ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaSpe tral Invariants of Operators of Dira Typeon Partitioned ManifoldsDavid Blee kerBernhelm Booss{Bavnbek

Vienna, Preprint ESI 1293 (2003) Mar h 18, 2003Supported by the Austrian Federal Ministry of Edu ation, S ien e and CultureAvailable via http://www.esi.a .at

Spe tral Invariants of Operators of Dira Typeon Partitioned ManifoldsDavid Blee ker and Bernhelm Booss{BavnbekAbstra t. We review the on epts of the index of a Fredholm operator, thespe tral ow of a urve of self-adjoint Fredholm operators, the Maslov in-dex of a urve of Lagrangian subspa es in symple ti Hilbert spa e, and theeta invariant of operators of Dira type on losed manifolds and manifoldswith boundary. We emphasize various (o asionally overlooked) aspe ts ofrigorous de�nitions and explain the quite di�erent stability properties. More-over, we utilize the heat equation approa h in various settings and show howthese topologi al and spe tral invariants are mutually related in the study ofadditivity and nonadditivity properties on partitioned manifolds.Contents1. Basi Notations and Results 41.1. Index of Fredholm Operators and Spe tral Flow of Curves ofSelf{Adjoint Fredholm Operators 41.2. Symmetri Operators and Symple ti Analysis 131.3. Operators of Dira Type and their Ellipti ity 201.4. Weak Unique Continuation Property 212. The Index of Ellipti Operators on Partitioned Manifolds 292.1. Examples and the Hellwig{Vekua Index Theorem 292.2. The Index of Twisted Dira Operators on Closed Manifolds 382.3. Dira Type Operators on Manifolds with Boundary 522.4. The Atiyah{Patodi{Singer Index Theorem 622.5. Symple ti Geometry of Cau hy Data Spa es 722.6. Non-Additivity of the Index 752.7. Pasting of Spe tral Flow 773. The Eta Invariant 791991 Mathemati s Subje t Classi� ation. 53C21, 58G30.Key words and phrases. Determinant, Dira operators, eta invariant, heat equation, index,Maslov index, partitioned manifolds, pasting formulas, spe tral ow, symple ti analysis .Printed April 8, 2003. 1

2 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK3.1. Fun tional Integrals and Spe tral Asymmetry 833.2. The �{Determinant for Operators of In�nite Rank 863.3. Spe tral Invariants of Di�erent `Sensitivity' 883.4. Pasting Formulas for the Eta{Invariant - Outlines 903.5. The Adiabati Additivity of the Small-t Chopped �-invariant 963.6. Asymptoti Vanishing of Large-t Chopped �-invariant on Stret hedPart Manifold 1033.7. The Estimate of the Lowest Nontrivial Eigenvalue 1093.8. The Spe trum on the Closed Stret hed Manifold 1153.9. The Additivity for Spe tral Boundary Conditions 123Referen es 126Introdu tionFor many de ades now, workers in di�erential geometry and mathemati alphysi s have been in reasingly on erned with di�erential operators (exterior dif-ferentiation, onne tions, Lapla ians, Dira operators, et .) asso iated to under-lying Riemannian or spa e{time manifolds. Of parti ular interest is the interplaybetween the spe tral de omposition of su h operators and the geometry/topologyof the underlying manifold. This has be ome a large, diverse �eld involving indextheory, the distribution of eigenvalues, zero sets of eigenfun tions, Green fun -tions, heat and wave kernels, families of ellipti operators and their determinants, anoni al se tions, et .. Moreover, Donaldson's analysis of moduli of solutions ofthe nonlinear Yang{Mills equations and Seiberg{Witten theory have led to pro-found insights into the lassi� ation of four{manifolds, whi h were not a essibleby te hniques that are e�e tive in higher dimensions.We shall tou h upon many of these topi s, but we fo us on three invariants hara terizing the asymmetry of the spe trum of operators of Dira type: the indexwhi h gives the hiral asymmetry of the kernel or null spa e (i.e., the di�eren e inthe number of independent left and right zero modes) of a total Dira operator; thespe tral ow of a urve of Dira operators that ounts the net number of eigenvaluesmoving from the negative half line over to the positive; and the eta invariantwhi h des ribes the overall asymmetry of the spe trum of a Dira operator. Thethree invariants appear both for Dira operators and urves of Dira operatorson a losed manifold or on a smooth ompa t manifold with boundary subje t tosuitable boundary onditions.Index and spe tral ow an be des ribed in general fun tional analyti terms,namely for bounded and unbounded, losed Fredholm operators and urves ofbounded and unbounded self{adjoint Fredholm operators. Correspondingly, sta-bility properties of index and spe tral ow and their topologi al and geometri

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 3meaning are relatively well understood. There is, however, not an easily identi�-able operator lass for eta invariant, and its behavior under perturbations is ratherdeli ate.Most signi� ant di�eren es between the behavior of these three invariants aremet when we address splitting properties on partitioned manifolds (with produ tmetri s assumed near the partitioning hypersurfa e). It is well known that theindex an be des ribed in lo al terms. This is one aspe t of the Atiyah{SingerIndex Theorem. So, splitting formulas for the index are relatively easily obtainable,on e one spe i�es and understands boundary and transmission onditions for thegluing of the parts of the underlying manifold. In this pro ess, pre ise additivityis obtained only for a few invariants, Euler hara teristi and signature, whi ha tually an be hara terized by utting and pasting invarian e. In general, an errorterm appears whi h an be expressed either by spe tral ow or, equivalently, by theindex of a boundary value problem on a ylinder over the separating hypersurfa eor, alternatively, by the index of a suitable Fredholm pair.Surprisingly, simple splitting formulas an be obtained also for the spe tral ow and eta invariant. This is parti ularly surprising for the eta invariant wherewe only have a lo al formula for the �rst derivative. Now the integer error termsare expressed by the Maslov index for urves of Cau hy data spa es. These for-mulas relate the symmetri ategory of self{adjoint Dira operators over losedpartitioned manifolds (and self{adjoint boundary value problems over ompa tmanifolds with boundary) to the symple ti analysis of Lagrangian subspa es (theCau hy data spa es).One message of this review is, that index, spe tral ow, and eta invariant, inspite of their quite di�erent appearan e, share various features whi h be ome mostvisible on partitioned manifolds. Roughly speaking, one reason for that is that thespe tral ow of a path of Dira type operators (say, on a losed manifold) withunitarily equivalent ends A1 = gA0g�1 equals the index of the indu ed suspensionoperator �t+At on the underlying mapping torus (see [BoWo93, Theorems 17.3,17.17, and Proposition 25.1℄). Another reason is that the integer part of the deriv-ative of the eta invariant along a path of Dira operators or boundary problems an be expressed by the spe tral ow (see, e.g., [DoWo91℄, [LeWo96, p. 39℄, and[KiLe00, Se tion 3℄).We also summarize re ent dis ussion on additivity and non{additivity of thezeta regularized determinant. It should be mentioned that the various splitting for-mulas all depend de isively on the well{established unique ontinuation property(UCP) for operators of Dira type. We give a full proof of weak UCP below in Se -tion 1.4. Moreover (modulo some te hni alities for the omputation of the indexdensity form, to appear in our forth oming book [BlBo03℄), we provide proofs ofthe Atiyah{Singer and Atiyah{Patodi{Singer Index Theorems in important spe ial ases using heat equation methods. That we explain on rete al ulations only forthe ase of the Atiyah{Patodi{Singer (spe tral { \APS") boundary ondition isno big loss of generality sin e ea h admissible boundary ondition for an operator

4 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKof Dira type D with tangential part B an be written as the APS proje tion ofa perturbed operator B0 (see below Lemma 2.29, following a re ent result, GerdGrubb [Gr02℄).The se ond author wants to thank K.P. Woj ie howski (Indiana University {Purdue University Indianapolis) for many dis ussions about the subje t(s) of thisreview and the Erwin S hr�odinger International Institute for Mathemati al Physi sat Vienna for generous hospitality during the �nalizing phase of this review. Weboth thank the referees for their orre tions, thoughtful omments, and helpfulsuggestions whi h led to many improvements. They learly went beyond the allof duty, and we are in their debt.1. Basi Notations and Results1.1. Index of Fredholm Operators and Spe tral Flow of Curves ofSelf{Adjoint Fredholm Operators.1.1.1. Notation. Let H be a separable omplex Hilbert spa e. First let usintrodu e some notation for various spa es of operators in H:C(H) := losed, densely de�ned operators on H;B(H) := bounded linear operators H ! H;U(H) := unitary operators H ! H;K(H) := ompa t linear operators H ! H;F(H) := bounded Fredholm operators H ! H;CF(H) := losed, densely de�ned Fredholm operators on H:If no onfusion is possible we will omit \(H)" and write C; B;K; et .. ByCsa;Bsa et ., we denote the set of self{adjoint elements in C; B; et ..1.1.2. Operators With Index { Fredholm Operators. The topology of the op-erator spa es U(H), F(H), and F sa(H) is quite well understood. The key resultsare (1) the Kuiper Theorem whi h states that U(H) is ontra tible;(2) the Atiyah{J�ani h Theorem whi h states that F(H) is a lassifying spa efor the fun tor K. Expli itly, the onstru tion of the index bundle ofa ontinuous family of bounded Fredholm operators parametrized overa ompa t topologi al spa e X yields a homomorphism of semigroups,namely index : [X;F ℄ ! K (X) : In parti ular, the index is a homotopyinvariant, and it provides a one{to{one orresponden e of the onne ted omponents [point;F ℄ of F with K (point) =Z:(3) the orresponding observation that F sa(H) onsists of three onne ted omponents, the ontra tible subsets F sa(H)+ and F sa(H)� of essen-tially positive, respe tively essentially negative, Fredholm operators, andthe topologi ally nontrivial omponent F sa� (H) whi h is a lassifyingspa e for the fun tor K�1 . In parti ular, the spe tral ow gives an iso-morphism of the fundamental group �1 (F sa� (H)) onto the integers.

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 5Full proofs an be found in [BlBo03℄ for items 1 and 2, and in [AtSi69℄,[BoWo93℄, and [Ph96℄ for item 3.So mu h for the bounded ase. Of ourse, the Dira operators of interest tous are not bounded in L2. On losed manifolds, however, they an be onsideredas bounded operators from the �rst Sobolev spa e H1 into L2, and by identifyingthese two Hilbert spa es we an onsider a Dira operator as a bounded operatorof a Hilbert spa e in itself. The same philosophy an be applied to Dira operatorson ompa t manifolds with boundary when we onsider the domain not as densesubspa e in L2 but as Hilbert spa e (with the graph inner produ t) and thenidentify.Stri tly speaking, the on ept of unbounded operator is dispensable here. How-ever, for varying boundary onditions it is ne essary to keep the distin tion andto follow the variation of the domain as a variation of subspa es in L2.Following [CorLab℄, we generalize the on ept of Fredholm operators to theunbounded ase.Definition 1.1. Let H be a omplex separable Hilbert spa e. A linear (notne essarily bounded) operator F with domain Dom(F ), null{spa e Ker(F ), andrange Im(F ) is alled Fredholm if the following onditions are satis�ed.(i) Dom(F ) is dense in H.(ii) F is losed.(iii) The range Im(F ) of F is a losed subspa e of H.(iv) Both dimKer(F ) and odimIm(F ) = dimIm(F )? are �nite. The di�er-en e of the dimensions is alled index(F ).So, a losed operator F is hara terized as a Fredholm operator by the sameproperties as in the bounded ase. Moreover, as in the bounded ase, F is Fredholmif and only if F � is Fredholm (proving the losedness of Im(F �) is deli ate: see[CorLab, Lemma 1.4℄) and ( learly) we haveindexF = dimKerF � dimKerF � = � indexF �:In parti ular, indexF = 0 in ase F is self{adjoint.The omposition of (not ne essarily bounded) Fredholm operators yields againa Fredholm operator. More pre isely, we have the following omposition rule. Forthe proof, whi h is onsiderably more involved than that in the bounded ase, werefer to [GoKr57℄, [CorLab, Lemma 2.3 and Theorem 2.1℄.Theorem 1.2. (Gohberg, Krein) If F and G are (not ne essarily bounded)Fredholm operators then their produ t GF is densely de�ned withDom(GF ) = Dom(F ) \ F�1Dom(G)and is a Fredholm operator. Moreover,indexGF = indexF + indexG :

6 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK1.1.3. Metri s on the Spa e of Closed Operators. For S; T 2 C(H) the or-thogonal proje tions PG(S), PG(T ) onto the graphs of S; T in H �H are boundedoperators and (S; T ) := kPG(T ) � PG(S)kde�nes a metri for C(H), the proje tion metri .It is also alled the gap metri and it is (uniformly) equivalent with the metri given by measuring the distan e between the ( losed) graphs, namelyd(G(S);G(T )) := supfx2G(S):kxk=1g d (x;G(T )) + supfx2G(T ):kxk=1g d (x;G(S)) :For details and the proof of the following Lemma and Theorem, we refer to[CorLab, Se tion 3℄.Lemma 1.3. For T 2 C(H) the orthogonal proje tion onto the graph of T inH �H an be written (where RT := (I+T �T )�1) asPG(T ) = � RT RTT �TRT TRTT �� = � RT T �RT�TRT TT �RT�� = � RT T �RT�TRT I�RT�� :Theorem 1.4. (Cordes, Labrousse) a) The spa e B(H) of bounded operatorson H is dense in the spa e C(H) of all losed operators in H. The topology indu edby the proje tion (�= gap) metri on B(H) is equivalent to that given by the operatornorm.b) Let CF(H) denote the spa e of losed (not ne essarily bounded) Fredholm op-erators. Then the index is onstant on the onne ted omponents of CF(H) andyields a bije tion between the integers and the onne ted omponents.Example 1.5. Let H be a Hilbert spa e with e1; e2; : : : a omplete orthonor-mal system (i.e., an orthonormal basis forH). Consider the multipli ation operatorMid, given by the domainD := Dom(Mid) := nX1j=1 jej j X1j=1 j2j jj2 < +1oand the operationD 3 u =X jej 7! Mid(u) :=X j jej :It is a densely de�ned losed operator whi h is inje tive and surje tive. Let Pndenote the orthogonal proje tion of H onto the linear span of the n-th orthonor-mal basis element en . Clearly the sequen e (Pn) does not onverge in B(H) inthe operator norm. However, the sequen e Mid � 2nPn of self{adjoint Fredholmoperators onverges in C(H) with the proje tion metri to Mid.This an be seen by the following argument: On the subset of self{adjoint(not ne essarily bounded) operators in the spa e Csa(H) the proje tion metri isuniformly equivalent to the metri given by (A1; A2) := k(A1 + i)�1 � (A2 + i)�1k ;

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 7(see below Theorem 1.10). Then for Tn :=Mid � 2nPn, (Tn + i)�1 � (Mid + i)�1 = (�n + i)�1 en � (n+ i)�1 en = 2nn2+1 ! 0:Remark 1.6. The results by Heinz Cordes and Jean{Philippe Labrousse mayappear to be rather ounter{intuitive. For (a), it is worth mentioning that theoperator{norm distan e and the proje tion metri on the set of bounded opera-tors are equivalent but not uniformly equivalent sin e the operator norm is om-plete while the proje tion metri is not omplete on the set of bounded operators.A tually, this is the point of the �rst part of (a); see also the pre eding example.Assertion (b) says two things: (i) that the index is a homotopy invariant,i.e., two Fredholm operators have the same index if they an be onne ted by a ontinuous urve in CF(H); (ii) that two Fredholm operators having the sameindex always an be onne ted by a ontinuous urve in CF (H). Note that thetopologi al results are not as far rea hing as for bounded Fredholm operators.1.1.4. Self{Adjoint Fredholm Operators and Spe tral Flow. We investigate thetopology of the subspa e of self{adjoint (not ne essarily bounded) Fredholm oper-ators. Many users of the notion of spe tral ow feel that the de�nition and basi properties are too trivial to bother with. However, there are some diÆ ulties bothwith extending the de�nition of spe tral ow from loops to paths and from urvesof bounded self{adjoint Fredholm operators to urves of not ne essarily boundedself{adjoint Fredholm operators.To over ome the se ond diÆ ulty, the usual way is to apply the Riesz trans-formation whi h yields a bije tion(1.1) R : Csa �! fS 2 Bsa j kSk � 1 and S � I both inje tiveg; whereR (T ) := T (I + T 2)�1=2In [BoFu98℄ the following theorem was proved:Theorem 1.7. Let S be a self{adjoint operator with ompa t resolvent in areal separable Hilbert spa e H and let C be a bounded self{adjoint operator. Thenthe sum S +C also has ompa t resolvent and is a losed Fredholm operator. Wehave kR(S + C) �R(S)k � kCk ;where the onstant does not depend on S or on C.The pre eding theorem is applied in the following form:Corollary 1.8. Curves of self{adjoint (unbounded) Fredholm operators ina separable real Hilbert spa e of the form fS + Ctgt2I are mapped into ontin-uous urves in F sa by the transformation R when S is a self{adjoint operatorwith ompa t resolvent and fCtgt2I is a ontinuous urve of bounded self{adjointoperators.

8 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKRemark 1.9. De�neTS : Bsa �! CF sa by TS (C) := S +CThis is translation by S, mapping bounded self{adjoint operators on H into self{adjoint Fredholm operators in H. On CF sa, the gap topology is de�ned by themetri g(A1; A2) :=pkRA1 � RA2k2 + kA1RA1 � A2RA2k2 ;where RA: := (I+A2: )�1 as before (see Cordes and Labrousse, [CorLab℄ and alsoKato, [Kat℄). Theorem 1.7 says that the omposition R Æ TS is ontinuous.Bsa TS�! CF sa&RÆTS #RF saFurther, we an prove that the translation operator TS is a ontinuous operatorfrom Bsa onto the subspa e Bsa + S � CF sa.The pre eding arguments permit to treat ontinuous urves of Dira operatorsin the same way as ontinuous urves of self{adjoint bounded Fredholm operatorsunder the pre ondition that the domain is �xed and the perturbation is only bybounded self{adjoint operators. That pre ondition is satis�ed when we have a urve of Dira operators on a losed manifold whi h di�er only by the underlying onne tion. It is also satis�ed for urves of Dira operators on a manifold withboundary as long the perturbation is bounded. In parti ular, this demands thatthe domain remains �xed.A loser look at Example 1.5 shows that the pre eding argument annot begeneralized: The sequen e of the Riesz transforms R(Tn) of Tn := Mid � 2nPndoes not onverge, sin e R(Tn) is not a Cau hy sequen e:kR(Tn) �R (Tn+1)k � kR(Tn)en �R (Mid) enk= n�2np1+(n�2n)2 en � np1+n2 en = 2np1+n2 ! 2 as n!1:In parti ular, the Riesz transformation is not ontinuous on the whole spa e Csa,nor on CF sa, neither on the whole spa e of self{adjoint operators with ompa tresolvent. Other methods are needed for working with varying domains.Of ourse, we an de�ne a di�erent metri in Csa ; e.g. the metri whi h makesthe Riesz transformation a homeomorphism. That approa h was hosen by L.Ni olaes u in [Ni00℄. He shows that quite a large lass of naturally arising urvesof Dira operators with varying domain are ontinuous under this `Riesz metri ',as opposed to the aforementioned example.Here, we hoose a di�erent approa h and adopt the gap metri . Note that ontinuity in the gap metri is mu h easier to establish than ontinuity in the

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 9Riesz metri . We follow [BoLePh01℄ where the proofs an be found. A feature ofthis approa h is the use of the Cayley Transform:Theorem 1.10. (a) On Csa the gap metri is (uniformly) equivalent to themetri given by (T1; T2) = k(T1 + i)�1 � (T2 + i)�1k:(b) Let � : R! S1 n f1g; x 7! x�ix+i denote the Cayley transform. Then � indu esa homeomorphism� : Csa(H) �! fU 2 U(H) j U � I is inje tive g =: UinjT 7! �(T ) = (T � i)(T + i)�1:(1.2)More pre isely, the gap metri is (uniformly) equivalent to the metri ~Æ de�ned by~Æ(T1; T2) = k�(T1)� �(T2)k.We note some immediate onsequen es of the Cayley pi ture:Corollary 1.11. (a) With respe t to the gap metri the set Bsa(H) is densein Csa(H).(b) For � 2 R the setsfT 2 Csa(H) j � =2 spe Tg and fT 2 Csa(H) j � 62 spe ess Tgare open in the gap topology.( ) The set CF sa = fT 2 Csa j 0 =2 spe ess Tg = ��1(FU), where FU := fU 2U j �1 =2 spe ess Ug = fU 2 U j U + I Fredholm operatorg, of (not ne essarilybounded) self{adjoint Fredholm operators is open in Csa.The pre eding Corollary implies that the set F sa is dense in CF sa with respe tto the gap metri .Contrary to the bounded ase, we have the following somewhat surprising result inthe unbounded ase: In parti ular, it shows that not every gap ontinuous path inCF sa with endpoints in F sa an be ontinuously deformed into an operator norm ontinuous path in F sa, in spite of the density of F sa in CF sa.Theorem 1.12. (a) CF sais path onne ted with respe t to the gap metri .(b) Moreover, its Cayley imageFUinj := fU 2 U j U + I Fredholm and U � I inje tiveg = �(CF sa)is dense in FU .Proof. (a) On e again we look at the Cayley transform pi ture. Note thatso far we have introdu ed three di�erent subsets of unitary operatorsUinj := fU 2 U j U � I inje tiveg = �(Csa)(1.3) FU := fU 2 U j U + I Fredholmg ; and(1.4) FUinj := FU \ U inj = �(CF sa):(1.5)

10 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKLet U 2 FUinj. Then H is the dire t sum of the spe tral subspa es H� of U orresponding to [0; �) and [�; 2�℄ respe tively and we may de ompose U = U+ �U�. More pre isely, we havespe (U+) � feit j t 2 [0; �)g and spe (U�) � feit j t 2 [�; 2�℄g :Note that there is no interse tion between the spe tral spa es in the endpoints: if�1 belongs to spe (U ), it is an isolated eigenvalue by our assumption and hen ebelongs only to spe (U�); if 1 belongs to spe (U ), it an belong both to spe (U+)and spe (U�), but in any ase, it does not ontribute to the de omposition of Usin e, by our assumption, 1 is not an eigenvalue at all.By spe tral deformation (\squeezing the spe trum down to +i and �i") we ontra t U+ to iI+ and U� to �iI� , where I� denotes the identity on H� . We dothis on the upper half ar and the lower half ar , respe tively, in su h a way that1 does not be ome an eigenvalue under the ourse of the deformation: a tually itwill no longer belong to the spe trum; neither will �1 belong to the spe trum.That is, we have onne ted U and iI+ � �iI� within � (CF sa).We distinguish two ases: If H� is �nite-dimensional, we now rotate �iI� upthrough �1 into iI�_. More pre isely, we onsider fiI+ � ei(�=2+(1�t)�)I�gt2[0;1℄ .This proves that we an onne t U with iI+ � iI� = iI within � (CF sa) in this�rst ase.If H� is in�nite-dimensional, we \dilate" �iI� in su h a way that no eigen-values remain. To do this, we identify H� with L2([0; 1℄). Now multipli ation by�i on L2([0; 1℄) an be onne ted to multipli ation by a fun tion whose values area short ar entered on �i and so that the resulting operator V� on H� has noeigenvalues. This will at no time introdu e spe trum near +1 or �1. We then ro-tate this ar up through +1 (whi h keeps us in the right spa e) until it is enteredon +i. Then we ontra t the spe trum on H� to be +i. That is, also in this asewe have onne ted our original operator U to +iI. To sum up this se ond ase(see also Figure 1):U � iI+ � �iI� � iI+ � V� � iI+ � eit�V� for t 2 [0; 1℄� iI+ ��V� � iI+ � �(�iI�) � iI :To prove (b), we just de ompose any V 2 FU into V = U � I1, where U 2FUinj(H0) and I1 denotes the identity on the 1-eigenspa e H1 = Ker(V � I) of Vwith H = H0�H1 an orthogonal de omposition. Then for " > 0, U�ei"I1 2 FUinjapproa hes U for "! 0. �Remark 1.13. The pre eding proof shows also that the two subsets of CF saCF sa� = fT 2 CF sa j spe ess(T ) � C�g;the spa es of all essentially positive, resp. all essentially negative, self-adjoint Fred-holm operators, are no longer open. The third of the three omplementary subsetsCF sa� = CF sa n �CF sa+ [ CF sa� �

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 11U U= + © {U

» ©{ {iI+ iI

» © {iI+ iI iI=

» © {iI+ iI iI=

» © {iI+ V

» © {{iI+ V

Case IICase I

Figure 1. Conne ting a �xed U in FUinj to iI. Case I (�nite rankU�) and Case II (in�nite rank U�)is also not open. We do not know whether the two \trivial" omponents are on-tra tible as in the bounded ase nor whether the whole spa e is a lassifying spa efor K1 as the nontrivial omponent in the bounded ase. Independently of Exam-ple 1.5, the onne tedness of CF sa and the dis onne tedness of F sa show that theRiesz map is not ontinuous on CF sa in the gap topology.In analogy to [Ph96℄, we an give an expli it des ription of the winding num-ber (spe tral ow a ross �1) wind(f) of a urve f in FU . Alternatively, it an beused as a de�nition of wind:

12 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKProposition 1.14. Let f : [0; 1℄! FU be a ontinuous path.(a) There is a partition f0 = t0 < t1 < � � � < tn = 1g of the interval and positivereal numbers 0 < "j < �, j = 1; : : : ; n, su h that Ker(f(t) � ei(��"j)) = f0g fortj�1 � t � tj .(b) Then wind(f) = nXj=1 k(tj ; "j) � k(tj�1; "j);where k(t; "j) := X0��<"j dimKer(f(t) � ei(�+�)):( ) In parti ular, this al ulation of wind(f) is independent of the hoi e of thepartition of the interval and of the hoi e of the barriers.Proof. In (a) we use the ontinuity of f and the fa t that f (t) 2 FU . (b)follows from the path additivity of wind. ( ) is immediate from (b). �This idea of a spe tral ow a ross �1 was introdu ed �rst in [BoFu98, Se .1.3℄, where it was used to give a de�nition of the Maslov index in an in�nitedimensional ontext (see also De�nition 1.24 further below).After these explanations the de�nition of spe tral ow for paths in CF sa isstraightforward:Definition 1.15. Let f : [0; 1℄ ! CF sa(H) be a ontinuous path. Then thespe tral ow SF(f) is de�ned bySF(f) := wind(� Æ f):>From the properties of � and of the winding number we infer immediately:Proposition 1.16. SF is path additive and homotopy invariant in the fol-lowing sense: let f1; f2 : [0; 1℄ ! CF sa be ontinuous paths and let h : [0; 1℄ �[0; 1℄ ! CF sa be a homotopy su h that h(0; t) = f1(t); h(1; t) = f2(t) and su hthat dimKer h(s; 0); dimKerh(s; 1) are independent of s. Then SF(f1) = SF(f2).In parti ular, SF is invariant under homotopies leaving the endpoints �xed.>From Proposition 1.14 we getProposition 1.17. For a ontinuous path f : [0; 1℄ ! F sa our de�nition ofspe tral ow oin ides with the de�nition in [Ph96℄.Note that also the onventions oin ide for 0 2 spe f(0) or 0 2 spe f(1).Corollary 1.18. For any S 2 CF sa with ompa t resolvent and any ontin-uous path C : [0; 1℄! Bsa we have SF(S + C) = SF(R (S +C)) where R denotesthe Riesz transformation of (1.1).

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 13Note that the urve S+C is in CF sa, so that SF(S +C) is de�ned via Cayleytransformation, whereas the urve R (S + C) of the Riesz transforms is in F sa.Remark 1.19. The spe tral ow indu es a surje tion of �1 (CF sa) onto Z.Be ause Zis free, there is a right inverse of SF and a normal subgroup G of�1 (CF sa) su h that we have a split short exa t sequen e0! G! �1 (CF sa)!Z! 0:For now, an open question is whether G is trivial: does the spe tral owdistinguish the homotopy lasses? That is, the question is whether ea h loop withspe tral ow 0 an be ontra ted to a onstant point, or equivalently, whether two ontinuous paths in CF sa with same endpoints and with same spe tral ow anbe deformed into ea h other? Or is �1 (CF sa) �= Z�j G the semi-dire t produ tof a nontrivial fa tor G with Z? In that ase, homotopy invariants of a urve inCF sa are not solely determined by the spe tral ow ( ontrary to the folklore behindparts of the topology and physi s literature). For now, we an only spe ulate aboutthe existen e of an additional invariant and its possible de�nition. For example,one an try to de�ne a spe tral ow at in�nity. Then, ontinuity of the Riesztransformation R on a sub lass S � CF sa would imply vanishing spe tral ow atin�nity. Non-vanishing spe tral ow at in�nity will typi ally appear with familiesof the type dis ussed in Example 1.5 (and after Remark 1.9). One may also expe tit with urves of di�erential operators of se ond order. However, the results of[Ni00℄, though only obtained under quite restri tive onditions, may indi ate thatperhaps spe tral ow at in�nity will not be exhibited for ontinuous urves ofDira operators. If this is true, it will also explain why the mentioned unfoundedfolklore has not yet led to lear ontradi tions.1.2. Symmetri Operators and Symple ti Analysis. In an interviewwith Vi tor M. Bu hstaber in the Newsletter of the European Mathemati al So- iety [No01, p. 20℄, Sergej P. Novikov re alls his idea of the late 1960's and theearly 1970's, whi h were radi ally di�erent of the main stream in topology at thattime, \that the explanation of higher signatures and of other deep properties ofmultiply onne ted manifolds had a symple ti origin... In 1971 I. Gelfand wentinto my algebrai ideas: they impressed him greatly. In parti ular, he told me ofhis observation that the so- alled von Neumann theory of self-adjoint extensionsof symmetri operators is simply the hoi e of a Lagrangian subspa e in a Hilbertspa e with symple ti stru ture." Closely following [BoFu98℄, [BoFu99℄, and[BoFuOt01℄ (see also the Krein{Vishik{Birman theory summarized in [AlSi80℄and [LaSnTu75℄, for �rst pointing to symple ti aspe ts), we will elaborate onthat thought.

14 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK1.2.1. Symple ti Hilbert Spa e, Fredholm Lagrangian Grassmannian, and Mas-lov Index. We �x the following notation. Let (H; h:; :i) be a separable real Hilbertspa e with a �xed symple ti form !, i.e., a skew-symmetri bounded bilinear formon H�H whi h is nondegenerate. We assume that ! is ompatible with h:; :i in thesense that there is a orresponding almost omplex stru ture J : H ! H de�nedby(1.6) !(x; y) = hJx; yiwith J2 = � I; tJ = �J , and hJx; Jyi = hx; yi. Here tJ denotes the transpose of Jwith regard to the (real) inner produ t hx; yi. Let L = L(H) denote the set of allLagrangian subspa es � of H (i.e., � = (J�)?, or equivalently, let � oin ide withits annihilator �0 with respe t to !). The topology of L is de�ned by the operatornorm of the orthogonal proje tions onto the Lagrangian subspa es.Let �0 2 L be �xed. Then any � 2 L an be obtained as the image of �?0under a suitable unitary transformation� = U (�?0 )(see also Figure 2a). Here we onsider the real symple ti Hilbert spa e H asa omplex Hilbert spa e via J . The group U(H) of unitary operators of H a tstransitively on L; i.e., the mapping(1.7) � : U(H) �! LU 7! U (�0?)is surje tive and de�nes a prin ipal �bre bundle with the group of orthogonaloperators O(�0) as stru ture group.Example 1.20. (a) In �nite dimensions one onsiders the spa e H := Rn�Rnwith the symple ti form! ((x; �); (y; �)) := �hx; �i+ h�; yi for (x; �); (y; �) 2 H:To emphasize the �niteness of the dimension we write Lag(R2n) := L(H). Forlinear subspa es of R2n one hasl 2 Lag(R2n) () dim l = n and l � l0 := !-annihilator of l;i.e., Lagrangian subspa es are true half-spa es whi h are maximally isotropi (`isotropi ' means l � l0).Note that R2n = Rn C with the Hermitian produ t((x; �); (y; �))C = (x+ i�)(y � i�) := hx; yi + h�; �i+ i h�; yi � i hx; �i :Then every U 2 U(n) an be written in the form U = A + iB = � A �BB A �with tAB = tBA, AtB = BtA, and AtA+ BtB = I, andO(n) 3 A! � A 00 A � = A+ 0i 2 U(n)

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 15U

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a) b)Figure 2. a) The generation of L = �� = U (�?0 )jU 2 U (H)b) One urve and two Maslov y les in FL�0 = FLb�0gives the embedding of O(n) in U(n). One �nds Lag(R2n) �= U(n)=O(n) with thefundamental group �1(Lag(R2n)); �0) �=Z:The mapping is given by the `Maslov index' of loops of Lagrangian subspa eswhi h an be des ribed as an interse tion index with the `Maslov y le'. There isa ri h literature on the subje t, see e.g. the seminal paper [Ar67℄, the systemati review [CaLeMi94℄, or the ohomologi al presentation [Go97℄.(b) Let f'kgk2Znf0g be a omplete orthonormal system for H. We de�ne an almost omplex stru ture, and so by (1.6) a symple ti form, byJ'k := sign(k)'�k :Then the spa es H� := spanf'kgk<0 and H+ := spanf'kgk>0 are omplementaryLagrangian subspa es of H.In in�nite dimensions, the spa e L is ontra tible due to Kuiper's Theorem(see [BlBo03℄, Part I) and therefore topologi ally not interesting. Also, we needsome restri tions to avoid in�nite-dimensional interse tion spa es when ountinginterse tion indi es. Therefore we repla e L by a smaller spa e. This problem anbe solved, as �rst suggested in Swanson [Sw78℄, by relating symple ti fun tionalanalysis with the spa e Fred(H) of Fredholm operators, we obtain �nite dimensionsfor suitable interse tion spa es and at the same time topologi ally highly nontrivialobje ts.

16 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKDefinition 1.21. (a) The spa e of Fredholm pairs of losed in�nite-dimensionalsubspa es of H is de�ned byFred2(H) := f(�; �) j dim(� \ �) < +1 and �+� � H losedand dimH=(�+�) < +1g:Then index(�; �) := dim(� \ �)� dimH=(�+�)is alled the Fredholm interse tion index of (�; �).(b) The Fredholm-Lagrangian Grassmannian of a real symple ti Hilbert spa e Hat a �xed Lagrangian subspa e �0 is de�ned asFL�0 := f� 2 L j (�; �0) 2 Fred2(H)g:( ) The Maslov y le of �0 in H is de�ned asM�0 := FL�0 n FL(0)�0 ;where FL(0)�0 denotes the subset of Lagrangians interse ting �0 transversally; i.e.,� \ �0 = f0g.Re all the following algebrai and topologi al hara terization of general andLagrangian Fredholm pairs (see [BoFu98℄ and [BoFu99℄, inspired by [BoWo85℄,Part 2, Lemma 2.6).Proposition 1.22. ( a) Let �; � 2 L and let ��; �� denote the orthogonalproje tions of H onto � respe tively �. Then(�; �) 2 Fred2(H)() �� + �� 2 Fred(H)() �� � �� 2 Fred(H):( b) The fundamental group of FL�0 isZ, and the mapping of the loops in FL�0(H)onto Zis given by the Maslov indexmas : �1(FL�0(H))) �=�!Z:To de�ne the Maslov index, one needs a systemati way of ounting, addingand subtra ting the dimensions of the interse tions �s\�0 of the urve f�sg withthe Maslov y leM�0 . We re all from [BoFu98℄, inspired by [Ph96℄, a fun tionalanalyti al de�nition for ontinuous urves without additional assumptions.First we re all from (1.7) that any � 2 FL�0(H) an be obtained as the imageof �?0 under a suitable unitary transformation � = U (�?0 ). As noted before, theoperator U is not uniquely determined by �. A tually, from � �= �0C we obtain a omplex onjugation so that we an de�ne the transpose by the following formulaTU := U�and obtain a unitary operator W (�) := UTU whi h an be de�ned invariantly asthe omplex generator of the Lagrangian spa e � relative to �0.

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 17Proposition 1.23. The omposed mapping WFL�0(H) 3 � 7�! U 7�!W (�) := UTU 2 FU (H)is well de�ned. In parti ular, we haveKer (I +W ) = (� \ �0) C = (� \ �0) + J (� \ �0)Proof. The operator I + W is a Fredholm operator be ause (�; �0) is aFredholm pair. �Definition 1.24. Let H be a symple ti Hilbert spa e and �0 2 L(H). Let[0; 1℄ 3 s 7�! �s 2 FL�0(H)be a ontinuous urve. ThenW Æ� is a ontinuous urve in FU (H), and the Maslovindex an be de�ned by mas (�; �0) := wind (W Æ �) ;where wind is de�ned as in Proposition 1.14.Remark 1.25. To de�ne the Maslov (interse tion) index mas (f�sg ; �0), we ount the hange of the eigenvalues of Ws near �1 little by little. For example,between s = 0 and s = s0 we plot the spe trum of the omplex generator Ws loseto ei� . In general, there will be no parametrization available of the spe trum near�1. For suÆ iently small s0, however, we an �nd barriers ei(�+�) and ei(���) su hthat no eigenvalues are lost through the barriers on the interval [0; s0℄. Then we ount the number of eigenvalues (with multipli ity) of Ws between ei� and ei(�+�)at the right and left end of the interval [0; s0℄ and subtra t. Repeating this pro- edure over the length of the whole s-interval [0; 1℄ gives the Maslov interse tionindex mas (f�sg ; �0) without any assumptions about smoothness of the urve,`normal rossings', or non-invertible endpoints.It is worth mentioning that the onstru tion an be simpli�ed for a omplex sym-ple ti Hilbert spa e H. Then ea h Lagrangian subspa e of H is the graph of auniquely determined unitary operator fromKer (J � iI) to Ker (J + iI). Moreover,a pair (�; �) of Lagrangian subspa es is a Fredholm pair if and only if U�1V is Fred-holm, where � = G (U ) and � = G (V ). We have mas (f�tg ; �0) = SF �U�1t V; 1�where the 1 indi ates that the spe tral ow is taken at the eigenvalue 1.Remark 1.26. (a) By identifying H �= �0 C �= �0 � p�1�0 , we split in[BoFuOt01℄ any U 2 U(H) into a real and imaginary partU = X +p�1Ywith X;Y : �0 ! �0 . Let U(H)Fred denote the subspa e of unitary operatorswhi h have a Fredholm operator as real part. This is the total spa e of a prin ipal�bre bundle over the Fredholm Lagrangian Grassmannian FL�0 as base spa e and

18 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKwith the orthogonal group O(�0) as stru ture group. The proje tion is given bythe restri tion of the trivial bundle � : U(H)! L of (1.7) . This bundleU(H)Fred ��! FL�0may be onsidered as the in�nite-dimensional generalization of the familiar bundleU(n) ! Lag(R2n) for �nite n and provides an alternative proof of the homotopytype of FL�0 .(b) The Maslov index for urves depends on the spe i�ed Maslov y le M�0. Itis worth emphasizing that two equivalent Lagrangian subspa es �0 and b�0 (i.e.,dim�0=(�0 \ b�0) < +1) always de�ne the same Fredholm Lagrangian Grassman-nian FL�0 = FLb�0 but may de�ne di�erent Maslov y les M�0 6= Mb�0 . Theindu ed Maslov indi es may also be ome di�erent(1.8) mas(f�sgs2[0;1℄; �0)�mas(f�sgs2[0;1℄; b�0) in general6= 0(see [BoFu99℄, Proposition 3.1 and Se tion 5). However, if the urve is a loop,then the Maslov index does not depend on the hoi e of the Maslov y le. >Fromthis property it follows that the di�eren e in (1.8), beyond the dependen e on �0and b�0, depends only on the initial and end points of the path f�sg and maybe onsidered as the in�nite-dimensional generalization �H�or(�0; �1;�0; b�0) of theH�ormander index. It plays a part as the transition fun tion of the universal overingof the Fredholm Lagrangian Grassmannian (see also Figure 2b).1.2.2. Symmetri Operators and Symple ti Analysis. Let H be a real sepa-rable Hilbert spa e and A an (unbounded) losed symmetri operator de�ned onthe domain Dmin whi h is supposed to be dense in H. Let A� denote its adjointoperator with domain Dmax. We have that A�jDmin = A and that A� is the maxi-mal losed extension of A in H. Note that Dmax is a Hilbert spa e with the graphs alar produ t (x; y)G := (x; y) + (A�x;A�y) ;and Dmin is a losed subspa e of this Dmax sin e A is losed (ea h sequen e inDmin whi h is Cau hy relative to the graph norm de�nes a sequen e in the graphG (A) whi h is Cau hy relative to the simple norm in H �H).We form the spa e � of natural boundary values with the natural tra e map in the following way: Dmax �! Dmax=Dmin =: �x 7�! (x) = [x℄ := x+Dmin:The spa e � be omes a symple ti Hilbert spa e with the indu ed s alar produ tand the symple ti form given by Green's form! ([x℄ ; [y℄) := (A�x; y) � (x;A�y) for [x℄ ; [y℄ 2 �:

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 19We de�ne the natural Cau hy data spa e � := (KerA�). It is a Lagrangiansubspa e of � under the assumption that A admits at least one self-adjoint Fred-holm extension AD. A tually, we shall assume that A has a self-adjoint extensionAD with ompa t resolvent. Then (�; (D)) is a Fredholm pair of subspa es of �;i.e., � 2 FL (D) (�).We onsider a ontinuous urve fCsgs2[0;1℄ in the spa e of bounded self-adjointoperators onH. We assume that the operators A�+Cs�r have no `inner solutions';i.e., they satisfy the weak inner unique ontinuation property (UCP)(1.9) Ker (A� +Cs � r) \Dmin = f0gfor s 2 [0; 1℄ and jrj < "0 with "0 > 0. For a dis ussion of UCP see Se tion 1.4below.Clearly, the domains Dmax and Dmin are un hanged by the perturbation Csfor any s. So, � does not depend on the parameter s. Moreover, the symple ti form ! is invariantly de�ned on � and so also independent of s. It follows (see[BoFu98, Theorem 3.9℄) that the urve f�s := (Ker (A� +Cs))g is ontinuousin FL (D) (�).We summarize the basi �ndings:Proposition 1.27. (a) Assume that there exists a self{adjoint Fredholm ex-tension AD of A with domain D. Then the Cau hy data spa e �(A) is a losedLagrangian subspa e of � and belongs to the Fredholm{Lagrangian GrassmannianFL (D) (�).(b) For arbitrary domains D with Dmin � D � Dmax and (D) Lagrangian, theextension AD := AmaxjD is self{adjoint. It be omes a Fredholm operator, if andonly if the pair ( (D);�(A)) of Lagrangian subspa es of � be omes a Fredholmpair.( ) Let fCtgt2I be a ontinuous family (with respe t to the operator norm) ofbounded self{adjoint operators. Here the parameter t runs within the interval I =[0; 1℄. Assume the weak inner UCP for all operators A� + Ct . Then the spa es (Ker(A� +Ct; 0)) of Cau hy data vary ontinuously in �.Given this, the family fAD + Csg an be onsidered at the same time in thespe tral theory of self{adjoint Fredholm operators, de�ning a spe tral ow, andin the symple ti ategory, de�ning a Maslov index. Under the pre eding assump-tions, the main result obtainable at that level is the following general spe tral ow formula (proved in [BoFu98, Theorem 5.1℄ and inviting to generalizationsfor varying domains Ds instead of �xed domain D):Theorem 1.28. Let AD be a self-adjoint extension of A with ompa t resolventand let fAD + Csgs2[0;1℄ be a family satisfying the weak inner UCP assumption.Let �s denote the Cau hy data spa e (Ker (A� + Cs)) of A� + Cs. ThenSF fAD + Csg =mas (f�sg ; (D)) :

20 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK1.3. Operators of Dira Type and their Ellipti ity. There are di�erentnotions of operators of Dira type. We shall not dis uss the original hyperboli Dira operator (in the Minkowski metri ), but restri t ourselves to the ellipti ase related to Riemannian metri s.Re all that, if (M; g) is a ompa t smooth Riemannianmanifold (with or with-out boundary) with dimM = m, we denote by Cl(M ) = fCl(TMx; gx)gx2M thebundle of Cli�ord algebras of the tangent spa es. For S ! M a smooth om-plex ve tor bundle of Cli�ord modules, the Cli�ord multipli ation is a bundlemap : Cl(M ) ! Hom(S; S) whi h yields a representation : Cl(TMx; gx) !HomC(Sx; Sx) in ea h �ber. We may assume that the bundle S is equipped witha Hermitian metri whi h makes the Cli�ord multipli ation skew-symmetri h (v)s; s0i = �hs; (v)s0i for v 2 TMx and s 2 Sx:We note that it is not ne essary to assume that (M; g) admits a spin stru turein order that Cl(M ) and S exist. Indeed, one spe ial ase is obtained by takingS = �� (TM ) and letting be the extension of (v) (�) = v ^ � � vx� for v 2TMx � Cl(TMx; gx) and � 2 �� (TMx) ; where \x" denotes interior produ t (i.e.,the dual of the exterior produ t ^). The extension is guaranteed by the fa t that (v)2 = �gx (v; v) I. However, we do need a spin stru ture in the ase where S isa bundle of spinors.Any hoi e of a smooth onne tionr : C1(M ;S)! C1(M ;T �M S)de�nes an operator of Dira type D := Ær under the Riemannian identi� ationof the bundles TM and T �M . In lo al oordinates we have D := Pmj=1 (ej)rejfor any orthonormal base fe1; : : : ; emg of TMx. A tually, we may hoose a lo alframe in su h a way thatrej = ��xj + zero order termsfor all 1 � j � m. So, lo ally, we have(1.10) D :=Xmj=1 (ej ) ��xj + zero order terms :It follows at on e that the prin ipal symbol �1(D)(x; �) is given by Cli�ord mul-tipli ation with i�, so that any operator of Dira type is ellipti with symmetri prin ipal symbol. If the onne tionr is ompatible with Cli�ord multipli ation (i.e.r = 0), then the operator D itself be omes symmetri . We shall, however, ad-mit in ompatible metri s. Moreover, the Dira Lapla ian D2 has prin ipal symbol�2(D2)(x; �) given by s alar multipli ation by k�k2 using the Riemannian metri .So, the prin ipal symbol of D2 is a real multiple of the identity, and D2 is ellip-ti . In the spe ial ase above where S = �� (TM ) and we identify �� (TM ) with�� (T �M ) by means of the metri g, D be omes d+Æ : � (M; C ) ! � (M; C ) andD2 = dÆ + Æd is the Hodge Lapla ian on the spa e � (M; C ) of omplex-valuedforms on M .

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 21On a losed manifoldM , a key result for any operator D of Dira type (a tu-ally, for any linear ellipti operator of �rst order) is the a priori estimate (G�arding'sinequality)(1.11) k k1 � C (kD k0 + k k0) for all 2 H1 (M ;S) :Here k � k0 denotes the L2 norm and H1 (M ;S) denotes the �rst Sobolev spa ewith the norm k � k1. Note that the same symbol D is used for the original operator(de�ned on smooth se tions) and its losed L2 extension with domain H1 (M ;S).Combined with the simple ontinuity relation kD k0 � C 0 k k1, inequality(1.11) shows that the �rst Sobolev norm k � k1 and the graph norm oin ide onH1 (M ;S).1.4. Weak Unique Continuation Property. A linear or non-linear oper-ator D, a ting on fun tions or se tions of a bundle over a ompa t or non- ompa tmanifoldM has the weak Unique Continuation Property (UCP) if any solution of the equation D = 0 has the following property: if vanishes on a nonemptyopen subset of M , then it vanishes on the whole onne ted omponent of M ontaining . Note that weak inner UCP, as de�ned in (1.9), follows from weakUCP, but not vi e versa.There is also a notion of strong UCP, where, instead of assuming that a solution vanishes on an open subset, one assumes only that vanishes `of high order'at a point. The on epts of weak and strong UCP extend a fundamental propertyof analyti fun tions to some ellipti equations other than the Cau hy-Riemannequation.Up to now, (almost) all work on UCP goes ba k to two seminal papers [Ca33℄,[Ca39℄ by Torsten Carleman, establishing an inequality of Carleman type (see ourinequality 1.16 below). In this approa h, the di�eren e between weak and strongUCP and the possible presen e of more deli ate nonlinear perturbations are relatedto di�erent hoi es of the weight fun tion in the inequality, and to whether L2estimates suÆ e or Lp and Lq estimates are required.The weak UCP is one of the basi properties of an operator of Dira type D.Contrary to ommon belief, UCP is not a general fa t of life for ellipti operators.See [Pl61℄ where ounter-examples are given with smooth oeÆ ients. La k ofUCP invalidates the ontinuity of the Cau hy data spa es and of the Calder�onproje tion (Proposition 1.27 and Theorem 2.28b) and of the main ontinuitylemma (Lemma 2.27). It orrupts the invertible double onstru tion (Se tion 2.3.4)and threatens Bojarski type theorems (like Proposition 2.33 and Theorem 2.37).For partitioned manifoldsM =M1 [�M2 (see Subse tion 2.3), it guarantees thatthere are no ghost solutions of D = 0; that is, there are no solutions whi h vanishon M1 and have nontrivial support in the interior of M2. This property is also alled UCP from open subsets or a ross any hypersurfa e. For Eu lidean ( lassi al)Dira operators (i.e., Dira operators on Rmwith onstant oeÆ ients and without

22 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKperturbation), the property follows by squaring dire tly from the well-establishedUCP for the lassi al ( onstant oeÆ ients and no potential) Lapla ian.>From [BoWo93, Chapter 8℄ we re all a very simple proof of the weak UCPfor operators of Dira type, inspired by [Ni73, Se tions 6-7, in parti ular theproof of inequality (7.11)℄ and [Tr80, Se tion II.3℄. We refer to [Boo00℄ for afurther slight simpli� ation and a broader perspe tive, and to [BoMaWa02℄ forperturbed equations.The proof does not use advan ed arguments of the Aronszajn/Cordes type (see[Ar57℄ and [Co56℄) regarding the diagonal and real form of the prin ipal symbolof the Dira Lapla ian nor any other redu tion to operators of se ond order (like[We82℄), but only the following produ t property of Dira type operators (besidesG�arding's inequality).Lemma 1.29. Let � be a losed hypersurfa e of M with orientable normalbundle. Let u denote a normal variable with �xed orientation su h that a bi ollarneighborhood N of � is parameterized by [�";+"℄��. Then any operator of Dira type an be rewritten in the form(1.12) DjN = (du)� ��u +Bu +Cu� ;where Bu is a self-adjoint ellipti operator on the parallel hypersurfa e �u, andCu : Sj�u ! Sj�u a skew-symmetri operator of 0-th order, a tually a skew-symmetri bundle homomorphism.Proof. Let (u; y) denote the oordinates in a tubular neighborhood of �.Lo ally, we have y = (y1; : : : ; ym�1). Let u; 1; : : : ; m�1 denote Cli�ord multi-pli ation by the unit tangent ve tors in normal, resp. tangential, dire tions. By(1.10), we haveD = u ��u +Xm�1k=1 k ��yk + zero order terms = u� ��u + Bu� ; whereBu :=Xm�1k=1 � u k ��yk + zero order terms.We shall all Bu the tangential operator omponent of the operator A. Clearly itis an ellipti di�erential operator of �rst order over �u. From the skew-hermi ityof u and k, we have� u k ��yk�� = �� ��yk� (� k)(� u) = � k u ��yk + zero order terms= u k ��yk + zero order terms:So, B�u = Bu + zero order terms:

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 23Hen e, the prin ipal symbol of Bu is self-adjoint. Then the assertion of the lemmais proved by setting(1.13) Bu := 12 (Bu + B�u) and Cu := 12 (Bu � B�u) �Remark 1.30. (a) It is worth mentioning that the produ t form (1.12) isinvariant under perturbation by a bundle homomorphism. More pre isely: Let Dbe an operator on M whi h an be written in the form (1.12) lose to any losedhypersurfa e � , with Bu and Cu as explained in the pre eding Lemma. Let R bea bundle homomorphism. Then(D+ R) jN = (du)� ��u + Bu + Cu�+ (du)SjNwith T jN := (du)�RjN . Splitting T = 12(T + T �) + 12(T � T �) into a symmetri and a skew-symmetri part and adding these parts to Bu and Cu, respe tively,yields the desired form of (D+ R) jN .(b) For operators of Dira type, it is well known that a perturbation by a bundlehomomorphism is equivalent to modifying the underlying onne tion of the oper-ator. This gives an alternative argument for the invarian e of the form (1.12) foroperators of Dira type under perturbation by a bundle homomorphism.( ) By the pre eding arguments (a), respe tively (b), establishing weak UCP forse tions belonging to the kernel of a Dira type operator, respe tively an operatorwhi h an be written in the form (1.12), implies weak UCP for all eigense tions.Warning: for general linear ellipti di�erential operators, weak UCP for \zero-modes" does not imply weak UCP for all eigense tions.To prove the weak UCP, in ombination with the pre eding lemma, the stan-dard lines of the UCP literature an be radi ally simpli�ed, namely with regard tothe weight fun tions and the integration order of estimates. These simpli� ationsmake it also very easy to generalize the weak UCP to the perturbed ase.We repla e the equation D = 0 by(1.14) eD := D +PA( ) = 0 ;where PA is an admissible perturbation, in the following sense.Definition 1.31. A perturbation is admissible if it satis�es the followingestimate:(1.15) jPA( )jxj � P ( ; x)j (x)j for x 2Mfor a real-valued fun tion P ( ; �) whi h is lo ally bounded on M for ea h �xed .

24 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKExample 1.32. Some typi al examples of perturbations satisfying the admis-sibility ondition of De�nition 1.31 are:1. Consider a nonlinear perturbationPA( )jx := !( (x)) � (x) ;where !( (x))jx2M is a (bounded) fun tion whi h depends ontinuously on (x),for instan e, for a �xed (bounded) spinor se tion a(x) we an take!( (x)) := h (x); a(x)iwith h�; �i denoting the Hermitian produ t in the �ber of the spinor bundle overthe base point x 2M . This satis�es (1.15) .2. Another interesting example is provided by (linear) nonlo al perturbations with!( ; x) = ����Z k(x; z) (z)dz����with suitable integration domain and integrability of the kernel k. These alsosatisfy (1.15) .3. Clearly, an unbounded perturbation may be both nonlinear and global at thesame time. This will, in fa t, be the ase in our main appli ation. In all these asesthe only requirement is the estimate (1.15) with bounded !( (�)).We now show that (admissible) perturbed Dira operators always satisfy theweak Unique Continuation Property. In parti ular, we show that this is true forunperturbed Dira operators.Theorem 1.33. Let D be an operator of Dira type and PA an admissibleperturbation. Then any solution of the perturbed equation (1.14) vanishes iden-ti ally on any onne ted omponent of the underlying manifold M if it vanisheson a nonempty open subset of the onne ted omponent.Proof. Without loss of generality, we assume thatM is onne ted. Let be asolution of the perturbed (or, in parti ular) unperturbed equation whi h vanisheson an open, nonempty set . First we lo alize and onvexify the situation andwe introdu e spheri al oordinates (see Figure 3). Without loss of generality wemay assume that is maximal, namely the union of all open subsets on whi h vanishes; i.e., =M n supp .Sin eM is onne ted, to prove that =M it suÆ es to show that is losed(i.e., = ). If 6= ; then let y0 2 � := n; and let B be an open, normal oordinate ball about y0. Let p 2 \ B and let x0 2 supp be a point of thenon-empty, ompa t set B n = B \ supp whi h is losest to p. Letr := d (p; x0) = minx2Bn d (p; x) :For z 2 B, let u (z) := d (p; z)�r be a \radial oordinate". Note that u = 0 de�nesa sphere, say Sp;0, of radius r about x0. We have larger hyperspheres Sp;u � M

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 25¶W

W

M \W

u =Tu = 0

r

p

x0

Sp,0

Sp,TFigure 3. Lo al spe i� ation for the Carleman estimatefor 0 � u � T with T > 0 suÆ iently small. In su h a way we have parameterizedan annular region NT := fSp;ugu2[0;T ℄ around p of width T and inner radius r,ranging from the hypersphere Sp;0 whi h is ontained in , to the hypersphereSp;T . Note that NT ontains some points where 6= 0, for otherwise x0 2 . Let ydenote a variable point in Sp;0 and note that points in NT may be identi�ed with(u; y) 2 [0; T ℄� Sp;0. Next, we repla e the solution jNT by a uto�v(u; y) := '(u) (u; y)with a smooth bump fun tion ' with '(u) = 1 for u � 0:8T and '(u) = 0 foru � 0:9T . Then supp v is ontained in NT . More pre isely, it is ontained inthe annular region N0:9T . Now our proof goes in two steps: �rst we establish aCarleman inequality for any spinor se tion v in the domain of D whi h satis�essupp(v) � NT . More pre isely, we are going to show that for T suÆ iently smallthere exists a onstant C, su h that(1.16)RZ Tu=0 ZSp;u eR(T�u)2 jv(u; y)j2 dy du � C Z Tu=0 ZSp;u eR(T�u)2 jDv(u; y)j2 dy duholds for any real R suÆ iently large. In the se ond step we apply (1.16) to our uto� se tion v and on lude that then is equal 0 on NT=2 .Step 1. First onsider a few te hni al points. The Dira operator D has theform G(u)(�u + Bu) on the annular region [0; T ℄� Sp;0, and it is obvious that wemay onsider the operator (�u + Bu) instead of D. Moreover, we have by Lemma1.29 that Bu = Bu + Cu with a self-adjoint ellipti di�erential operator Bu andan anti-symmetri operator Cu of order zero, both on Sp;u. Note that the metri stru tures depend on the normal variable u.

26 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKNow make the substitutionv =: e�R(T�u)2=2v0whi h repla es (1.16) by(1.17)R Z T0 ZSp;u jv0(u; y)j2 dy du � C Z T0 ZSp;u �����v0�u + Buv0 +R(T � u)v0����2 dy du:We denote the integral on the left side by J0 and the integral on the right side byJ1. Now we prove (1.17). De ompose ��u +Bu+R(T � u) into its symmetri partBu + R(T � u) and anti-symmetri part �u + Cu. This givesJ1 = Z T0 ZSp;u �����v0�u + Buv0 + R(T � u)v0����2 dy du= Z T0 ZSp;u �����v0�u +Cuv0����2 dy du+ Z T0 ZSp;u j(Bu +R(T � u)) v0j2 dy du+ 2< Z T0 ZSp;u ��v0�u + Cuv0; Buv0 + R(T � u)v0� dy du :Integrate by parts and use the identity for the real part< hf; Pfi = 12 hf; (P + P �)fiin order to investigate the last and riti al term whi h will be denoted by J2. Thisyields (where we drop domains of integration)J2 = 2< Z Z ��v0�u +Cuv0; Buv0 + R(T � u)v0� dy du= 2< Z Z ��v0�u ;Buv0 + R(T � u)v0� dy du+ 2< Z Z hCuv0; Buv0i dy du= �2< Z Z �v0;� ��u (Bu +R(T � u))� v0� dy du� 2< Z Z hv0; CuBuv0i dy du= 2 Z Z �v0;��Bu�u v0 +Rv0� dy du+ Z Z (v0; [Bu; Cu℄v0) dy du= 2R Z T0 kv0k20 du+ Z Z �v0;�2�Bu�u v0 + [Bu; Cu℄v0� dy du= 2RJ0 + J3;where k � km denotes the m-th Sobolev norm on EjSp;u and J3 requires a arefulanalysis. It follows from the pre eding de ompositions of J1 and J2 that the proofof (1.17) will be ompleted with C = 12 when J3 � 0. If J3 < 0 and C = 12 , it

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 27suÆ es to show that(1.18) jJ3j � 12 R Z T0 kv0k20 du+ Z T0 k(Bu +R(T � u))v0k2 du! :Sin e the operators Bu are ellipti of order 1, G�arding's inequality (1.11) yieldskfk1 � (kfk0 + kBufk0)for any se tion f of E on Sp;u (and 0 � u � T ). Then, also using the fa t that�2�Bu�u + [Bu; Cu℄ is a �rst-order di�erential operator on EjSp;u, we obtainjJ3j � Z T0 kv0k0 �2�Bu�u v0 + [Bu; Cu℄v0 0 du � 1 Z T0 kv0k0 kv0k1 du� 1 Z T0 kv0k0 (kBuv0k0 + kv0k0) du� 1 Z T0 kv0k0 fk(Bu +R(T � u))v0k0 + (R(T � u) + 1)kv0k0g du� 1 (RT + 1) Z T0 kv0k20 du+ 1 Z T0 k(Bu +R(T � u))v0k0 kv0k0 du :The integrand of the se ond summand is equal to(1.19) k(Bu +R(T � u))v0k0p 1 �p 1 kv0k0�� 12 n 1 1 k(Bu +R(T � u))v0k20 + 1 kv0k20owith the inequality due to the estimate ab � 12 (a2+ b2). By inserting (1.19) in thepre eding inequality for jJ3j we obtainjJ3j � 1 (RT + 1) Z T0 kv0k20 du+ 1 Z T0 �12 n 1 1 k(Bu + R(T � u))v0k20 + 1 kv0k20o� du= 1 (RT + 1) Z T0 kv0k20 du+ 1 Z T0 12 1 kv0k20 du+ Z T0 �12 k(Bu + R(T � u))v0k20 � du= Z T0 12 k(Bu +R(T � u))v0k20 du+ 1 �(RT + 1) + 12 1 � Z T0 kv0k20 du= 12 Z T0 k(Bu +R(T � u))v0k20 du+R 1 �T + 1 + 22R �Z T0 kv0k20 duSo (1.18) holds for T and 1R suÆ iently small, and we then have the Carlemaninequality (1.16) for C = 12 .

28 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKStep 2. To begin with, we have(1.20)eRT2=4 Z T20 ZSp;u j (u; y)j2 dy du � Z T0 ZSp;u eR(T�u)2 j (' ) (u; y)j2 dy du =: IWe apply our Carleman type inequality (1.16):(1.21)I = Z T0 ZSp;u eR(T�u)2 j (' ) (u; y)j2 dy du � CR Z Tu=0 ZSp;u eR(T�u)2 jD(' )(u; y)j2 dy du:Assuming that is a solution of the perturbed equation D +PA( ) = 0, we getD(' ) = 'D + (du)'0 = �'PA( ) + (du)'0 :Using this in (1.21) and noting that (a+ b)2 � 2 �a2 + b2�, yieldsI � 2CR Z T0 ZSp;u eR(T�u)2 �j'(u)PA( )(u; y)j2 + j (du)'0(u) (u; y)j2� dy du:Now we exploit our assumption(1.22) jPA( )(x)j � P ( ; x)j (x)j for x 2Mabout the perturbation with lo ally bounded P ( ; �), sayjP ( ; (u; y))j � C0 := maxx2K jP ( ; x)j for all y 2 Sp;u; u 2 [0; T ℄where K is a suitable ompa t set. We obtain at on e�1� 2CC0R � I � 2CR Z T0 ZSp;u eR(T�u)2 j (du)'0(u) (u; y)j2 dy du� 2CR eRT2=25 Z T0 ZSp;u j (du)'0(u) (u; y)j2 dy du:Here we use that '0(u) = 0 for 0 � u � 0:8T so that we an estimate the expo-nential and pull it in front of the integral. Using (1.20),Z T20 ZSp;u j (u; y)j2 dy du � e�RT2=4I� 2CR1� 2CC0R eRT2( 125� 14 ) Z T0 ZSp;u j (du)'0(u) (u; y)j2 dy du:As R ! 1, we get R T20 RSp;u j (u; y)j2 dy du = 0 whi h ontradi ts x0 2 supp .�

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 292. The Index of Ellipti Operators on Partitioned Manifolds2.1. Examples and the Hellwig{Vekua Index Theorem. We begin withsome elementary examples.Example 2.1. Consider the (trivially ellipti ) ordinary di�erential operatoron the unit interval I = [0; 1℄ de�ned byP : C1(I) �C1(I)! C1(I) � C1(I);where (f; g) 7! (f 0;�g0) ;with the boundary onditions C1(I)�C1(I)! C1(�I) �= C � C (where �I :=f0; 1g) (i) R1 : (f; g) 7! (f � g) j�I(ii) R2 : (f; g) 7! f j�I(iii) R3 : (f; g) 7! (f + g0)j�IWe determine the index of the operators (for i = 1; 2; 3)(P;Ri) : C1(I) �C1(I) ! C1(I) � C1(I) �C1(�I):Clearly, dim Ker(P;Ri) = 1. To determine the okernel, one writes P (f; g) =(F;G) and Ri(f; g) = h, with F;G 2 C1(I) and h = (h0; h1) 2 C � C obtaining,f(t) = Z t0 F (� )d� + 1; g(t) = � Z t0 G(� )d� + 2and two more equations for the boundary ondition. The dimension of Coker(P;Ri)is then the number of linearly independent onditions on F ,G, and hwhi h must beimposed in order to eliminate the onstants of integration. For ea h i 2 f1; 2; 3g,there is only one ondition, namely h0 = h1 � R 10 F (� )d� � R 10 G(� )d� ; h0 =h1� R 10 F (� )d� ; resp., h0 = h1� R 10 F (� )d� �G (0)+G(1). So, the index vanishesin all three ases.For a more omprehensive treatment of the existen e and uniqueness of boundary-value problems for ordinary di�erential equations (in luding systems), we refer to[CodLev℄ and [Har64, p. 322-403℄.We now onsider the Lapla e operator � := �2=�x2 + �2=�y2, as a linearellipti di�erential operator from C1(X) to C1(X), where X is the unit diskfz = x+ iy j jzj � 1g � C .Example 2.2. For the Diri hlet boundary onditionR : C1(X) ! C1(�X); with R (u) = uj�X (�X := fz 2 C j jzj = 1g );we show that a) Ker(�; R) = f0g and b) Im(�; R)? = f0g ;

30 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKwhere ? is orthogonal omplement in L2(X) � L2(�X).1 In parti ular, it followsthat index(�; R) = 0.For (a): Ker(�; R) onsists of fun tions of the form u+iv, where u and v are real-valued. Sin e the oeÆ ients of the operators � and R are real, we may assumev = 0 without loss of generality. Thus, onsider a real solution u with �u = 0 inX and u = 0 on �X. Then (where ru := (ux; uy) := (�u�x ; �u�y ))(2.1) 0 = � ZX u�u dxdy = ZX jruj2 dxdy;when e ru = 0. Thus, u is onstant, and indeed zero sin e u = 0 on �X. Thetri k lies in the equality (2.1), whi h follows from Stokes' formula, namely RX d! =R�X !, where ! is a 1-form. Indeed, setting ! := u ^ �du, where � is the Hodgestar operator (�du = � (uxdx+ uydy) := uxdy � uydx), we obtaind! = du ^ �du+ u ^ d � du = jruj2 dx ^ dy + (u�u)dx ^ dy:Using Stokes' formula and uj�X = 0, we haveZX jruj2 dxdy + ZX (u�u)dxdy = ZX d! = Z�X ! = Z�X u ^ �du = 0:>From this and �u = 0, we on lude that ru = 0 and u is onstant.For (b): Choose L 2 C1 (X) and l 2 C1 (�X) with (L; l) orthogonal to Im(�; R),when e (relative to the usual measures on X and �X)(2.2) ZX (�u)L+ Z�X ul = 0 for all u 2 C1(X):Using a 2-fold integration by parts (in the exterior al ulus), we obtain(2.3)ZX u�L� ZX (�u)L = ZX (u(d � dL)� d (�du)L) = Z�X (u � dL� L � du) :First we onsider u with support supp (u) := the losure of fz 2 X j u(z) 6= 0g ontained in the interior of X. ThenZX u�L = ZX (�u)L = � Z�X ul = 0;and so �L = 0. Now for u 2 C1(X) we apply (2.2) and (2.3) to dedu e thatZ�X ul = � ZX (�u)L = Z�X (u � dL� L � du)= Z�X (u (xLx + yLy)� L (xux + yuy)) :1Here, onsider that the interse tion of the orthogonal omplement of the range Im(�; R)relative to the usual inner produ t in L2(X) � L2(�X) with the spa e C1(X) � C1(�X) isisomorphi to Coker(�; R). This is true, sin e it turns out that the image of the natural Sobolevextension of (�; R) is losed in the L2-norm, and its L2-orthogonal omplement is ontained inC1(X)� C1(�X).

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 31Thus, l = xLx+yLy and Lj�X = 0, and we �nally apply (a). Details are in [Ho63,p. 264℄.The pre eding result index(�; R) = 0 (for Ru = uj�X) an also be obtainedby proving the symmetry of � and that the L2 extension on the domain de�nedby Ru = 0 is a self-adjoint Fredholm extension.We now onsider a C1 ve tor �eld � : �X ! C on the boundary �X =fz 2 C j jzj = 1g. For u 2 C1(X), z 2 �X, and �(z) = �(z) + i�(z), the \dire -tional derivative" of the fun tion u relative to � at the point z is�u�� (z) := �(z)�u�x (z) + �(z)�u�y (z) :>From the standpoint of di�erential geometry it is better, either to denote theve tor �eld by ��� or to write the dire tional derivative as simply as � [u℄ (z), sin e��x and ��y an be regarded as ve tor �elds. The pair (�; ��� ) de�nes a linearoperator (�; ��� ) : C1(X)! C1(X) � C1(�X) given by u 7! (�u; �u�� ):Theorem 2.3. (I. N. Vekua 1952). For p 2 Zand �(z) := zp, we have that(�; ��� ) is an operator with �nite-dimensional kernel and okernel, andindex(�; ��� ) = 2 (1� p) :Remark 2.4. The theorem of Vekua remains true, if we repla e zp by anynonvanishing \ve tor �eld" � : �X ! C n f0g with \winding number" p:Moreover, in pla e of the disk, we an take X to be any simply- onne teddomain in C with a \smooth" boundary �X. The reason is the homotopy invarian eof the index.Remark 2.5. In the theory of Riemann surfa es (e.g., in Riemann{Ro h The-orem), one also en ounters the number 2(1 � p) as the Euler hara teristi of a losed surfa e of genus p. This is no a ident, but rather it is onne ted with therelation between ellipti boundary-value problems and ellipti operators on losedmanifolds. Spe i� ally, there is a relation between the index of (�; ��� ) and theindex of Cau hy-Riemann operator for omplex line bundles over S2 = P1 (C ) withChern number 1� p (e.g., see Example 2.19 for a start).Remark 2.6. Motivated by the method of repla ing a di�erential equation bydi�eren e equations, David Hilbert and Ri hard Courant expe ted \linear prob-lems of mathemati al physi s whi h are orre tly posed to behave like a systemof N linear algebrai equations in N unknowns... If for a orre tly posed problemin linear di�erential equations the orresponding homogeneous problem possessesonly the trivial solution zero, then a uniquely determined solution of the generalinhomogeneous system exists. However, if the homogeneous problem has a nontriv-ial solution, the solvability of the inhomogeneous system requires the ful�llment of

32 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKFigure 4. The ve tor �eld � : �X ! C with winding number p = 0; 1; 2

º

Xp =2

@X Figure 5. Another ve tor �eld � with winding number 2 ertain additional onditions." This is the \heuristi prin iple" whi h Hilbert andCourant saw in the Fredholm Alternative [CoHi℄. G�unter Hellwig [He52℄ (ni elyexplained in [Ha52℄) in the real setting and Ilya Nestorovi h Vekua [Ve56℄ in omplex setting disproved it with their independently found example where theprin iple fails for p 6= 1.We remark that in addition to these \oblique-angle" boundary-value problems,\ oupled" os illation equations, as well as restri tions of boundary-value prob-lems, even with vanishing index, to suitable half-spa es, furnish further more or

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 33less elementary examples for index 6= 0. The simplest example of a system of �rstorder di�erential operators on the dis is provided in Example 2.7 below. A worldof more advan ed, and for di�erential geometry mu h more meaningful examples,is approa hed by the Atiyah{Patodi{Singer Index Theorem, see Se tion 2.4 below.Proof of Theorem 2.3. (After [Ho63, p. 266 f.℄) Sin e the oeÆ ients ofthe di�erential operators (�; ��� ) are real, we may restri t ourselves to real fun -tions. Thus, u 2 C1(X) denotes a single real-valued fun tion, rather than a omplex-valued fun tion.Ker(�; ��� ): It is well-known that Ker(�) onsists of real (or imaginary) partsof holomorphi fun tions on X (e.g., see [Ah53, p. 175 �.℄). Hen e, u 2 Ker(�),exa tly when u = Re(f) where f = u+iv is holomorphi ; i.e., the Cau hy-Riemannequation �f��z = 0 holds, where ���z := 12( ��x + i ��y ). Expli itly,0 = �f��z = 12 � ��x + i ��y� (u+ iv) = 12 ��u�x � �v�y�+ i2 ��u�y + �v�x� :Every holomorphi (= omplex di�erentiable) fun tion f is twi e omplex di�er-entiable and its derivative is given by�f�z := 12 � ��x � i ��y� (u+ iv)= 12 ��u�x + �v�y�+ i2 ���u�y + �v�x� = �u�x � i�u�y :In this way we have a holomorphi fun tion � := f 0 for ea h u 2 Ker(�). Sin e�u�� = Re (zp) �u�x + Im (zp) �u�y = Re���u�x � i�u�y� zp� = Re (� (z) zp) ;the boundary ondition �u�� = 0 (� = zp) then means that the real part Re(�(z)zp)vanishes for jzj = 1. For p � 0, �(z)zp is holomorphi as well as �, and hen e for�(z) := �u�x � i�u�y ; we haveu 2 Ker ��; ��� � with � = zp; p � 0) Re (�(z)zp) 2 Ker (�; R) where R (�) = (�) j�X :Thus, we su eed in asso iating with the \oblique-angle" boundary-value problemfor u a Diri hlet boundary-value problem forRe (�(z)zp), whi h has only the trivialsolution by Example 2.2a. Sin e �(z)zp is holomorphi with Re (�(z)zp) = 0,the partial derivatives of the imaginary part vanish, and so there is a onstantC 2 R su h that �(z)zp = iC for all z 2 X. If p > 0, then we have C = 0 (setz = 0). Hen e � = 0, and (by the de�nition of �) the fun tion u is onstant (i.e.,dimKer ��; ��� � = 1). If p = 0, then �u�x � i�u�y = �(z) = iC, and so u(x; y) =�Cy + eC, when e dimKer ��; ��� � = 2 in this ase.We now ome to the ase p < 0, whi h uriously is not immediately redu ibleto the ase q > 0 where q := �p. One might try to look for a solution by simplyturning ��� around to � ��� , but this is futile sin e the winding numbers of � and�� about 0 are the same. Besides, if �p (z) = zp, we do not have ����p = � ���p . Inorder to redu e the boundary-value problem with p < 0 to the elementary Diri hlet

34 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKproblem, we must go through a more areful argument. Note that �(z)zp an havea pole at z = 0, when e Re (�(z)zp) is not ne essarily harmoni . We write theholomorphi fun tion �(z) as�(z) =Xqj=0 ajzj + g (z) zq+1where q := �p and g is holomorphi . We de�ne a holomorphi fun tion by (z) := g (z) z +Xq�1j=0 ajzq�j ;with (0) = 0. Then one an write�(z)zp = aq +Xq�1j=0 �ajzj�q � ajzq�j�+ (z)The boundary ondition �u�� = 0 implies Re (�(z)zp) = 0 for jzj = 1. By theabove equation, we have 0 = Re (�(z)zp) = Re( (z) + aq) for jzj = 1 sin e thenz�1 = �z: Sin e is holomorphi , we have again arrived at a Diri hlet boundaryvalue problem; this time for the fun tion Re( (z) + aq). >From Example 2.2a, itfollows again that (z) + aq is an imaginary onstant, when e (z) = (0) = 0and aq is pure imaginary. We have�(z) = �(z)zpzq = aqzq +Xq�1j=0 �ajzj � ajz2q�j�for arbitrary a0; a1; :::; aq�1 2 C and aq 2 iR. As a ve tor spa e over R, the setn�u�x � i�u�y j u 2 Ker ��; ��� �ohas dimension 2q+ 1; here we have restri ted ourselves to real u, a ording to our onvention above. Sin e u is uniquely determined by � up to an additive onstant,it follows that for � = zp and p < 0,dimKer ��; ��� � = 2q + 2 = 2� 2pCoker ��; ��� �: As Example 2.2b shows the equation �u = F has a solution forea h F 2 C1(X). In view of this we an showCoker ��; ��� � = C1(X) � C1(�X)Im��; ��� � �= C1(�X)��� (Ker�)as follows. We assign to ea h representative pair (F; h) 2 C1(X) � C1(�X) the lass of h � �u�� 2 C1(�X), where u is hosen so that �u = F . This map is learly well de�ned on the quotient spa e of pairs, and the inverse map is givenby h 7! (0; h). Hen e, we have found a representation for Coker(�; ��� ) in termsof the \boundary fun tions" ��u�� j u 2 Ker�, rather than the umbersome pairsin Im��; ��� �. (This tri k an always be applied for the boundary-value problems(P;R), when the operator P is surje tive.)We therefore investigate the existen e of solutions of the equation �u = 0with the \inhomogeneous" boundary ondition �u�� = h, where h is a given C1fun tion on �X. A ording to the tri k introdu ed in the �rst part of our proof, it is

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 35equivalent to ask for the existen e of a holomorphi fun tion � with the boundary ondition Re (�(z)zp) = h, jzj = 1, i.e., for a solution of a Diri hlet problem forRe (�(z)zp). By Example 2.2b, there is a unique harmoni fun tion whi h restri tsto h on the boundary �X; hen e, we have a (unique up to an additive imaginary onstant) holomorphi fun tion � with Re�(z) = h for jzj = 1.In the ase p < 0, the boundary problem for � is always solvable; namely, set�(z) := z�p�(z). Hen e, we havedimCoker ��; ��� � = 0 for �(z) = zp and p � 0:For p > 0, we an onstru t a solution of the boundary-value problem in for � from�, if and only if there is a onstant C 2 R, su h that (�(z)� iC)=zp is holomorphi (i.e., the holomorphi fun tion �(z) � iC has a zero of order at least p at z = 0.Using the Cau hy Integral Formula, these onditions on the derivatives of � atz = 0 orrespond to onditions on line integrals around �X. In this way, we have2p � 1 linear (real) equations that h must satisfy in order that the boundary-value problem have a solution. We summarize our results in the following table(�(z) = zp) and Figure 6:p dimKer ��; ��� � dimCoker ��; ��� � index ��; ��� �> 0 1 2p� 1 2� 2p� 0 2� 2p 0 2� 2pFigure 6. The dimensions of kernel, okernel and the index ofthe Lapla ian with boundary ondition given by � (z) := zp forvarying p �

36 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKNote 1: We already noted in the proof the pe uliarity that the ase p < 0 annot simply be played ba k to the ase p > 0. This is re e ted here in theasymmetry of the dimensions of kernel and okernel and the index. It simplyre e ts the fa t that there are \more" rational fun tions with pres ribed polesthan there are polynomials with \ orresponding" zeros.Note 2: In ontrast to the Diri hlet Problem, whi h we ould solve via inte-gration by parts (i.e., via Stokes' Theorem), the above proof is fun tion-theoreti in nature and annot be used in higher dimensions. This is no loss in our spe ial ase, sin e the index of the \oblique-angle" boundary-value problem must vanishanyhow in higher dimensions for topologi al reasons; see [Ho63, p. 265 f.℄. Thea tual mathemati al hallenge of the fun tion-theoreti proof arises less from therestri tion dimX = 2 than from a ertain arbitrariness, namely the tri ks anddevi es of the de�nitions of the auxiliary fun tions �; ; �, by means of whi h theoblique-angle problem is redu ed to the Diri hlet problem.Note 3: The theory of ordinary di�erential equations easily onveys the im-pression that partial di�erential equations also possess a \general solution" in theform of a fun tional relation between the unknown fun tion (\quantity") u, theindependent variables x and some arbitrary onstants or fun tions, and that every\parti ular solution" is obtained by substituting ertain onstants or fun tionsf; h; et . for the arbitrary onstants and fun tions. (Corresponding to the higherdegree of freedom in partial di�erential equations, we deal not only with onstantsof integration but with arbitrary fun tions.) The pre eding al ulations, regardingthe boundary value problem of the Lapla e operator, learly indi ate how limitedthis notion is whi h was on eived in the 18th entury on the basis of geomet-ri intuition and physi al onsiderations. The lassi al re ipe of �rst sear hing forgeneral solutions and only at the end determining the arbitrary onstants andfun tions fails. For example, the spe i� form of boundary onditions must enterthe analysis to begin with.Example 2.7. Let X := fz = x+ iy j jzj < 1g be the unit disk and de�ne anoperator T : C1(X) �C1(X) ! C1(X) �C1(X) �C1(�X) byT (u; v) := ��u��z ; �v�z ; (u� v) j�X�where ��z = 12 ( ��x � i ��y ) is omplex di�erentiation and ���z = 12 ( ��x + i ��y ) is theCau hy-Riemann di�erential operator \formally adjoint" to ��z . We show thatdim(Ker T ) = 1 and Coker(T ) = f0g and hen e that index(T ) = 1. Suppose that(u; v) 2 Ker T . Then �u��z = 0 and �v�z = 0 in whi h ase u and v are harmoni . Thensin e (u� v) j�X = 0; we have u = v on X. Now �v��z = �u��z = 0) v is holomorphi ,and v0 (z) = �v�z = 0 ) v (= u) is onstant. Thus, dim(Ker T ) = 1. To show thatCoker(T ) = f0g, or more pre isely (ImT )? = f0g; see the footnote to Example2.2, we hoose arbitrary f; g 2 C1(X) and h 2 C1(�X) and prove that f , g and

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 37h must identi ally vanish, if(2.4) ZX ��u��z f + �v�z g� + Z�X(u � v)h = 0 for all u; v 2 C1(X):Note that for P;Q 2 C1(X)d (Pdz +Qdz) = �P��z dz ^ dz + �Q�z dz ^ dz = ��P��z � �Q�z � dz ^ dz= ��P��z � �Q�z � (dx� idy) ^ (dx+ idy) = 2i��P��z � �Q�z � dx^ dyThus we obtain the omplex version of Stokes' Theorem,Z�X Pdz +Qdz = ZX d (Pdz +Qdz) = 2i ZX ��P��z � �Q�z � :>From this, we getZX �u��z f = ZX ���z (uf) � ZX u�f��z = 12i Z�X uf dz � ZX u�f��z andZX �v�z g = ZX ��z (vg) � ZX v �g�z = �12i Z�X vgd�z � ZX v �g�z :Hen e, ZX ��u��z f + �v�z g� = � ZX �u�f��z + v �g�z�+ 12i Z�X (uf dz � vgd�z) :Assuming (2.4), we have0 = ZX ��u��z f + �v�z g� + Z�X(u� v)h= � ZX �u�f��z + v �g�z�+ 12i Z�X (uf dz � vg d�z) + Z�X(u� v)hBy onsidering u and v with ompa t support inside the open disk, we dedu e that�f��z = 0 and �g�z = 0 (i.e., f and g are analyti and onjugate analyti respe tively).Thus, (2.4) implies 0 = 12i Z�X (uf dz � vg d�z) + Z�X(u� v)h;for all u; v 2 C1(X). Choosing v = u; we have0 = 12i Z�X u (f dz � g d�z) for all u) f dz = g d�z on �X) f �ei�� iei�d� = �g �ei�� ie�i�d�) f �ei�� ei� = �g �ei�� e�i�:However, sin e f is analyti , the Fourier series of f �ei�� ei� has a nonzero oeÆ- ient for eim� only when m > 0; and sin e g is onjugate analyti , g �ei�� e�i� only

38 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKhas a nonzero oeÆ ient for eim� only when m < 0. Thus, f = g = 0. Choosingv = �u, (2.4) then yields0 = Z�X(u� v)h = 2 Z�X uh for all u 2 C1(X)) h = 0:Remark 2.8. In engineering one alls a system of separate di�erential equa-tions Pu = fQv = g;whi h are related by a \transfer ondition" R(u; v) = h, a \ oupling problem";when the domains of u and v are di�erent, but have a ommon boundary (orboundary part) on whi h the transfer ondition is de�ned, then we have a \trans-mission problem"; e.g., see [Boo72, p.7 �℄. Thus, we may think of T as an operatorfor a problem on the spheri al surfa e X [�X X with di�erent behavior on theupper and lower hemispheres, but with a �xed oupling along the equator.2.2. The Index of Twisted Dira Operators on Closed Manifolds.Re all that the index of a Fredholm operator is a measure of its asymmetry: itis de�ned by the di�eren e between the dimension of the kernel (the null spa e)of the operator and the dimension of the kernel of the adjoint operator (= the odimension of the range). So, the index vanishes for self-adjoint Fredholm opera-tors. For an ellipti di�erential or pseudo-di�erential operator on a losed manifoldM , the index is �nite and depends only on the homotopy lass of the prin ipalsymbol � of the operator over the otangent sphere bundle S�M . It follows (see[LaMi, p. 257℄) that the index for ellipti di�erential operators always vanisheson losed odd-dimensional manifolds. On even-dimensional manifolds one has theAtiyah{Singer Index Theorem whi h expresses the index in expli it topologi alterms, involving the Todd lass de�ned by the Riemannian stru ture of M andthe Chern lass de�ned by gluing two opies of a bundle over S�M by �.The original approa h in proving the Index Theorem in the work of Atiyahand Singer [AtSi69℄ is based on the following lever strategy. The invarian e of theindex under homotopy implies that the index (say, the analyti index ) of an ellipti operator is stable under rather dramati , but ontinuous, hanges in its prin ipalsymbol while maintaining ellipti ity. Moreover, the index is fun torial with respe tto ertain operations, su h as addition and omposition. Thus, the indi es of ellipti operators transform predi tably under various global operations (or relations) su has dire t sums, embedding and obordism. Using K-theory, a topologi al invariant(say, the topologi al index ) with the same transformation properties under theseglobal operation is built from the symbol of the ellipti operator. It turns outthat the global operations are suÆ ient to onstru t enough ve tor bundles andellipti operators to dedu e the Atiyah{Singer Index Theorem (i.e., analyti index= topologi al index ). With the aid of Bott periodi ity, it suÆ es to he k that thetwo indi es are the same in the trivial ase where the base manifold is just a single

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 39point. A parti ularly ni e exposition of this approa h is found in E. Guentner'sarti le [Gu93℄ following an argument of P. Baum.Not long after this �rst proof (given in quite di�erent variants), there emergeda fundamentally di�erent means of proving the Atiyah{Singer Index Theorem,namely the heat kernel method. This is outlined here in the important ase ofthe hiral half D+ of a twisted Dira operator D. (In the index theory of losedmanifolds, one usually studies the index of a hiral half D+ instead of the to-tal Dira operator D, sin e D is symmetri for ompatible onne tions and thenindex D = 0.) The heat kernel method had its origins in the late 1960s (e.g., in[M K-Si℄) and was pioneered in the works [Pa71℄, [Gi73℄, [ABP73℄, et .. In the�nal analysis, it is debatable as to whether this method is really mu h shorter orbetter. This depends on the ba kground and tastes of the beholder. Geometersand analysts (as opposed to topologists) are likely to �nd the heat kernel methodappealing, be ause K-theory, Bott periodi ity and obordism theory are avoided,not only for geometri operators whi h are expressible in terms of twisted Dira operators, but also largely for more general ellipti pseudo-di�erential operators,as Melrose has done in [Me93℄. Moreover, the heat method gives the index of a\geometri " ellipti di�erential operator naturally as the integral of a hara ter-isti form (a polynomial of urvature forms) whi h is expressed solely in terms ofthe geometry of the operator itself (e.g., urvatures of metri tensors and onne -tions). One does not destroy the geometry of the operator by taking advantage ofthe fa t that it an be suitably deformed without altering the index. Rather, in theheat kernel approa h, the invarian e of the index under hanges in the geometryof the operator is a onsequen e of the index formula itself rather than a meansof proof. However, onsiderable analysis and e�ort are needed to obtain the heatkernel for e�tD2 and to establish its asymptoti expansion as t! 0+. Also, it anbe argued that in some respe ts the K-theoreti al embedding/ obordism methodsare more for eful and dire t. Moreover, in [LaMi℄, we are autioned that the in-dex theorem for families (in its strong form) generally involves torsion elementsin K-theory that are not dete table by ohomologi al means, and hen e are not omputable in terms of lo al densities produ ed by heat asymptoti s. Nevertheless,when this diÆ ulty does not arise, the K-theoreti al expression for the topologi alindex may be less appealing than the integral of a hara teristi form, parti ularlyfor those who already understand and appre iate the geometri al formulation of hara teristi lasses. All disputes aside, the student who learns both approa hesand formulations will be more a omplished (and probably older).The lassi al geometri operators su h as the Hirzebru h signature operator,the de Rham operator, the Dolbeaut operator and even the Yang-Mills operator an all be lo ally expressed in terms of hiral halves of twisted Dira operators.Thus, we will fo us on index theory for su h operators. The index of any of these lassi al operators (and their twists) an then be obtained from the Lo al IndexTheorem for twisted Dira operators. This theorem supplies a well-de�ned n-formon M , whose integral is the index of the twisted Dira operator. This n-form (or

40 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK\index density") is expressed in terms of forms for hara teristi lasses whi h arepolynomials in urvature forms. The Index Theorem thus obtained then be omesa formula that relates a global invariant quantity, namely the index of an operator,to the integral of a lo al quantity involving urvature. This is in the spirit of theGauss-Bonnet Theorem whi h an be onsidered a spe ial ase.Definition 2.9. LetM be an oriented Riemanniann-manifold (n = 2m even)with metri h, and oriented orthonormal frame bundle FM . Assume thatM has aspin stru ture P ! FM , where P is a prin ipal Spin (n)-bundle and the proje tionP ! FM is a two-fold over, equivariant with respe t to Spin (n) ! SO (n).Furthermore, let E ! M be a Hermitian ve tor bundle with unitary onne tion". The twisted Dira operator D asso iated with the above data is(2.5) D := (1 ) Æ r : C1 (E � (M ))! C1 (E � (M )) :Here, � (M ) is the spin bundle over M asso iated to P ! FM ! M via thespinor representation Spin (n)! End (�n), : C1 (� (M ) TM�)! C1 (� (M ))is Cli�ord multipli ation, andr : C1 (E � (M ))! C1 (E � (M ) TM�)is the ovariant derivative determined by the onne tion " and the spinorial lift toP of the Levi-Civita onne tion form, say �, on FM .Note that D here is a spe ial ase of an operator of Dira type introdu edin Subse tion 1.3 with Cl(M ) a ting on the se ond fa tor of E � (M ) whi his playing the role of S. Let �� (M ) denote the �1 eigenbundles of the omplexCli�ord volume element in C1 (Cl(M )), given at a point x 2 M by ime1 � � �en,where e1; : : : ; en is an oriented, orthonormal basis of TxM . The �� (M ) are theso- alled hiral halves of � (M ) = �+ (M )��+ (M ). Sin er �C1 �E �� (M )�� � C1 �E �� (M ) TM�� and(1 ) �C1 �E �� (M ) TM��� � C1 �E �� (M )� ;we haveD = D+ � D�, where D� : C1 �E �� (M )�! C1 �E �� (M )� :The symbol of the �rst-order di�erential operator D is omputed as follows. For� 2 C1 (M ) with � (x) = 0 and 2 C1 (E � (M )) ; we have at x(1 ) Æ r (� ) = (1 ) Æ ((d�) + �r ) = (1 ) Æ (d�) = (1 (d�)) :Thus, the symbol � (D) : TxM� ! End (� (M )) at the ove tor �x 2 TxM� isgiven by � (D) (�x) = 1 (�x) 2 End (Ex �x) :

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 41For �x 6= 0, � (D) (�x) is an isomorphism, sin e� (D) (�x) Æ � (D) (�x) = 1 (�x)2 = � j�xj2 I :Thus, D is an ellipti operator. Moreover, sin e � (D+) and � (D�) are restri tionsof � (D), it follows that D+ and D� are ellipti . It an be shown that D is formallyself-adjoint, and D+ and D� are formal adjoints of ea h other (see [LaMi℄). Wealso have a pair of self-adjoint ellipti operatorsD2+ := D2jC1 �E �+ (M )� = D� Æ D+D2� := D2jC1 �E �� (M )� = D+ Æ D�:For � 2 C , let V� �D2�� := � 2 C1 �E �� (M )� j D2� = � :>From the general theory of formally self-adjoint, ellipti operators on ompa tmanifolds, we know thatSpe �D2�� = �� 2 C j V� �D2�� 6= f0g . onsists of the eigenvalues of D2� and is a dis rete subset of [0;1), the eigenspa esV� �D2�� are �nite-dimensional, and an L2 (E �� (M ))- omplete orthonormal setof ve tors an be sele ted from the V� �D2��. Note that D+ �V� �D2+�� � V� �D2��,sin e for 2 V� �D2+�D2� �D+ � = �D+ Æ D�� �D+ � = D+ ��D� Æ D+� ( )�= D+ �D2+ ( )� = D+ (� ) = �D+ ( ) ;and similarly D� �V� �D2��� � V� �D2+�. For � 6= 0,D�jV� �D2�� : V� �D2��! V� �D2��is an isomorphism, sin e it has inverse 1�D�. Thus the set of nonzero eigenvalues(and their multipli ities) of D2+ oin ides with that of D2�. However, in generaldimV0 �D2+� � dimV0 �D2�� = dimKer �D2+�� dimKer �D2�� = index �D+� 6= 0:Sin e dimV� �D2+�� dimV� �D2�� = 0 for � 6= 0, obviouslyindex �D+� = dimV0 �D2+�� dimV0 �D2��=X�2Spe (D2+) e�t� �dimV� �D2+�� dimV� �D2��� :This may seem like a very ineÆ ient way to write index (D+), but the point isthat the sum an be expressed as the integral of the supertra e of the heat kernelfor the spinorial heat equation � �t = �D2 , from whi h the Lo al Index Theorem(Theorem 2.13 below) for D+ will eventually follow. However, �rst the existen eof the heat kernel needs to be established.Let the positive eigenvalues of D2� be pla ed in a sequen e 0 < �1 � �2 � �3 �: : : where ea h eigenvalue is repeated a ording to its multipli ity. Let u�1 ; u�2 ; : : :be an L2-orthonormal sequen e in C1 (E �+ (M )) with D2� �u�j � = �ju�j (i.e.,

42 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKu�j 2 V�j �D2��). We let u+01 ; : : : ; u+0n+ be an L2-orthonormal basis of KerD2+ =KerD+, and u�01 ; : : : ; u�0n� be an L2-orthonormal basis of KerD2� = KerD�. We an pull ba k the bundleE�� (M ) via either of the proje tionsM�M�(0;1)!M given by �1 (x; y; t) := x and �2 (x; y; t) := y and take the tensor produ t of theresults to form a bundleK� := ��1 �E �� (M )� ��2 �E �� (M )�!M �M � (0;1) :Note that for x 2M , the Hermitian inner produ t h ; ix on (E �� (M ))x givesus a onjugate-linear map 7! � (�) := h�; ix from (E �� (M ))x to its dual(E �� (M ))�x. Thus, we an (and do) make the identi� ations��1 �E �� (M )� ��2 �E �� (M )� �= ���1 �E �� (M )��� ��2 �E �� (M )��= Hom���1 �E �� (M )� ; ��2 �E �� (M )�� :The full proof of the following Proposition 2.10 will be found in [BlBo03℄, but itis already ontained in [Gi95℄ for readers of suÆ ient ba kground.Proposition 2.10. For t > t0 > 0, the series k0�; de�ned byk0� (x; y; t) := 1Xj=1 e��jtu�j (x) u�j (y) ; onverges uniformly in Cq(K�jM � M � (t0;1)) for all q � 0. Hen e k0� 2C1 (K�), and (for t > 0)(2.6) ��tk0� (x; y; t) = � 1Xj=1 �je��j tu�j (x) u�j (y) = �D2�k0� (x; y; t) :Definition 2.11. The positive and negative twisted spinorial heat kernels (orthe heat kernels for D2�) k� 2 C1 (K�) are given byk� (x; y; t) :=Xn�i=1 u�0i (x) u�0i (y) + k0� (x; y; t)=Xn�i=1 u�0i (x) u�0i (y) +X1j=1 e��jtu�j (x) u�j (y) for t > 0.The total twisted spinorial heat kernel (or the heat kernel for D2) isk = �k+; k�� 2 C1 �K+� �C1 �K�� �= C1 �K+ � K�� � C1 (K) ;where K := K+ �K� = Hom(��1 (E � (M )) ; ��2 (E � (M ))) :(2.7)The terminology is justi�ed in view of the following, whose proof is to be foundin [BlBo03℄.Proposition 2.12. Let �0 2 C1 (E �� (M )) and let � (x; t) = ZM k� (x; y; t) ; �0 (y)�y �y:

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 43Then for t > 0, � solves the heat equation with initial spinor �eld �0 :� ��t = �D2� andlimt!0+ � (�; t) = �0 in Cq for all q � 0.Moreover, for 0 2 C1 (E � (M )) and (x; t) := ZM hk (x; y; t) ; 0i �y;we have � �t = �D2 and limt!0+ (�; t) = 0 (�) in Cq for all q � 0.For any �nite dimensional Hermitian ve tor spa e (V; h�; �i) with orthonormalbasis e1; : : : ; eN , we have (for v 2 V )Tr (v� v) =XNi=1 h(v� v) (ei) ; eii =XNi=1 hv� (ei) v; eii=XNi=1 hhei; vi v; eii =XNi=1 hei; vi hv; eii =XNi=1 jhei; vij2 = jvj2 :In parti ular, k� (x; x; t) 2 End ((E �� (M ))x) andTr �k� (x; x; t)� =Xn�i=1 ��u�0i (x)��2 +X1j=1 e��j t ��u�j (x)��2 :Sin e this series onverges uniformly and u�0i 2;0 = u�j 2;0 = 1, we haveZM Tr �k� (x; x; t)� �x = n� +X1j=1 e��j t <1:For t > 0; we de�ne the bounded operator e�tD2� 2 End �L2 (E �� (M ))� bye�tD2� � �� =Xn�i=1 �u�0i; �0 �u�0i +X1j=1 e��jt �u�j ; �0 �u�j :Note that e�tD2� is of tra e lass, sin eTr�e�tD2�� = n� +X1j=1 e��j t = ZM Tr �k� (x; x; t)� �x <1:Now, we haveindex �D+� = dimV0 �D2+�� dimV0 �D2��= n+ � n� +X1j=1 �e��j t � e��j t�= n+ +X1j=1 e��jt � �n� +X1j=1 e��j t�= ZM �Tr �k+ (x; x; t)��Tr �k� (x; x; t)�� �x:(2.8)

44 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKSin e D2 = D2+�D2�, we also have the tra e- lass operator e�tD2 2 End �L2 (E � (M ))�,whose tra e is given byTr�e�tD2� = ZM Tr (k (x; x; t)) �x = ZM �Tr �k+ (x; x; t)�+ Tr �k� (x; x; t)�� �x:The supertra e of k (x; x; t) is de�ned byStr (k (x; x; t)) := Tr �k+ (x; x; t)�� Tr �k� (x; x; t)� ;and in view of (2.8), we have(2.9) index �D+� = ZM Str (k (x; x; t)) �x:The left side is independent of t and so the right side is also independent of t.The main task now is to determine the behavior of Str (k (x; x; t)) as t ! 0+.We suspe t that for ea h x 2 M , as t ! 0+, k (x; x; t) and Str (k (x; x; t)) arein uen ed primarily by the geometry (e.g., urvature form � of M with metri hand Levi-Civita onne tion �, and the urvature " of the unitary onne tion forE) near x, sin e the heat sour es of points far from x are not felt very strongly atx for small t. Indeed, we will give an outline a proof of the following Lo al IndexFormula, the full proof of whi h will appear in [BlBo03℄.Theorem 2.13 (The Lo al Index Theorem). In the notation of De�nitions2.9 and 2.11, let D : C1 (E � (M )) ! C1 (E � (M )) be a twisted Dira operator and let k 2 C1 (K) be the heat kernel for D2. If " is the urvature formof the unitary onne tion " for E and � is the urvature form of the Levi-Civita onne tion � for (M;h) with volume element �, then(2.10) limt!0+ Str (k (x; x; t)) = *Tr�ei"=2�� ^ det� i�=4�sinh (i�=4�)�12 ; �x+ :Remark 2.14. As will be explained below, the right side is really the innerprodu t, with volume form � at x, of the anoni al form h (E; ") ` bA (M; �)(depending on the onne tions " for E and the Levi-Civita onne tion � for themetri h) whi h represents h (E) ` bA (M ). As a onsequen e, we obtain the IndexTheorem for twisted Dira operators from the Lo al Index Formula in Corollary2.15 below. Thus the Lo al Index Formula is stronger than the Index Theorem fortwisted Dira operators. Indeed, the Lo al Index Formula yields the Index Theoremfor ellipti operators whi h are lo ally expressible as twisted Dira operators ordire t sums of su h.Corollary 2.15 (Index formula for twisted Dira operators). For an orientedRiemannian n-manifold M (n even) with spin stru ture, and a Hermitian ve torbundle E ! M with unitary onne tion, let D = D+ � D� be the twisted Dira operator, with D+ : C1 (E �+ (M ))! C1 (E �� (M )). We have(2.11) index �D+� = � h (E) ` bA (M )� [M ℄ ;

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 45where h (E) is the total Chern hara ter lass of E and bA (M ) is the total bA lass of M , both de�ned below. In parti ular, we obtain:n = 2) index (D+) = h1 (E) [M ℄ = 1 (E) [M ℄ andn = 4)8>><>>: index (D+) = � h (E) ` bA (M )� [M ℄= �� dimE � 124p1 (TM ) + h2 (E)� [M ℄= ��dimE24 p1 (TM ) + 12 1 (E)2 � 2 (E)� [M ℄ :Proof. By (2.9), (2.10) and the above Remark, we haveindex �D+� = ZM Str (k (x; x; t)) �x= limt!0+ ZM Str (k (x; x; t)) �x = ZM limt!0+ Str (k (x; x; t)) �x= ZM D h (E; ")x ` bA (M; �)x ; �xE �x = � h (E) ` bA (M )� [M ℄ :�We now explain the meaning of the formTr�ei"=2�� ^ det� i�=4�sinh (i�=4�)�12 :The �rst part Tr �ei"=2�� is relatively easy. We have (re all 2m = dimM )(2.12) ei"=2� := 1Xk=0 1k! � i2��k " ^ k� � � ^" = mXk=0 1k! � i2��k " ^ k� � � ^",where " ^ k� � � ^" 2 2k (End (E)). Also Tr�ik" ^ k� � � ^"� 2 2k (M ) andTr�ei"=2�� 2 mMk=12k (M ) :This (by one of many equivalent de�nitions) is a representative of the total Chern hara ter h (E) 2 Lmk=1H2k (M;Q). The urvature � of the Levi-Civita on-ne tion � for the metri h has values in the skew-symmetri endomorphisms ofTM ; i.e., � 2 2 (End (TM )). A skew-symmetri endomorphism of R2m, sayB 2 so (n), has pure imaginary eigenvalues �irk; where rk 2 R (1 � k � m).Thus, iB has real eigenvalues �rk. Now z=2sinh(z=2) is a power series in z with ra-dius of onvergen e 2�. Thus, isB=2sinh(isB=2) is de�ned for s suÆ iently small and has

46 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKeigenvalues rks=2sinh(rks=2) ea h repeated twi e. Hen edet� isB=2sinh (isB=2)� = mYk=1� rks=2sinh (rks=2)�2 anddet� isB=2sinh (isB=2)�12 = mYk=1 rks=2sinh (rks=2) :The last produ t is a power series in s of the form(2.13) mYk=1 rks=2sinh (rks=2) = 1Xk=0ak �r21; : : : ; r2m� s2k;where the oeÆ ient ak �r21; : : : ; r2m� is a homogeneous, symmetri polynomial inr21; : : : ; r2m of degree k. One an always express any su h a symmetri polynomialas a polynomial in the elementary symmetri polynomials�1; : : : ; �m in r21; : : : ; r2m,where �1 =Xmi=1 r2i ; �2 =Xmi<j r2i r2j ; �2 =Xmi<j<k r2i r2j r2k; : : : :These in turn may be expressed in terms of SO (n)-invariant polynomials in theentries of B 2 so (n) viadet (�I � B) = mYj=1 (� + irj) (� � irj) = mYj=1 ��2 + r2j �= mXk=1�k �r21; : : : ; r2m��2(m�k):On the other hand,det (�I � B) = mXk=10� 1(2k)! X(i);(j)Æj1���j2ki1 ���i2k Bi1j1 � � �Bi2kj2k1A �2(m�k), and so�k �r21; : : : ; r2m� = 1(2k)! X(i);(j) Æj1���j2ki1���i2kBi1j1 � � �Bi2kj2k ;where (i) = (i1; � � � ; i2k) is an ordered 2k-tuple of distin t elements of f1; : : : ; 2mgand (j) is a permutation of (i) with sign Æj1���j2ki1���i2k . If we repla e Bij with the 2-form12� ���ij relative to an orthonormal frame �eld, we obtain the Pontryagin formspk ��� := 1(2�)2k (2k)! X(i);(j)Æj1���j2ki1 ���i2k �i1j1 ^ � � � ^�i2kj2k ;whi h represent the Pontryagin lasses of the SO (n) bundle FM:Note that pk ���is independent of the hoi e of framing by the ad-invarian e of the polynomials

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 47�k. If we express the ak �r21; : : : ; r2m� as polynomials, say Ak (�1; : : : ; �k), in the �j(j � k), we an ultimately writedet� isB=2sinh (isB=2)�12 = 1Xk=0Ak (�1; : : : ; �k) s2k:Formally repla ing B by 12��, we �nally have the reasonable de�nitiondet� i�=4�sinh (i�=4�)�12 := 1Xk=0Ak �p1 ��� ; : : : ; pk ���� ;where the pj ��� are multiplied via wedge produ t when evaluating the termsin the sum; the order of multipli ation does not matter sin e pj ��� is of evendegree 4j. Also, sin e Ak �p1 ��� ; : : : ; pk ���� is a 4k-form, there are only a �nitenumber of nonzero terms in the in�nite sum. Abbreviating pj ��� simply by pj ,one �ndsdet� i�=4�sinh (i�=4�)�12 = 1� 124p1 + 15760 �7p21 � 4p2�� 1967 680 �31p31 � 44p1p2 + 16p3�+ � � � :(2.14)This (by one de�nition) represents the total bA- lass of M , denoted by(2.15) bA (M ) 2 mMk=1H2k (M;Q) ;where a tually bA (M ) has only nonzero omponents in H2k (M;Q) when k is even(or 2k � 0mod4). In (2.10) the multi-degree forms (2.12) and (2.14) have beenwedged, and the top (2m-degree) omponent (relative to the volume form) hasbeen harvested.We now turn to our outline of the proof of Theorem 2.13. The well-knownheat kernel (or fundamental solution) for the ordinary heat equation ut = �u inEu lidean spa e Rn, is given by(2.16) e (x; y; t) = (4�t)�n=2 exp �� jx� yj2 =4t� :Sin e H (x; y; t) only depends on r = jx� yj and t; it is onvenient to write(2.17) e (x; y; t) = E (r; t) := (4�t)�n=2 exp ��r2=4t� :We do not expe t su h a simple expression for the heat kernel k = (k+; k�) ofDe�nition 2.11. However, it an be shown that for x; y 2 M (of even dimensionn = 2m) with r = d(x; y) := Riemannian distan e from x to y suÆ iently small,we have an asymptoti expansion as t! 0+ for k (x; y; t) of the form(2.18) k (x; y; t) � HQ (x; y; t) := E (d(x; y); t)XQj=0 hj (x; y) tj ;

48 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKfor any �xed integer Q > m+ 4, wherehj (x; y) 2 Hom�(E � (M ))x ; (E � (M ))y� ; j 2 f0; 1; : : : ; Qg :The meaning of k (x; y; t) � HQ (x; y; t) is that for d(x; y) and t suÆ iently small,jk (x; y; t)�HQ (x; y; t)j � CQE (d(x; y); t) tQ+1 � CQtQ�m+1;where CQ is a onstant, independent of (x; y; t). We then have(2.19) k (x; x; t) � (4�t)�m QXj=0 hj (x; x) tj = (4�)�m QXj=0 hj (x; x) tj�m:Using (2.9), i.e., RM Str (k (x; x; t)) �x = index (D+) and (2.19), we dedu e thatZM Str (hj (x; x)) �x = 0 for j 2 f0; 1; : : : ;m� 1g ; while(4�)�m ZM Str (hm (x; x)) �x = ZM Str (k (x; x; t)) �x = index �D+� :Thus, to prove the Lo al Index Formula, it suÆ es to show that(4�)�m Str (hm (x; x)) = *Tr�ei"=2�� ^ det� i�=4�sinh (i�=4�)�12 ; �x+ :While this may not be the intelle tual equivalent of limbing Mount Everest, it isnot for the faint of heart.We hoose a normal oordinate system �y1; : : : ; yn� in a oordinate ball B entered at the �xed point x 2 M , so that �y1; : : : ; yn� = 0 at x. The oordinate�elds �1 := �=�y1; : : : ; �n := �=�yn are orthonormal at x, and for any �xedy0 2 B with oordinates �y10; : : : ; yn0 �, the urve t 7! t �y10 ; : : : ; yn0 � is a geodesi through x. By parallel translating the frame (�1; : : : ; �n) at x along these radialgeodesi s, we obtain an orthonormal frame �eld (E1; : : : ; En) on B whi h generallydoes not oin ide with (�1; : : : ; �n) at points y 2 B other than at x. The framing(E1; : : : ; En) de�nes a parti ularly ni e se tion B ! FM jB and we may lift thisto a se tion B ! P jB of the spin stru ture, whi h enables us to view the spa eC1 (� (M ) jB) of spinor �elds on B as C1 (B;�n), i.e., fun tions on B with valuesin the �xed spinor representation ve tor spa e �n = �+n � ��n . By similar radialparallel translation (with respe t to the onne tion ") of an orthonormal basis ofthe twisting bundle �ber Ex, we an identify C1 (EjB) with C1 �B; CN �, whereN = dimCE. The oordinate expressions for the urvatures �, " and D2 are assimple as possible in this so- alled radial gauge.With the above identi� ations, we pro eed as follows. For 0 � Q 2 Z, letQ 2 C1 �B � (0;1) ; CN �2m� be of the formQ (y; t) := E (r; t)XQk=0Uk (y) tk;

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 49where Uk 2 C1 �B; CN �2m�. If U0 (0) 2 CN �2m is arbitrarily spe i�ed, weseek a formula for Uk (y), k = 0; : : : ; Q, su h that(2.20) �D2 + �t�Q (y; t) = E (r; t) tQD2 (UQ) (y) ;where the square D2 of the Dira operator D an be written (where \�" is Cli�ordmultipli ation) asD2 = �� + 12Xj;k"jkEj �Ek � + 14S ;by virtue of the generalized Li hnerowi z formula (see [LaMi, p. 164℄). It is on-venient to de�ne the 0-th order operator F on C1 �B; CN �2m� viaF [ ℄ := 12Xj;k"jkEj �Ek � ; so thatD2 = �� + �F + 14S� [ ℄ :The desired formula for the Uk (y) involves the operator A on C1 �B; CN �2m�given by A [ ℄ := �h1=4D2[h�1=4 ℄ = h1=4�[h�1=4 ℄� �F + 14S� [ ℄ ;where h1=4 := (pdeth)1=2. For s 2 [0; 1℄, letAs [ ℄ (y) := A [ ℄ (sy) :As is proved in [Ble92℄ or in the forth oming [BlBo03℄, we haveProposition 2.16. Let U0 (0) 2 CN �2m, and let V0 2 C1 �B; CN �2m�be the onstant fun tion V0 (y) � U0 (0). Then the Uk (y) whi h satisfy (2.20) aregiven by Uk (y) = h (y)�1=4 Vk (y) ; whereVk (y) = ZIkYk�1i=0 (si)i �Ask�1 Æ � � � ÆAs0 [V0℄� (y) ds0 : : : dsk�1;(2.21)and where Ik = f(s0; : : : sk) : si 2 [0; 1℄ ; i 2 f0; : : : ; k � 1g g :Note that U0 (0) 2 CN �2m may be arbitrarily spe i�ed, and on e U0 (0) is hosen, the Um (y) are uniquely determined via (2.21). Let hk (y) 2 End �CN �2m�be given by(2.22) hk (y) (U0 (0)) := Uk (y)(in parti ular, h0 (0) = I 2 End �CN �2m�), andHQ (0; y; t) := E (r; t) QXk=0hk (y) tk 2 C1 �B;End �CN �2m�� :We may regard HQ (0; y; t) asHQ (x; y; t) 2 Hom�Ex � (M )x ; Ey � (M )y� ;

50 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKwhere we re all that x 2 M is the point about whi h we have hosen normal oordinates. For y suÆ iently lose to x, we set(2.23) HQ (x; y; t) := E (d (x; y) ; t) QXk=0hk (x; y) tk:Of ourse, one expe ts that HQ (x; y; t) provides the desired asymptoti expansion(2.18). Although this is very plausible, it is not at all easy to prove honestly. Whenproofs are attempted in the literature, often steps are skipped, hands are waved,and errors are made. An extremely areful (and hen e nearly unbearable) proofwill be provided in [BlBo03℄, but we must forgo this here. Thus, we will only statehere without proof thatk (x; y; t) � HQ (x; y; t) := E (d (x; y) ; t)XQj=0 hj (x; y) tjfor d (x; y) suÆ iently small, where the hj are given in (2.22).Using normal oordinates �y1; : : : ; y2m� 2 B (r0; 0) about x 2 M and theradial gauge, and sele ting V0 2 CN �2m, by Proposition 2.16, we have(2.24)hm (x; x) (V0) = ZImYm�1i=0 (si)i ��Asm�1 Æ � � � ÆAs0� heV0i� (0) ds0 : : :dsm�1;where eV0 2 C1 �B (r0; 0) ; CN �2m� is the onstant extension of V0. Re all thatfor 2 C1 �B (r0; 0) ; CN �2m�, we haveAs [ ℄ (y) := A [ ℄ (sy) ; where A [ ℄ := h1=4�[h�1=4 ℄ � �F + 14S� [ ℄ :While the right side of (2.24) may seem unwieldy, there is substantial simpli� ationdue to fa ts that �Asm�1 Æ � � � ÆAs0� [eV0℄ (y) is evaluated at y = 0 in (2.24). Also,if 1; : : : ; n denote the so- alled gamma matri es for Cli�ord multipli ation by�1; : : : ; �n, only those terms of Asm�1 Æ � � � ÆAs0 [eV0℄ (0) whi h involve the produ t n+1 := 1 � � � n will survive when the supertra e Str (hm (x; x)) is taken. As a onsequen e, we have the following simpli� ation (essentially ontained in [Ble92℄,or better yet, to appear in [BlBo03℄)Proposition 2.17. Let Rklji (0) = h �� (�i; �j) �l; �k� denote the ompo-nents of the Riemann urvature tensor of h at x; and let "ij (0) := " (�i; �j) atx. Set e�1 (�j) := 18Xk;l;iRklji (0) k lyi;F0 := 12Xi;j Fij i j = 12Xi;j "ij (0) i j ; andA0 :=Xi ��2i + e�1 (�i)2�� F0:(2.25)

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 51For V0 2 CN �2m, de�ne(2.26)h0m (0; 0) (V0) := ZIkYm�1i=0 (si)i ��A0sm�1 Æ � � � ÆA0s0� heV0i� (0)ds0 : : :dsm�1;where eV0 2 C1 �B (r0; 0) ; CN �2m� is the onstant extension of V0 2 CN �2m.Then(2.27) Str (hm (0; 0)) = Str �h0m (0; 0)� :In other words, in the omputation of Str (hm (0; 0)) given by (2.24), we may re-pla e A by A0.This is a substantial simpli� ation, not only in that A0 is a se ond-orderdi�erential operator with oeÆ ients whi h are at most quadrati in y, but it alsoshows that Str (hm (x; x)) only depends on the urvatures � and " at the pointx. One might regard the gist of the Index Formula for twisted Dira operator asexhibiting the global quantity index (D+) as the integral of a form whi h maybe lo ally omputed. From this perspe tive, Proposition 2.17 does the job. Also,knowing in advan e that index (D+) is insensitive to perturbations in h and ", onesuspe ts that Str (hm (x; x)) �x an be expressed in terms of the standard formswhi h represent hara teristi lasses for TM and E. The Lo al Index Formula on�rms this. Moreover, for low values of m; say m = 1 or 2 (i.e., for 2 and4-manifolds), one an dire tly ompute Str �h0m (0; 0)� using (2.26), and therebyverify Theorem 2.13 and hen e obtain Corollary 2.15 rather easily. For readers whohave no use for the Lo al Index Theorem beyond dimension 4, this is suÆ ient.It requires more e�ort to prove Theorem 2.13 for general m. For la k of spa e,we annot go into the details of this here, but they an be found in [Ble92℄ or[BlBo03℄. It is well worth mentioning that the appearan e of the sinh fun tion inthe Lo al Index Formula has its roots in Mehler's formula for the heat kernel(2.28)ea (x; y; t) = 1q4� sinh(2at)2a exp � 14 sinh(2at)2a � osh (2at) �x2 + y2�� 2xy�! :of the generalized 1-dimensional heat problemut = uxx � a2x2u; u (x; t) 2 R; (y; t) 2 R�(0;1) ;u (x; 0) = f (x) ;where 0 6= a 2 R is a given onstant. A solution of this problem is given byu (x; t) = Z 1�1 ea (x; y; t) f (y) dy;and this redu es to the usual formula as a ! 0. The ni e idea of using Mehler'sformula in a rigorous derivation of the Lo al Index Theorem appears to be dueto Getzler in [Get83℄ and [Get86℄, although it was at least impli itly involved in

52 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKFigure 7. The partition of M =M1 [�M2earlier heuristi supersymmetri path integral arguments for the Index Theorem.In the same vein, further simpli� ations and details an be found in [BeGeVe92℄,[Yu01℄, and [Ble92℄, but the treatment to be found in [BlBo03℄ will be moreself- ontained and less demanding.While the Lo al Index Theorem (Theorem 2.13) is stated for twisted Dira operators, the same proof may be applied to obtain the index formulas for anellipti operator, possibly on a nonspin manifold, whi h is only lo ally of the formof twisted Dira operator D+. Indeed, if A is su h an operator and k is the heatkernel for A�A�AA�, then from the spe tral resolution of A, we an still dedu efrom the asymptoti expansion of k thatindex (A) = (4�)�m ZM Str (hm (x; x)) �x;where the supertra e Str is de�ned in the natural way. The ru ial observation isthat sin e A is lo ally in the form of a twisted Dira operator, we an omputeStr (hm (x; x)) in exa tly the same way (i.e., lo ally) as we have done. Sin e it isnot easy to �nd �rst-order ellipti operators of geometri al signi� an e whi h arenot expressible in terms of lo ally twisted Dira operators (or 0-th order perturba-tions thereof), the Lo al Index Formula for twisted Dira operators is mu h more omprehensive than it would appear at �rst glan e.2.3. Dira Type Operators on Manifolds with Boundary . We �x thenotation and re all basi properties of operators of Dira type on manifolds withboundary.2.3.1. The General Setting for Partitioned Manifolds. Let M be a ompa tsmooth Riemannian partitioned manifoldM =M1 [�M2 :=M1 [M2 ; where M1 \M2 = �M1 = �M2 = �and � a hypersurfa e (see Figure 7). We assume thatM n� does not have a losed

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 53 onne ted omponent (i.e. � interse ts any onne ted omponent of M1 and M2).Let D : C1(M ;S)! C1(M ;S)be an operator of Dira type a ting on se tions of a Hermitian bundle S of Cli�ordmodules overM , i.e. D = Ær where denotes the Cli�ord multipli ation and ris a onne tion for S. Unlike the more general ase introdu ed in Subse tion 1.3,we assume here that r is ompatible with in the sense that r = 0. From this ompatibility assumption it follows that D is symmetri and essentially self-adjointover M .For even n = dimM , the splitting Cl(M ) = Cl+(M )� Cl�(M ) of the Cli�ordbundles indu es a orresponding splitting of S = S+ � S� and a hiral de ompo-sition D = � 0 D� = (D+)�D+ 0 �of the total Dira operator. The hiral Dira operators D� are ellipti but notsymmetri , and for that reason they may have nontrivial indi es whi h provide uswith important topologi al and geometri invariants, as was found in the spe ial ase of twisted Dira operators in Subse tion 2.2.Here we assume that all metri stru tures of M and S are produ t in a ollarneighborhood N = [�1; 1℄� � of �. If u denotes the normal oordinate (runningfromM1 to M2), thenDjN = �(�u + B) ; where � := (du) , �u := ��u;and B denotes the anoni ally asso iated Dira operator over �, alled the tan-gential operator. We have a similar produ t formula for the hiral Dira operator.Here the point of the produ t stru ture is that then � and B do not depend on thenormal variable. Note that � (Cli�ord multipli ation by du) is a unitary mappingL2(�;Sj�)! L2(�;Sj�) with �2 = � I and �B = �B�. In the non-produ t ase,there are ertain ambiguities in de�ning a `tangential operator' whi h we shall notdis uss here (but see also Formula 1.12).2.3.2. Analysis tools: Green's Formula. For notational e onomy, we set X :=M2. For greater generality, we onsider the hiral Dira operatorD+ : C1(X;S+)! C1(X;S�)and write D� for its formally adjoint operator. The orresponding results followat on e for the total Dira operator.Lemma 2.18. Let h:; :i� denote the s alar produ t in L2(X;S�). Then we haveD+f+; f��� � f+;D�f��+ = � Z�(� 1f+; 1f�) d vol�for any f� 2 C1(X;S�).

54 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKHere(2.29) 1 : C1(X;S�)! C1(�;S�j�)denotes the restri tion of a se tion to the boundary �.2.3.3. Cau hy Data Spa es and the Calder�on Proje tion. To explain the L2Cau hy data spa es we re all three additional, somewhat deli ate and not widelyknown properties of operators of Dira type on ompa t manifolds with boundaryfrom [BoWo93℄:(1) the invertible extension to the double;(2) the Poisson type operator and the Calder�on proje tion; and(3) the twisted orthogonality of the Cau hy data spa es for hiral and totalDira operators whi h gives the Lagrangian property in the symmetri ase (i.e., for the total Dira operator).The idea and the properties of the Calder�on proje tion were announ ed inCalder�on [Ca63℄ and proved in Seeley [Se66℄ in great generality. In the follow-ing, we restri t ourselves to onstru ting the Calder�on proje tion for operators ofDira type (or, more generally, ellipti di�erential operators of �rst order) whi hsimpli�es the presentation substantially.2.3.4. Invertible Extension. First we onstru t the invertible double. Cli�ordmultipli ation by the inward normal ve tor gives a natural lut hing of S+ overone opy of X with S� over a se ond opy of X to a smooth bundle fS+ over the losed double eX . The produ t forms of D+ and D� = (D+)� �t together over theboundary and provide a new operator of Dira type, namely(2.30) gD+ := D+ [D� : C1 � eX; fS+�! C1 � eX; fS�� :Clearly (D+ [D�)� = D� [ D+; and so indexgD+ = 0. It turns out that gD+ isinvertible with a pseudo-di�erential ellipti inverse (gD+)�1. Clearly, lo al solutionsof the homogenous equation (here, solutions on one opy of X) do not extend toglobal solutions on eX in general. As a matter of fa t, the operator D+, even beingthe restri tion of an invertible, lo ally de�ned di�erential operator, is not invertiblein general, and we have r+(gD+)�1e+D+ 6= I, where e+ : L2 (X;S+)! L2( eX; fS+)denotes the extension-by-zero operator and r+ : Hs � eX; fS+� ! Hs (X;S+) thenatural restri tion operator for Sobolev spa es for s real. The pre ise de omposi-tion L2( eX ; fS+) = L2 (X;S+)� L2 (X;S�) gives a di�erent pi ture in ase that agiven operator T on one omponent an be extended to an invertible operator eT onthe whole spa e. This, indeed, would imply T invertible with T�1 = r+ �eT��1 e+.The L2 extension of gD+, however, is not a pre ise extension of the L2 extensionof D+. Therefore, in our ase, the L2-argument breaks down.

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 55Example 2.19. In the simplest possible two-dimensional ase we onsiderthe Cau hy-Riemann operator � : C1(D2) ! C1(D2) over the dis D2, where� = 12(�x + i�y). In polar oordinates, this operator has the form 12ei'(�r + ir �').Therefore, after some small smooth perturbations (and modulo the fa tor 12 ), weassume that � has the following form in a ertain ollar neighborhood of theboundary: � = ei'(�r + i�')Now we onstru t the invertible double of �. By Ek, k 2Z, we denote the bundle,whi h is obtained from two opies of D2�C by the identi� ation (z; w) = (z; zkw)near the equator. We obtain the bundle E1 by gluing two halves of D2 � C by�(') = ei' and E�1 by gluing with the adjoint symbol. In su h a way we obtainthe operator e� := � [ (�)� : C1(S2;E1)! C1(S2;E�1)over the whole 2-sphere.Let us analyze the situation more arefully. We �x N := (�";+")� S1, a bi ollarneighborhood of the equator. The operator formally adjoint to � has the form(�)� = e�i' (��u + i�' + 1)(u = r � 1) in this ylinder. A se tion of E1 is a ouple (s1; s2) su h that in Ns2 (u; ') = ei's1 (u; ') :The ouple ��s1; (�)�s2� is a smooth se tion of E�1. To show this, we he k that(�)�s2 = e�i'�s1. In the neighborhood N , we have(�)�s2 = (�)� �ei's1� = e�i' (��u + i�' + 1) �ei's1�= �us1 + ie�i'�' �ei's1�+ s1 = (�u + i�') s1= e�i'ei' (�u + i�') s1 = e�i' ��s1� :Then the operator � [ �� be omes inje tive and index � [ �� = 0.2.3.5. The Poisson Operator and the Calder�on Proje tion. Next we investigatethe solution spa es and their tra es at the boundary. For a total or hiral operatorof Dira type over a smooth ompa t manifold with boundary � and for any reals we de�ne the null spa eKer(D+; s) := ff 2 Hs(X;S+) j D+f = 0 in X n�g:The null spa es onsist of se tions whi h are distributional for negative s; byellipti regularity they are smooth in the interior; in parti ular they possess asmooth restri tion on the hypersurfa e �" = f"g � � parallel to the boundary �of X at a distan e " > 0. By a Riesz operator argument they an be shown toalso possess a tra e over the boundary. Of ourse, that tra e is no longer smoothbut belongs to Hs� 12 (�;S+j�). More pre isely, we have the following well-knownGeneral Restri tion Theorem (for a proof see e.g. [BoWo93℄, Chapters 11 and13):

56 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKTheorem 2.20. (a) Let s > 12 . Then the restri tion map 1 of (2.29) extendsto a bounded map(2.31) s : Hs(X;S+)! Hs�12 (�;S+j�):(b) For s � 12 , the pre eding redu tion is no longer de�ned for arbitrary se tionsbut only for solutions of the operator D+: let f 2 Ker(D+; s) and let (")f denotethe well-de�ned tra e of f in C1(�";S+j�). Then, as " ! 0+, the se tions (")f onverge to an element sf 2 Hs�12 (�;S+j�).( ) Let gD+ denote the invertible double of D+, r+ denote the restri tion operatorr+ : Hs( eX ; eS+) ! Hs(X;S+) and let �1 be the dual of 1 in the distributionalsense. For any s 2 R the mapping (Poisson type operator)K := r+ �gD+��1 �1� : C1(�;S+j�)! C1(X;S+)extends to a ontinuous map K(s) : Hs�1=2(�;S+j�)! Hs(X;S+) withrangeK(s) = Ker(D+; s):For s = 0, Theorem 2.20 an be reformulated in the following way:Corollary 2.21. For a onstant C independent of f , we havek (f)k� 12 � C � D+f 0 + kfk0� for all f 2 Dmax �D+� ;where Dmax (D+) := �f 2 L2 (X;S) j D+f 2 L2 (X;S). So, the restri tion : Dmax �D+� �! H�1=2(�;S+j�)is well de�ned and bounded.Proofs of Theorem and Corollary an be found, e.g., in Booss{Bavnbek andWoj ie howski [BoWo93℄, Theorems 13.1 and 13.8 for our situation (D+ is oforder 1); and in H�ormander [Ho66℄ in greater generality (Theorem 2.2.1 and theEstimate (2.2.8), p. 194).The omposition(2.32) P(D+) := 1 Æ K : C1(�;S+j�)! C1(�;S+j�)is alled the (Szeg�o{)Calder�on proje tion. It is a pseudo-di�erential proje tion(idempotent, but in general not orthogonal). We denote by P(D+)(s) its extensionto the s-th Sobolev spa e over �. It has the following geometri meaning.We now have three options of de�ning the orresponding Cau hy data (orHardy) spa es:Definition 2.22. For all real s we de�ne�(D+; s) := s(Ker(D+; s));� los(D+; s) := 1ff 2 C1(X;S+) j D+f = 0 in X n�gHs� 12 (�;S+ j�); and�Cald(D+; s) := rangeP(D+)(s� 12 ) :

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 57The range of a proje tion is losed; the in lusions of the Sobolev spa es aredense; and rangeP(D+) = 1ff 2 C1(X;S+) j D+f = 0 in Xn�g, as shownin [BoWo93℄. So, the se ond and the third de�nition of the Cau hy data spa e oin ide. Moreover, for s > 12 one has �(D+; s) = �Cald(D+; s). This equality anbe extended to the L2 ase (s = 12 , see also Theorem 2.31 below), and remainsvalid for any real s, as proved in Seeley, [Se66℄, Theorem 6. For s � 12 , the result issomewhat ounter-intuitive (see also Example 2.24b in the following Subse tion).We have:Proposition 2.23. For all s 2 R�(D+; s) = � los(D+; s) = �Cald(D+; s):2.3.6. Calder�on and Atiyah{Patodi{Singer Proje tion . The Calder�on proje -tion is losely related to another proje tion determined by the `tangential' part ofD+, des ribed as follows. Let B denote the tangential symmetri ellipti di�erentialoperator over � in the produ t formD+ = �(�u + B) :C1 �N;S+ jN�! C1 �N;S�jN�in a ollar neighborhood N of � in X. It has dis rete real eigenvalues and a omplete system of L2 orthonormal eigense tions. Let P�(B) denote the spe -tral (Atiyah{Patodi{Singer) proje tion onto the subspa e L+(B) of L2(�;S+j�)spanned by the eigense tions orresponding to the nonnegative eigenvalues of B.It is a pseudo-di�erential operator and its prin ipal symbol p+ is the proje tiononto the eigenspa es of the prin ipal symbol b(y; �) of B orresponding to nonneg-ative eigenvalues. It turns out that p+ oin ides with the prin ipal symbol of theCalder�on proje tion.We all the spa e of pseudo-di�erential proje tions with the same prin ipalsymbol p+ the Grassmannian Grp+ and equip it with the operator norm or-responding to L2(�;S+j�). It has ountably many onne ted omponents; twoproje tions P1, P2 belong to the same omponent, if and only if the virtual odi-mension(2.33) i(P2; P1) := index fP2P1 : rangeP1 ! rangeP2gof P2 in P1 vanishes; the higher homotopy groups of ea h onne ted omponentare given by Bott periodi ity.Example 2.24. (a) For the Cau hy-Riemann operator on the dis D2 = fjzj �1g, the Cau hy data spa e is spanned by the eigenfun tions eik� of the tangentialoperator �� over S1 = [0; 2�℄=f0; 2�g for nonnegative k. So, the Calder�on proje -tion and the Atiyah{Patodi{Singer proje tion oin ide in this ase.(b) Next we onsider the ylinder XR = [0; R℄��0 with DR = �(�u+B). Here Bdenotes a symmetri ellipti di�erential operator of �rst order a ting on se tionsof a bundle E over �0 , and � a unitary bundle endomorphism with �2 = � I and

58 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKFigure 8. Cylinder of length R�B = �B�. Let B be invertible (for the ease of presentation). Let f'k; �kg denoteB's spe tral resolution of L2(�0;E) with: : :��k � :::��1 < 0 < �1 � :::�k � : : :Then(2.34) B'k = �k'k for all k 2Zn f0g ;��k = ��k ; �('k) = '�k ; and �('�k) = �'k for k > 0:We onsiderf 2 Ker(DR; 0) = spanfe��ku'kgk2Znf0g in L2(XR)= KerDRmax (kernel of maximal extension):It an be written in the form(2.35) f(u; y) = f>(u; y) + f<(u; y); u 2 [0; R℄ ;where f<(u; y) =Xk<0ake��ku'k(y) and f>(u; y) =Xk>0ake��ku'k(y):Be ause ofhf; fiL2(XR) < +1 () hf<; f<i < +1 and hf>; f>i < +1 ;the oeÆ ients ak satisfy the onditions(2.36) Xk<0 jakj2 e�2�kR � 12j�kj < +1 or, equivalently, Xk<0 jakj2 e2j�kjRj�kj < +1and(2.37) Xk>0 jakj21� e�2�kR2�k < +1 or, equivalently, Xk>0 jakj2=�k < +1 :We onsider the Cau hy data spa e �(DR; 0) onsisting of all (f) with f 2Ker(DR; 0). Here (f) denotes the tra e of f at the boundary� = �XR = ��0 t�R ;where �R denotes a se ond opy of �0 . A ording to the spe tral splitting f =f> + f<, we have (f) = (s0<; sR<) + (s0>; sR>)

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 59where s0> = f>(0); s0< = f<(0); sR> = f>(R); sR< = f<(R):Be ause of (2.36) and (2.37), we have(s0<; sR>) 2 C1(�0 [�R) and (s0>; sR<) 2 H�1=2(�0 [�R):Re all that X ak'k 2 Hs(�0) () X jakj2jkj2s=(m�1) < +1and j�kj � jkj 1m�1 for k !�1, where m � 1 denotes the dimension of �0.One noti es that the estimate (2.36) for the oeÆ ients of s0< is stronger thanthe assertion that Pk<0jakj2j�kjN < +1 for all natural N . Thus our estimates on�rm that not every smooth se tion an appear as initial value over �0 of asolution of DRf = 0 over the ylinder.To sum up the example, the spa e �(DR; 0) an be written as the graph ofan unbounded, densely de�ned, losed operator T : DomT ! H�12 (�R), mappings0< + s0> =: s0 7! sR := sR< + sR> with DomT � H�12 (�0) . To obtain a losedsubspa e of L2(�) one takes the range �(DR; 12 ) of the L2 extension P(DR)(0) ofthe Calder�on proje tion. It oin ides with �(DR; 0) \ L2(�) by Proposition 2.23.In Theorem 2.31 we show without use of the pseudo-di�erential al ulus why theinterse tion �(DR; 0)\L2(�) must be losed in L2(�). See [S Wo00, p. 1214℄ foranother des ription of the Cau hy data spa e �(DR; 12), namely as the graph of aunitary ellipti pseudo-di�erential operator of order 0.Sin e � = ��0 t�R , the tangential operator takes the form B = B � (�B)and we obtain from (2.34)rangeP>(B)(0) = L+(B) = spanL2(�)f('k; �('k))gk>0 :For omparison, we have in this examplerangeP(DR)(0) = �(DR; 12) = spanL2(�)f('k; e��kR'k)gk>0 ;hen e L+(B) and �(DR; 12 ) are transversal subspa es of L2(�). On the semi-in�nite ylinder [0;1)��, however, we have only one boundary omponent �0 . Hen erangeP>(B)(0) = spanL2(�0)f'kgk>0 = limR!1 rangeP(DR)(0):One an generalize the pre eding example: For any smooth ompa t manifoldX with boundary � and any real R � 0, let XR denote the stret hed manifoldXR := ([�R; 0℄��) [�X:Assuming produ t stru tures with D = �(�u + B) near � gives a well-de�nedextension DR ofD. Ni olaes u, [Ni95℄ proved that the Calder�on proje tion and theAtiyah{Patodi{Singer proje tion oin ide up to a �nite-dimensional omponent inthe adiabati limit (R ! +1 in a suitable setting). Even for �nite R and, inparti ular for R = 0, one has the following interesting result. It was �rst proved in

60 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKS ott, [S o95℄ (see also Grubb, [Gr99℄ and Woj ie howski, [DaKi99℄, Appendixwho both o�ered di�erent proofs).Lemma 2.25. For all R � 0, the di�eren e P(DR) � P�(B) is an operatorwith a smooth kernel.2.3.7. Twisted Orthogonality of Cau hy Data Spa es. Green's formula (in par-ti ular the Cli�ord multipli ation � in the ase of Dira type operators) providesa symple ti stru ture for L2(�;Sj�) for linear symmetri ellipti di�erential op-erators of �rst order on a ompa t smooth manifold X with boundary �. Forellipti systems of se ond-order di�erential equations, various interesting resultshave been obtained in the 1970s by exploiting the symple ti stru ture of orre-sponding spa es (see e.g. [LaSnTu75℄). Restri ting oneself to �rst-order systems,the geometry be omes very lear and it turns out that the Cau hy data spa e�(D; 12) is a Lagrangian subspa e of L2(�;Sj�).More generally, in [BoWo93℄ we des ribed the orthogonal omplement of theCau hy data spa e of the hiral Dira operator D+ by(2.38) ��1(�(D�; 12)) = (�(D+; 12 ))? :We obtained a short exa t sequen e0! ��1(�(D�; s)) ,!Hs� 12 (�;S+j�) K(s)! Ker(D+; s)! 0:For the total (symmetri ) Dira operator this means:Proposition 2.26. The Cau hy data spa e �(D; 12 ) of the total Dira oper-ator is a Lagrangian subspa e of the Hilbert spa e L2(�;Sj�) equipped with thesymple ti form !('; ) := (�'; ).2.3.8. \Admissible" Boundary Value Problems. We refer to [BoWo93, Chap-ter 18℄, [BrLe99℄, and [S 01℄ for a rigorous de�nition and treatment of large lasses of admissible boundary value problems de�ned by pseudo-di�erential pro-je tions. Prominent examples belong to the Grassmannian Grp+ (introdu ed abovein Se tion 2.3.6 before Example 2.24). On even{dimensional manifolds, otherprominent examples are lo al hiral proje tions and unitary modi� ations as ex-plained in [BoWo93, p. 273℄.For all admissible boundary onditions de�ned by a pseudo-di�erential proje -tion R over � the following features are ommon (For simpli ity, we suppress thedistin tion between total and hiral spinor bundle in this paragraph and denotethe bundle by E):(1) We have an estimate(2.39) kfk1 � C � D+f 0 + kfk0 + kR Æ i Æ (f)k 12� for f 2 H1 (X;E) :(2) De�ning a domain by(2.40) D = Dom(D+R) := �f 2 H1 (X;E) j R (f j�) = 0 ;we obtain a losed Fredholm extension.

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 61(3) The restri tion of Dom(D+R) to the boundary makes a Fredholm pair withthe Cau hy data spa e of D+.(4) The omposition RPD+ de�nes a Fredholm operator from the Cau hydata spa e to the range of R (see also Proposition 3.34).(5) The spa e Ker �D+R� onsists only of smooth se tions.Warning: A new feature of operators of Dira type on manifolds with boundaryis that the index of admissible boundary value problems an jump under ontinuousor even smooth deformation of the oeÆ ients. E.g., this is the ase for the Atiyah-Patodi-Singer boundary problem, as follows from the next subse tion.A total ompatible (and so symmetri ) Dira operator D and an orthogonal ad-missible proje tion R de�ne a self-adjoint extension D+D = D+�jD, if the proje tionR de�nes the domainD and satis�es the symmetry ondition I �R = ��1R�. Wedenote by Grsap+ or, shortly, Grsa the subspa e of Grp+ of orthogonal proje tionswhi h satisfy the pre eding ondition and di�er from the Atiyah{Patodi{Singerproje tion only by an in�nitely smoothing operator.Posing a suitable well-posed boundary value problem provides for a ni elyspa ed dis rete spe trum near 0. Then, varying the oeÆ ients of the di�erentialoperator and the imposed boundary ondition suggests the use of the powerfultopologi al on ept of spe tral ow. From (2.39) and a areful analysis of the or-responding parametri es we see in [BoLePh01, Se tion 3℄ under whi h onditionsthe urves of the indu ed self-adjoint L2 extensions be ome ontinuous urves inCF sa �L2(X;E)� in the gap topology so that their spe tral ow is well-de�ned andtruly homotopy invariant.We summarize the main results. They depend strongly on the weak unique ontinuation property (either in the form of Se tion 1.4 or in the weaker form of(1.9) whi h is suÆ ient here) and the invertible extension (Se tion 2.3.4).Lemma 2.27. For �xed D the mappingGrsa (D) 3 P 7�! DP 2 CF sa �L2(X;E)�is ontinuous from the operator norm to the gap metri .Theorem 2.28. Let X be a ompa t Riemannian manifold with boundary. LetfDsgs2M ,M a metri spa e, be a family of ompatible operators of Dira type. Weassume that in ea h lo al hart, the oeÆ ients of Ds depend ontinuously on s.Then we have(a) The Poisson operator Ks : L2 (�;Ej�)! H1=2 (X;E) of Ds depends ontinu-ously on s in the operator norm.(b) The Calder�on proje tor P+(s) : L2 ��;Ej��! L2 ��;Ej�� of Ds depends on-tinuously on s in the operator norm.( ) The family M 3 s 7�! (Ds)P+(s) 2 CF sa �L2(X;E)�is ontinuous.(d) Let Ptft2Y g be a norm- ontinuous path of orthogonal proje tions in L2(X;E).

62 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKIf Pt 2 \s2M Grsa (Ds) ; t 2 Y;then M � Y 3 (s; t) 7�! (Ds)Pt 2 CF sa �L2(X;E)�is ontinuous.Note that (b) is a pseudo-di�erential reformulation of the ontinuity of Cau hydata spa es whi h is valid in mu h greater generality (see our Se tion 1.2.2).We lose this Se tion with a re ent result of [Gr02℄:Lemma 2.29. (G. Grubb 2002) Let D be an operator of Dira type over a om-pa t manifold with boundary. Let P be an orthogonal proje tion whi h de�nes an`admissible' self{adjoint boundary ondition for D. Then there exists an invertibleoperator of Dira type B0 over the boundary su h that P = P>(B0).Grubb's Lemma shows that the Atiyah{Patodi{Singer boundary proje tion isthe most general admissible self{adjoint boundary ondition, in the spe i�ed sense.2.4. The Atiyah{Patodi{Singer Index Theorem. Let X be a ompa t,oriented Riemannian manifold with boundary Y = �X with dimX = n = 2meven. Let D : C1 (X;S) ! C1 (X;S) be a ompatible operator of Dira typewhere S ! X is a bundle of Cli�ord modules. Relative to the splitting S =S+�S� into hiral halves, we have the operators D+ : C1 (X;S+)! C1 (X;S�)and D� : C1 (X;S�) ! C1 (X;S+) whi h are formal adjoints on se tions withsupport in X nY . We assume that all stru tures (e.g., Riemannian metri , Cli�ordmodule, onne tion) are produ ts on some ollared neighborhood N �= [�1; 1℄�Yof Y . Then D+jN := D+ : C1 (N;S+ jN )! C1 (N;S�jN ) has the formD+jN = �(�u + B):Here u 2 [�1; 1℄ is the normal oordinate (i.e., N = f(u; y) j y 2 Y; u 2 [�1; 1℄g)with �u = ��u the inward normal), � = (du) is the (unitary) Cli�ord multipli a-tion by du with � (S+jN ) = S�jN , andB : C1 �Y; S+jY �! C1 �Y; S+jY �denotes the anoni ally asso iated (ellipti , self-adjoint) Dira operator over Y , alled the tangential operator. Note that due to the produ t stru ture, � andB do not depend on u. Let P�(B) denote the spe tral (Atiyah{Patodi{Singer)proje tion onto the subspa e L+(B) of L2(Y; S+j�X) spanned by the eigense tions orresponding to the nonnegative eigenvalues of B. LetC1 �X;S+;P�� := � 2 C1 �X;S+� j P�(B) ( jY ) = 0 ; andD+P� := D+jC1(X;S+ ;P�) : C1 �X;S+;P��! C1 �X;S�� :

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 63The eta fun tion for B is de�ned by�B (s) :=X�2spe B�f0g (sign�)m� j�j�s ,for R (s) suÆ iently large, where m� is the multipli ity of �. Impli it in the fol-lowing result (originating in [AtPaSi75℄) is that �B extends to a meromorphi fun tion on all C , whi h is holomorphi at s = 0 so that �B (0) is �nite.Theorem 2.30 (Atiyah-Patodi-Singer Index Formula). The above operatorD+P� has �nite index given byindexD+P� = ZX � h (S; ") ^ eA (X; �)�� m0 + �B (0)2 :where m0 = dim(KerB) ; h (S; ") 2 � (X;R) is the total Chern hara ter form ofthe omplex ve tor bundle S with ompatible, unitary onne tion ", and eA (X; �) 2� (X;R) is losely related to the total bA (X; �) form relative to the Levi-Civita onne tion �, namely eA (X; �)4k = 22k�m bA (X; �)4k.Proof. (outline) The proof found in [AtPaSi75℄ or [BoWo93℄ is based onthe heat kernel method for omputing the index, but the pro ess is less straightfor-ward than in the losed ase be ause of the boundary ondition. The appropriateheat kernel is onstru ted by means of Duhamel's method. Namely, an exa t kernelis obtained from an approximate one by an iterative pro ess initiated by writingthe error as the integral of a derivative of the onvolution of the true and approx-imate kernel; see (2.50) below. The initial approximate heat kernel is obtained bypat hing together two heat kernels, denoted by E and Ed. Here, E is a heat kernelfor a Dira operator over an in�nite extension [0;1)�Y of the ollared neighbor-hood N = [0; 1℄�Y of Y in X, for whi h the boundary ondition P�(B) ( jY ) = 0is imposed. The heat kernel Ed is the usual one (without boundary onditions) fore�tgD�gD+ , wheregD� are hiral halves of the invertible Dira operator (see (2.30)),namely gD� := D� [D� : C1 � eX; fS��! C1 � eX; fS��over the double eX , a losed manifold without boundary.We begin with the onstru tion of E . We letD := �u + B : C1 �N;S+jN�! C1 �N;S+jN� ,whi h has formal adjoint D� := ��u + B. De�neD := DjDomD, whereDomD := �f 2 H1 �R+� Y; �� �S+jY �� j P� �f jf0g�Y � = 0 , andD� = D�jDomD�, whereDomD� := �f 2 H1 �R+� Y; �� �S�jY �� j P< �f jf0g�Y � = 0 :

64 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKWe also have the Lapla ians given by� := D�DjDomD�D, whereDomD�D := � f 2 H1 (R+� Y; �� (S+jY )) jP� �f jf0g�Y � = 0; P< �Df jf0g�Y � = 0 � ; and� � := DD�jDomDD�, whereDomDD� := � f 2 H1 (R+� Y; �� (S+jY )) jP< �f jf0g�Y � = 0; P� �D�f jf0g�Y � = 0 � :Let '� (y) 2 C1 (Y; S+jY ) be an eigense tion of B with eigenvalue � 2 R. Notethat g� (t;u; y) = f� (t; u)'� (y) is a solution of the heat equation0 = (�t +� ) g� = �tg� + (�u + B) (��u + B) g� = ��t � �2u + B2� g�= ��tf� � �2uf� + �2f��'� (y) ;with g� (t; �; �) 2 DomD�D, when f� : (0;1)� [0;1)! R solves the heat problem�tf� = �2uf� � �2f� with boundary onditionf� (t; 0) = 0 if � � 0�uf� (t; 0) + �f� (t; 0) = Df� = 0 if � < 0:(2.41)Re all that the omplementary error fun tion iserf (x) := 2p� Z 1x e��2d�.Of use to us are the fa tserf 0 (x) = �2p�e�x2 anderf (x) � 2p� e�x2x+qx2 + 4� � e�x2 for x � 0:(2.42)For � � 0; the heat kernel for the problem (2.41) is (via the method of images)e� (t;u; v) := e��2t2p�t �e� (u�v)24t � e� (u+v)24t � for u; v � 0 and t > 0:For � < 0; and u; v � 0 and t > 0; the heat kernel is (using Lapla e transforms)e� (t;u; v) := e��2t2p�t �e� (u�v)24t + e� (u+v)24t + �e�(u+v) erf �u+v2pt � �pt�� :The heat kernel for � (i.e., the kernel for e�t� ) is then(2.43) E (t;u; y; v; z) =X� e� (t;u; v)'� (y) '� (z)�Here � ranges over spe B and (by a onvenient abuse of notation) '� (y)'� (z)�is really a sum Pk '�;k (y) '�;k (z)� where f'�;1; : : : ; '�;kg is an orthonormal

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 65eigenbasis of the eigenspa e for �, and n'��;1; : : : ; '��;ko is the dual basis. Similarly,the heat kernel for � � is(2.44) E � (t;u; y; v; z) =X� e�� (t;u; v)'� (y) '� (z)� ;where e�� is the heat kernel for the problem�tf� = �2uf� � �2f� with boundary onditionf� (t; 0) = 0 if � < 0�uf� (t; 0)� �f� (t; 0) = D�f� = 0 if � � 0;(2.45)where f� : (0;1)� [0;1)! R. Thus, for u; v � 0 and t > 0, we havee�� (t;u; v) := e��2t2p�t �e� (u�v)24t � e� (u+v)24t � for � < 0;while for � � 0 (note the sign hange in passing from (2.41) to (2.45))e�� (t;u; v) := e��2t2p�t �e� (u�v)24t + e� (u+v)24t � �e�(u+v) erf �u+v2pt + �pt�� :Combining (2.43) and (2.44), we obtain the tra e of kernel E � E � for e�t� �e�t� � evaluated at the point (u; y;u; y) of the diagonalK (t;u; y) := TrX� (e� (t;u; u)� e�� (t;u; u))'� (y) '�� (y)=X��0 �e��2t2p�t e�u2=t � e��2t2p�te�u2=t + �e2�u erf � upt + �pt�! j'� (y)j2+X�<0 e��2t2p�te�u2=t + �e�2�u erf � upt � �pt�+ e��2t2p�te�u2=t! j'� (y)j2=X� sign (�) �e��2te�u2=tp�t + j�j e2j�ju erf � upt + j�jpt�! j'� (y)j2=X� sign (�) ��u �12e2j�ju erf � upt + j�jpt�� j'� (y)j2 :We used erf 0 (x) = �2p� e�x2 to obtain the last equality, and we have set sign (0) := 1for onvenien e. We ompute (where the inter hange of the sum and integral an

66 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKbe justi�ed)K (t) := Z 10 ZY K (t;u; y) dydu= Z 10 ZY X� sign (�) ��u �12e2j�ju erf � upt + j�jpt�� j'� (y)j2 dydu=X� m� sign (�) Z 10 ��u �12e2j�ju erf � upt + j�jpt�� du=X� m� sign (�) 12e2j�ju erf � upt + j�jpt����1u=0= �X� m� sign�2 erf �j�jpt� = �12m0 �X�6=0m� sign�2 erf �j�jpt� :Re all that m� is the multipli ity of �. Thus,K (t) + 12m0 = �X�6=0 m� sign�2 erf �j�jpt�For j�j > 0, one veri�es using integration by parts and substitution thatZ 10 erf �j�jpt� ts�1dt = j�j�2ssp� � �s+ 12� :Using this and the fa t that K (t) + 12h � Ce��t for onstants C and � > 0,Z 10 �K (t) + 12m0� ts�1dt= � Z 10 �X�6=0 m� sign�2 erf �j�jpt�� ts�1dt= �X�6=0 m� sign�2 Z 10 erf �j�jpt� ts�1dt= �X�6=0 m� sign�2 j�j�2ssp� � �s + 12�= �� �s + 12�2sp� X�6=0m� sign (�) j�j�2s= �� �s + 12�2sp� �B (2s) :Suppose that we have an asymptoti expansionK (t) �XNk=�n+1 aktk=2 as t! 0+; i.e.,K (t) �XNk=�n+1 aktk=2 = O(t 12 (N+1)) as t! 0+:

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 67In (2.57) below, su h an asymptoti expansion will eventually be produ ed (as wasdone in [BoWo93, p. 239f℄) from the known asymptoti expansion (see [Gi95,p. 68℄) of the heat kernel on a losed manifold, namely the double of M . Sin eR (s) > �N+12 ) R �12 (N + 1) + s� 1� > �1, we then have thatf1 (s) := Z 10 �K (t) �XNk=�n+1 aktk=2� ts�1dt = Z 10 O �t 12 (N+1)+s�1�dsis holomorphi for R (s) > �N+12 . We also have the entire fun tionf1 (s) := Z 11 �K (t) + 12h� ts�1dtWe laim that�� �s + 12�2sp� �B (2s) = m02s +XNk=�n+1 aks + 12k + f1 (s) + f1 (s) :Indeed, we have� � �s + 12�2sp� �B (2s)= Z 10 �K (t) + 12h� ts�1dt = Z 10 �K (t) + 12h� ts�1dt+ f1 (s)= Z 10 �12m0 +XNk=�n+1 aktk=2� ts�1dt+ f1 (s) + f1 (s)= Z 10 12m0ts�1dt+XNk=�n+1 ak Z 10 t(s+ 12k)�1dt+ f1 (s) + f1 (s)= 12m0 tss ����t=1t=0!+XNk=�n+1 ak ts+ 12ks + 12k �����t=1t=0 + f1 (s) + f1 (s)= m02s +XNk=�n+1 aks + 12k + f1 (s) + f1 (s) :Thus, where �N (s) = f1 (s) + f1 (s) is holomorphi for R (s) > �N+12 ,(2.46) �B (2s) = � 2sp�� �s+ 12� �m02s +XNk=�n+1 ak12k + s + �N (s)� :The heat kernels, say Ed for gD�gD+ and Ed� for gD+gD�, over the double eX aremore familiar. For t > 0,Tr e�tgD�gD+ � Tr e�tgD+gD� = Z eX Tr (Ed (t;x; x)� Ed� (t;x; x)) dx;and there is the asymptoti expansionF (t;x) := Tr (Ed (t;x; x)� Ed� (t;x; x)) �Xk��n�k (x) tk=2:

68 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKFor 0 < a < b < 1, let �(a;b) : [0; 1℄! [0; 1℄ be C1 and in reasing with�(a;b) (u) = � 0 for u � a1 for u � b:Thinking of u as the normal oordinate fun tion on N = [0; 1℄ � Y � X, andextending by the onstant values 0 and 1, we an (and do) regard �(a;b) as afun tion on X. LetQ (t;x; x0) := ' (x) E (t;x; x0) (x0) + 'd (x) Ed (t;x; x0) d (x0):= �1� �(5=7;6=7) (x)� E (t;x; x0) �1� �(3=7;4=7) (x0)�+ �(1=7;2=7) (x) Ed (t;x; x0) �(3=7;4=7) (x0) :We have that + d = 1 and f ; dg is a partition of unity for the over�u�1 �37 ;1� ; u�1 �0; 47� of X. Moreover, for j = ; d(2.47) 'jj supp j = 1 and dist �supp �ku'j; supp j� � 17 (for k � 1).Of ourse, we do not expe t Q (t;x; x0) to equal the exa t kernel for �t + D�Dthroughout X. However, note that for x; x0 2 [0; 1=7)�Y , Q (t;x; x0) = E (t;x; x0),so that Q (t;x; x0) meets the APS boundary ondition. Let x0 2 X be �xed.For x 2 X n [0; 6=7) � Y; we have Q (t;x; x0) = Ed (t;x; x0) d (x0) ; while forx 2 [0; 1=7)� Y; Q (t;x; x0) = E (t;x; x0) (x0). Thus, for x 2 X n ��17 ; 67� � Y �,� (�t + D�D) Q (�; �; x0)j(t;x;x0) = 0. Moreover, for d (x; x0) < 1=7,' (x) (x0) + 'd (x) d (x0)= �1� �(5=7;6=7) (x)� �1� �(3=7;4=7) (x0)�+ �(1=7;2=7) (x) �(3=7;4=7) (x0)= 1� �(5=7;6=7) (x) � �(3=7;4=7) (x0)+ �(5=7;6=7) (x) �(3=7;4=7) (x0) + �(1=7;2=7) (x)�(3=7;4=7) (x0)= 1� �(5=7;6=7) (x) � �(3=7;4=7) (x0) + �(5=7;6=7) (x) + �(3=7;4=7) (x0) = 1Note that d (x; x0) < 1=7 ) �(5=7;6=7) (x) �(3=7;4=7) (x0) = �(5=7;6=7) (x) ; et .. Be- ause of this, we expe t thatlimt!0+Q (t;x; x0)= limt!0+ (' (x) E (t;x; x0) (x0) + 'd (x) Ed (t;x; x0) d (x0))= ' (x) Æ (x; x0) (x0) + 'd (x) Æ (x; x0) d (x0)= (' (x) (x0) + 'd (x) d (x0)) Æ (x; x0) = Æ (x; x0) :Thus, on the operator level, we expe t that limt!0+ Qt = I; i.e.,limt!0+Qt (f) (x) = limt!0+ ZX Q (t;x; x0) f (x0) dx0 = ZX Æ (x; x0) f (x0) dx0 = f (x) :The extent to whi h Q (�; �; x0) fails to satisfy the heat equation in �17 ; 67� � Y isgiven by C (t;x; x0) := � (�t + D�D) Q (�; �; x0)j(t;x;x0) :

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 69For x0 2 X �xed and x = (u; y) 2 N = [0; 1℄� Y , we have�C (t;x; x0)(2.48) = (�t +D�D)Q (t;x; x0)= (�t +D�D)�Xj2f ;dg 'j (x) Ej (t;x; x0) j (x0)�= Xj2f ;dg �2u ('j (x)) Ej (t;x; x0) j (x0) + 2�u ('j (x)) �uEj (t;x; x0) j (x0)For d (x; x0) < 17 , we have C (t;x; x0) = 0, sin e (2.47) implies that (x0) = 0when �ku ('j (x)) 6= 0 and d (x; x0) < 17 . Using this and the estimatesEj (t;x; x0) � At�n=2e�Bd(x;x0)2=t and�uEj (t;x; x0) � At�(n+1)=2e�Bd(x;x0)2=tfor positive onstants A and B, it follows from (2.48) thatjC (t;x; x0)j � 1t�(n+1)=2e�Bd(x;x0)2=t� 1t�(n+1)=2e�B7�2=t = 1e� 2=t; for 0 < t < T0 <1;(2.49)for positive onstants 1 and 2.Let E (t;x; x0) be the exa t heat kernel for D�D and let the orrespondingoperator be E (t) = exp (�tD�D). On the operator level, we haveE (t) �Q (t) = E (t)Q (0)� E (0)Q (t)= Z t0 dds (E (s)Q (t� s)) ds= Z t0 dEdsQ (t� s) + E (s) ddsQ (t� s) ds= Z t0 �D�DE (s)Q (t � s) + E (s) ddsQ (t � s) ds= Z t0 E (s) ��D�D � dd(t�s)�Q (t� s) ds= Z t0 E (s)C (t� s) ds:(2.50)Hen e on the kernel level,E (t;x; x0) = Q (t;x; x0) + Z t0 ZX E (s;x; z)C (t� s; z; x0) dzds= Q (t;x; x0) + (E �C) (t;x; x0) ;(2.51)

70 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKwhere the onvolution operation is de�ned by(� � �) (t;x; x0) := Z t0 ZX � (s;x; z)� (t� s; z; x0) dzds:Note that (2.51) an be written asE (I� �C) = E � E �C = Q:Thus, at least formally,E = Q (I� �C (t))�1 = Q�I+X1k=1 (�C)k�= Q+Q �X1k=1 Ck = Q+ Q � C; where(2.52) C1 := C = � (�t +D�D)Q; Ck+1 := C1 �Ck, and C :=X1k=1 Ck.The proof of the validity of the Levi sum (2.52) for E rests on the estimate (2.49)and we refer to [BoWo93, Ch. 22℄ for the details. The above onstru tions an bealso be applied to obtain a kernel E� (t;x; x0) for exp (�tDD�) starting fromQ� (t;x; x0) := ' (x) E � (t;x; x0) (x0) + 'd (x) Ed� (t;x; x0) d (x0) :Re all that for (u; y) 2 [0;1)� Y ,K (t;u; y) := Tr (E (t; (u; y) ; (u; y))� E � (t; (u; y) ; (u; y))) ; andK (t) := Z 10 ZY K (t;u; y) dydu:(2.53)For x 2 eX , let F (t;x) := Tr (Ed (t;x; x)� Ed� (t;x; x)) :Now, indexD+P� = Tr�e�tD�D � e�tDD�� = Tr (E (t)� E� (t)) :We use the following notation for fun tions whi h are \exponentially lose" ast! 0+ :(2.54) f (t) �exp g (t) , jf (t)� g (t)j � ae�b=t for t 2 (0; ")for some positive onstants a; b; ". We laim (but must omit important details here{ see [BoWo93, Ch. 22℄) thatTr (E (t)� E� (t)) �exp Tr (Q (t)�Q� (t)) :

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 71Sin e E = Q + Q � C (resp., E� = Q� + Q� � C�), this result ultimately rests on(2.49) (resp., the orresponding result for C�). Sin e ' = and 'd d = d,Tr (Q (t)�Q� (t)) = ZX TrQ (t;x; x) dx= ZX Tr� ' (x) (E � E �) (t;x; x) (x)+'d (x) (Ed � E �) (t;x; x) d (x) � dx= Z 10 ZY K (t;u; y) (u) dydu+ ZX F (t;x) d (x) dx:Let Ka (t) := Z a0 ZY K (t;u; y) dydu:Note thatK (t)� Ka (t) = Z 1a ZY K (t;u; y)dydu=X�m� sign (�) 12e2j�ju erf � upt + j�jpt����1u=a= �12X�m� sign (�) e2j�ja erf � apt + j�jpt� :Thus (using (2.42)) for some onstant > 0jK (t) �Ka (t)j �X�m�e2j�ja exp��� apt + j�jpt�2�� 12e�a2=tX�m�e�j�j2t �t!0+ e�a2=tt�n=2;whi h says that K (t) �exp Ka (t). Hen e,indexD+P� = Tr (E (t)� E� (t)) �exp Tr (Q (t)� Q� (t))= Z 10 ZY K (t;u; y) (u) dydu+ ZX F (t;x) d (x) dx�exp �Z 10 ZY K (t;u; y) dydu+ ZX F (t;x) d (x) dx�= K (t) + ZX F (t;x) dx:(2.55)The last equality follows from (2.53) and the fa t that(2.56) F (t;x) = Tr (Ed (t;x; x)� Ed� (t;x; x)) = 0 for x 2 N = [0; 1℄� Y;whi h is seen as follows. Sin e �� = ��,gD+jN = �(�u + B) : C1 �N; fS+�! C1 �N; fS�� impliesgD�jN = (�(�u + B))� = (�u � B)� : C1 �N; fS��! C1 �N; fS+� ;

72 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKThus, over N ,gD�gD+ = (�u � B)��(�u + B) = ���2u + B2� ���C1 �N; fS+� andgD+gD� = �(�u + B)(�u � B)� = �2(�u � B)(�u + B)= ���2u + B2� ���C1 �N; fS�� :Sin e ����2u + B2� ; �� = 0, � maps the eigenspa es of gD�gD+ pointwise isometri- ally (over N ) onto those ofgD+gD�. Hen e, we obtain (2.56) upon taking the tra eof the eigense tion expansion of Ed (t;x; x)� Ed� (t;x; x) for x 2 N . From (2.55),we obtainK (t) = indexD+P� � ZX F (t;x) dx� indexD+P� � ZXXk��n�k (x) tk=2 dx= indexD+P� �Xk��nAktk=2 (for Ak := ZX �k (x) dx);(2.57)where F (t;x) �Pk��n�k (x) tk=2 is the asymptoti expansion of the tra e F (t;x)of the kernel for e�tgD�gD+ � e�tgD+gD� , whi h is known for ellipti di�erential op-erators (e.g., gD�gD+ and gD+gD�) over losed manifolds su h as eX (see [Gi95, p.68℄). Thus, a ording to (2.46) with ak = �Ak for k 6= 0 and a0 = indexD+P��A0,�B (2s) = � 2sp�� �s + 12� �m02s +XNk=�n+1 ak12k + s + �N (s)��B (2s) = � 2sp�� �s + 12� m0 + 2 indexD+P�2s �XNk=�n+1 Ak12k + s + �N (s)! :Setting s = 0, yields �B (0) = ��m0 + 2 indexD+P� � 2A0� orindexD+P� = A0 � 12 (m0 + �B (0)) : �2.5. Symple ti Geometry of Cau hy Data Spa es. As we have seen inour Se tion 1.2.2 there exists a on ept of Cau hy data spa es whi h solely is basedon the on epts of minimal and maximal domain and whi h is more elementaryand more general than the de�nitions provided in Se tion 2.3.5 whi h are basedon pseudo-di�erential al ulus.In this Se tion we stay in the real ategory and do not assume produ t stru -ture near � = �X unless otherwise stated. The operator D need not be of Dira type. We only assume that it is a linear, ellipti , symmetri , di�erential operatorof �rst order.

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 732.5.1. The Natural Cau hy Data Spa e. Let D0 denote the restri tion of Dto the spa e C10 (X;S) of smooth se tions with support in the interior of X. Asmentioned above, there is no natural hoi e of the order of the Sobolev spa esfor the boundary redu tion. Therefore, a systemati treatment of the boundaryredu tion may begin with the minimal losed extension Dmin := D0 and the adjointDmax := (D0)� of D0. Clearly, Dmax is the maximal losed extension. This givesDmin := Dom(Dmin) = C10 (X;S)G = C10 (X;S)H1(X;S)andDmax := Dom(Dmax)= fu 2 L2(X;S) j Du 2 L2(X;S) in the sense of distributionsg:Here, the supers ript G means the losure in the graph norm whi h oin ides withthe �rst Sobolev norm on C10 (X;S). We form the spa e � of natural boundaryvalues with the natural tra e map as des ribed in Se tion 1.2.There we de�ned also the natural Cau hy data spa e �(D) := (KerDmax)under the assumption that D has a self-adjoint L2 extension with a ompa tresolvent. Su h an extension always exists. Take for instan e DP(D), the operatorD with domain DomP(D) := ff 2 H1(X;S) j P(D)( 12 )(f j�) = 0g;where P(D) denotes the Calder�on proje tion de�ned in (2.32).Clearly, Dmax and Dmin are C1(X) modules, and so the spa e � is a C1(�)module. This shows that � is lo al in the following sense: If � de omposes into r onne ted omponents � = �1 t � � � t�r ; then � de omposes into� = rMj=1 �j ;where �j := (ff 2 Dmax j supp f � Njg)with a suitable ollared neighborhood Nj of �j . Note that ea h �j is a losedsymple ti subspa e of � and therefore a symple ti Hilbert spa e.By Theorem 2.20a and, alternatively and in greater generality, by H�ormander[Ho66℄ (Theorem 2.2.1 and the Estimate (2.2.8), p. 194), the spa e � is naturallyembedded in the distribution spa e H� 12 (�;Sj�). Under this embedding we have�(D) = �(D; 0), where the last spa e was de�ned in De�nition 2.22.If the metri s are produ t lose to �, we an give a more pre ise des riptionof the embedding of �, namely as a graded spa e of distributions. Let f'k;�kg bea spe tral resolution of L2(�) by eigense tions of B. (Here and in the following wedo not mention the bundle S). On e again, for simpli ity, we assume KerB = f0gand have B'k = �k'k for all k 2 Zn f0g, and ��k = ��k , �('k) = '�k , and

74 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK�('�k) = �'k for k > 0. In [BoFu99℄, Proposition 7.15 (see also [BrLe99℄ for amore general setting) it was shown that� = �� � �+ with�� := [f'kgk<0℄H 12 (�) and �+ := [f'kgk>0℄H� 12 (�) :(2.58)Then �� and �+ are Lagrangian and transversal subspa es of �.2.5.2. Criss{ ross Redu tion. Let us de�ne two Lagrangian and transversalsubspa es L� of L2(�) in a similar way, namely by the losure in L2(�) of thelinear span of the eigense tions with negative, resp. with positive eigenvalue. ThenL+ is dense in �+ , and �� is dense in L�. This anti{symmetri relation mayexplain some of the well{observed deli a ies of dealing with spe tral invariants of ontinuous families of Dira operators.Moreover, (Daps) = �� , where(2.59) Daps := ff 2 H1(X) j P>(f j�) = 0gdenotes the domain orresponding to the Atiyah{Patodi{Singer boundary on-dition. Note that a series Pk<0 k'k may onverge to an element ' 2 L2(�)without onverging in H 12 (�). Therefore su h ' 2 L� an not appear as tra e atthe boundary of any f 2 Dmax.Re all Proposition 1.27 and note that (��;�(D)) is a Fredholm pair.This an all be a hieved without the symboli al ulus of pseudo-di�erentialoperators. Therefore one may ask how the pre eding approa h to Cau hy dataspa es and boundary value problems via the maximal domain and our symple ti spa e � is related to the approa h via the Calder�on proje tion, whi h we reviewedin the pre eding se tion. How an results from the distributional theory be trans-lated into L2 results?To relate the two approa hes we re all a fairly general symple ti `Criss{Cross'Redu tion Theorem from [BoFuOt01℄ (Theorem 1.2). Let � and L be symple ti Hilbert spa es with symple ti forms !� and !L , respe tively. Let� = �� u �+ and L = L� u L+be dire t sum de ompositions by transversal (not ne essarily orthogonal) pairs ofLagrangian subspa es. We assume that there exist ontinuous, inje tive mappingsi� : �� �! L� and i+ : L+ �! �+with dense images and whi h are ompatible with the symple ti stru tures, i.e.!L(i�(x); a) = !�(x; i+(a)) for all a 2 L+ and x 2 �� :Let � 2 FL��(�), e.g. � = (� \ ��) u � with a suitable losed �. Let us de�ne(see also Figure 9)(2.60) � (�) := i�(� \ ��) + graph('�) ;

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 75Figure 9. The mapping � : FL�� (�)! FLL� (L)where '� : i�1+ (F�) �! L�x 7! i� Æ f� Æ i+(x) :Here F� denotes the image of � under the proje tion �+ from � to �+ along ��and f� : F� �! �� denotes the uniquely determined bounded operator whi hyields � as its graph. Then:Theorem 2.31. The mapping (2.60) de�nes a ontinuous mapping� : FL�� (�) �! FLL� (L)whi h maps the Maslov y le M��(�) of �� into the Maslov y leML�(L) of L�and preserves the Maslov indexmas(f�sgs2[0;1℄; ��) =mas(f� (�s)gs2[0;1℄; L�)for any ontinuous urve [0; 1℄ 3 s 7! �s 2 FL��(�).In the produ t ase, the `Criss{Cross' Redu tion Theorem implies for our twotypes of Cau hy data that all results proved in the theory of natural boundaryvalues (� theory) remain valid in the L2 theory. In parti ular we have:Corollary 2.32. The L2(�) part �(D) \ L2(�) of the natural Cau hy dataspa e �(D) is losed in L2(�). A tually, it is a Lagrangian subspa e of L2(�) andit forms a Fredholm pair with the omponent L� , de�ned at the beginning of thissubse tion.2.6. Non-Additivity of the Index.

76 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK2.6.1. The Bojarski Conje ture. The Bojarski Conje ture gives quite a dif-ferent des ription of the index of an ellipti operator over a losed partitionedmanifoldM =M1 [�M2 . It relates the `quantum' quantity index with a ` lassi- al' quantity, the Fredholm interse tion index of the Cau hy data spa es from bothsides of the hypersurfa e �. It was suggested in [Bo79℄ and proved in [BoWo93℄for operators of Dira type.Proposition 2.33. LetM be a partitioned manifold as before and let �(D+j ; 12)denote the L2 Cau hy data spa es, j = 1; 2 (see De�nition 2.22). ThenindexD+ = index �� �D+1 ; 12� ;� �D+2 ; 12�� :Re all thatindex �� �D+1 ; 12� ;� �D+2 ; 12�� := dim�� �D+1 ; 12� \ � �D+2 ; 12��� dim L2 (�;Sj�)� �D+1 ; 12�+� �D+2 ; 12�! :It is equal to i(I�P(D+2 );P(D+1 )), where P(D+j ) denotes the orresponding Calder�onproje tions, and i (�; �) was de�ned in (2.33).The proof of the Proposition depends on the unique ontinuation propertyfor Dira operators and the Lagrangian property of the Cau hy data spa es, morepre isely the hiral twisting property (2.38) .2.6.2. Generalizations for Global Boundary Conditions. On a smooth om-pa t manifold X with boundary �, the solution spa es Ker(D; s) depend on theorder s of di�erentiability and they are in�nite-dimensional. To obtain a �niteindex one must apply suitable boundary onditions (see [BoWo93℄ for lo al andglobal boundary onditions for operators of Dira type). In this report, we restri tourselves to boundary onditions of Atiyah{Patodi{Singer type (i.e., P 2 Gr(D)),and onsider the extension(2.61) DP : Dom(DP ) �! L2(X;S)of D de�ned by the domain(2.62) Dom(DP ) := ff 2 H1(X;S) j P (0)(f j�) = 0g:It is a losed operator in L2(X;S) with �nite-dimensional kernel and okernel. Wehave an expli it formula for the adjoint operator(2.63) (DP )� = D�(I�P )�� :In agreement with Proposition 1.27b, the pre eding equation shows that an ex-tension DP is self-adjoint, if and only if KerP (0) is a Lagrangian subspa e of thesymple ti Hilbert spa e L2(�;Sj�).Let us re all the Boundary Redu tion Formula for the Index of (global) el-lipti boundary value problems dis ussed in [BoWo85℄ (inspired by [Se69℄, see[BoWo93℄ for a detailed proof for Dira operators). Like the Bojarski Conje ture,the point of the formula is that it gives an expression for the index in terms of the

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 77geometry of the Cau hy data in the symple ti spa e of all (here L2) boundarydata.Proposition 2.34.indexDP = indexnPP (D) : �(D; 12)! range(P (0))o2.6.3. Pasting Formulas. We shall lose this se tion by mentioning a slightmodi� ation of the Bojarski Conje ture/Theorem, namely a non-additivity for-mula for the splitting of the index over partitioned manifolds.>From the Atiyah{Singer Index Theorem (here the expression of the indexon the losed manifold M by an integral of the index density) and the Atiyah{Patodi{Singer Index Theorem (Theorem 2.30), we obtain at on eindexD = index (D1)P� + index (D2)P� � dimKer(B):Then an Agranovi {Dynin type orre tion formula (based on Proposition 2.34)yields:Theorem 2.35. Let Pj be proje tions belonging to Gr(Dj), j = 1; 2. ThenindexD = index (D1)P1 + index (D2)P2 � i (P2; I�P1)It turns out that the boundary orre tion term i(P2; I�P1) equals the indexof the operator �(�t + B) on the ylinder [0; 1℄� � with the boundary onditionsP1 at t = 0 and P2 at t = 1. A dire t proof of Theorem 2.35 an be derived fromProposition 2.34 by elementary operations with the virtual indi es i (P1;P (D1))and i (P2;P (D2)); see, e.g., [DaZh96℄ in a more general setting.Remark 2.36. (a) In this se tion we have not always distinguished betweenthe total and the hiral Dira operator be ause all the dis ussed index formulasare valid in both ases.(b) Important index formulas for (global) ellipti boundary value problems foroperators of Dira type an also be obtained without analyzing the on ept andthe geometry of the Cau hy data spa es (see e.g. the elebrated Atiyah{Patodi{Singer Index Theorem 2.30 or [S 01℄ for a re ent survey of index formulas wherethere is no mention of the Calder�on proje tion). The basi reason is that theindex is an invariant represented by a lo al density inside the manifold plus a orre tion term whi h lives on the boundary and may be lo al or non-lo al aswell. However, these formulas do not explain the simple origin of the index orthe spe tral ow, namely that all index information is naturally oded by thegeometry of the Cau hy data spa es. To us it seems ne essary to use the Calder�onproje tion in order to understand (not al ulate) the index of an ellipti boundaryproblem and the reason for the lo ality or non-lo ality.2.7. Pasting of Spe tral Flow.

78 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK2.7.1. Spe tral Flow and the Maslov Index. Let fDtgt2[0;1℄ be a ontinuousfamily of (from now on always total) Dira operators with the same prin ipalsymbol and the same domainD. To begin with, we do not distinguish between the ase of a losed manifold (when D is just the �rst Sobolev spa e and all operatorsare essentially self-adjoint) and the ase of a manifold with boundary (when D isspe i�ed by the hoi e of a suitable boundary value ondition).We onsider the spe tral ow SF fDt;Dg (see Se tion 1.1.4). We want a pastingformula for the spe tral ow. To a hieve that, one repla es the spe tral ow of a ontinuous one-parameter family of self-adjoint Fredholm operators, whi h is a`quantum' quantity, by the Maslov index of a orresponding path of LagrangianFredholm pairs, whi h is a `quasi- lassi al' quantity. The idea is due to Floer andwas worked out subsequently by Yoshida in dimension 3, by Ni olaes u in all odddimensions, and pushed further by Cappell, Lee and Miller, Daniel and Kirk, andmany other authors. For a survey, see [BoFu98℄, [BoFu99℄, [DaKi99℄, [KiLe00℄.In this review we give two spe tral ow formulas of that type. To begin with,we onsider the ase of a manifold with boundary. Then weak UCP for Dira type operators (established in Se tion 1.4) implies weak inner UCP in the senseof (1.9), if the manifold is onne ted. A tually, it would suÆ e that there is no onne ted omponent without boundary. Let us �x the spa e � for the family.By Proposition 1.27 the orresponding family f�(Dt)g of natural Cau hy dataspa es is ontinuous. Applying the General Boundary Redu tion Formula for thespe tral ow (Theorem 1.28) gives a family version of the Bojarski onje ture (ourProposition 2.33):Theorem 2.37. The spe tral ow of the family fDt;Dg and the Maslov indexmas (f�(Dt)g; (D)) are well-de�ned and we have(2.64) SF fDt;Dg =mas (f�(Dt)g; (D)) :We have various orollaries for the spe tral ow on losed manifolds with �xedhypersurfa e (see [BoFu99℄). Note that a partitioned manifold M = M1 [� M2 an be onsidered as a manifoldM# =M1 tM2 with boundary �M# = ��t�.Here t denotes taking the disjoint union. Then any ellipti operator D over Mde�nes an operator D# over M# with natural Fredholm extension D#D by �xingthe domain D := f(f1; f2) 2 H1(M1) �H1(M2) j (f1j�; f2j�) 2 �g;where � denotes the diagonal of L2 (��;Sj�)�L2 (�;Sj�). By ellipti regularitywe have KerD#D = KerD and SFnD#t;Do = SF fDtg :For produ t stru tures near �, one an apply Theorem 2.31 and obtain an L2version of the pre eding Theorem whi h gives a new proof and a slight generaliza-tion of the Yoshida{Ni olaes u Formula (for details see [BoFuOt01℄, Se tion 3).For safety reasons, we assume that Dt is a zero-order perturbation of D0, indu ed

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 79by a ontinuous hange of the de�ning onne tion, or alternatively, Dt := D0+Ctwhere Ct is a self-adjoint bundle morphism.Theorem 2.38.SFfDtg =mas�n�1t \ L2 (��) �+ �2t \ L2 (�)o ;��=:mas �f�1t \ L2(��)g; f�2t \L2(�)g� ;where the Cau hy data spa es �jt are taken on ea h side j = 1; 2 of the hypersurfa e�. Remark 2.1. Here it is not ompelling to use the symple ti geometry ofthe Cau hy data spa es (see Remark 2.36b). A tually, deep gluing formulas anand have been obtained for the spe tral ow by oding relevant information notin the full in�nite-dimensional Cau hy data spa es but in families of Lagrangiansubspa es of suitable �nite-dimensional symple ti spa es, like the kernel of thetangential operator (see [CaLeMi96℄ and [CaLeMi00℄).2.7.2. Corre tion Formula for the Spe tral Flow. Let D; D0 with Dmin <D;D0 < Dmax be two domains su h that both fDt;Dg and fDt;D0g be ome familiesof self-adjoint Fredholm operators. We assume that D andD0 di�er only by a �nitedimension, more pre isely that(2.65) dim (D) (D) \ (D0) = dim (D0) (D) \ (D0) < +1 :Then we �nd from Theorem 2.37 (for details see [BoFu99℄, Theorem 6.5):sf fDt;Dg � sf fDt;D0g(2.66) =mas (f� (Dt)g ; (D0)) �mas (f� (Dt)g ; (D))= �H�or (�(D0);�(D1); (D0); (D))(see Remark 1.26b). The assumption (2.65) is rather restri tive. The pair of do-mains, for instan e, de�ned by the Atiyah{Patodi{Singer proje tion and the Calder�onproje tion, may not always satisfy this ondition. For the present proof, however,it seems indispensable. 3. The Eta InvariantAs mentioned in the Introdu tion, a systemati fun tional analyti al frame ismissing for the eta invariant in ontrast to the index and the spe tral ow (see,however, [CoMo95℄ for an ambitious approa h to establish analogues of Sobolevspa es, pseudo-di�erential operators, and zeta and eta fun tions in the ontextof non ommutative spe tral geometry). Basi ally, however, the on ept of the etainvariant of a (total and ompatible, hen e self-adjoint) Dira operator is rather animmediate generalization of the index. Instead of measuring the hiral asymmetryof the zero eigenvalues we now measure the asymmetry of the whole spe trum.

80 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKLet D be an operator of Dira type; i.e., roughly speaking, an operator witha real dis rete spe trum whi h is ni ely spa ed without �nite a umulation pointsand with an in�nite number of eigenvalues on both sides of the real line. In loseanalogy with the de�nition of the zeta-fun tion for essentially positive ellipti operators like the Lapla ian, we set�D(s) := X�2spe (D)nf0g sign(�)��s :Clearly, the formal sum �D(s) is well de�ned for omplex s with <(s) suÆ ientlylarge, and it vanishes for a symmetri spe trum (i.e., if for ea h � 2 spe (D) also�� 2 spe (D)).In Subse tions 2.2 and 2.4, we expressed the index by the di�eren e of thetra es of two related heat operators. Similarly,we also have a heat kernel expressionfor the eta fun tion. Let D be any self-adjoint operator with ompa t resolventand let f�kg denote its eigenvalues ordered so that � � � � �k�1 � �k � �k+1 � � � � ,ea h repeated a ording its multipli ity. Formally, we have�D(s) � �(s + 12 ) =X�k 6=0 sign(�k) � j�kj�s � Z 10 r s�12 e�r dr(3.1) =X�k 6=0 �k(�2k)� s+12 Z 10 (t�2k) s�12 e�t�2k d(t�2k)=X�k 6=0 Z 10 t s�12 �ke�t�2k dt = Z 10 t s�12 Tr e�tD2 dt:For omparison we give the orresponding formula for the zeta fun tion of the(positive) Dira Lapla ian D2:(3.2) �D2(s) := Tr(D2)�s = 1�(s) Z 10 ts�1Tr e�tD2 dt:For the zeta fun tion, we must assume that D has no vanishing eigenvalues (i.e. D2is positive). Otherwise the integral on the right side is divergent. (The situation,however, an be ured by subtra ting the orthogonal proje tion onto the kernel ofD2 from the heat operator before taking the tra e.) For the eta fun tion, on the ontrary, it learly does not matter whether there are 0{eigenvalues and whetherthe summation is over all or only over the nonvanishing eigenvalues.The derivation of (3.1) and (3.2) is ompletely elementary for <(s) > 1+dimX2 ,resp. <(s) > dimX2 , where X denotes the underlying manifold. It follows at on ethat �(s) (and �(s)) admit a meromorphi extension to the whole omplex plane.However, it is not lear at all how to hara terize the operators for whi h the etafun tion (and the zeta fun tion) have a �nite value at s = 0, the eta invariant(resp. the zeta invariant).Histori ally, the eta invariant appeared for the �rst time in the 1970s as anerror term showing up in the index formula for the APS spe tral boundary valueproblem of a Dira operator D on a ompa t manifold X with smooth boundary� (see our Subse tion 2.4). More pre isely, what arose was the eta invariant of the

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 81tangential operator (i.e., the indu ed Dira operator over the losed manifold �).Even in that ase it was hard to establish the existen e and �niteness of the etainvariant.Stri tly speaking, one an de�ne the eta invariant as the onstant term in theLaurent expansion of the eta fun tion around the point s = 0. For various appli a-tions this suÆ es. Many pra ti al al ulations, however, are mu h fa ilitated whenwe know the regularity a priori.Basi ally, there are three di�erent approa hes to establish it: the original proofby Atiyah, Patodi, and Singer, worked out in [BoWo93, Corollary 22.9℄ and sum-marized in our Subse tion 2.4; it is based on an assumption about the existen e ofa suitable asymptoti expansion for the orresponding heat kernel on the in�nite ylinder R+��. An intrinsi proof an be found in Gilkey [Gi95, Se tion 3.8℄. Itdoes not exploit that � bounds X, but requires strong topologi al means.For a ompatible (!) Dira operator over a losed manifold Bismut and Freed[BiFr86℄ have shown that the eta fun tion is a tually a holomorphi fun tion of sfor <(s) > �2. They used the heat kernel representation (3.1) whi h implies thatthe eta invariant, when it exists, an be expressed as�D(0) = 1p� Z 10 1pt �Tr�De�tD2� dt:It follows from (3.1) that the estimatejTrDe�tD2 j < ptimplies the regularity of the eta fun tion at s = 0. In fa t, Bismut and Freed proveda sharper result, using nontrivial results from sto hasti analysis. Inspired by al- ulations presented in Bismut and Cheeger [BiCh89, Se tion 3℄, Woj ie howskigave a purely analyti reformulation of the details of their proof. This provides athird and ompletely elementary way of proving the regularity of the eta fun tionat s = 0. The essential steps are:Theorem 3.1. Let D : C1(�;E) ! C1(�;E) denote a ompatible Dira operator over a losed manifold � of odd dimension m. Let e(t;x; x0) denote theintegral kernel of the heat operator e�tD2 . Then there exists a positive onstant Csu h that jTr (Dxe(t;x; x0))jx=x0 j < Cptfor all x 2 � and 0 < t < 1.We re all the de�nition of the `lo al' � fun tion.

82 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKDefinition 3.2. Let ffk;�kgk2Zbe a dis rete spe tral resolution of D. Thenwe de�ne �D(s;x) := X�k 6=0 sign(�k)j�kj�s hfk(x); fk(x)iEx= 1�( s+12 ) Z 10 t s�12 0�X�k 6=0�ke�t�2k hfk(x); fk(x)i1A dt= 1�( s+12 ) Z 10 t s�12 TrD e(t;x; x)dt:Corollary 3.3. Under the assumptions of the pre eding theorem the `lo al'� fun tion �D(s;x) is holomorphi in the half plane <(s) � �2 for any x 2 �.In the de ade or so following 1975, it was generally believed that the existen eof a �nite eta invariant was a very spe ial feature of operators of Dira typeon losed manifolds whi h are boundaries, and then, more generally, of Dira type operators on all losed manifolds. Only after the seminal paper by Douglasand Woj ie howski [DoWo91℄ was it gradually realized that globally ellipti self-adjoint boundary value problems for operators of Dira type also have a �nite etainvariant. On e again, there are quite di�erent approa hes to obtain that result.The work by Woj ie howski and ollaborators is based on the Duhamel Prin- iple and provides omplete asymptoti expansions of the heat kernels for theself-adjoint Fredholm extension DP where P belongs to the smooth self-adjointGrassmannian. We re all from our review of the Atiyah{Patodi{Singer Index The-orem that the Duhamel Prin iple allows one to study the interior ontribution andthe boundary ontribution separately and identify the singularities aused by theboundary ontribution. It seems that Woj ie howski's method is only appli ableif the metri stru tures lose to the boundary are produ t.Based on joint work with Seeley, Grubb [Gr99℄ also obtained ni e asymptoti expansions of the tra e of the heat kernels in (3.1) and (3.2). Contrary to thequalitative arguments of the Duhamel{Woj ie howski approa h, Grubb's approa hrequires the expli it omputation of ertain oeÆ ients in the expansion. Some ofthem have to vanish to guarantee the desired regularity.For a slightly larger lass of self-adjoint Fredholm extensions, Br�uning andLes h [BrLe99a℄ also studied the eta invariant. However, the authors had to dealwith the residue of the eta fun tion at s = 0 whi h is not present in the Duhamel{Woj ie howski approa h.In spite of the di�eren es between the various approa hes and types of resultsit seems that one onsequen e an be drawn immediately, namely there must bea deeper meaning of the eta invariant beyond its role as error term in the APSindex formula. Happily, su h a meaning was found by Singer in [Si85℄ for theeta invariant on losed manifolds and su essively generalized by Woj ie howskiand ollaborators for manifolds with boundary, namely the identi� ation of the

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 83eta invariant as the phase of the zeta fun tion regularized determinant. We shallexplain this now, and postpone our main topi , the pasting formulas for the etainvariant on partitioned manifolds.3.1. Fun tional Integrals and Spe tral Asymmetry. Several importantquantities in quantum me hani s and quantum �eld theory are expressed in termsof quadrati fun tionals and fun tional integrals. The on ept of the determinantfor Dira operators arises naturally when one wants to evaluate the orrespondingpath integrals. As Itzykson and Zuber report in the hapter on Fun tional Methodsof their monograph [ItZu80℄: \The path integral formalism of Feynman and Ka provides a uni�ed view of quantum me hani s, �eld theory, and statisti al models.The original suggestion of an alternative presentation of quantum me hani al am-plitudes in terms of path integrals stems from the work of Dira (1933) and wasbrilliantly elaborated by Feynman in the 1940s. This work was �rst regarded withsome suspi ion due to the diÆ ult mathemati s required to give it a de ent status.In the 1970s it has, however, proved to be the most exible tool in suggesting newdevelopments in �eld theory and therefore deserves a thorough presentation."We shall restri t our dis ussion to the easiest variant of that omplex matterby fo using on the partition fun tion of a quadrati fun tional given by the Eu- lidean a tion of a Dira operator whi h is assumed to be ellipti with imaginarytime due to Wi k rotation and oupled to ontinuously varying ve tor potentials(sour es, �elds, onne tions), for the ease of presentation in va uum. We refer toBertlmann, [Be96℄ and S hwarz, [S h93℄ for an introdu tion to the quantum the-oreti language for mathemati ians and for a more extensive treatment of generalaspe ts of quadrati fun tionals and fun tional integrals involving the relations tothe Lagrangian and Hamiltonian formalism.There are various alternative notions of \path integral" around, some moresophisti ated than others. A mathemati ally rigorous formulation of the on eptof \path integrals", as physi ists typi ally \understand" it in quantum �eld the-ory is imsy at best. For �elds on Minkowski spa e, they are not mathemati alintegrals at all, be ause no measure is de�ned. For �elds on Eu lidean spa e (withimaginary time), one an onstru t genuine measures in limited settings whi h arenot entirely realisti . Even then there is the issue of ontinuing the integrals ba kto real time whi h is generally ill-de�ned on a urved spa e-time. Many physi istsdo not are about su h matters. Indeed, su h physi ists use path integrals primar-ily as ompa t generators of re ipes to produ e Feynman integrals whi h whenregularized and renormalized yield oeÆ ients in a formal power series in oupling onstants for physi al quantities of interest. However, no one has ever proved thatthese series onverge, even for quantum ele trodynami s (QED). Indeed the on-sensus of those who are is that these are only asymptoti series. Adding �rstfew terms of these renormalized perturbation series yields remarkable 11-de imalpoint agreement with experiment, and QED is thereby hailed as a huge su ess.For many mathemati ians and a few physi ists, the great tragedy is that these

84 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKre ipes work so well without a genuine mathemati al foundation. Although pathintegrals may not make pre ise sense per se, not only do they generate su ess-ful re ipes in physi s, but they an motivate fruitful ideas and pre ise on epts.In mathemati s, path integrals have motivated the �-regularized determinant forthe (Eu lidean) Dira operator, as a mathemati ally genuine anoni al obje t, in-dependent of parti ular hoi es made for regularization, whi h an be pre isely al ulated in prin iple.A spe ial feature of Dira operators is that their determinants involve a phase,the imaginary part of the determinant's logarithm.As we will see now, this is a on-sequen e of the fa t that, unlike se ond-order semi-bounded Lapla ians, �rst-orderDira operators have an in�nite number of both positive and negative eigenval-ues. Then the phase of the determinant re e ts the spe tral asymmetry of the orresponding Dira operator.The simplest path integral we meet in quantum �eld theory takes the form ofthe partition fun tion and an be written formally as the integral(3.3) Z(�) := Z� e��S(!) d! ;where d! denotes fun tional integration over the spa e � := �(M ;E) of se tionsof a Eu lidean ve tor bundle E over a Riemannian manifoldM .In quantum theoreti language,M is spa e or spa e-time; a ! 2 � is a positionfun tion of a parti le or a spinor �eld. The s aling parameter � is a real or omplexparameter, most often � = 1. The fun tional S is a quadrati real-valued fun tionalon � de�ned by S(!) := h!; T!i with a �xed linear symmetri operator T : �! �.Typi ally T := D is a Dira operator and S(!) is the a tion S(!) = RM h!;D!i.Mathemati ally speaking, the integral (3.3) is an os illating integral like theGaussian integral. It is ill-de�ned in general be ause(I) as it stands, it is meaningless when dim�(M ;E) = +1 (i.e., whendimM � 1); and,(II) even when dim�(M ;E) <1 (i.e. when dimM = 0 and M onsists of a�nite set of points), the integral Z(�) diverges unless �S(!) is positiveand nondegenerate.Nevertheless, these expressions have been used and onstrued in quantum �eldtheory. As a matter of fa t, re onsidering the physi ists' use and interpretation ofthese mathemati ally ill-de�ned quantities, one an des ribe ertain formal ma-nipulations whi h lead to normalizing and evaluating Z(�) in a mathemati allypre ise way.We begin with a few al ulations in Case II, inspired by Adams and Sen,[AdSe96℄, to show how spe tral asymmetry is naturally entering into the al ula-tions even in the �nite-dimensional ase and how this suggests a de�nition of thedeterminant in the in�nite-dimensional ase for the Dira operator.

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 85Then, let dim� = d <1 and, for a symmetri endomorphism T , letS(!) := h!; T!i for all ! 2 �:Case 1. We assume that S positive and nondegenerate, i.e. T is stri tly positive,say spe T = f�1; : : :�dg with all �j > 0. This is the lassi al ase. We hoosean orthonormal system of eigenve tors (e1; : : : ; ed) of T as basis for �. We haveS(!) =P�jx2j for ! =Pxjej and, for real � > 0; we getZ(�) = ZG e��S(!) d! = ZRd dx1 : : : dxd e��P�jx2j= Z 1�1 e���1x21dx1 Z 1�1 e���2x22dx2 : : :Z 1�1 e���dx2ddxd=r ���1 r ���2 : : :r ���d = �d=2 � ��d=2 � (det T )�1=2 :In that way the determinant appears when evaluating the simplest quadrati in-tegral.Case 2. If the fun tional S is positive and degenerate, T � 0, the partitionfun tion is given byZ(�) = ��=2 � ���=2 � (det !T )�1=2 �Vol(Ker T );where � := dim��dimKer T and eT := T j(KerT )?, but, of ourse Vol(Ker T ) =1.For approa hes to \renormalizing" this quantity in quantum hromodynami s, werefer to [AdSe96℄, [BMSW97℄, [S h93℄. One approa h ustomary in physi s isto take ��=2���=2(det eT )�1=2 as the de�nition of the integral by setting the fa torVol(Ker T ) equal to 1.Case 3. Now we assume that the fun tional S is nondegenerate, i.e. T invert-ible, but S is neither positive nor negative. We de ompose � = �+ � �� andT = T+ � T� with T+;�T� stri tly positive on ��. Formally, we obtainZ(�) = Z�+ d!+e��h!+;T+!i! Z�� d!�e�(��)h!�;�T�!i!= �d+=2��d+=2(det T+)�1=2 �d�=2(��)�d�=2(det�T�)�1=2= ��=2��d+=2(��)�d�=2(det jT j)�1=2where d� := dim��, hen e � = d+ + d� and jT j :=peT 2 = T+ � �T�.Case 4. In the pre eding formula, the term (�)�d+=2(��)�d�=2 is unde�nedfor � 2 R�. We shall repla e it by a more intelligible term for � = 1 by �rstexpanding Z(�) in the upper omplex half plane and then formally setting � = 1.More pre isely, let � 2 C+ = fz 2 C j =z > 0g and write � = j�jei� with � 2 [0; �℄,hen e �� = j�jei(���) with � � � 2 [��; 0℄. We set �a := j�jaei�a and get��d+=2(��)�d�=2 = (j�jei�)�d+=2(j�jei(���))�d�=2= j�j��=2 e�i d+2 � e�i d�2 � ei� d�2 :

86 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKMoreover,�d+2 � � d�2 � + �d�2 = ��2 (d+ + d�) + �2 �d�2 + d+2 + d�2 � d+2 �= ��2� + �4 (� � �) = ��4 �2��� + (� � �)� ;where � := d+ + d� is the �nite-dimensional equivalent of the � invariant, ount-ing the eigenvalues, and � := d+ � d� the �nite-dimensional equivalent of the �invariant, measuring the spe tral asymmetry of T . We obtainZ (�) = ��=2 j�j��=2 e�i�4 ( 2��� +(���)) (det jT j)�1=2and, formally, for � = 1, i.e. � = 0,(3.4) Z(1) = ��=2 e�i�4 (���) (det jT j)�1=2| {z }=:detT :Equation (3.4) suggests a nonstandard de�nition of the determinant for the in�nite-dimensional ase.Remark 3.4. (a) The methods and results of this se tion also apply to real-valued quadrati fun tionals on omplex ve tor spa es. Sin e the integration in(3.3) in this ase is over the real ve tor spa e underlying �, the expressions for thepartition fun tions in this ase be ome the square of those above.(b) In the pre eding al ulations we worked with ordinary ommuting numbers andfun tions. The resulting Gaussian integrals are also alled bosoni integrals. If we onsider fermioni integrals, we work with Grassmannian variables and obtain thedeterminant not in the denominator but in the nominator (see e.g. [BeGeVe92℄or [Be96℄).3.2. The �{Determinant for Operators of In�nite Rank. On e again,our point of departure is �nite-dimensional linear algebra. Let T : C d ! C d be aninvertible, positive operator with eigenvalues 0 < �1 � �2 � ::: � �d . We havethe equality detT =Y�j = expfX ln�je�s ln�j js=0g= exp(� dds (X��sj )js=0) = e� dds �T (s)js=0 ;where �T (s) :=Pdj=1 ��sj .We show that the pre eding formula generalizes naturally, when T is repla edby a positive de�nite self-adjoint ellipti operator L (for the ease of presentation,of se ond order, like the Lapla ian) a ting on se tions of a Hermitian ve tor bundleover a losed manifoldM of dimensionm. Then L has a dis rete spe trum spe L =f�jgj2Nwith 0 < �1 � �2 � : : : , satisfying the asymptoti formula �n � Cnm=2

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 87for a onstant C > 0 depending on L (see e.g. [Gi95℄, Lemma 1.6.3). We extend�L(s) :=P1j=1 ��sj in the omplex plane by�L(s) := 1�(s) Z 10 ts�1 Tr e�tL dtwith �(s) := R10 ts�1e�tdt. Note that e�tL is the heat operator transforming anyinitial se tion f0 into a se tion ft satisfying the heat equation ��tf + Lf = 0.Clearly Tr e�tL =P e�t�j .One shows that the original de�nition of �L(s) yields a holomorphi fun -tion for <(s) large and that its pre eding extension is meromorphi in the entire omplex plane with simple poles only. The point s = 0 is a regular point and�L(s) is a holomorphi fun tion at s = 0. From the asymptoti expansion of�(s) � 1s + + sh(s) lose to s = 0 with the Euler number and a suitableholomorphi fun tion h we obtain an expli it formula�0L(0) � Z 10 1t Tr e�tL dt� �L(0) :This is explained in great detail, e.g., in [Wo99℄. Therefore, Ray and Singer in[RaSi71℄ ould introdu e detz(L) by de�ning:detzL := e� dds �L(s)js=0 = e��0L(0) :The pre eding de�nition does not apply immediately to the main hero here, theDira operator D whi h has in�nitely many positive �j and negative eigenvalues��j . Clearly by the pre eding argumentdetzD2 = e��0D2 and detzjDj = e��0jDj = e� 12 �0D2 :For the Dira operator we setlndetD := � dds�D(s)js=0with, hoosing 2 the bran h (�1)�s = ei�s,�D(s) =X��sj +X(�1)�s��sj =X��sj + ei�sX��sj= P��sj +P��sj2 + P��sj �P��sj2+ ei�s(P��sj +P��sj2 � P��sj �P��sj2 )= 12 n�D2( s2) + �D(s)o + 12ei�s n�D2( s2)� �D(s)o ;2Choosing the alternative representation, namely (�1)�s = e�i�s yields the opposite signof the phase of the determinant whi h may appear to be more natural for some quantum �eldmodels and also in view of (3.4). However, we follow the more ommon onvention introdu edby Singer in [Si85, p. 331℄ when de�ning the determinant of operators of Dira type.

88 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKwhere �D(s) :=P��sj �P��sj . Later we will show that �D(s) is a holomorphi fun tion of s for <(s) large with a meromorphi extension to the whole omplexplane whi h is holomorphi in the neighborhood of s = 0 . We obtain� 0D(s) = 14� 0D2( s2 ) + 12�0D(s) + 12 i�ei�sf�D2( s2)� �D(s)g+ 12ei�sf12�0D2( s2 )� �0D(s)g:It follows that �0D(0) = 12�0D2(0) + i�2 f�D2 (0)� �D(0)gand detzD = e� 12 �0D2 (0) e� i�2 f�D2 (0)��D(0)g = e� i�2 f�jDj(0)��D(0)g e��0jDj(0)= e� i�2 f�jDj(0)��D(0)g detzjDj :So, the Dira operator's `partition fun tion' in the sense of (3.3) be omesZ(1) = ��jDj(0)(det�D)� 12 :3.3. Spe tral Invariants of Di�erent `Sensitivity'. In the pre eding for-mulas four spe tral invariants of the Dira operator D enter:3.3.1. The Index. First re all that the index of arbitrary ellipti operators on losed manifolds and the spe tral ow of 1-parameter families of self-adjoint ellip-ti operators are topologi al invariants and so stable under small variation of the oeÆ ients and, by de�nition, solely depending on the multipli ity of the eigen-value 0. In the theory of bounded or losed (not ne essarily bounded) Fredholmoperators and bounded or not ne essarily bounded self-adjoint Fredholm opera-tors and the related K and K�1 theory, we have a powerful fun tional analyti aland topologi al frame for dis ussing these invariants. Moreover, index and spe tral ow are lo al invariants, i.e., an be expressed by an integral where the integrandis lo ally expressed by the oeÆ ients of the operator(s). Consequently, we havesimple, pre ise pasting formulas for index and spe tral ow on partitioned mani-folds where the error term is lo alized along the separating hypersurfa e.On manifolds with boundary the Calder�on proje tion and its range, the Cau hydata spa e, hange ontinuously when we vary the Dira operator as shown in Se -tion 2.3.8, exploiting the unique ontinuation property of operators of Dira type.The same is not true for the Atiyah{Patodi{Singer proje tion: it an jump fromone onne ted omponent of the Grassmannian to another omponent under small hanges of the Dira operator. Correspondingly, the index of the APS boundaryproblem an jump under small variation of the oeÆ ients (i.e., of the de�ning onne tion or the underlying Riemannian or Cli�ord stru ture).Regarding parity of the manifold, the index density (of all ellipti di�erential op-erators) vanishes on odd-dimensional manifolds for symmetry reason. Then, in the losed ase the index vanishes, and on manifolds with boundary the APS IndexTheorem takes the simple form indexDP� = � dimKerB+ where we have thetotal Dira operator on the left and a hiral omponent of the indu ed tangential

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 89operator on the right. On e again, the formula shows the instability of the indexunder small hanges of the Dira operator. This is no ontradi tion to the stabilityof the index on the spa es F , respe tively CF , dis ussed in Se tion 1.1.3 be ausethe graph norm distan e between two APS realizations DP� and D0P� an remainbounded away from zero when D runs to D0. This is the ase if and only if thedimension of the kernel of the tangential operator hanges under the deformation.3.3.2. The �{invariant. Similarly, �D2(0) is also lo al; i.e., it is given by theintegral RM �(x)dx, where �(x) denotes a ertain oeÆ ient in the heat kernelexpansion and is lo ally expressed by the oeÆ ients of D. In parti ular, �D2(0)remains un hanged for small hanges of the spe trum. A tually, �L(0) vanisheswhen L is the square of a self-adjoint ellipti operator on a losed manifold of odddimension. It an be de�ned (and it vanishes, see [PaWo02a, Appendix℄) for alarge lass of squares of operators of Dira type with globally ellipti boundary onditions on ompa t, smooth manifolds (of odd dimension) with boundary. So,there are no nontrivial pasting formulas at all in su h ases.3.3.3. The �{invariant. Unlike the index and spe tral ow on losed mani-folds, we have neither an established fun tional analyti al nor a topologi al framefor dis ussing �D(0), nor is it given by an integral of a lo ally de�ned expression.On the ir le, e.g., onsider the operator(3.5) Da := �i ddx + a = e�ixaD0eixa;so that Da and D0 have the same total symbol (i.e., oin ide lo ally), but �Da(0) =�2a depends on a.The �{invariant depends, however, only on �nitely many terms of the symbol ofthe resolvent (D��)�1 and the real part (inR=Z)will not hange when one hangesor removes a �nite number of eigenvalues. The integer part hanges a ording tothe net sign hange o urring under removing or modifying eigenvalues. Moreover,the �rst derivative of the eta invariant of a smooth 1-parameter family of Dira type operators is lo al, namely the spe tral ow, as noted in our Introdu tion.This leads again to pre ise (though not so simple) pasting formulas for the etainvariant on partitioned manifolds.In even dimensions, the eta invariant vanishes on any losed manifold � for anyDira type operator whi h is the tangential operator of a Dira type operator ona suitable manifold whi h has � as its boundary be ause of the indu ed pre isesymmetry of the spe trum due to the anti- ommutativity of the tangential operatorwith Cli�ord multipli ation. For the study of eta of boundary value problems oneven-dimensional manifolds see [KlWo96℄.3.3.4. The Modulus of the Determinant. The number � 0D2(0) is the most del-i ate of the invariants involved: It is neither a lo al invariant, nor does it dependonly on the total symbol of the Dira operator. Even small hanges of the eigen-values will hange the � 0 invariant and hen e the determinant. Moreover, no pre- ise pasting formulas are obtained but only adiabati ones (i.e., by inserting a

90 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKlong ylinder around the separating hypersurfa e (see [PaWo02a℄, [PaWo02b℄,[PaWo02 ℄).Without proof we present the main results by Woj ie howski and ollabora-tors, based on [Wo99℄ where the �{fun tion regularized determinant was estab-lished for pseudo-di�erential boundary value onditions belonging to the smooth,self-adjoint Grassmannian. The �rst is a boundary orre tion formula, proved in[S Wo00℄ (see also the re ent [S o02℄):Theorem 3.5. (S ott, Woj ie howski). Let D be a Dira operator over an odd{dimensional ompa t manifold M with boundary � and let P 2 Grsa(D) . Then therange of the Calder�on proje tion P(D) (the Cau hy data spa e �(D; 12)) and therange of P an be written as the graphs of unitary, ellipti operators of order0, K, resp. T whi h di�er from the operator (B+B�)�1=2B+ : C1(�;S+j�) !C1(�;S�j�) by a smoothing operator. Moreover,(3.6) det�DP = det�DP(D) � detFr 12(I+KT�1) :The se ond result, in most simple form, is found in [PaWo00℄:Theorem 3.6. (Park, Woj ie howski) Let R 2 R be positive, let MR denotethe stret hed partitioned manifold MR =M1 [� [�R; 0℄�� [� [0; R℄�� [� M2 ,and let DR, D1;R, D2;R denote the orresponding Dira operators. We assume thatthe tangential operator B is invertible. ThenlimR!1 det�D 2R�det�(D1;R) 2I�P>� � �det�(D2;R) 2P>� = 2��B2 (0) :Although Felix Klein in [Kl27℄ rated the determinant as simplest exampleof an invariant, today we must give an inverse rating. For the present authors,it is not the invariants that are stable under the largest transformation groupswhi h deserve the highest interest, but rather (a ording to Dira 's approa h toelementary parti le physi s) the �nest invariants whi h exhibit anomalies undersmall perturbations. Correspondingly, the determinant and its amplitude are themost subtle and the most fas inating obje ts of our study. They are mu h morediÆ ult to omprehend than the �-invariant 3.3.2; and the �-invariant is mu hmore diÆ ult to omprehend than the index.3.4. Pasting Formulas for the Eta{Invariant - Outlines. In the rest ofthis review, i.e. over the next 33 pages, we shall prove a strikingly simple (to state)additivity property of the �-invariant. We �x the assumptions and the notation.3.4.1. Assumptions and Notation. (a) Let M be an odd{dimensional losedpartitioned Riemannian manifoldM =M1[�M2 withM1;M2 ompa t manifoldswith ommon boundary �. Let S be a bundle of Cli�ord modules over M .(b) To begin with we assume that D is a ompatible (= true) Dira operator overM . Thus, in parti ular, D is symmetri and has a unique self-adjoint extension inL2(M ;S).

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 91( ) We assume that there exists a bi ollared ylindri al neighborhood (a ne k)N ' (�1; 1) � � of the separating hypersurfa e �; su h that the Riemannianstru ture on M and the Hermitian stru ture on S are produ t in N ; i.e., they donot depend on the normal oordinate u, when restri ted to �u = fug � �. Our onvention for the orientation of the oordinate u is that it runs fromM1 to M2;i.e., M1 \N = (�1; 0℄� � and N \M2 = [0; 1)� �. Then the operator D takesthe following form on N :(3.7) DjN = �(�u +B);where the prin ipal symbol in u{dire tion � : Sj� ! Sj� is a unitary bundleisomorphism (Cli�ord multipli ation by the normal ve tor du) and the tangentialoperator B : C1(�;Sj�) ! C1(�;Sj�) is the orresponding Dira operator on�. Note that � and B do not depend on the normal oordinate u in N and theysatisfy the following identities(3.8) �2 = � I ; �� = �� ; � �B = �B � � ; B� = B:Hen e, � is a skew-adjoint involution and S, the bundle of spinors, de omposesin N into �i{eigenspa es of �, SjN = S+ � S�. It follows that (3.7) leads to thefollowing representation of the operator D in NDjN = �i 00 �i� ���u +� 0 B� = B�+B+ 0 �� ;where B+ : C1(�;S+) ! C1(�;S�) maps the spinors of positive hirality intothe spinors of negative hirality.(d) To begin with we onsider only the ase of KerB = f0g. That implies that Bis an invertible operator. More pre isely, there exists a pseudo-di�erential ellipti operator L of order �1 su h that BL = IS = LB (see, for instan e, [BoWo93℄,Proposition 9.5).(e) For real R > 0 we study the losed stret hed manifold MR whi h we obtainfrom M by inserting a ylinder of length 2R, i.e. repla ing the ollar N by the ylinder (�2R� 1;+1)� �MR =M1 [ ([�2R; 0℄� �) [M2 :We extend the bundle S to the stret hed manifold MR in a natural way. Theextended bundle will be also denoted by S. The Riemannian stru ture on M andthe Hermitian stru ture on S are produ t on N . Hen e we an extend them tosmooth metri s onMR in a natural way and, at the end, we an extend the oper-ator D to an operator DR onMR by using formula (3.7). ThenMR splits into twomanifolds with boundary: MR = MR1 [MR2 with MR1 = M1 [ ((�2R;�R℄� �),MR2 = ([�R; 0)� �) [M2, and �M1 = �MR2 = f�Rg � �. Consequently, theoperator DR splits into DR = DR1 [ DR2 . We shall impose spe tral boundary on-ditions to obtain self-adjoint operators D1;P< , DR1;P< , D2;P> , and DR2;P> in the orresponding L2 spa es on the parts (see (3.9)).

92 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK(f) We also introdu e the omplete, non ompa t Riemannian manifold with ylin-dri al end M12 := ((�1; 0℄� �) [M2by gluing the half{ ylinder (�1; 0℄ � � to the boundary � of M2. Clearly, theDira operator D extends also to C1(M12 ; S).Remark 3.7. Our presentation is somewhat simpli�ed by our assumption (b)that D is ompatible and assumption (d) that the tangential operator B is invert-ible. Both assumptions an be lifted. This is done in the literature; see [Wo95℄and [Wo99℄.We re all the following ideas in the big s heme from Se tion 2.5 of this review.Let P> (respe tively P<) denote the spe tral proje tion of B onto the subspa e ofL2(�;Sj�) spanned by the eigense tions orresponding to the positive (respe tivelynegative) eigenvalues. Then P> is a self-adjoint ellipti boundary ondition for theoperator D2 = DjM2 (see [BoWo93℄, Proposition 20.3). This means that theoperator D2;P> de�ned by(3.9) D2;P> = DjM2Dom(D2;P> ) = fs 2 H1(M2;SjM2) j P>(sj�) = 0gis an unbounded self-adjoint operator in L2(M2;SjM2) with ompa t resolvent. Inparti ular, D2;P> : Dom(D2;P>)! L2(M2;SjM2)is a Fredholm operator with dis rete real spe trum and the kernel of D2;P> onsistsof smooth se tions of SjM2 .As mentioned before, the eta fun tion of D2;P> is well de�ned and enjoys allproperties of the eta fun tion of the Dira operator de�ned on a losed manifold.In parti ular, �D2;P> (0), the eta invariant of D2;P> , is well de�ned. Similarly, P<is a self-adjoint boundary ondition for the operator DjM1, and we de�ne theoperator D1;P< using a formula orresponding to (3.9). To keep tra k of the variousmanifolds, operators, and integral kernels we refer to the following table where wehave olle ted the major notations.manifolds operators integral kernelsM =M1 [�M2 e�tD2; De�tD2 E(t;x; x0)MR =MR1 [�MR2 e�t(DR)2 ; DRe�t(DR)2 ER(t;x; x0)M2 e�tD 22;P> ; D2e�tD 22;P> E2(t;x; x0)MR2 = ([�R; 0℄� �) [M2 e�t(DR2;P> )2 ; DR2 e�t(DR2;P> )2 ER2 (t;x; x0)M12 = ((�1; 0℄� �) [M2 e�t(D12 )2 ; D12 e�t(D12 )2 E12 (t;x; x0)�1 yl = (�1;+1)�� e�tD2 yl ; D yle�tD2 yl E yl(t;x; x0)�1 yl =2 = [0;+1)� � e�tD2aps ; Dapse�tD2aps Eaps(t;x; x0)

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 93In addition, on MR2 we have the operator QR2 (t) with kernel QR2 (t;x; x0) andCR(t) = �(DR2;P>)2 + ddt�QR2 (t) with kernel CR(t;x; x0).3.4.2. The Gluing Formulas. The most basi results for pasting � are thefollowing theorem on the adiabati limits of the � invariants and its additivity orollary:Theorem 3.8. Atta hing a ylinder of length R > 0 at the boundary of themanifold M2, we an approximate the eta invariant of the spe tral boundary on-dition on the prolonged manifold MR2 by the orresponding integral of the `lo al'eta fun tion of the losed stret hed manifold MR:limR!1(�DR2;P> (0)� ZMR2 �DR(0;x) dx) � 0modZ:Corollary 3.9. �D(0) � �D1;P< (0) + �D2;P> (0)modZ:Remark 3.10. (a)With hindsight, it is not surprising that modulo the integersthe pre eding additivity formula for the �-invariant on a partitioned manifold ispre ise. An intuitive argument runs as follows. \Almost all" eigense tions andeigenvalues of the operator D on the losed partitioned manifold M = M1 [M2 an be tra ed ba k, either to eigense tions 1;k and eigenvalues �1;k of the spe tralboundary problem D1;P< on the part M1; or to eigense tions 2;` and eigenvalues�2;` of the spe tral boundary problem D2;P> on the part M2. While we have noexpli it exa t orresponden e, due to the produ t form of the Dira operator in aneighborhood of the separating hypersurfa e, eigense tions on one part M1 orM2of the manifoldM an be extended to smooth se tions on the whole of M . Theseare not true eigense tions of D, but they have a relative error whi h is rapidlyde reasing as R!1 when we atta h ylinders of length R to the part manifoldsor, equivalently, insert a ylinder of length 2R in M . There is also a residualset f�0;jg of eigenvalues of D whi h an neither be tra ed ba k to eigenvalues ofD1;P< nor to those of D2;P> . These eigenvalues an, however, be tra ed ba k tothe kernel of the Dira operators D11 and D12 on the part manifoldsM11 andM12with ylindri al ends. Be ause of Fredholm properties the residual set is �nite and,hen e (as noti ed in Se tion 3.3) an be dis arded for al ulating the eta invariantmoduloZ.Therefore, no R (i.e., no prolongation of the bi ollar neighborhood N ) enters theformula. Nevertheless, our arguments rely on an adiabati argument to separatethe spe trum of D into its three parts(3.10) spe D � f�0;jg [ f�1;kg [ f�2;`g:For the most part, however, we need not make all arguments expli it on the levelof the single eigenvalue. It suÆ es to work on the level of the eta invariant for thefollowing reason. Unlike the index, the eta invariant annot be des ribed by a lo alformula, as explained in Se tion 3.3. Nevertheless, it an be des ribed by an integral

94 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKover the manifold. The integrand, however, is not de�ned in lo al terms solely. Inparti ular, when writing the eta fun tion in integral form and de omposing the �integral�D(s) = ZM �D(s;x; x) dx = ZM1 �D(s;x1; x1) dx1 + ZM2 �D(s;x2; x2) dx2there is no geometri al interpretation of the integrals on the right over the twoparts of the manifold. This is very unfortunate. But for suÆ iently large R, theintegrals be ome intelligible and an be read as the � invariants of DR1;P< andDR2;P> . That is the meaning of the adiabati limit.(b) Theorem 3.40 an be generalized to larger lasses of boundary onditions byvariational argument (see [LeWo96℄) yielding the general gluing formula (in R=Z)�D(0) = �D1;P1 (0) + �D2;P2 (0) + �NP1;I�P2 (0);where �NP1;I�P2(0) denotes the �{invariant on the ylinder N , see also [BrLe99℄,[DaFr94℄, [MaMe95℄, and the re ent review [PaWo02e℄. G. Grubb's new re-sult of [Gr02℄, mentioned in our Introdu tion, may open an alternative route forproving the general gluing formula. The integer jump was al ulated in [KiLe00℄yielding (among other formulas)�D(0) = �D1;I�P (0) + �D2;P (0) + 2 SFfD1;I�Ptg+ 2SFfD2;Ptg;where fPtg is a smooth urve in the Grassmannian from P to the Calder�on pro-je tion P(D2).( ) An interesting feature of [KiLe00℄ is that the pasting formula is derived fromthe S ott{Woj ie howski Comparison Formula (Theorem 3.5). This is quite anal-ogous to the existen e of two ompletely di�erent proofs of the pasting formula forthe index (Theorem 2.35), where also the derivation from the boundary redu tionformula is mu h, mu h shorter than arguing via the Atiyah-Singer Index Theoremand the Atiyah-Patodi-Singer Index Theorem plus the Agranovi -Dynin Formula.However, for this review we prefer the long way be ause of the many interestinginsights about the gluing on the eigenvalue level whi h an be gained.3.4.3. Plan of the Proof. Let ER2 (t) denote the integral kernel of the operatorDR2 e�t(DR2;P> )2 de�ned on the manifoldMR2 = ([�R; 0℄��)[M2. Without proof,we mentioned before that the eta invariant of the self-adjoint operator DR2;P> iswell de�ned and we have�DR2;P> (0) = 1p� Z 10 dtpt ZMR2 TrER2 (t;x; x) dx= 1p� Z pR0 dtpt ZMR2 Tr ER2 (t;x; x) dx(3.11) + 1p� Z 1pR dtpt ZMR2 Tr ER2 (t;x; x) dx:(3.12)

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 95We �rst deal with the integral of (3.11) and show that it splits into an interior on-tribution and a ylinder ontribution as R!1. This will be done in Subse tion3.5 by �rst in paragraph 3.5.1 applying the Duhamel method whi h we intro-du ed before in the proof of Theorem 2.30 (pp. 63�). By Lemma 3.11, Lemma3.12, Proposition 3.13, and Corollary 3.14, we an repla e the heat kernel ER2 ofthe Atiyah-Patodi-Singer boundary problem on the prolonged manifoldMR2 withboundary � by an arti� ially glued integral kernel QR2 whi h, near the bound-ary, is equal to the heat kernel of the APS problem on the half-in�nite ylinderand in the interior equal to the heat kernel on the stret hed losed manifold. We an do it in su h a way that the original small-t integral (i.e., the integral from0 to pR in (3.11)) an be approximated by the new integral up to an error oforder O(e� R). Then we shall show in Lemma 3.15 and Lemma 3.16 (paragraph3.5.2) that the ylinder ontribution is tra eless. More pre isely, we obtain thatthe tra e TrQR2 (T ;x; x) an be repla ed pointwise (for x 2 MR2 ) by the tra eTr ER(t;x; x) of the integral kernel of the operator DRe�t(DR)2 whi h is de�nedon the stret hed manifold MR. Consequently, the small-t hopped �-invariant ofthe losed stret hed manifoldMR oin ides with the sum of the small-t hopped�-invariants of the APS problems on the two prolonged manifolds MR1 and MR2up to O �e� R�.Then we will show that the integral of (3.12) vanishes as R ! 1. This isin Lemma 3.28 (p. 114f) a dire t onsequen e of Theorem 3.17 whi h states thatthe eigenvalues of DR2;P> are uniformly bounded away from 0. Theorem 3.17 isof independent interest. The proof is a long story stret hing over Subse tions3.6 and 3.7. First, in paragraph 3.6.1 we shall onsider the operator D yl on thein�nite ylinder �1 yl in De�nition 3.19 and Lemma 3.20. We obtain that D ylhas no eigenvalues in the interval (��1; �1) where �1 denotes the smallest positiveeigenvalue of the tangential operator B on the manifold�. Next, in paragraph 3.6.2we investigate the operator D12 on the manifoldM12 with in�nite ylindri al endin Lemma 3.21, Lemma 3.22, Lemma 3.23, and Proposition 3.24. Proposition 3.24states that D12 has only �nitely many eigenvalues in the aforementioned interval(��1; �1), ea h of �nite multipli ity. Its proof is somewhat deli ate and involvesLemma 3.25 and Corollary 3.26. Then, in Subse tion 3.7 the proof of Theorem3.17 follows with Lemma 3.27.By then we will have that the sum of the �-invariants of the APS boundaryproblems on the prolonged manifoldsMR1 and MR2 an be repla ed by the small-t hopped �-invariant on the stret hed losed manifoldMR with an error exponen-tially vanishing as R ! 1. We then would like to repeat the pre eding hain ofarguments around Theorem 3.17 to show that the large-t hopped �-invariant ofthe operator DR onMR also vanishes. However, this an be done only in R=Zbe- ause in general the eigenvalues of DR are not bounded away from 0. So we shallpresent a di�erent hain of arguments in Subse tion 3.8. The main te hni al resultis Theorem 3.29, on e again of independent interest. It des ribes a partition ofthe eigenvalues into two subsets, the exponentially small ones and the eigenvalues

96 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKbounded away from 0. The key for our arguments is a gluing onstru tion (De�ni-tion 3.30). Then we �rst show by Lemma 3.31, Lemma 3.32 and Proposition 3.33that DR has at least q := dimKerD11 + dimKerD12 exponentially small eigen-values belonging to eigense tions whi h we an approximate by pasting togetherL2 solutions. Then we shall show in Lemma 3.34, Theorem 3.35, Lemma 3.36 andProposition 3.37 that this makes the list of eigenvalues approa hing 0 as R !1 omplete.It follows in Lemma 3.38 (at the beginning of the losing Subse tion 3.7) thatthe large-t hopped �-invariant on MR vanishes asymptoti ally up to an integererror. This establishes Theorem 3.8. To arrive at Corollary 3.9, we must get rid ofthe adiabati limit. This is a simple onsequen e of the lo ality of the derivative inR-dire tion of the �-invariants on the stret hed part manifoldswith APS boundary ondition and on the stret hed losed manifold (Proposition 3.39).3.5. The Adiabati Additivity of the Small-t Chopped �-invariant.3.5.1. Applying Duhamel's Method to the Small-t Chopped �-invariant. Thesimplest onstru tion of a parametrix for ER2 (t) (i.e., of an approximate heat kernel)is the following: we glue the kernel E of the operator De�tD2 (given on the whole, losed manifoldM ) and the kernel E1aps of the L2 extension of the operator �(�u+B)e�t(�(�u+B))2 , given on the semi-in�nite ylinder [�R;1)�� and subje t to theAtiyah{Patodi{Singer boundary ondition at the end u = �R. In that onstru tionthe gluing happens on the ne k N = [0; 1)�� with suitable uto� fun tions.Lo ally, the heat kernel is always of the form (4�t)�m=2e 1te�jx�x0j2=4t . ByDuhamel's Prin iple we get after gluing a similar global result for the kerneleR2 (t;x; x0) of the operator e�t(DR2;P> )2 and, putting a fa tor t�1=2 in front, forthe kernel of the ombined operator De�t(DR2;P> )2 (e.g., see Gilkey [Gi95℄, Lemma1.9.1). That yields two ru ial estimates:Lemma 3.11. There exist positive reals 1, 2, and 3 whi h do not depend onR, su h that for all x; x0 2MR2 and any t > 0 and R > 0,jeR2 (t;x; x0)j � 1 � t�m2 � e 2t � e� 3 d2(x;x0)t ;(3.13) jER2 (t;x; x0)j � 1t�1+m2 � e 2t � e� 3 d2(x;x0)t :(3.14)Here d(x; x0) denotes the geodesi distan e.Noti e that exa tly the same type of estimate is also valid for the kernelER(t;x; x0) on the stret hed losed manifold MR and for the kernel E1aps(t;x; x0)on the in�nite ylinder. For details see also [BoWo93℄, Theorem 22.14. There,however, the term e 2t was suppressed in the �nal formula be ause the emphasiswas on small time asymptoti s.As mentioned before, as R!1, we want to separate the ontribution to thekernel ER2 whi h omes from the ylinder and the ontribution from the interior

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 97by a gluing pro ess. Unfortunately, the inequality 3.14 does not suÆ e to showthat the ontribution to the eta invariant, more pre isely to the integral (3.11),whi h omes from the `error' term vanishes with R!1. Therefore, we introdu ea di�erent parametrix for the kernel ER2 .Instead of gluing over the �xed ne k N = [0; 1)� �; we glue over a segmentNR of growing length of the atta hed ylinder, say NR := (�47R;�37R) �� (thereason for hoosing these ratios will be lear soon). Thus, we hoose a smoothpartition of unity f�aps; �intg on MR2 suitable for the overing fUaps; Uintg withUaps := [�R;�37R)�� and Uint := �(�47R; 0℄���[M2, hen e Uaps\Uint = NR.Moreover, we hoose nonnegative smooth uto� fun tions f aps; intg su h that j � 1 on fx 2MR2 j dist(x; supp�j) < 17Rg and j � 0 on fx 2MR2 j dist(x; supp�j) � 27Rgfor j 2 faps;intg. We noti e(3.15) dist(supp 0j; supp�j) = dist(supp 00j ; supp�j) � 17R :Moreover, we may assume that �����k j�uk ���� � 0=Rfor all k, where 0 is a ertain positive onstant.For any parameter t > 0, we de�ne an operator QR2 (t) on C1(MR2 ;S) with asmooth kernel, given by(3.16) QR2 (t;x; x0) := aps(x)E1aps(t;x; x0)�aps(x0) + int(x)ER(t;x; x0)�int(x0):Re all that ER denotes the kernel of the operator DRe�t(DR)2 , given on thestret hed losed manifoldMR. Noti e that, by onstru tion, QR2 (t) mapsL2(MR2 ;S)into the domain of the operator DR2;P> .Then, for x0 2 Uaps with �aps(x0) = 1, we have by de�nition:(3.17) QR2 (t;x; x0) = � E1aps(t;x; x0) if d(x; supp�aps) < 17R, and0 if d(x; supp�aps) � 27R:Correspondingly, we have for x0 2 Uint with �int(x0) = 1,(3.18) QR2 (t;x; x0) = � ER(t;x; x0) if d(x; supp�int) < 17R, and0 if d(x; supp�int) � 27R:For �xed t > 0, we determine the di�eren e between the pre ise kernel ER2 (t;x; x0)and the approximate one QR2 (t;x; x0). Let CR(t) denote the operator �(DR2;P>)2 + ddt�ÆQR2 (t) and CR(t;x; x0) its kernel. By de�nition, we have �(DR2;P>)2 + ddt�ÆER2 (t) =0. Thus, CR(t) `measures' the error we make when repla ing the pre ise kernelER2 (t;x; x0) by the glued, approximate one.

98 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKMore pre isely, we have by Duhamel's FormulaER2 (t;x; x0)� QR2 (t;x; x0) = � Z t0 ds ZMR2 dz ER2 (s;x; z)CR(t� s; z; x0)withCR(t � s; z; x0) = �(DR2;(z))2 +D�QR2 (t � s; z; x0)= �(DR2 (z))2 � dds�QR2 (t� s; z; x0)= 00aps(z)ERaps(t � s; z; x0)�aps(x0) + 2 0aps(z) ��u (ERaps(t � s; z; x0))�aps(x0)+ aps(z)�D2(z) � dds� ERaps(t� s; z; x0)| {z }=0 �aps(x0)+ 00int(z)ER(t� s; z; x0)�int(x0) + 2 0int(z) ��u (ER(t� s; z; x0))�int(x0)+ int(z)�(DR(z))2 � dds� ER(t � s; z; x0)| {z }=0 �int(x0):Here, D(z) denotes the operator D a ting on the z variable; and in the partialderivative ��u the letter u denotes the normal oordinate of the variable z.As stated in (3.15), the supports of �j and 0j (and, equally, 00j ) are disjointand separated from ea h other by a distan e R=7 in the normal variable for j 2faps;intg. Then the error term CR(t � s; z; x0) vanishes both for the distan e inthe normal variable d(z; x0) < R=7 and, a tually, whenever z or x0 are outside thesegment [�67R; 17R℄� �.Let z and x0 be on the ylinder and ju � vj > R=7 where u and v denotetheir normal oordinates. We investigate the error term CR(t � s; z; x0) whi h onsists of six summands. Two of them vanish as we have pointed out above.The remaining four summands involve the kernels E1aps(t� s; z; x0) on the in�nite ylinder [�R;1) � � and ER(t � s; z; x0) on the stret hed losed manifold MR.We shall use that both kernels an be estimated a ording to inequality (3.14).We estimate the �rst summandj 00aps(z)E1aps(t� s; z; x0)�aps(x0)j � 0R 1(t� s)� 1+m2 e 2te� 3 d2(z;x0)t�s� 01e 02te� 03R2=t :Here we have used t � s � 0 and(t� s)�(1+m)=2e� 2 d2(z;x0)(t�s) � t�(1+m)=2e� 2 d2(z;x0)t � ~ e� 2 d2(z;x0)2t :

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 99Similarly we estimate the se ond summand2 ���� 0aps(z) ��uE1aps(t� s; z; x0)�aps(x0)����� 0R 1 (t � s)� 1+m2pt e 2te� 3 d2(z;x0)t�s � 01e 02te� 03R2=t ;where the fa tor 1=pt omes from the di�erentiation of the kernel as explainedbefore. The third and fourth summands, involving the kernel ER of the losedstret hed manifoldMR, are treated in exa tly the same way. Altogether we haveprovedLemma 3.12. The error kernel CR(t;u; v) vanishes for u =2 [�67R;�17R℄.Moreover, CR(t;u; v) vanishes whenever ju� vj � R=7. For arbitrary x; x0 2MR2we have the estimate jCR(t;x; x0)j � 1e 2te� 3R2=twith onstants 1; 2; 3 independent of x; x0; t; R.We onsider the pointwise errorER2 (t;x; x)� QR2 (t;x; x) = Z t0 ds ZMR2 dz ER2 (s;x; z)CR(t� s; z; x):We obtain the following proposition as a onsequen e of the pre eding lemma.Proposition 3.13. For all x 2MR2 and all t > 0 we haveTr ER2 (t;x; x)� TrQR2 (t;x; x) = Tr �ER2 (t;x; x)�QR2 (t;x; x)� :Moreover, there exist positive onstants 1; 2; 3, independent of R, su h that the`error' term satis�es the inequality��ER2 (t;x; x)� QR2 (t;x; x)�� � 1 � e 2t � e� 3(R2=t) :

100 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKProof. We estimate the error termjER2 (t;x; x)� QR2 (t;x; x)j� Z t0 ds ZMR2 dz jER2 (s;x; z)CR(t� s; z; x)j� 21e 2t � Z t0 ds ZMR2 dz �s� d+12 � e� 3 d2(x;z)s � � e� 3 d2(x;z)t�s� 21e 2t � Z t0 ds Zsuppz CR(t�s;z;x) dz e� 4 d2(x;z)s � e� 3 d2(x;z)t�s� 21e 2t � Z t0 ds Zsuppz CR(t�s;z;x) dz e� 5 t�R2s(t�s)� 21e 2t � R � Z t0 ds e� 5 t�R2s(t�s) � 21e 2t � 2 R � Z t=20 ds e� 5 t�R2s(t=2)= 21e 2t � 2 R � Z t=20 ds e�2 5 R2s :Here we have used that Vol(suppz CR(t�s; z; x)) � Vol(�)�R a ording to Lemma3.12. We investigate the last integral.Z t0 e� s ds = � Z t0 s2 � e� s � (� s2 ) ds< � Z t0 t2 � e� s � (� s2 ) ds = � t2 Z t1 e�r dr = t2 e� t :Thus we havejER2 (t;x; x)�QR2 (t;x; x)j � 21e 2t � 2 R � t2 6R2 e� 6R2t � 7e 2t � e� 8(R2=t) : �The pre eding proposition shows that, for t smaller thanpR, the tra e Tr ER2 (t;x; x)of the kernel of the operator DR2 e�t(DR2;P> )2 approa hes the tra e TrQR2 (t;x; x) ofthe approximative kernel pointwise as R!1. In parti ular, we have:Corollary 3.14. The following equality holds, as R!1,1p� Z pR0 dtpt ZMR2 Tr ER2 (t;x; x) dx= 1p� Z pR0 dtpt ZMR2 TrQR2 (t;x; x) dx+O(e� R):

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 101Proof. We have1p� Z pR0 dtpt ZMR2 Tr ER2 (t;x; x) dx= 1p� Z pR0 dtpt ZMR2 TrQR2 (t;x; x) dx+ 1p� Z pR0 dtpt ZMR2 Tr �ER2 (t;x; x)�QR2 (t;x; x)� dx;and we have to show that the se ond summand on the right side is O(e� R) asR!1. We estimate����� 1p� Z pR0 dtpt ZMR2 Tr �ER2 (t;x; x)� QR2 (t;x; x)� dx������ 1p� Z pR0 dtpt ZMR2 jER2 (t;x; x)�QR2 (t;x; x)j dx� 1p� Z pR0 dtpt ZMR2 1 � e 2t � e� 3(R2=t) dx� 1Vol(MR2 )p� Z pR0 e 2t � e� 3(R2=t)pt dt� 4R Z pR0 e 2pR � e� 5R3=2 dt � 4R3=2 � e� 6R � 7 � e� 8R : �3.5.2. Dis arding the Cylinder Contributions. Corollary 3.14 shows that theessential part of the lo al eta fun tion of the spe tral boundary ondition on thehalf manifold with atta hed ylinder of length R, i.e., the `small{time' integralfrom 0 to pR an be repla ed, as R ! 1, by the orresponding integral overthe tra e TrQR2 (t;x; x) of the approximate kernel, onstru ted in (3.16). Now weshow that TrQR2 (t;x; x) an be repla ed pointwise (for x 2 MR2 ) by the tra eTr ER(t;x; x) of the kernel of the operator DRe�t(DR)2 whi h is de�ned on thestret hed losed manifoldMR.Consider the Dira operator�(�u +B) : C1([0;1)� �;S)! C1([0;1)� �;S);on the semi-in�nite ylinder with the domainfs 2 C10 ([0;1)��;S) j P>(sjf0g��) = 0g:It has a unique self-adjoint extension whi h we denote by Daps. Re all that theintegral kernel E1aps of the operator Dapse�t(Daps)2 enters in the de�nition of the

102 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKapproximative kernel QR2 as given in (3.16). We show that E1aps(t;x; x) is tra elessfor all x 2 [0;1)� �. Then(3.19) TrQR2 (t;x; x) = Tr ER(t;x; x) for all x 2MR2 ;follows.To prove that a produ t TV is tra eless, the following easy result an be used.Lemma 3.15. Let � be unitary with �2 = � I. We onsider an operator Vof tra e lass whi h is `even', i.e. it ommutes with �. Moreover, T is odd, i.e. itanti ommutes with �. Then Tr(TV ) = 0:Proof. We have, by unitary equivalen e,Tr(TV ) = Tr(��(TV )�) = Tr(��T�V ) = Tr(�2TV ) = Tr(�TV ): �Lemma 3.16. Let � : [0;1)! R be a smooth fun tion with ompa t supportand t > 0. Then the tra e of the operator � �Dapse�t(Daps)2 vanishes. In parti ular,Z�Tr E1aps(t;u; y;u; y) dy = 0for all u 2 [0;1).Proof. Clearly, D2aps = (�(�u +B))2 = ��2u + B2 is even, hen e also thepower series e�t(Daps)2 is even. On the other hand, as with B, �B is also odd. So,Tr�� � �Be�t(Daps)2� = 0:To show that Tr�(� � ��ue�t(Daps)2� = 0;we need a slightly more spe i� argument: Let e1aps denote the heat kernel of theoperator Daps. For u; v 2 [0;1) and y; z 2 � it has the following form (see e.g.[BoWo93℄, Formulae 22.33 and 22.35):eaps(t;u; y; v; z) =Xk2Zek(t;u; v)'k(y) '�k(z)for an orthonormal system f'kg of eigense tions of B. Hen e,��u eaps(t;u; y; v; z) =Xk2Ze0k(t;u; v)�'k(y) '�k(z):But h�'k;'ki = 0 on � sin e � is skew-adjoint. �

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 1033.6. Asymptoti Vanishing of Large-tChopped �-invariant on Stret hedPart Manifold. So far we have found�DR2;P> (0) = 1p� Z pR0 dtpt ZMR2 Tr ER(t;x; x) dx+O(e� R)+ 1p� Z 1pR dtpt ZMR2 Tr ER2 (t;x; x) dxas R!1. To prove Theorem 3.8, we still have to show1p� Z 1pR dtpt ZMR2 Tr ER2 (t;x; x) dx = O(e� R) and(3.20) 1p� Z 1pR dtpt ZMR2 TrER(t;x; x) dx= O(e� R);(3.21)asR!1. Re all that ER2 (t;x; x0) denotes the kernel of the operator DR2 e�t(DR2;P> )2on the ompa t manifoldMR2 with boundary f�Rg��, and ER(t;x; x0) the kernelof the operator DRe�t(DR)2 on the losed stret hed manifoldMR.In the following we show (3.20), i.e. that we an negle t the ontribution tothe eta invariant of DR2;P> whi h omes from the large t asymptoti of ER2 (t;x; x0).The key to that is that the eigenvalue of DR2;P> with the smallest absolute valueis uniformly bounded away from zero.Theorem 3.17. Let �0(R) denote the smallest (in absolute value) nonvan-ishing eigenvalue of the operator DR2;P> on the manifold MR2 . Let us assume, asalways in this se tion, that KerB = f0g. Then there exists a positive onstant 0,whi h does not depend on R su h that �0(R) > 0 for R suÆ iently large.Remark 3.18. As we will dis over, this result indi ates that the behavior ofthe small eigenvalues on MR2 di�ers from that on the stret hed, losed manifoldMR. On the manifold with boundary MR2 with the atta hed ylinder of lengthR, the eigenvalues are bounded away from 0 when R ! 1 due to the spe tralboundary ondition. That is the statement of Theorem 3.17 whi h we are going toprove in the next two se tions. However, on MR, the set of eigenvalues splits intoone set of eigenvalues be oming exponentially small and another one of eigenvaluesbeing uniformlybounded away from 0 as R!1. This we are going to show furtherbelow. Roughly speaking, the reason for the di�erent behavior is that on MR2 theeigense tions must satisfy the spe tral boundary ondition. Therefore they areexponentially de reasing on the ylinder, and the eigenvalues are bounded awayfrom 0. But onMR we have to ope with eigense tions on a losed manifold whi hneed not de rease, but require part of the eigenvalues to de rease exponentially(for details see Theorem 3.29 below).

104 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK3.6.1. The Cylindri al Dira Operator. To prove Theorem 3.17 we �rst re alla few properties of the ylindri al Dira operator D yl := �(�u+B) on the in�nite ylinder �1 yl := (�1;+1)��. A spe ial feature of the ylindri al manifold �1 ylis that we may apply the theory of Sobolev spa es exa tly as in the ase ofRm. Thepoint is that we an hoose a overing of the open manifold �1 yl by a �nite numberof oordinate harts. We an also hoose a �nite trivialization of the bundle Sj�1 yl .Let fU�; ��gK�=1 be su h a trivialization, where �� : SjU� ! V� � CN is a bundleisomorphism and V� an open (possibly non{ ompa t) subset of Rm. Let ff�g be a orresponding partition of unity. We assume that for any � the derivatives of thefun tion f� are bounded.Definition 3.19. We say that a se tion (or distribution) s of the bundle Sover �1 yl belongs to the p-th Sobolev spa e Hp(�1 yl;S), p 2 R, if and only if f� � sbelongs to the Sobolev spa e Hp(Rm; CN ) for any �. We de�ne the p-th Sobolevnorm kskp :=XK�=1 (I+��)p=2 (f� � s) L2(Rm) ;where �� denotes the Lapla ian on the trivial bundle V� � CN � Rm� CN .Lemma 3.20. (a) For the unique self-adjoint L2 extension of D yl (denoted bythe same symbol) we have Dom(D yl) = H1(�1 yl;S):(b) Let �1 denote the smallest positive eigenvalue of the operator B on the manifold�. Then we have(3.22) (D yl)2s; s� � �21 ksk2for all s 2 Dom(D yl), and for any � 2 (��1;+�1) the operatorD yl � � : H1(�1 yl;S)! L2(�1 yl;S)is an isomorphism of Hilbert spa es.( ) Let R yl(�) denote the inverse of the operator D yl � �. Then the familyfR yl(�)g�2(��1;�1) is a smooth family of ellipti pseudo-di�erential operators oforder �1.Proof. (a) follows immediately from the orresponding result on the modelmanifold Rm. To prove (b) we onsider a spe tral resolution f'k; �kgk2Zn0 ofL2(�;S) generated by the tangential operator B. Be ause of (3.8), we have ��k =��k. We an assume '�k = �'k for k 2 N. We onsider a se tion s belongingto the dense subspa e C10 (�1 yl;S) of Dom(D yl), and expand it in terms of thepre eding spe tral resolutions(u; y) =Xk2Znf0g fk(u)'k(y):

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 105Sin e (D yl)2 = ��2u +B2, we obtain(D yl)2s =Xk(�2kfk � f 00k )'k;hen e (D yl)2s; s� =Xk Z 1�1(�2kfk(u) � f 00k (u)) �fk(u)du� �21 ksk2 �Xk Z 1�1 f 00k (u) �fk(u)du= �21 ksk2 +Xk Z 1�1 f 0k(u) �f 0k(u)du � �21 ksk2 :It follows that (D yl)2 (and therefore D yl) has bounded inverse in L2(�1 yl;S) and,more generally, that (D yl)2 � � is invertible for � 2 (��1; �1). To prove ( ) weapply the symboli al ulus and onstru t a parametrix S for the operator D yl;i.e., S is an ellipti pseudo-di�erential operator of order �1 su h that SD yl =I+T , where T is a smoothing operator. Thus D�1 yl = S � TD�1 yl . The operatorTD�1 yl is a smoothing operator, hen e D�1 yl is an ellipti pseudo-di�erential operatorof order �1. The same argument an be applied to the resolvent R yl(�) = (D yl��)�1 for arbitrary � 2 (��1; �1). The smoothness of the family follows by standard al ulation. �3.6.2. The Part Manifolds with Half-in�nite Cylindri al Ends. To prove The-orem 3.17 we need to re�ne the pre eding results on the in�nite ylinder �1 yl tothe Dira operator, naturally extended to the manifoldM12 = ((�1; 0℄��)[M2with ylindri al end. Let C10 (M12 ;S) denote the spa e of ompa tly supportedsmooth se tions of S over M12 . Then(3.23) D12 jC10 (M12 ;S) : C10 (M12 ;S)! L2(M12 ;S)is symmetri . Moreover, we haveLemma 3.21. Let s 2 C1(M12 ;S) be an eigense tion of D12 . Then there existC; > 0 su h that, on (�1; 0℄� �, one has js(u; y)j � Ce u.Proof. Let f'k; �kgk2Zn0 be a spe tral resolution of the tangential operatorB. Be ause of (3.8) we have ��k = ��k and we an assume that '�k = �'k fork 2 N. Then(3.24) n'�k = 1p2 ('k � �'k) ;��kok2Nis a spe tral resolution of the omposed operator �B on �. Noti e that we have(3.25) �'+k = �'�k and �'�k = '+k :Let s 2 C1(M12 ;S) and(3.26) D1L2 = �

106 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKwith � 2 R. We expand sj(�1;0℄�� in terms of the spe tral resolution of �B just onstru ted: s(u; y) = 1Xk=1 fk(u)'+k (y) + gk(u)'�k (y) :Be ause of (3.24), (3.25), and (3.26) the oeÆ ients fk; gk must satisfy the systemof ordinary di�erential equations� �k �=�u��=�u ��k ��fkgk� = ��fkgk�or, equivalently,�f 0kg0k� = A�fkgk� with A := � 0 �(� + �k)� � �k 0 � :Sin e s 2 L2, of the eigenvalues �p�2k � �2 of A, only those whi h are on thepositive real line enter in the onstru tion of s by solving the pre eding di�erentialequation. In parti ular, all oeÆ ients fk; gk must vanish identi ally for �k � �.Thus,(3.27)s(u; y) = X�k>� ak exp�q�2k � �2 u� '+k � �k � �p�2k � �2 exp�q�2k � �2 u� '�k !and, in parti ular,js(u; y)j � C exp�q�2k0 � �2 u2� ; u < 0;for some onstant C, where �k0 denotes the smallest positive eigenvalue of B with�k0 > j�j. �In spe tral theory we are looking for self-adjoint L2 extensions of a symmetri operator. We re all: on a losed manifold, the Dira operator is essentially self-adjoint ; i.e. its minimal losed extension is self-adjoint (and therefore there do notexist other self-adjoint extensions) and it is a Fredholm operator. On a ompa tmanifold with boundary, the situation is mu h more ompli ated. There is a hugevariety of dense domains to whi h the Dira operator an be extended su h thatit be omes self-adjoint; and there is a smaller, but still large variety where the ex-tension of the Dira operator be omes self-adjoint and Fredholm (see Se tion 1.2.2above and Boo�-Bavnbek and Furutani [BoFu98℄); a spe ial type of self-adjointand Fredholm domains are the domains spe i�ed by the boundary onditions be-longing to the Grassmannian of all self-adjoint generalized Atiyah{Patodi{Singerproje tions.Now we shall show that the situation on manifolds with (in�nite) ylindri alends resembles the situation on losed manifolds.We re all the following simple lemma (see also Reed and Simon [ReSi72℄,Theorem VIII.3, Corollary, p. 257).

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 107Lemma 3.22. Let A be a densely de�ned symmetri operator in a separable omplex Hilbert spa e H. We assume that range(A+ iI) is dense in H. Then A isessentially self-adjoint.Proof. Sin e A is symmetri , the operator A+i I is inje tive and the operator(A + i I)�1 is well de�ned and bounded on the dense subspa e range(A + i I) ofH. Then the losure Ri of (A+ i I)�1 has the whole spa e H as domain and Ri isbounded and inje tive. Now a standard argument of fun tional analysis (see e.g.Pedersen [Pe89℄, Proposition 5.1.7) says that the inverse R�1i of a densely de�ned, losed, and inje tive operator Ri has the same properties. Thus our R�1i is losed;and by onstru tion it is the minimal losed extension of A+i I. Therefore, R�1i �i Iis symmetri and the minimal losed extension of A, hen e self-adjoint and equalA�. �We apply the lemma for H = L2(M12 ;S) and take for A the operator of(3.23). To prove that the range (D12 + iI)(C10 (M12 ;S)) is dense in L2(M12 ;S) we onsider a se tion s 2 L2(M12 ;S) whi h is orthogonal to (D12 + iI)(C10 (M12 ;S));i.e. the distribution (D12 � iI)s vanishes when applied to any test fun tion, hen e(3.28) (D12 � iI)s = 0:Sin e D12 � i is ellipti , by ellipti regularity s is smooth at all interior points, thatis for our omplete manifold in all points. On the ylinder (�1; 0℄�� we expands in terms of the eigense tions of the omposed operator �B on �. It follows thats satis�es an estimate of the form(3.29) js(u; y)j � Ce u; (u; y) 2 (�1; 0℄� �;for some onstants C; > 0 (a ording to Lemma 3.21). On the manifoldMR2 with ylindri al end of �nite length R we apply Green's formula and get(3.30) DR2 sR; sR�� sR;DR2 sR� = � Zf�Rg��(�sjf�Rg�� dy; sjf�Rg��);where sR denotes the restri tion of s to the manifoldMR2 with boundary f�Rg��.For R!1, the right side of (3.30) vanishes; and the left side be omes 2i jsj2 by(3.28). Hen e s = 0. Thus we have provedLemma 3.23. The operator (3.23) is essentially self-adjoint.We denote the (unique) self-adjoint L2 extension by the same symbolD12 , andwe de�ne the Sobolev spa es on the manifoldM12 as in De�nition 3.19. On e again,the point is that manifolds with ylindri al ends, even if they are not ompa t butonly omplete, are like the in�nite ylinder suÆ iently simple to be overed by a�nite system of lo al harts. ClearlyDom(D12 ) = H1(M12 ;S) and D12 : H1(M12 ;S)! L2(M12 ;S)is bounded. There are, however, substantial di�eren es between the properties ofthe simple Dira operator D yl on the in�nite ylinder and the Dira operator D12on the manifold with ylindri al end. For instan e, from B the dis reteness of the

108 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKspe trum and the regularity at 0 (i.e., 0 is not an eigenvalue) are passed on toD yl, but not to D12 . Yet we an prove the following result:Proposition 3.24. The operatorD12 : Dom(D12 ) = H1(M12 ;S)! L2(M12 ;S)is a Fredholm operator and its spe trum in the interval (��1; �1) onsists of �nitelymany eigenvalues of �nite multipli ity. Here �1 denotes the smallest positive eigen-value of B.Note . A tually, using more advan ed methods one an show that the es-sential spe trum of D12 is equal to (�1;��1℄ [ [�1;1) (see for instan e M�uller[Mu94℄, Se tion 4).Before proving the proposition we shall olle t various riteria for the om-pa tness of a bounded operator between Sobolev spa es on an open manifold. LetX be a omplete (not ne essarily ompa t) Riemannian manifold with a �xedHermitian bundle. Re all the three ornerstones of the Sobolev analysis of Dira operators for X losed.Relli h Lemma: The in lusion H1(X) � L2(X) is ompa t.Compa t Resolvent: To ea h Dira operator D we have a parametrix Rwhi h is an ellipti pseudo-di�erential operator of order �1 with prin ipalsymbol equal to the inverse of the prin ipal symbol of D. So R is abounded operator from L2(X) to H1(X), and hen e ompa t in L2(X).In parti ular, for � in the resolvent set the resolvent (D��I)�1 is ompa tas operator in L2(X).Smoothing Operator: Any integral operator over X with smooth kernelis a smoothing operator, i.e. it maps distributional se tions of arbitrarylow order into smooth se tions. Moreover, it is of tra e lass and thus ompa t.In the general ase, i.e. for not ne essarily ompa t X, the Relli h Lemmaremains valid for se tions with ompa t support. A ompa t resolvent is not at-tainable, hen e the essential spe trum appears. Operators with smooth kernelremain smoothing operators, but in general they are no longer of tra e lass nor ompa t. We re all:Lemma 3.25. Let X be a omplete (not ne essarily ompa t) Riemannian man-ifold with �xed Hermitian bundle. Let K be a ompa t subset of X.(a) The inje tion H1(X) � L2(X) de�nes a ompa t operator when restri ted tose tions with support in K. In parti ular, for any uto� fun tion � with supportin K and any bounded operator R : L2(X)! H1(X) the operator �R is ompa tin L2(X).(b) Let T : L2(X)! L2(X) be an integral operator with a kernel k(x; y) 2 L2(X2).Then the operator T is a bounded, ompa t operator (in fa t it is of Hilbert{S hmidt lass).

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 109( ) Let T : L2(X) ! L2(X) be a bounded ompa t operator and H0 a losed sub-spa e of L2(X), e.g. H0 := L2(X 0) where X 0 is a submanifold of X of odimension0. Assume that T (H0) � H0. Then T jH0 is ompa t as operator from H0 to H0.Proof. (a) follows immediately from the lo al Relli h Lemma. (b) is the fa-mous Hilbert{S hmidt Lemma. Also ( ) is well known, see e.g. H�ormander [Ho85,Proposition 19.1.13℄ where ( ) is proved within the ategory of tra e lass opera-tors. �In general an integral operator T with smooth kernel is not ompa t even ifeither suppx k(x; x0) or suppx0 k(x; x0) are ontained in a ompa t subset K � X.Consider for instan e on �1 yl = (�1;+1) � � an integral operator T with asmooth kernel of the form k(x; x0) = �(x)d(x; x0);where d(x; x0) denotes the distan e and � is a fun tion with support in a ballof radius 1 (and equal 1 in a smaller ball). Then T is not a ompa t operatoron L2(�1 yl): hoose a sequen e fsng of L2 fun tions of norm 1 and with supp sn ontained in a ball of radius 1 su h that d(supp�; supp sn) = n. Then for any nwe have jTsnj > Cn. Thus T is not ompa t, in fa t not even bounded.For the bounded resolvent (see Lemma 3.20)R yl : L2(�1 yl;S)!H1(�1 yl;S)we have, however, the following orollary to the pre eding lemma. It provides anexample of a ompa t integral operator on an open manifold with a smooth kernelwhi h is ompa tly supported only in one variable.Corollary 3.26. Let � and be smooth uto� fun tions on �1 yl with sup-port ontained in the half- ylinder (�1; 0)��. Let supp� be ompa t. Then theoperators �R yl and R yl� are ompa t in L2(�1 yl;S).Proof. The operator �R yl is ompa t a ording to the pre eding lemma, laim (a). Its adjoint operator is R yl�, sin e R yl is self-adjoint. Thus it is also ompa t (even if its range is not ompa tly supported). �3.7. The Estimate of the Lowest Nontrivial Eigenvalue. In this se tionwe prove Theorem 3.17. Re all that the tangential operator B is assumed to benonsingular and that �1 denotes the smallest positive eigenvalue of B. So far, wehave established thatI: the operator D yl on the in�nite ylinder �1 yl has no eigenvalues in theinterval (��1;+�1), andII: the operator D12 on the manifoldM12 with in�nite ylindri al end hasonly �nitely many eigenvalues in the interval (��1;+�1), ea h of �nitemultipli ity.We have to show that

110 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKIII: the nonvanishing eigenvalues of (DR2 )P> are bounded away from 0 bya bound independent of R.Proof of Theorem 3.17 The idea of the proof is the following. We de�nea positive onstant �1 independent of R. Then let R be a positive real (morepre isely R > R0 for a suitable positive R0), and s 2 L2(MR2 ;S) any eigense tionwith eigenvalue � 2 (��1=p2;+�1=p2), i.e.s 2 Dom(DR2 )P> i.e. P>(sjf�Rg��) = 0 and DR2 s = �s:Then we show that �2 > �1=2 for a ertain real �1 > 0 whi h is independent of Rand s. A natural hoi e of �1 is(3.31) �1 = min( jD12 j2jj2 ����� 2 H1(M12 ;S) and ? KerD12 ) :Note that by II above (Proposition 3.24), KerD12 is of �nite dimension. We shallde�ne a ertain extension s1 2 H1(M12 ;S) of s.The reasoning would be easy, if we ould extend s to an eigense tion of D12on all ofM12 . Then it would follow at on e that the dis rete part of the spe trumof D12 is not empty, � belongs to it, p�1 is the smallest eigenvalue > 0, and hen ewe would have �2 > �1=2 as desired. In general, su h a onvenient extension ofthe given eigense tion s annot be a hieved. But due to the spe tral boundary ondition satis�ed by s in the hypersurfa e f�Rg��, the eigense tion s overMR2 an be ontinuously extended by a se tion over (�1;�R℄�� on whi h the Dira operator vanishes. By onstru tion, both the enlargement � of the L2 norm of s bythe hosen extension and the osine, say �, of the angle between s1 and KerD12 an be estimated independently of the spe i� hoi e of s and �. It turns out thatthey both de rease exponentially with growing R.Let fs1; : : : ; sqg be an orthonormal basis of KerD12 and setes := s1 � qXj=1 hs1; sji sj :Clearly, the se tion es belongs to H1(M12 ;S) and is orthogonal to KerD12 . Hen e,on the one hand,(3.32) jD12 esj2jesj2 � �1 :On the other hand, we have by onstru tionjD12 esj2 = jD12 s1j2 = ��DR2 s��2MR2 = �2 :Finally, we shall prove that(3.33) jesj ! 1 as R!1:

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 111Then the estimate(3.34) �2 > �12follows for suÆ iently large R. Sin e we have assumed that �2 < �212 , we have also�1 < �21, hen e �1 belongs to the dis rete part of the spe trum of (D12 )2 and, bythe Min{Max Prin iple (e.g., see [ReSi78℄), must be its smallest eigenvalue > 0.Thus, to prove the Theorem 3.17 we are left with the task of �rst onstru tinga suitable extension es of s and then proving (3.33).We expand sj[�R;0℄�� in terms of a spe tral resolutionf'k; �k;�'k;��kgk2Nof L2(�;S) generated by B:s(u; y) = 1Xk=1 fk(u)'k(y) + gk(u)�'k(y) :Sin e �(�u+B)sj[�R;0℄�� = 0, the oeÆ ients must satisfy the system of ordinarydi�erential equations(3.35) �f 0kg0k� = Ak�fkgk� with Ak := ���k ��� �k� :Moreover, sin e P>sjf�Rg�� = 0 we have(3.36) fk(�R) = 0 for any k � 1:Thus, for ea h k the pair (fk; gk) is uniquely determined up to a onstant ak. Moreexpli itly, sin e the eigenvalues of Ak are �(�2k � �2)1=2, a suitable hoi e of theeigenve tors of Ak givesfk(u) = ak �p�2k � �2 sinhq�2k � �2(R + u) andgk(u) = ak oshq�2k � �2(R+ u) + �kp�2k � �2 sinh(�2k � �2)1=2(R+ u)! :

112 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKWe assume jsjL2 = 1. Then we have, with v := (�2k � �2)1=2(R+ u):1 � Z[�R;0℄��js(u; y)j2 dudy = 1Xk=1Z 0�R �jfk(u)j2 + jgk(u)j2� du= 1Xk=1jakj2 1(�2k � �2)1=2 Z (�2k��2)1=2R0 � �2�2k � �2 � sinh2 v+ osh2 v + 2 �k(�2k � �2)1=2 � osh v � sinh v + �2k�2k � �2 � sinh2 v� dv= 1Xk=1jakj2�� �2k�2k � �2 �R+ (1=4) � �2(�2k � �)3=2 � sinh(2(�2k � �2)1=2R)+ (1=4) ��1 + �2k�2k � �2� � (�2k � �2)1=2 � sinh(2(�2k � �2)1=2R)+ �2k�2k � �2 � osh2((�2k � �2)1=2R)� :Sin e �2k � �21 > 2�2 we have 2(�2k � �2)1=2 > p2�k > �k. Moreover, we havefor all k � 1� �2k�2k � �2 �R+ �2k�2k � �2 � (�2k � �2)1=24 � sinh(2(�2k � �2)1=2R)> �R + (�21 � �2)1=24 � sinh(2(�21 � �2)1=2R)> �R + p28 �1 � sinh(p2�1R) > 0;if R � R0 for some positive R0 whi h depends only on �1 and not on �, s and k.Thus, for any k the sum in the bra es an be estimated in the following way:f : : :g > �2k�2k � �2 � osh2((�2k � �2)1=2R) > 14e2(�2k��2)1=2R > 14e�kR :Hen e, we have(3.37) 1Xk=1jakj2 � e�kR � 4 :Note that the pre eding estimate does not depend on R (provided that R > R0),k or the spe i� hoi e of s, and that R0 only depends on �1.A ording to (3.37) the absolute value of the oeÆ ients ak is rapidly de reas-ing in su h a way that, in parti ular, we an extend the eigense tion s of DR2;P> ,given on MR2 to a ontinuous se tion on M12 by the formulas1(x) := � s(x) for x 2MR2P1k=1 ake�k(R+u)�'k(y) for x = (u; y) 2 (�1;�r℄��:

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 113By onstru tion, s1 is smooth on M12 n (f�Rg � �) and belongs to the Sobolevspa e H1(M12 ;S). It follows from (3.37) thatjs1j2L2 = jsj2L2 + 1Xk=1jakj2 Z �R�1 e2�k(R+u) du = 1 + 1Xk=1jakj2 12�k� 1 + 12�1 � 1Xk=1jakj2 � 1 + 12�1 � �X1k=1jakj2 � e�kR� � e��1R� 1 + 2�1 � e��1R :(3.38)Next, let 2 KerD12 and assume that jj = 1. By (3.27), on (�1; 0℄� �the se tion has the form(3.39) ((u; y)) = 1Xk=1 bke�kuG(y)'k(y)with 1Xk=1 Z 0�1jbkj2e2�ku du = 1Xk=1 12�k � jbkj2 < +1:Set l := jMR2 . Then l satis�es the equationsDR2 l = 0 and P>(ljf�Rg��) = 0:Hen e, l belongs to KerDR2;P> . This implies the following equality:ZMR2 hs1(x); (x)idx = 1� � ZMR2 DR2 s1(x); l(x)�dx= 1� � ZMR2 s1(x);DR2 l(x)� dx� 1� � Z� h�s1(�R; y); l(�R; y)i dx:On the other hand,Z(�1;�r℄�� hs1(x); (x)i dx =Xk>0 akbk2� � e��kR � C1e��1R :Therefore(3.40) jhs1; ij � C1e��1R :Hen e, we have provedLemma 3.27. Any eigense tion s 2 H1(MR2 ;S) of DR2;P> with eigenvalue � 2(��1=p2; �1=p2) an be extended to a ontinuous se tion s1 on M12 whi h issmooth on M12 n (f�Rg � �) and belongs to the �rst Sobolev spa e H1(M12 ;S).Moreover, the enlargement of the norm of s by the extension and the osine of theangle between s1 and KerD12 are exponentially de reasing by formulae (3.38) and(3.40).

114 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKThe �nal step in proving the Theorem 3.17 follows at on e from the pre edinglemma. We re all: by de�nition of �1 we have jD12 esj2 = jesj2 � �1 and by onstru -tion of es we have jD12 esj2 = �2. Thus, we have �2 � jesj2 � �1. To get the desiredbound �2 > �1=2, it remains to show that jesj2 > 1=2 for suÆ iently large R.Sin e es is the orthogonal proje tion of s1 onto (KerD12 )? and the basisfs1; : : : ; sqg of KerD12 is orthonormal, we havejesj2 = js1j2 � qXj=1jhs1; sjij2 � 1 + 2�1 e��1R � qXj=1jhs1; sjij2 :Thus, Theorem 3.17 follows fromjjesj2 � 1j � 2�1 e��1R + qC1e�2�1R � C2e�C3R:We �nish this se tion by proving the asymptoti estimate (3.20). Re all thatER2 (t;x; x0) denotes the kernel of the operator DR2;P>e�t(DR2;P> )2 , where DR2;P> de-notes the Dira operator over the manifoldMR2 with the spe tral boundary on-dition at the boundary f�Rg ��. Then we have:Lemma 3.28. 1p� Z 1pR dtpt ZMR2 TrER2 (t;x; x) dx= O(e� R):Proof. For any eigenvalue � 6= 0 of DR2;P> and R > 0 suÆ iently large(R � 20 � 1, where 0 denotes the lower uniform bound for �2 of Theorem 3.17) wehave(3.41) jZ 1pR 1pt�e�t�2 dtj � Z 1pR 1pt j�je�t�2 dt = Z 1j�jR1=4 e��2 d�� Z 1j�jR1=4 �e��2 d� = ��12e��2�1j�jR1=4 12e��2pR ;whi h givesjZ 1pR dtpt ZMR2 Tr�DR2;P>e�t(DR2;P> )2�j � Z 1pR dtpt ZMR2 X�6=0j�je�t�2 dt� 12 �X�6=0 e��2pR = 12 �X�6=0 e�(pR�1)�2 � e��2� C1 � e�pR�20 Tr�(e�(DR2;P>)2� � C2 � e�pR�20 Vol(MR2 )� C3 � e�pR�20 � C3 � e�C4pR :

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 115Here we have exploited that the heat kernel eR2 (t;x; x0) of the operator DR2;P> anbe estimated by jeR2 (t;x; x0)j � 1 � t�m2 � e 2t � e� 3 d2(x;x0)ta ording to (3.13). Thus,(3.42) jTr�e�(DR2;P> )2�j � ZMR2 jTreR2 (1;x; x)j dx� 1 � e 2 � ZMR2 dx: �3.8. The Spe trum on the Closed Stret hed Manifold. Thus far, wehave proved the asymptoti equation1p� Z pR0 dtpt ZMR2 Tr ER(t;x; x) dx+O(e� R) = �DR2;P> (0)as R!1. It follows thatlimR!1 �R = limR!1 ��DR1;P< (0) + �DR2;P> (0)� ;where �R := 1p� Z pR0 dtpt ZMR TrER(t;x; x) dx:To prove Theorem 3.8, we still have to show (3.21), i.e., that we an extend theintegration from pR to in�nity:1p� Z 1pR dtpt ZMR2 Tr ER(t;x; x) dx= O(e� R) as R!1.Re all that ER(t;x; x0) denotes the kernel of the operator DRe�t(DR)2 on the losedstret hed manifoldMR.Formally, our task of proving the pre eding estimate is reminis ent of ourprevious task of proving the orresponding estimate for the kernel ER2 (t;x; x0) ofthe operator DR2 e�t(DR2;P> )2 (see Lemma 3.28). Both integrals are over the sameprolonged ompa t manifoldMR2 with boundary f�Rg��. However, the methodswe an apply are di�erent: In the previous ase, we had a uniform positive boundfor the absolute value of the smallest nonvanishing eigenvalue of the boundaryvalue problem DR2;P> for suÆ iently large R.As mentioned above in Remark 3.18, su h a bound does not exist for the Dira operator DR on the losed stret hed manifold MR. Moreover, for the spe tralboundary ondition we shall showdimKerDR2;P> = dimKerD2;P> and �DR2;P> (0) = �D2;P> (0)for any R (see Proposition 3.39 below). For DR, on the ontrary, the dimensionof the kernel an hange and, thus, �DR an admit an integer jump in value asR!1. This is due to the presen e of `small' eigenvalues reated by L2 solutions

116 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKof the operators D11 and D12 on the half-manifolds with ylindri al ends. We usea straightforward analysis of small eigenvalues inspired by the proof of Theorem3.17 to prove the following resultTheorem 3.29. There exist R0 > 0 and positive onstants a1; a2; and a3,su h that for any R > R0, the eigenvalue � of the operator DR is either boundedaway from 0 with a1 < j�j, or is exponentially small j�j < a2e�a3R. LetWR denotethe subspa e of L2(MR;S) spanned by the eigense tions of DR orresponding tothe exponentially small eigenvalues. Then dimWR = q, where q = dim(KerD11 )+dim(KerD12 ).Re all from Proposition 3.24 that the operator D1j , a ting on the �rst Sobolevspa e H1(M1j ;S), is an (unbounded) self-adjoint Fredholm operator in L2(M1j ;S)whi h has a dis rete spe trum in the interval (��1;+�1) where �1 denotes thesmallest positive eigenvalue of the tangential operator B. Thus, the spa e KerD1jof L2 solutions is of �nite dimension.To prove the theorem we �rst investigate the small eigenvalues of the operatorDR and the pasting of L2 solutions. Let R > 0. We reparametrize the normal oordinate u su h that MR1 = M1 [ ((�R; 0℄��) and MR2 = ([0; R)� �) [M2,and introdu e the subspa e VR � L2(MR;S) spanned by L2 solutions of theoperators D1j . We hoose an auxiliary smooth real fun tion fR = fR1 [fR2 onMRwith fR = 1 outside the ylinder [�R;R℄� �, and where fR is a fun tion of thenormal variable u on the ylinder. Moreover, we assume fR(�u) = fR(u) (i.e.,fR1 (�u) = fR2 (u)), and that fR2 is an in reasing fun tion of u withfR2 (u) = � 0 for 0 � u � R41 for R2 � u � R:We also assume that there exists a onstant > 0 su h that j�pfR2�up (u)j < R�p. Ifsj 2 C1(M1j ;S), we de�ne s1 [fR s2 by the formula�s1 [fR s2� (x) := � fR1 (x)s1(x) for x 2MR1fR2 (x)s2(x) for x 2MR2 :Clearly, we have s1 [fR s2 = s1 [fR 0 + 0 [fR s2DR(s1 [fR s2) = (D11 s1) [fR (D12 s2) + s1 [gR s2 and(3.43) ��s1 [fR s2��2 = ��s1 [fR 0��2 + �� 0 [fR s2��2 ;where gR := gR1 [ gR2 with gRj (u; y) = �(y)�fRj�u (u; y) and j�j denotes the L2 normon the manifoldMR.Definition 3.30. The subspa e VR � C1(MR;S) is de�ned byVR := spanfs1 [fR s2 j sj 2 KerD1j g:

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 117Let fs1;1; : : : ; s1;q1g be a basis of KerD11 and fs2;1; : : : ; s2;q2g a basis of KerD12 .Then the q = q1 + q2 se tions fs1;�1 [fR 0g [ f0 [fR s2;�2g form a basis of VR.We want to show that VR approximates the spa e WR of eigense tions of DR orresponding to the `small' eigenvalues, for R suÆ iently large. We begin with anelementary result:Lemma 3.31. There exists R0, su h that for any R > R0 and any s 2 VR, thefollowing estimate holds ��DRs�� � e��1R jsj :Proof. It suÆ es to prove the estimate for basis se tions of VR. Thus, lets = s1 [fR 0 with s1 2 KerD11 . By (3.43) we have(3.44) DRs(x) = ( 0 for x 2M1 [M2�(y)�fR1�u (u; y) � s1(u; y) for x = (u; y) 2 [�R;R℄��:Here fR1 is ontinued in a trivial way on the whole ylinder [�R;R℄ � �. Now,s1 is a L2 solution of D11 , hen e s1(u; y) =Pk ke�(R+u)�k'k(y) on this ylinderwhere f'k; �k;�'k;��kgk2Nis, as above, a spe tral resolution of L2(�;S) for B.We estimate the norm of DRs:��DRs��2 = �����fR1�u � s1����2=Xk Z �R4�R2 Z���fR1�u �2 � j kj2 � e�2(R+u)�k('k(y);'k(y)) dy du= Z �R4�R2 ��fR1�u �2 �Xk j kj2 � e�2(R+u)�k � 1 � du� 2R2 �Xk j kj2 � Z �R4�R2 e�2(R+u)�k du!= 2R2 �Xk j kj2 � Z 32R�kR�k e�v dv2�k!� 2R2 �Xkj kj2 � e�R�k � e� 32R�k2�k� 2R2 �Xk e�R�k2�k j kj2 � 2R2 e�R�1 �Xk j kj22�k :On the other hand, we have the elementary inequalityjsj2 = ��s1 [fR 0��2 � Z �R+1�R Z�js1(u; y)j2 dy du=Xj kj2 � 1� e�2�k2�k � d �X j kj22�k ;

118 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKwith 0 < d � 1 � e�2�1 . Thus, we have the following estimate for any s 2 VR ofthe form s1 [fR 0 and for suÆ iently large R��DRs��2 � 2R2de�R�1 � d �Xk j kj22�k � 2R2de�R�1 � jsj2 � e�R�1 � jsj2 :For s = 0 [fR s2, we estimate the norm of DRs in the same way, in view of thefa t that s2 has the form s2(u; y) =Pk dke(u+R)�k�(y)'k(y) on the ylinder. �Let f�k; kg denote a spe tral de omposition of the spa e L2(MR;S) gener-ated by the operator DR. For a > 0, let Pa denote the orthogonal proje tion ontothe spa e Ha := spanf k j j�kj > ag.Lemma 3.32. For suÆ iently large R, we have the estimatej(I�Pe�R�1=4 ) sj � e�R�1=2 � jsj , for all s 2 VR:Proof. We represent s as the series s =Pk ak k . We havej(I�Pe�R�1=4) sj = X�2k>e�R�1=2 a2k � X�2k>e�R�1=2 eR�12 � �2ka2k�Xk eR�1=2 � �2ka2k = eR�1=2 ��DRs��2� eR�1=2e�R�1 jsj2 = e�R�1=2 jsj2 : �Proposition 3.33. The spe tral proje tion Pe�R�1=4 restri ted to the subspa eVR is an inje tion. In parti ular, DR has at least q eigenvalues � su h that j�j �e�R�1=4, where q is the sum of the dimensions of the spa es KerD1j of L2 solutionsof the operators D11 and D12 .Proof. Let s 2 VR, and assume that Pe�R�1=4(s) = 0. We havejsj = j(I�Pe�R�1=4 ) sj � e�R�12 � jsj � 12 jsj ;for R suÆ iently large. �The proposition shows that the operator DR has at least q exponentially smalleigenvalues with orresponding eigense tions, whi h we an approximate by past-ing together L2 solutions. Now we will show that this makes the list of eigenvaluesapproa hing 0 as R! +1 omplete.Let be an eigense tion of DR orresponding to an eigenvalue �, where j�j <�1. As in the proof of Theorem 3.17, we expand j[�R;R℄�� in terms of a spe tralresolution f'k; �k;�'k;��kgk2N

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 119of L2(�;S) generated by B: (u; y) = 1Xk=1 fk(u)'k(y) + gk(u)�'k ;where the oeÆ ients satisfy the system of ordinary di�erential equations of (3.35)�f 0kg0k� = Ak�fkgk� with Ak := ���k ��� �k� :For the eigenvalues �p�2k � �2 of Ak and the eigenve tors��k +p�2k � �2� � and � ��k +p�2k � �2� ;we get a natural splitting of (u; y) in the form (u; y) = +(u; y)+ �(u; y) with +(u; y) =Xk ake�p�2k��2u���k +q�2k � �2�'k(y) + ��(y)'k(y)� ; and �(u; y) =Xk bkep�2k��2u��'k(y) + ��k +q�2k � �2��(y)'k(y)� :Then we have the following estimate of the L2 norm of in the y dire tionon the ylinder:Lemma 3.34. Assume that j j = 1. There exist positive onstants 1; 2 su hthat �� jfug���� � 1e� 2R for �34R � u � 34R.Proof. We have�� jf�R+rg����2� e�2rp�2k��2 � ����Xk ake�Rp�2k��2 ���k +q�2k � �2� fk + ��fk�����2= e�2rp�2k��2 � �� jf�Rg����2 :In the same way we get�� jfR�rg����2 � e�2rp�2k��2 � �� jfRg����2 :Let us observe that, in fa t, the argument used here proves that�� +jfrg���� � e�(r�s)p�2k��2 � �� +jfsg���� ; and�� �jfsg���� � e�(r�s)p�2k��2 � �� �jfrg���� ;for any �R < s < r < R. We also have another elementary inequality�� jfrg����2 � �� +jfrg����2 � 2 � �� +jfrg���� � �� �jfrg���� :

120 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKThis helps estimate the L2 norm of � in the y dire tion. We havej j2 � Z �R+1�R �� jfug����2 du� Z �R+1�R ��� +jfug����2 � 2 �� +jfug���� �� �jfug����� du� �� +jf�Rg����2� 2 Z �R+1�R �� +jf�Rg���� e�2Rp�2k��2 �� �jfRg���� du� �� +jf�Rg����2 � 2e�2Rp�2k��2 �� +jf�Rg���� �� �jfRg���� :In the same way we obtainj j2 � �� �jfRg����2 � 2e�2Rp�2k��2 �� +jf�Rg���� �� �jfRg���� :We add the last two inequalities and use2 �� +jf�Rg���� �� �jfRg���� � �� +jf�Rg����2 + �� �jfRg����2to obtain 2 j j2 � �1� e�2Rp�2k��2���� +jf�Rg����2 + �� �jfRg����2� :This gives us the inequality we need, namely�� �jf�Rg����2 � 4 j j2 :Now we �nish the proof of the lemma.�� jfug���� = �� +jfug�� + �jfug����� e�(u+R)p�2k��2 �� +jf�Rg����+ e�(R�u)p�2k��2 �� �jfRg����� 2�e�(u+R)p�2k��2 + e�(R�u)p�2k��2� j j � 1e� 2R ;for ertain positive onstants 1; 2 when �34R � u � 34R. �We are ready to state the te hni al main result of this se tion.Theorem 3.35. Let denote an eigense tion of the operator DR orrespond-ing to an eigenvalue �, where j�j < �1 . Assume that is orthogonal to the sub-spa e Pe�R�1=4VR � L2(MR;S). Then there exists a positive onstant , su h thatj�j > .To prove the theorem we may assume that j j = 1. We begin with an elemen-tary onsequen e of Lemma 3.32.Lemma 3.36. For any s 2 VR we havejh ; sij � e�R�1=2 jsj :

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 121Proof. We havejh ; sij = jh ;Pe�R�1=4(s) + (s � Pe�R�1=4 (s))ij = jh ;Pe�R�1=4 (s)ij� j j jPe�R�1=4 (s)j � e�R�1=2 jsj : �We want to ompare with the eigense tions on the orresponding mani-folds with ylindri al ends. We use to onstru t a suitable se tion on M12 =((�1; R℄� �) [M2 (Note the reparametrization ompared with the onvention hosen in the beginning of this hapter). Let h :M12 ! R be a smooth in reasingfun tion su h that h is equal to 1 onM2 and h is a fun tion of the normal variableon the ylinder, equal to 0 for u � 12R, and equal to 1 for 34R � u. We also assume,as usual, that j�ph�up j � R�p for a ertain onstant > 0. We de�ne 12 (x) := � h(x) (x) for x 2MR20 for x 2 (�1; 0℄��.Proposition 3.37. There exist positive onstants 1; 2, su h thatjh 12 ; sij � 1e� 2R jsjfor any s 2 KerD12 .Proof. For a suitable uto� fun tion fR2 we havejh 12 ; sij = �����ZM12 ( 12 (x); s(x)) dx����� = �����ZMR2 �h(x) (x); fR2 (x)s(x)� dx������ �����ZMR2 � (x); fR2 (x)s(x)� dx�����+ �����ZMR2 �(1� h(x)) (x); fR2 (x)s(x)� dx����� :We use Lemma 3.36 to estimate the �rst summand:�����ZMR2 � (x); fR2 (x)s(x)� dx����� = �����ZMR2 � (x); �0 [fR s� (x)� dx�����= j ; 0 [fR s�j � e�R�1=2 jsj :

122 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKWe use Lemma 3.34 to estimate the se ond summand:�����ZMR2 �(1� h(x)) (x); fR2 (x)s(x)� dx������ ZMR2 ���(1� h(x)) (x); fR2 (x)s(x)��� dx� ZMR2 j((1� h(x)) (x))j ��fR2 (x)s(x)�� dx� ZMR2 j((1� h(x)) (x))j2 dx!12 jsj� Z 34R0 �� jfug����2!12 jsj du� � 21e�2 2R 34R� 12 jsj � 3e� 4R jsj : �Proof of Theorem 3.35. Now we estimate �2 from below by following theproof of Theorem 3.17. We hoose fskgq2k=1an orthonormal basis of the kernel ofthe operator D12 . Let us de�nee := 12 � q2Xk=1 h 12 ; ski sk:Then e is orthogonal to KerD12 , and it follows from Proposition 3.37 that��� e ��� � 13 j 12 j > � > 0;for R large enough, where � is independent of R, of the spe i� hoi e of theeigense tion , and of the uto� fun tion h. Let �21 denote the smallest nonzeroeigenvalue of the operator (D12 )2. On e again, it follows from the Min-Max Prin- iple, that D(D12 )2 e ; e E � �2�2. We have�2 = D�DR�2 ; E � ZMR ��DR (x)��2 dx= ZMR ��DR (h(x) (x) + (1� h(x)) (x)))��2 dx� ZMR ��DRh(x) (x)��2 dx� ZMR ��DR((1� h(x)) (x))��2 dx

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 123It is not diÆ ult to estimate the �rst term from below. We haveZM12 j(D12 12 ) (x)j2 dx = D(D12 )2 12 ; 12 E = D(D12 )2 e ; e E � �21�2We estimate the se ond term as follows:ZMR ��DR(1 � h(x) (x))��2 dx = ZMR2 ��(1� h(x)) �DR � (x)) � � (x) �h�u(x) (x)��2 dx� ZMR2 �j� (1� h(x)) (x))j2 + 2 j� (1� h(x)) (x))j+ ��� (x) �h�u(x) (x)��2� dxNow we use Lemma 3.34 su essively to estimate ea h summand on the right sideby 1e� 2R. This gives usZMR2 ��DR ((1� h) ) (x)��2 dx � 3e� 4R;and �nally we have �2 � �21�2 � 3e� 4R � 12�21�2 for R large enough. �Theorem 3.29 is an easy onsequen e of Theorem 3.35.3.9. The Additivity for Spe tral Boundary Conditions. We �nish theproof of Theorem 3.8. We still have to show equation (3.21), i.e.Lemma 3.38. We have �R = O(e� R) modZwhere�R := 1p� Z 1pR 1pt Tr�DRe�tD2R� dt:Proof. It follows fromTheorem 3.29 that we have `exponentially small' eigen-values orresponding to the eigense tions from the subspa e WR and the eigen-values � bounded away from 0, with j�j � a1 , orresponding to the eigense tionsfrom the orthogonal omplement of WR . First we show that we an negle t the ontribution due to the eigenvalues that are bounded away from 0.We are pre iselyin the same situation as with the large t asymptoti of the orresponding integralfor the Atiyah{Patodi{Singer boundary problem on the half manifold with the ylinder atta hed. Literally, we an repeat the proof of Lemma 3.28 by repla ingDR2;P> by DR and the uniform bound for the smallest positive eigenvalue of DR2;P>by our present bound a1. Thus, we have����Z 1pR 1pt Tr�DRe�tD2R j(WR)?� dt���� � Z 1pR 1pt �Xj�j�a1 j�je�t�2� dt� Z 1pR 1pt �Xj�j�a1 e�(t�1)�2� dt � Z 1pR 1pt �Xj�j�a1 e��2� e�(t�2)a21 dt� e2a21 Tr�e�tD2R�Z 1pR 1pte�ta21 dt = e2a21 Tr�e�tD2R� 1a1 Z 1pR 1pte�ta21 a1 dt� e2a212a1 Tr�e�tD2R� e�a21pR :

124 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEKFor the last inequality see (3.41). A standard estimate on the heat kernel of theoperator DR gives (as in (3.42)) the inequality Tr�e�tD2R� � b3 �Vol(MR) � b4R,whi h implies that(3.45) ����Z 1pR 1pt Tr�DRe�tD2R j(WR)?� dt���� � b5e�b6pR :That proves that the ontribution from the large eigenvalues disappears as R!1.The essential part of �R omes from the subspa e WR:1p� Z 1pR 1pt Tr�DRe�tD2R jWR� dt =Xj�j<a1 1p� Z 1pR 1pt�e�t�2 dt = Xj�j<a1 sign(�) 2p� Z 1j�jR1=4 e�v2 dv:(3.46)It follows from Theorem 3.29 that limR!1 j�jR1=4 = 0. Thus, the right side of(3.46) is equal to(3.47) signR(D) := Xj�j<a1 sign(�)plus the smooth error term whi h is rapidly de reasing as R!1. �Thus we have proved Theorem 3.8. In parti ular, we have provedlimR!1 �DR(0) � limR!1n�DR1;P< (0) + �DR2;P> (0)omodZ:To establish the true additivity assertion of Corollary 3.9, we show that the pre- eding � invariants do not depend on R modulo integers.Proposition 3.39. (W. M�uller.) The eta invariant �DR2;P> (0) 2 R=Zis inde-pendent of the ylinder length R.Proof. Near to the boundary of MR2 we parametrize the normal oordinateu 2 [�R; 1) with the boundary at u = �R. First we show that dimKerDR2;P> isindependent of R. Let s 2 KerDR2;P> , namely(3.48) s 2 C1(MR2 ;S); DR2 s = 0; and P>(sjf�Rg��) = 0:As in equation (3.27) (and in equation (3.39) of the proof of Theorem 3.17) wemay expand sj[�R;0℄�� in terms of the eigense tions of the tangential operator B:s(u; y) = 1Xk=1 e�ku�(y)'k(y):Let R0 > R. Then s an be extended in the obvious way to es 2 KerDR02;P> , and themap s 7! es de�nes an isomorphism of KerDR2;P> onto KerDR02;P> . Next, observe

SPECTRAL INVARIANTS AND PARTITIONED MANIFOLDS 125that there exists a smooth family of di�eomorphisms fR : [0; 1)! [�R; 1) whi hhave the following uto� propertiesfR(u) = � u for 23 < u < 1u+ R for 0 � u < 13 :Let R : [0; 1) � � ! [�R; 1) � � be de�ned by R(u; y) := (fR(u); y), andextend R to a di�eomorphism R : M2 ! MR2 in the anoni al way, i.e., Rbe omes the identity on M2 n ((0; 1)� �). There is also a bundle isomorphismwhi h overs R. This indu es an isomorphism �R : C1(MR2 ;S) ! C1(M2;S).LetgDR2 := �R Æ DR2 Æ ( �R)�1 . Then fgDR2 gR is a family of Dira operators on M2,and gDR2 = �(�u + B) near �. We pi k the self-adjoint L2 extension de�ned byDom D̂R2;P> := �R �DomDR2;P>�. Hen e,�D̂R2;P> (s) = �DR2;P> (s) and Ker D̂R2;P> = �RKerDR2;P> :In parti ular, dim D̂R2;P> is onstant, and we apply variational al ulus to getddR ��DR2;P> (0)� = � 2p� m(R);where m(R) is the oeÆ ient of t�1=2 in the asymptoti expansion ofTr _ARe�tA2R � 1Xj=0 j(R)t(j�m�1)=2with AR := D̂R2;P> and m := dimM . Now let S1R denote the ir le of radius 2R. Welift the Cli�ord bundle from � to the torus TR := S1R � �. We de�ne the a tionof dDR : C1(TR; S) ! C1(TR; S) by dDR = �(�u + B). Sin e m(R) is lo ally omputable, it follows in the same way as above thatddR ��dDR (0)� = � 2p� m(R):But a dire t omputation shows that the spe trum of dDR is symmetri . Hen e�dDR(s) = 0 and, therefore, m(R) = 0. �In the same way we show that �DR is independent of R. This proves theadditivity assertion of Corollary 3.9. In fa t, we have proved a little bit more:Theorem 3.40. The following formula holds for R large enough�D(0) = �D1;P< (0) + �D2;P> (0) + signR(D);where signR(D) :=Pj�j<a1 sign(�); see (3.47).Theorem 3.40 has an immediate orollary whi h des ribes the ase in whi hour additivity formula holds in R, not just in R=Z.

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130 DAVID BLEECKER AND BERNHELM BOOSS{BAVNBEK[ReSi78℄ |, |, Methods of Modern Mathemati al Physi s, IV. Analysis of Operators, A a-demi Press, NewYork, 1978.[S 01℄ B.W. S hulze, An algebra of boundary value problems not requiring Shapiro{Lopatinskij onditions, J. Fun t. Anal. 197 (2001), 374{408.[S 56℄ L. S hwartz E ua iones di�eren iales elipti as, Bogota, Universidad na ional deColombia, 1956.[S h93℄ A.S. S hwarz, Quantum Field Theory and Topology, Springer, Berlin{Heidelberg{New York, 1993. (Russian original: Kvantovaya teoriya polya i topologiya, Nauka,Mos ow 1989).[S o95℄ S.G. S ott, Determinants of Dira boundary value problems over odd{dimensionalmanifolds, Comm. Math. Phys. 173 (1995), 43{76.[S o02℄ |, Zeta determinants on manifolds with boundary, J. Fun t. Anal. 192 (2002),112{185.[S Wo00℄ S.G. S ott and K.P.Woj ie howski, The &{determinant and Quillen determinantfor a Dira operator on a manifold with boundary,GAFA { Geom. Fun t. Anal. 10(2000), 1202{1236.[Se66℄ R.T. Seeley, Singular integrals and boundary value problems, Amer. J. Math. 88(1966), 781{809.[Se67℄ |: Complex powers of an ellipti operator, AMS Pro . Symp. Pure Math. X. AMSProviden e, 1967, 288{307.[Se69℄ |: Topi s in pseudo-di�erential operators, in: CIME Conferen e on Pseudo-Di�erential Operators (Stresa 1968). Ed. Cremonese, Rome, 1969, pp. 167{305.[Si85℄ I.M. Singer, Families of Dira operators with appli ations to physi s, Asterisque,hors s�erie (1985), 323{340.[Sw78℄ R.C. Swanson, Fredholm interse tion theory and ellipti boundary deformationproblems, I, J. Di�erential Equations 28 (1978), 189{201.[Ta96℄ M.E. Taylor, Partial Di�erential Equations I - Basi Theory, Springer, New York,1996.[Tr80℄ F. Treves, Pseudodi�erential and Fourier Integral Operators I , Plenum Press, NewYork, 1980.[Ve56℄ I.N. Vekua, Systeme von Di�erentialglei hungen erster Ordnung vom elliptis henTypus und Randwertaufgaben (transl. from Russian), VEB Deuts her Vlg. der Wis-sens haften, Berlin, 1956.[We82℄ N. We k, Unique ontinuation for some systems of partial di�erential equations,Appli able Analysis 13 (1982), 53{63.[Wo94℄ K.P. Woj ie howski, The additivity of the �-invariant: The ase of an invertibletangential operator, Houston J. Math. 20 (1994), 603{621.[Wo95℄ |, The additivity of the �-invariant. The ase of a singular tangential operator,Comm. Math. Phys. 169 (1995), 315{327.[Wo99℄ |, The �-determinant and the additivity of the �-invariant on the smooth, self-adjoint Grassmannian, Comm. Math. Phys. 201 (1999), 423{444.[Yu01℄ Y.L. Yu, The Index Theorem and the Heat Equation Method, Nankai Tra ts inMathemati s, vol. 2, World S ienti� Publishing Co., River Edge, NJ, 2001.David Blee ker, Department of Mathemati s, University of Hawaii, Honolulu, HI96822, USAE-mail address: blee ker�math.hawaii.eduBernhelm Booss{Bavnbek, Institut for matematik og fysik, Roskilde Universitets- enter, Postboks 260, DK-4000 Roskilde, DenmarkE-mail address: booss�ru .dk