —, = H_) + index(P_ — :
See also Theorem 24.9 below.
Proof. From Lemma 24.3 we obtain
H_) = index{(Id =
—'
:
4iH÷
H
—,
differs from (Id —P.,.) — But (Id —'P.,.) — = only by a compact operator which proves our assertion.
H}. —
9
24. Bojarski's Theorem. General Linear Conjugation Problems
Remark 24.5.
265
Lemma 24.3 has a series of nice consequences for the
homotopy type of the space of Fredholm pairs of subspaces when this space is endowed with a metric inherited from the operator norm of the corresponding orthogonal projections by the formula (24.8)
p((Hi,H2),
:= lIP1
—
+ flP2 —
All the appropriate technical results were proved in J3ooB & Wojciechowski [1986, Section 2]. For abbreviation, we shall write Fred2(H) for the space of Fredhoim pairs of subspaces of a Hubert space H, and Fred2 H1 (H) for the subspace of all Ftedholm pairs (H1, 112) with fixed H1. Then we have:
(a) A pair (H1, H2) of closed infinite-dimensional subspaces is Fredhoim if and only if the difference (Id —P2) — P1 of the corresponding projections is a compact operator. In fact, we rewrite Id—P2 = g'P1g with a suitable unitary operator g, and in par= 9'H1 and ticular, we obtain (Id —P2)P1 = g'P19P1
:
H1
gH1.
Hence (H1, H2) is a Fredholm pair if and only if the operator P19P1: H1 is Fredhoim. That is the case if and only if g (and hence H1
g') commutes with P1 modulo compact operators. Hence (Id—P2)—P1 =g'Pig—Pi =(g'P1
a compact operator. We also see that then (Id —P1) + 9P1 is a Fredhoim operator (with (Id —F1) + P1g' as parametrix) and obtain as in the proof of Proposition 24.4 is
(24.9)
index(Hi, (g'Hi)1) =
index ((Id —P1) + gPi).
a fixed orthogonal projection with infinitedimensional range and kernel. We have seen that any Fredhoim pair is of the form (Hf, (Id (H1, 112) with = 1.'÷H =: for some unitary operator g which commutes with P+ modulo compact operators. This provides an identification of Fred2H+ with the (b)
Now let 1'.4.
be
homogeneous space (24.10)
/U(H÷)
III. Applications
266
where U denotes the spaces of unitary operators and the space of unitary operators on H which commute with
denotes modulo
compact operators. The space of (24.10) has the homotopy type of the space of Fred-
hohn operators on H. This follows from the fact that the natural projection Up÷
-,
Up÷/U(H+)
is a principal fibre bundle with contractible fibre and structural group U(H+) We do not reproduce all the details of the calculations here, but only mention that the total space is homotopically equivalent to the space of all bounded invertible operators in H which commute with P+ modulo compact operators. And that group is of the same homotopy type as the space of Fredhoim operators. Finally, we recognize that the whole space Fred2(H) of Fredhoim pairs of subspaces is homeomorphic to the product space
Proj°° (H) X where Proj°°(H) denotes the space of projections in H with infiniteis contractible by dimensional kernel and range. Since Kuiper's theorem, we have a homotopy equivalence between Fred2(H) and the space of Fredholin operators on H.
(c) Clearly the total Grassmannian Gr(P+) of (not necessarily pseudo-differential) projections in H, which differ from P-4. by a comof Fredhoim pairs of pact operator, is nothing but the space subspaces with fixed first component = P÷H. The identification is given by the map P '—p (H.,., (Id —P)H).
Theorem 24.1 can be reformulated in the language of Fredhoim pairs by applying Lemma 24.3:
Theorem 24.1. (Reformulated). The pair of the L2-closures of the Cauchy data spaces
of any Dirnc operator A over a partitioned
dosed manifold M = X_ U taken at the cut subrnanifold Y = X. fl X.,. = OX_ = OX.,., is a &edholm pair of subapaces of the corresponding L2(Y) section space and we have (24.11)
indexA = in
the rest of this chapter H(A±)
denotes the closure in L2(Y; Ely) of (24.12)
I u±
and
= 0 in
\ Y).
24. Bojarski's Theorem. General Linear Conjugation Problems
267
Proof. It follows from the unique continuation property for Dirac operators (Theorem 8.2) that kerA
(24.13)
H(A÷) fl H(A_),
see the intersection lemma 12.3. That is the trivial part of Theorem 24.1.
Now we apply our Corollary 12.6 and get (24.14)
cokerA = kerA*
=
fl H(A)
G (H(A+)-'-) n C (H(A_)-1-)
H(A+)-'- n
The last space is evidently the orthogonal complement of H(A+) + H(A_) in L2(Y;EIy). 0
Remark 24.6. As long as we are only interested in the index of Dirac operators, we may restrict ourselves to the case of even-dimensional manifolds, since the index of differential operators vanishes over (closed) odd-dimensional manifolds. This fits nicely with formula (24.11) because Dime operators over odd-dimensional manifolds are seif-adjoint, so (24.15)
H(A*) =
= C (H(A±)-'-)
n H(A....) =
n
and hence (24.16)
fl H(A_)-'- = (H(A÷) + which means
=0.
To obtain a truly odd variant of Bojarski's theorem, some replacements must be made on both sides of formula (24.11): The left side must be replaced by the spectral flow of a suitable family of selfadjoint elliptic boundary problems over a fixed cylindrical collar of Y in M. The right side must be replaced by the Maslov index of the corresponding family of Cauchy data spaces. This was worked out by L. Nicolaescu [1993].
More generally, we shall investigate the linear conjugation problem U A_ : C°°(M; E) —, C°°(M; F) over a for a Dirac operator A = U X_ along fl X... = Y. partitioned manifold M =
III. Applications
268
Definition 24.7. (a) Under the preceding conditions we denote by LCP(A, 4)) the following problem: We look for couples (u+, u_), where
E H1(X±;Elx±) such that (24.17) Here
=0
in X1, \ Y
and
= 4)(u.4.Iy).
4) shall be an automorphism of Ely with the consistency condi-
tion (24.18)
of 4) with the symbol is unitary. (b) We let
To simplify the notation we assume that 4) indexLcp(A, 4))
denote the difference between the dimensions of the kernel of the original linear conjugation problem (24.17) and of the kernel of the adjoint linear conjugation problem, where we look for couples (v+,v_) with v± E H1 (Xi; satisfying (24.19)
=
0
in
and
v_ly =
In the case of a Dirac operator with coefficients in an auxiliary bundle V, we have E := 0 V and F := 5 0 V, and condition (24.18) is satisfied by 4) := Id Og for any automorphism g: V on Y. This raises the following:
Problem 24.8. Describe the subgroup (Ely) of the automorphLsm group Aut(EIy) of the vector bundle Ely (with shift permitted in the base) which preserves the symbol or, in other words, acts nicely on the Grassmannian Grp+ by
(Ely) 4)
Now we turn to the index problem for linear conjugation problems. There we are confronted with two methodological problems: First, there is no well-defined operator in L2(M; E) which contains all information about a given linear conjugation problem. Second, the obvious question is whether we have a topological formula for the index of linear conjugation problems. The answer to both problems is given in the next theorem.
24. Bojarski's Theorem. CeneraJ Linear Conjugation Problems
269
E) -,
Theorem 24.9. Let A :
C°°(M; F) be a Dime operator over a partitioned manifold M = X÷ U X_ and E (Ely), where Y denotes the cutting submanifold. Then the pair (.1H(A÷), H(A...)) is a &edholm pair of subspaces of L2 (Y; Ely), and we get the following equalities for the index of the linear conjugation problem i.cp(A,4)): indexLcp(A, = index(4'H(A+), H(A_)) (24.20)
= index A + index(P(A_) —
H(A_)) is Proof. It is not difficult to see that the pair a Fredhoim pair. This is a consequence of the pt-invariance of which implies that the projection of H := L2(Y; Ely) H(A_)) Therefore, since onto belongs to is a Fredhoim pair by Theorem 24.1, so is the compact perturbation (see also Remark = 24.5a).
By the 1-1 correspondence between solutions and boundary data (established in Theorem 12.4a), fl H(A_) can be identified with the space of solutions of (24.17) and alternatively with the kernel of the operator (24.21)
T, :=
H(A_)1.
:
The cokernel of this generalized Toeplitz operator is equal to the space namely: +
(24.22) cokerT, =
=
This proves (24.23)
index(4H(A+), H(A_)) = indexT,.
We elaborate (24.22) further. Recall from Corollary 12.6 the gauge complementarities
H(A..)1 =
and
H(A+)1 = G'H(A÷).
Ill. Applications
270
is unitary, we have = first equality of (24.21) yields the identity
Then the
Since
=
fl
which is isomorphic to the space of solutions of the adjoint linear conjugation problem defined in (24.19). The isomorphy follows from the 1-1 correspondence between solutions and boundary data. Thus the first two equations of (24.20) are proved.
To prove the third equation of (24.20), we recall that the index really identifies the connected components in the Grassmannian, and that, by Proposition 15.15, for any R, S, T index{RT: rangeT —* rangeR} = index{RS : range S range R}+index{ST: range T
(24.24)
range S}.
From that we obtain
= (24.24)
:
—'
H(A_)1}
index{(Id—P(A_))2(A÷) : H(A÷) —'
+ index A The operator of the second summand can first be extended to (Id +
over the whole Hubert space, and then compact per-
turbated to (Id
+
+ (Id
= (Id
—
without changing the index. Here we have used the of which gives that commutes with modulo compact operators. El
Remarks 24.10. (a) The main point of Theorem 24.9 is that (24.20) separates the index of the linear conjugation problem into two contributions, one which comes from the global data (the operator A on the manifold M) and one which comes from data that live on the cutting submanifold Y. (b) The first summand in the final formula of (24.20), the index of A, is given by the Atiyah-Singer theorem. In the next chapter we
24. Bojarski's Theorem. General Linear Conjugation Problems
271
explain why the second summand, the index of the Toeplitz operator is also or equivalently the index of given by the index theorem, namely as the index of a corresponding Dirac operator on the mapping torus. (c) If A is invertible, its index vanishes, its Cauchy data spaces are orthogonally complementary, and we recover Proposition 24.4 from Theorem 24.9.
(d) Over the closed manifold M, the index of the adjoint problem equals the index of the following orthogonal Fredholm pairs (H(A* Ix÷), H(A*Ix_)) =
Similar calculations can be made for the linear conjugation problems.
We give an example which belongs to the sphere of prominent
classical problems of mathematics and is called, at least in the Russian mathematical literature, the Riemann-Hilbert problem, see [1962], Bitsadze [1966], Bojarski [1979], and Meister [1983, pp 123—172].
Example 24.11. Our model situation is the two-dimensional case, where we deal with the Cauchy-Riemann operator. We are looking for a couple of functions
onD2,
f+
f....
onC\D2
which satisfy
=
(24.25a)
0,
= 0, f_(oo) = 0,
and (24.25b)
f_(z) =
for z E S'.
We assume that g is a continuous complex-valued function on S1 such
that g(y) 0 for any y E S'. This problem seems to differ from the problem (24.17), since our manifold C is not compact. We explain how to transform (24.25) into (24.17) in Chapter 26. Clearly (24.25) with g(z) = has no solutions for k 0, and it has a (—k)-dimensional space of solutions for k < 0. More generally our Theorem 24.9 gives (24.26)
indexLcp (b, g) =
— deg g.
III. Applications
272
Now consider the special non-shift case of our situation. Assume that M is an even-dimensional manifold, that E := 0 V, F := 0 V, and that g is a unitary automorphism of VIy. In this case is a pseudo-differential operator and its index is a T 1'.... — topological invariant given by the Atiyah-Singer formula. Let us restrict ourselves to an even more special situation, where g is a unitary automorphism of the auxiliary bundle V. Then we can collect a number of useful formulas:
Proposition 24.12. We have the following equalities, if the linear conjugation problem LCP(A, g) is defined by an automorphism p of the auxzliary bundle V: indexLcp(A,g) — index A = index{(Id — P(B))
—
g
P(B)}
= index P(B)g P(B) = = index(P(A_) —
(24.27)
=
jch([B]. [g])Ur(Y).
Here the integration is over the Thom space of the cotangent bundle of Y; [B] denotes the class of the principal symbol of the tangenis the class in defined by tial operator B in K1(TY);
g; r(Y)
H(TY) the (lifted) Todd class of Y; ch : K(TY) Q) the Chern character; and
K(T*Y)
P := P denote, as the usual multiplication of K-theory. before, the Calderón projectors and the spectral projection. The formulas (24.27) are all consequences of the fact that P(A_) — gP(A÷) is an elliptic pseudo-differential operator of order 0. We introduce one more formula which makes a direct connection with the index theory of a certain boundary problem on X := X÷.
Theorem 24.13. Let A : C°°(X; E = S
®V) be a Dirac operator over a compact manifold X with boundary
Y, where A takes the form + B) with unitary C. Let p be a unitary automorphism of V. Denote by A the operator
A :=
:
C°°(X;E)€C°°(X;F) -,
24. Bojarski's Theorem. Genera' Linear Conjugation Problems
273
Denote by R the boundary condition
R: C°°(Y; Ely)
C°°(Y; F'y) - C°°(Y; Fly)
defined by
R(s, r) := G(gs) —
(24.28)
r.
Then (A, R) is an elliptic boundary problem of local type which has the same index as the corresponding linear conjugation problem: (24.29)
index(A, R) = ifldexLCp(A U A, Cg).
Note. In combination with Proposition 24.12 and from some lines
of the following proof, we get again a number of formulas to insert in (24.29): (24.30)
= index AR = index((Id — P(B)) — g. P(B)) = index((Id —P(A)) — gP(A)).
Proof. The ellipticity is obvious. R is a surjective condition; hence by Corollary 20.14 we have the equality index(A, R) = index AR. It is aiso obvious that (24.31)
kerAR = kerP(A)gP(A),
(see, however, the calculations of the cokernel below). Now we can rewrite the boundary condition R as a projection P such that Ap = Id Id
/
This shows that the adjoint condition is (by Proposition 20.3): (24.33)
T_(G G')' 0 "'Id —
o'\_1( Id 0
Gg Id
We can now calculate the cokernel of which is equal to the kernel of the operator (A*)T. Therefore it is the set of pairs (r, s)
C°°(X; F)
C°°(X; E)
III. Applications
274
which satisfy the,following equations:
A'r = 0, As = 0, and sly = —gG'(rjy).
(24.34)
The first two equations show that E H(A') and sly E H(A). Now we use Corollary 12.6 and the unique continuation property for Dirac
operators. That allows us to identify ker AZ with the kernel of the operator P(A)g'7'(A). Therefore we have: index(A, R) = index AR = index 1'(A)gP(A) = index P(A)G'CgP(A) = index(CP(A)G')Cgl'(A) = A and A_ =
with
A.
This ends the proof of Theorem 24.13.
At the end of this chapter we present a calculation which shows the difference between the indices of two problems of this type.
Theorem 24.14. Under the assumptions of the preceding theorem,
let A8 denote the problem A with boundarij condition sly = where we assume that E = ® V and g, i = 1,2 are automorphisms of the auxiliary bundle V. Then we have index A1 — index A2 = index{(Id —P(A)) +
Proof. Theorem 21.2 it follows that this difference of indices is equal to the index of the boundary integral R2P(A)R, where once again:
A
=
hence
1.'(A) = (P(A)
and Since
C is unitary —
(—gj'c' 1
P(A*))'
24. Bojarski's Theorem. Genera' Linear Conjugation Problems
275
and hence
R2P(A)R = (—C92
1)
(P(A
)
= + P(A') = G{g2P(A)gj1 + (Id —P(A))}G'. This shows that indexA1 — indexA2 = index{(Id—P(A))
+g22(A)gj'}
= index{(Id —P(A)) + g2gj'P(A)} since
and therefore also gj' commute with P(A) modulo compact
operators.
[]
25. Cutting and Pasting of Elliptic Operators
We discuss the process of the cutting and pasting of elliptic operators and show that some operators, like the signature operator, are stable in a suitable sense under this operation.
Essentially this chapter shows how to obtain new operators from operator pieces under the process of cutting and pasting of elliptic operators. We distinguish two extreme situations: (1) On the one hand, we have the flexible case: We may obtain complicated operators from simple pieces solely by cutting and pasting. We shall show later in Chapter 26 that the process is highly non-trivial even in the simplest possible case of Dirac operators on S2. (2) On the other hand, we have the rigid case: classes of operators which are invariant under cutting and pasting. We shall show
in this chapter that the signature operator is in fact stable in a suitable sense under this operation. Assume that we are in the situation of Chapter 24. We have an operator A on a closed partitioned manifold M which is essentially
the sum of two operators A± on suitable pieces X±. We use the symbolic notation (see Chapter 9 for a similar situation or Douglas & Wojciechowski [19891 for exactly this situation): (25.1)
Assume once again that we have a unitary automorphism
Ely of the bundle Ely
(possibly
: Ely
—.
with shift in the basis Y). Let a
denote the principal symbol of the operator A. In a bicollar neighbourhood of Y, the symbol a has the form: (25.2)
a(u, y; ii, () = G(y)(iv + b(y; ()).
We assume a consistency of 4 and C, namely that for all y E Y and CE
4(y)G(y) = G(f(y))4(y) and (25.3)
øF(y)b(y; C) =
b
(1(y); (f')(C))
25. Cutting and Pasting of Elliptic Operators
277
Here f is a diffeomorphism of Y covered by & We introduce the notation := and 1FF := G4EG', and then we can reformulate (25.3) as the condition: (25.4)
y; ii,
() = a (0,1(y); (f_I)*(p), (f')(C)) 'FE(y).
Now we can introduce a new manifold and new bundles:
M' (25.5)
and = (EIx+) F = (FIx+)U,F (FIx_),
with the obvious identifications. For instance, to get M', identify the point (0, y) with the point (O,f(y)). Actually, to have everything smooth, we have to be much more careful and must work as in Chapter 9. We extend the manifold to X+u((—l,0] x Y) and the manifold X_ to X_ U ([0,1) x Y), and then we identify (u, y) with (u, 1(y)) on the collar N, where the normal variable it E (—1, 1). Now we can define a new elliptic symbol in this situation: (25.6)
a
: 7r(E)
—
by the formula
:=a(x;e). Although the algebraic formula for the symbol is unchanged, the sym-
bol generally behaves in a different way, since it acts between the sections of the new bundles over the new manifold M1. Choose an arbitrary elliptic operator A with principal symbol a. The problem is to find a formula for the difference: (25.7)
index A — index A.
Our solution of the problem is based on an explicit realization of the operator A. An alternative approach, which deals with a more complicated analytical situation and therefore uses the intricate machinery of Boutet de Monvel, was offered in Booi3 & Rempel [1981), [1982J. There, however, the explicit topological formula of Theorem 25.2 was not obtained for the difference (25.7). The main point in our explicit realization of the operator A is the use of (25.3). We cannot directly paste A÷ = AIx+ to the operator A_ = AIx_, but must modify the tangential operator B. If we
III. Applications
278
substitute B with the operator B =
then everything works fine. The sections of the bundle E are couples (se, 82) such that on some collar neighbourhood of Y we have
s2(u,f(y)) =
(25.8)
We show that any operator, which is equal to A_ on X_ and which, restricted to the same collar on takes the form (25.9)
+
G
maps sections of in sections of F which are just couples (83,34) satisfying the condition (25.10)
84(U, 1(y)) = G(f(y))
C' (y) 33(u, y).
Essentially we used this argument before in Chapter 9, where we presented our construction of the operator A U A*. Let (81,82) denote a section of E. We have to show that A_s2) is a section of F. This is reduced to the equality: (25.11)
•E{G
= G(OU + B)s2.
+
The left side of (25.11) is equal to
s2+GBs2 = A_(s2).
=
(25.12)
Now the crucial point is the assumption (25.3) which tells us that the principal symbol of is equal to the principal symbol of B. The second equality in (25.3) can be written as: (25.13)
b(y;() =
Therefore there exists a family {Bt}tElo,1J of Dirac operators on Y, such that B0 = B and B1 = B dIE. Take for instance: (25.14)
:=
+ (1 —
is a smooth real valued function on the interval [0,11 equal to 0 in some neighbourhood of 0, and equal to 1 in a certain neighbourhood of 1. We introduce an operator on in the following where
way:
,._fA+ (
)
onX+\N onN.
25. Cutting and Pasting of Elliptic Operators
279
Now, we can explicitly define an operator A C°°(M1; E) —, C00(Mf; F4') with the required principal symbol UA• = a4' by :
putting: (25.16)
A4' := {
::
To compute the index difference (25.7) observe that we can localize
the problem as follows. The index of the Dirac operator is given by a local formula. This means that (25.17)
index A
=
J
where o(x) is a density given at the point x by a complicated algebraic
formula in terms of coefficients of the operator A in x (accounted for e.g. in Cilkey [1984], Getzler [1986], or Berline, Cetzler & Vergne [1992]). It is a consequence of the locality of the index expressed in (25.17) that we have the following formula: (25.18)
index A =
index A
+ index T.
In (25.18) the operator 7'4' : C°°(Yf; W,) —* C°°(Yf; W2) is given by the formula G(OU + and acts between sections of the bundles
W, := EIN/
and W2 := FIN /G4G-' over the mapping torus
yf := (I x Y)/ f. The index of the operator is given by the Atiyah-
Singer formula. We are looking for the desuspension of this formula to the manifold Y. With this purpose in mind, we must exploit the spectral flow which was introduced earlier in Chapter 17. Recall the following: We have a family of seif-adjoint operators parametrized over S'. It is explained that such families have only one topological invariant, It was noticed in Atiyah, Patodi & Singer [1976) that the spectral flow is in fact the analytical index of the family with values in K' (S') = Z. The following result is a consequence of the index theorem for families:
Proposition 25.1. Under the preceding assumptions we have
= indexT. An elementary argument was offered in Vafa & Witten [1984]. We have included a proof of Proposition 25.1 in Chapter 17, see Theorem 17.13. To obtain the following result, we just use a formula presented in Theorem 17.17.
III. Applications
280
Theorem 25.2. Assume that 4' is a unitary automorphism of Ely which satisfies the consistency condition (25.3). Let denote the Calderón projector and the Cauchy data space. Then the operator := P(A+)4P(A÷) : H(A+) —* H(A÷) is a Flredholm operator and we have the following equalities: (25.19)
indexA — index A = indexT =
= index((Id
—
= indexP.
Corollary 25.3. The index of the general linear conjugation problem LCP(A, 4) (see Chapter 24) on the manifold M equals the index of the
operator A. Let us give an important topological example of the situation in which cutting and pasting operations have no influence on the index of the corresponding operator:
Theorem 25.4. Let sign M denote the signature, and
the
U X_ with Euler characteristic of a partitioned manifold M = = oX..... Then an automorphism f : Y Y, where Y = sign M' = sign M and = Proof. As discussed in Chapter 7, the signature of a manifold is equal to the index of a special Dirac operator
-* C°°(M;
DM : C°°(M;
Therefore we only have to show that
indexDM = index DM1.
(25.20)
Fix a Riemannian structure g Ofl M and let gy denote its restriction Y carries the same metric to the boundary (in fact any the cutting and pasting Under gy). This defines the operator DM. of the manifold M, we have to modify the Riemannian structure in order to get a metric on the manifold M1. Replace the metric fly by a family of metrics: (25.21)
+ (1 —
:
Now any metric flu defines a boundary signature operator
follows from Theorem 25.2 that: (25.22)
signM'
—
signM =
and it
25. Cutting and Pasting of Elliptic Operators
281
As follows from Chapter 7, the kernel of any of the operators consists of all harmonic forms on the manifold Y. Therefore the dimension of ker is constant. This argument shows that there is no spectral flow, since there are no eigenvalues changing the sign. That concludes the proof for the signature operator. We use the same argument for the operator which gives the Euler characteristic. [J The preceding example illustrates a situation in which the topolog-
ical constraints prohibit any change of the index under cutting and pasting operations. On the other hand, in one of the most fundamental examples of the index theory, the index is completely determined by the cutting and pasting operations. This will be discussed in the next chapter.
26. Dirac Operators on the Two-Sphere
We show that the index of any elliptic operator on any closed Riemann surface or even-dimensional sphere is completely determined by the pasting of trivial pieces of the operator. Our example gives a full analysis of the index on S2 and determines the index of the generalized Riemann-Hilbert problem.
At the end of Chapter 25 we discussed an important example of a situation in which the cutting and pasting of elliptic operators have no influence on the value of the index of the operator. Here we discuss another extreme case in which the index is completely determined by
the pasting of the trivial pieces of the operator. The argument presented below gives the index of any elliptic operator on any closed Riemann surface and any even-dimensional sphere. Since the generalization is easy, we concentrate on the simplest and most important case of the two-dimensional sphere.
First we introduce the basic operator. It is not the CauchyRiemann operator f i— Of/Oz dz, as is usually assumed, but a total Dirac operator which acts on true spinors. Although we do not discuss K-homology groups here, we shall mention that the operator, which we construct, provides a generator of the group Ko(D2, S1). As explained in Wojciechowski [1985] (see also Douglas & Wojciechowski [19891), the cycles for the relative Ko-homology group of a manifold X with boundary Y are given by operators of the form A U A on the double of X. We have already used the construction of the double of elliptic operators in Chapter 12.
In the simplest possible two-dimensional case, we consider the Cauchy-Riemann operator 0 C°°(D2) C°°(D2), where 0 = :
(0/Ox + iO/Oy). In polar coordinates out of the origin, this operator has the form Therefore, after some small smooth + perturbations (and modulo the factor we assume that has the following form in a certain collar neighbourhood of the boundary: (26.1)
=
+ iOtp).
Now we can apply the construction from Chapter 9 directly. Remem-
ber that in this case
= e". By Hk we denote the bundle,
26. Dirac Operators on the Two-Sphere
283
which is obtained from two copies of D2 x C by the identification (z, w) = (z, near the equator, k E Z. We apply the construction from Chapter 9 and obtain the operator A0 =
:
Let us analyse the situation more carefully. We fix N := (—e, +e) x S', a bicollar neighbourhood of the equator. The formally adjoint operator to has the following form:
=
(26.2)
+
+ 1)
(u = r — 1) in this cylinder. A section of W is a couple (81,82) that in N:
such
=
(26.3)
is a smooth section of H'. To show that, we check the equality (8)s2 = We have in the neighbourhood
The couple N:
= = Ohs, + =e
(26.4)
= +
+ + =
+ = (& +
= 0. We produce an operator with a nontrivial index. Modify the operator A0 in such a way that it will act from sections of the bundle H2 to sections of the bundle H° = S2 x C. The resulting operator A1 has index equal to 1. The index can be computed by using Theorem 25.2. In this particular case, we can Here index
U
actually make a direct calculation. First we give the construction of the operator A,: Any section of H2 is a couple (Si, 52) such that we have in N: 52(u,W) =
Following the procedure from Chapter 25 we do not change the op-
erator (8), but modify the operator 0. We have to substitute the operator 0 with 0,. In order to obtain a section of the trivial bundle, we must have the following equality: (26.5)
=
III. Applications
284
Calculations similar to those in (26.4) allow us to conclude that the operator 0 outside of N. On N we define: :=
(26.6)
+
—
e and 1 in a
t,b(u) 0 is a smooth function equal 0 near u
where
small neighbourhood of u = 0. Hence, on a smaller cylinder (on which = 1), we have the following equalities, for any couple (81, 82), such that 82 = on this cylinder: (a)*82 =
(26.7)
=
+
=
+
=
=
These elementary calculations show that the operator A1
U
is well-defined. The only difference from the operator 0 U (0) is the modification on the collar N. It follows from the discussion in Chapter 25 that (26.8) index A1 = index A1—index A0 =
where the operator on the right side is a well-defined elliptic opera-
tor on S' x S1. We know that the index of an elliptic operator is unchanged under small continuous deformations, and we can calculate the index of the operator on the right side directly. After a small deformation we see that we must calculate the index of the elliptic operator T := + — u acting on functions f(u, p) from C°° (R x S1) which satisfy the periodicity condition
f(u + 1,
(26.9)
= e1"°f (u,
Any such f can be represented by a series (26.10)
f(u, ça) = kEZ
with fk+I(u) = fk(U + 1) because of fk(U +
= f(u + 1,
ço) = e2"°
k
> Ic
=
=
26. Dirac Operators on the Two-Sphere
285
Since
= the
equation Tf
—
(k +
= 0 gives the following series of equations for = (k + u)fk(u)
for all k
E Z.
As a result we get
fk(u) = Moreover, Ck does not depend on k because of
+ 1) =
= fk+1(u)
This means that the coefficients fk(u) of the series (26.10) explode for
0. So there is no non-trivial solution of Tf = 0 with the required periodicity. — u yield The same calculations for the operator + = k
oo,
if C =
Ck
=
+ (k + u)fk(u)} = 0
fk(u) = Ck again independent of k because of fk(u +1) = fk+1 (u). Thus
the solution of T*f =
0
has the form C
which converges nicely; hence the kernel of T with the required pe-
riodicity is one-dimensional. This gives the following formula:
indexA1 =
(26.11)
—1.
More generally, this argument proves the following result:
Theorem 26.1. Let
Ak denote an operator over S2 which is equal to 8 on D2 \ N, (O)* on the second copy of D2, and + ki,b(u)) on N. The operator Ak is in fact the operator A0 ® IdHk
C00(S2; orem: (26.12)
—*
C00(S2 ;
and we have the following index the-
index Ak = —k.
III. Applications
286
Remarks 26.2. (a) Alternatively, we can prove (26.11) by Propo= sition 25.1 which gives index A1 = — u}. We can also calculate the spectral flow of the family directly. We consider } as a family of ordinary differential operators over the circle S', parametrized by u E S1 = 1/ {O, 1). We have a spectral decomposiof eigenfunctions of tion of L2(S') by the system
(ii-
—
u)h = Ah,
h E C°°(S1)
with corresponding eigenvalues {A = —(k + U)}kEZ, i.e. the spectrum of is given by the graph in Figure 26.1 and hence = —1. A further analysis of this example (with opposite signs) has been given in Example 17.8b. A
U
Fig. 26.1 The spectrum of
—
(b) Direct generalization yields the corresponding index theorem for Dirac operators on even-dimensional spheres which gives the Atiyah-
Singer index theorem on these spheres by stable homotopy. That result, together with the computation of the signature operator on complex projective spaces, was the basis of the first proof of the index theorem (see Palais [1965a]).
(c) Notice that the cutting and pasting procedure explained above both in general terms and specifically for the 2-sphere also provides a direct proof of the Atiyah-Singer index theorem for elliptic pseudodifferential operators on arbitrary closed Riemann surfaces.
26. Dirac Operators on the Two-Sphere
287
Example 26.3. The Cauchy-Riemann operator 9 : C°°(S2) is given by the formal formula (26.13) Now the bundle of (0,1)-forms is in fact isomorphic to the bundle H2. The transition function is z '—' l/z, which gives for the tangent bundle z —l/z2 as the glueing function on the equator, hence TS2 = H2
and TS2 =
=
= —z2 di. So it becomes obvious that 0' = (A1)*. That gives the following H2, or directly
'—*
well-known equation: index A1 = 1.
(26.14)
Now we explain how the index of the classical Riemann-Hilbert problem (24.25) fits into our construction. Recall that we have a fixed continuous (we assume g to be a smooth function) invertible function f...) which satisfy: g on the circle, and we are looking for couples
•
is holomorphic in D2, f... is holomorphic in C \ D2, and f.... (oo) =0.
15' /
It is obvious that has the form and f..., thanks to the normalization condition at oo, has the form We want to compute the index of the Fredhoim pair (11÷, gH_) of subspaces in L2(S'), where .1:1+
•
.
Actually, H... =
,.
r
JkO
ano
ii
LI_
=
.
i r k& L.CtZ
we can transfer the
and
Riemann-Hilbert problem into a linear conjugation problem for the operator A0 = U In this corresponding linear conjugation problem we are looking for couples (s+, s_) such that (26.16)
= 0,
=0
and
s_(z) =
z reflects the fact that the couple (8k, 8_) is a section of the
bundle H'. Therefore we have here the linear conjugation problem Lcp(Ao, g) of Definition 24.7. Then Theorem 24.9 and Proposition 24.12 give a formula for the index of the problem (26.17)
g) =
— deg(g).
III. Applications
288
Remark 26.4. Of course, in the linear conjugation problem we consider in and (a1)' smooth perturbations of the true operators a and It follows from the formula for the Calderón projection given in Chapter 12 that the Calderón projections of the new operators are continuous perturbations of the original Calderón projections. Therefore they do not change their connected components in the Grassmannian, and the index of the corresponding Riemann-Hilbert problem (for the operator &) is equal to the index of the original problem. Formula (26.17) trivially extends to the situation of vector valued
functions. The (26.15) with g:
in this situation have values in Ci" and satisfy —' CL(N, C) and we can formulate the solution
of the index problem for the vector-valued Riemann-Hilbert problem. The following theorem holds:
Theorem 26.5. Under the assumptions made above, the vectorvalued Riemann-Hilbert problem has an index equal to (26.18)
indexLcp(a; g) = indexLcp(Ao ®
;
g)
= (ch[gJ)[S'] = —(2rri)' Remark 26.6. See
j tr(g'dg).
[1962] for a general discussion of the Riemann-Hilbert problem from the classical point of view.
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Index adiabatic argument, 157-160 adjoint boundary problem, 191-196, 212 1
adjoint linear conjugation problem, 268
type formulas, 205 if, 253-258 analytical index, 146-149
approximate fundamental solution,
180 if, 262281
Cauchy-Riemann operator,
205,
271, 2R2-2RR
233-239
Atiyah, M.F., viii if, 42, 152, 212, 241, 259, 261, 219 Atiyah-Bott index theorem for local elliptic boundary problems, 204 Atiyah-Bott trace formula, 213 f Atiyah-Patodi-Singer boundary problem, 157, 163 1, 200, 211-261 Atiyah-Singer index theorem, 147-150, 270 if,
Calderón, A.P., viii if, 78 Calderón projector, 79 if, 102 f, 112, 166, 171-189, 200-212, 262-275 Calkin algebra, 129 if Carleman type estimate, 45-49 Carslaw, H.S., 226, 233 Cauchy data spaces, 76-91, 177,
286
Atiyah-Singer operator, 37 auxiliary bundle, 42, 268, 272-274 Baum, P., 106 Berline, N., 212 if, 240, 249 Bessaga, C., 131 Bincer, A.M., 31 Birman, M., xi, 173 Bismut, J.-M., 241 Bitsadze, A.V., 271 Bleecker, D.D., 164 l3ochner identity, 28 Bojarski, B., x, xiii, 262 f, 271 BooB, B., 164, 261] 1, 277
Borisov, N.y., 50 Bott, R., viii, xii, 204 boundary integral, 166, 116-184, 199-207, 274 boundary projection symbol, 88-90. 106-108, 111-126, 163, 171-176, 183
Branson, T.P., viii, 42 Brown, L.C., 119 Bunke, U.,
J., 241 Chern character, 146-151, 272 chirality, 40-42. 50 if, 76 f Cheeger,
Clifford
algebra, bundle, 10 module, 5, 10-42 multiplication, 5, 10 relations, 29 closed double, 51-58, 231-242 closed L2-extension, 189-191, 196-199 closed range operator, 161 1, 183, 186 1,2001 clutching of bundles, 146, 276-288 cobordism theorem, 164, 208-210 compatible connection, 13 if, 251 compatible Dirac operator, 2(1 complementarity of Cauchy data spaces, 200, 267-274 complementary error function, 227-230 connected components of Grassmannian, 123 connection, 12-18 connection Laplacian, 26 consistency condition, 254, 268, 276, 280
continuity of eigenvalues, 138 f continuous extension operators, 72 cut-off operator, 72 cutting and pasting, 145, 262, 276-288 cylinder problems, 214 -230 de Rham complex, 21 desuspension, 145, 153-156. 270.2R1
Index
304
Dirac, P.A.M., 30 distribution, 60. 224 Laplacian, 26-28 operator, 20 if Dirichiet boundary condition, 224-228
Doetsch, C., 216 Dolbeault complex, 21 Donaldson, S.K., 62 double layer potential, 79 Douglas, R.C., x, 119, 231, 239, 276, 282 Duhamel, J.M.C., 233 Duhamel's principle, 231-239
elementary index theory of elliptic boundary problems, 188-204 elliptic boundary conditions given by pseudo-differential projections, 164,
173
elliptic boundary problems, 163-288
elliptic fan, 173 if, 180 essentially positive, 151 eta-function, eta-invariant, 158, 228-230, 240 if, 249, 260
full Grassmannian, 127 if full mapping pair, 165, 180-187, 201-204, 206, 273 fundamental solution, 216-221, 232-239
Gamma-five matrix, 8,32 general linear conjugation problem, 262-275 generalized
Atiyah-Patodi-Singer boundary condition, 164, 173, 253-261 Dirac operator, 250 if Riemann-Hilbert problem, 2R2-2RR Toeplitz operator, 269, 271 Cetzler, E., 212 if, 240, 249, 229 Gilkey, P.B., vii f, 21 f, 36, 42, 212 if, 240 f, 249, 279 glueing constructions, 50-63, 254 -261, 283 if
Goldstone, J., 31 graph of the spectrum, 140 if, 156 f Grassmannian of pseudo-differential projections, 111-126, 145, 173, 200, 259 f, 288 Green's formula, 24 Gromov, M., 59 Crubb, C., xii, 185
Euclidean Dirac operators, 2936 Euler characteristic, 26(1 f exact Poisson formula, 224 excision principle for indices, 62 extended L2-solution, 243 if, 259-261 extension by zero operator, 72 extension of operators to the double, 50-58 exterior algebra, 8, 14. 31
families of self-adjoint operators,
half-bounded operator, 151-153 heat 214-242 kernel, 225-240 operators, 220, Hermitian Clifford-skew-adjoint metric, 11 Hirzebruch L-polynomial, 258 Hitchin, N., 249 Hopi bundles, 2R2-2R5 Hörmander, L., 204
138-160
Fillmore, P.A., 119 formal adjoint, 192 four-component spinors, 30 Fredhoim pair of subspaces, 262 if properties, 173, 116 f, 199-201 Freed, D.S., 241 Fuglede, B., 111 Fujii, K., 154
index, 119-126, 145-156, 199-288 bundle, 126 density, 158 f, 240, 260. 279 formulas for Atiyah-Patodi-Singer problems, 241 f, 248 f, 251 generalized Atiyah-Patodi-Singer problems, 253, 26(1
Index
index formulas for (continued) global elliptic boundary problems, 177 f,199 glueing constructions, 59-63, 266,
285
linear conjugation problems, 269, 280
local elliptic boundary problems, 201 if of Fredhoim pair, 263 if injectively (left) elliptic boundary problem, 165, 186 inner extension problem, 242 integral representation of a projection, 168-170
intersection lemma, 77 invertible extension, 50-58 irreducible Clifford modules, 5-9
Jaeger, J.C., Jaife, R.L., 31 Jänich, K., 260 Jimbo, M., x
305
local
elliptic boundary condition, 164, 173, 200-204, 207-210, 273 eta-density, 279 index theorem, 279 Lopatinskil, Y.B., xii Luck, W., 255 mapping torus, 271 Mazzeo, R.R., 255 McKean, H., 231, 234 Meister, E., 271 Melrose, R.B., xiii, 243, 255. 2611 f 164 Michelsohn, M.-L., viii, 36 if, Minakshisundaram, S., 234 Miwa, J., x modified Neumann boundary condition, 224-228 Muller, W., 55, 241, 243, 255 271, 288
226, 233
Juig, P., 511
K-theory, 55. 41$. 146-153, 272, 279. 282
Karma, U., 260 Kato, T., L39 f, 263 Klimek, S., 55, 231, 239, 255 till. 253 f Kori, T., Kronheimer, P.B., 62 L2-solutions, 158 if, 242-248, 259 if Laplace transformation, 215 f, 218 if, 226
Laplace-Beltrami operator, 21 Lawson, B.H., viii, 36 if, 164 Leibnizian (Riemannian) connection,
natural operators on manifolds with boundary, Li if, 92 Newtonian potential, 79 Nicolaescu, L., 267 non-additivity of index, 254-258 non-compact Riemannian elongation, 243-248 f non-homogeneous equation, non-homotopy invariance of index, 249
non-vanishing spectral flow, 142-144, 157-160
non-vanishing virtual codimension, 123 normalized cylindrical Atiyah-Patodi-Singer problem, 214 if normalized orientation, 6 Novikov additivity of signature, 2611
obstructions for deformation of projections, 91-93. 173, 118. 182-186, operator of Dirac type, 43 if 190 orthogonalisation of pseudo-differential Levi-Civita connection, 13 projections, 93 f, 122 Levi, E.E., 234 outer extension problem, 242 Levi's sum, 234-239 Lichnerowicz, A., 38 P+-invariant operators, 266 if limiting values, 243-248 286 Palais, R.S., xii, 243 ilL 13
306
Index
parametrix, 91-93. 173, 182-186, 190, 216-221, 232-239, 265 partial Dirac operator, 41 if partitioned manifold, 59-63, 75, 262-288 Patodi, V.K., x if, 42, 241, 259, 261, 229
152, 212,
Pauli matrices, 3kM Pelczynski, A., 131 Plejel, A., 234 Poisson formula, 79 type operator, 78 if, 102 f, 184 f, 190, 198 potential operators, 78-80 Pressley, A., x, 111 principal symbol, 19 product form close to boundary, 42, 44, 77 f, 163, 191 if, 214 1, 254, 273, 276-288
product metric close to boundary, 42, 78,211 f 51 if, pseudo-differential inverse, 57 projections, 78-90, 93 f, 105-126, 163- 179
realization of an elliptic operator as an unbounded operator in L2, 165,
ITh
180-182, 186-261
Reed, M., 193 Rees, H.D., 260 reflection operator, 72 regularity of solutions, 179-187, 189 relative index theorem, 59 if 277 Rempel, S., restriction operator, 71 Riemann-Hilbert problem, 271 1, 281 f R.iesz operator, Roe, J., 261 rotation of Cauchy data spaces, 200, 267-274
Z.Y., xii
conditions, xii Sato, M., x scalar heat kernel, 224
Schmidt, J.R., 31 Schrader, R., 59 Seeley, R.T., viii if, 163,
78,
158,
199
Segal, G.B., x, 11.1 seif-adjoint Fredholm operators, 127-137, 140-142
semi-Fredhoim operator, 165 separation of variables, 44, 217 Shapiro, A., viii signature deficiency formula, operator, 41, 258 if, 280 f Simon, B., 193 simple module, 9 Singer, jjyj, x if, 42, 152, 212, 231, 234, 241, 259, 261, 279 Sobolev spaces, 67 if Solomyak, A., xi, 173 special group of invertible elliptic operat.ors, 112
spectral 138-160, 286 if iiwariants, 128 projection, 105-110, 134, 145, 151-156, 164, 200, 206, 211-261 resolution, spin bundle, 37 connection, 15 manifold, 36-39 spinor representations, split Dirac operators, 41 if stable symbol class, 148-153 flow,
standard trace operator, 68 Steenrod, N., 117 f surface potential, 59 surjective boundary operator, 203, 273 surjectively (right) elliptic boundary problem, 165, 185 suspension, 144, 148-151 symbol class, 147-153 spectrum, 169 symmetric heat kernel, 228 if
integral, p4., 228 if trace, 59 system of Dirac operators, 164
Index
tangential part operator, 44 f Taubes, xui, 252
theorem of Bartle and Graves, 131 Todd class, 150 f, 272 topological index, 147-150 total Dirac operator, if, 41 trace of operators of trace class, 213-242 operator for distributional solutions (restriction to boundary) 95-104 theorem, twisted complementarity of Cauchy data spaces, 200, 274
twisted signature operator, 25(1 if
unique continuation property, 4349 unitary equivalence (gauge invariance), 151-157
unitary retracts, 1Th 132 if Vain, C., 279 Vergne, M., 212 if, 240, 249, 279 virtual codimension, 119-126, 177, 199, 212, 253-258, 262-275 volume potential, 79
Wall, C.T.C., 260 Wall non-additivity of signature, 260 wave equation, 233 weighted spectral projection, Weitzenböck formula, 21 Weyl's theorem, 224 f Witten, E., 279 Wojciechowski, x,
231,,
276, 282
Yoshida, T., xiii,
252
307
