On the Gluing Problem for the η-Invariant

Ulrich Bunke^{∗} Humboldt-Universität zu Berlin, Institut für Reine Mathematik (SFB288), Ziegelstr. 13a, 10099 Berlin. E-mail:ubunke@mathematik.hu-berlin.de

Abstract

We solve the gluing problem for the η-invariant. Consider a generalized Dirac operator D over a compact Riemannian manifold M that is partitioned by a compact hypersurface N such that M := M_{1} ∪_{N} M_{2}. We assume that the Riemannian metric of M and D have a product structure near N, i.e. D = I(∂/∂r +D_{N}) with some Dirac operator D_{N} on N. Using boundary conditions of Atiyah-Patodi-Singer-type parametrized by Lagrangian subspaces L_{i} of ker D_{N} we define selfadjoint extensions D_{i}, i = 1, 2 over M_{i}. We express the η-invariant of D in terms of the η-invariants of D_{i}, an invariant m(L_{1}, L_{2}) of the pair of the Lagrangian subspaces L_{1}, L_{2} which is related to the Maslov index and an integer valued term J. In the adiabatic limit, i.e. if a tubular neighbourhood of N is long enough, the vanishing of J is shown under certain regularity conditions.

We apply this result in order to prove cutting and pasting formulas for the η-invariant, a Wall non-additivity result for the index of Atiyah-Patodi-Singer boundary value problems and a splitting formula for the spectral flow.

Contents

1 The gluing problem and applications 2 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The η-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The gluing problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 The η-invariant of cylinders . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 The gluing formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Outline of the proof of the gluing formula . . . . . . . . . . . . . . . 11 1.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.8 The adiabatic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.9 Applications in the adiabatic limit . . . . . . . . . . . . . . . . . . . . 16 1.10 A Splitting formula for the spectral flow . . . . . . . . . . . . . . . . 19

∗This work has been supported by the Sonderforschungsbereich 288, which was founded by the Deutsche Forschungsgemeinschaft

2 The Maslov index 21 2.1 Cochains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 The Maslov index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 The equality μ = m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 The twisted Maslov index . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 The signature defect . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 The proof of the gluing formula 27 3.1 The relative index of projections . . . . . . . . . . . . . . . . . . . . . 27 3.2 Construction of the isometry U . . . . . . . . . . . . . . . . . . . . . 29 3.3 The difference of the positive spectral projections . . . . . . . . . . . 29 3.4 Finite propagation speed and comparison of heat kernels . . . . . . . 31 3.5 Comparison away from the cylinder . . . . . . . . . . . . . . . . . . . 32 3.6 Comparison on the cylinder . . . . . . . . . . . . . . . . . . . . . . . 34 3.7 The proof of the gluing formula . . . . . . . . . . . . . . . . . . . . . 36

4 Vanishing of the relative index term 39 4.1 The space of limiting values is Lagrangian . . . . . . . . . . . . . . . 39 4.2 Adjusting U in the adiabatic limit . . . . . . . . . . . . . . . . . . . . 40 4.3 The operator B(u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.4 The structure of the eigenvectors on the cylinder . . . . . . . . . . . . 41 4.5 Definition of the map K . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.6 The small eigenvectors of D(u) . . . . . . . . . . . . . . . . . . . . . 44 4.7 The construction of the map A . . . . . . . . . . . . . . . . . . . . . 48 4.8 Comparison of the spectral flows of D(u) and B(u) . . . . . . . . . . 50

1 The gluing problem and applications

1.1 Introduction

We solve the gluing problem for the η-invariant of generalized Dirac operators. Consider a closed, compact Riemannian manifold carrying a Dirac bundle with a generalized Dirac operator. Assume that this manifold is separated into two pieces by a compact hypersurface. The gluing problem for the η-invariant consists in expressing the η-invariant of the original Dirac operator in terms of the η-invariants of the Dirac operators living on the pieces.

The selfadjoint operators on the two components with boundary depend on a boundary condition given by Lagrangian subspaces L_{1}, L_{2} of a certain symplectic vector space. The gluing formula also contains an additional real valued term m(L_{1}, L_{2}). This term is nicely related to symplectic geometry and the Maslov index.

Moreover, there will be an integer-valued term which is not very manageable. Hence it is interesting to have conditions under that this term vanishes. In fact, we will give such conditions under certain regularity assumptions. This leads to a very explicit version of the gluing formula for the η-invariant.

We apply our gluing formula in order to derive a cutting and pasting formula for the η-invariant. Thus, we compute how the η-invariant changes if the manifold and the Dirac bundle are cut into two pieces by a compact hypersurface and glued together again using a different identification.

As another application we derive a Wall non-additivity result for Atiyah-Patodi-Singer index problems. This generalizes the special case of the signature operator considered by Wall in a topological context.

A gluing problem for the η-invariant appeared first in Cheeger [20] in the context of manifolds with conical singularities. The main idea by now to study the gluing problem has been to use the adiabatic limit. Let D_{N} be the Dirac operator on the induced Dirac bundle over the separating hypersurface. The easiest case is if dim ker D_{N} = 0. Under this assumption the additivity of the η-invariant mod Z follows easily from the results of Douglas-Wojciechowski [24]. Moreover, the author was informed by K. Wojciechowski about a real valued version of the sum formula [40]. Without the regularity assumption it contains a non-explicit integer depending on the exponentially small (in the adiabatic limit) eigenvalues of D.

The case when ker D_{N} ≠ {0} was studied by Mazzeo-Melrose and Müller. Using the b-pseudodifferential calculus, Mazzeo-Melrose [33] defined a b − η-invariant and proved a gluing formula. Without the regularity assumption it involves the signature for the exponentially small eigenvalues. The author was informed by W. Müller that he has proved the gluing formula in full generality [36].

In the present section we give a detailed introduction into the subject and present the main results and applications.

The remainder of the present paper consists of three Sections. In Section 2 we discuss the Maslov index, the signature defect and the function m(L_{1}, L_{2}).

In Section 3 we provide the details of the proof of the gluing formula, Theorem 1.9. We study the relative index of projections. Then we review the finite propagation speed method, which is applied to compare the heat kernels of different Dirac operators. We carry out the program sketched in Subsection 1.6.

In Section 4 we study the behaviour of the small eigenvalues in the adiabatic limit and prove Theorem 1.17.

The results of the present paper have been announced in the series of preprints [13],[14],[15].

I want to thank W. Müller for introducing me to the gluing problem for the ηinvariant, for showing me related literature and for many interesting discussions. I am also grateful to M. Lesch for showing me his computation on the cylinder, which was very helpful for guessing the right formula.

1.2 The η-invariant

The η-invariant first appeared in the study of index problems for generalized Dirac operators on compact manifolds with boundary (Atiyah-Patodi-Singer [1]). Let (X, M) be a compact Riemannian manifold X of even dimension with boundary M. We assume that there is a metric product collar (−ϵ, 0] × M → X with a metric g^{X} = (dr)^{2} +g^{M}, where g^{M} is a Riemannian metric on M independent of the normal coordinate r. Let D_{X} : Γ(E_{X}) → Γ(F_{X}) be an elliptic first order differential operator. We assume that, on the product collar, there exist identifications E_{X} ≅ pr_{M}^{∗}E and F_{X} ≅ pr_{M}^{∗}F for certain real or hermitian vector bundles E, F over M, and that D_{X} = σ(∂/∂r + D) for a bundle isomorphism σ ∈ C^{∞}(M, Hom(E, F)) and a selfadjoint first order differential operator D : Γ(E) → Γ(E). Here, pr_{M} : (−ϵ, 0] × M ⊂ X → M is the projection of the collar (−ϵ, 0] × M to M. Note that X is considered to be on the left-hand side of M. Thus, M is a right boundary and ∂r points in the outward normal direction. Let P_{≤} be the projection onto the subspace of L^{2}(M, E) spanned by the eigenvectors corresponding to non-positive eigenvalues of D. Let C^{∞}(X, E_{X}, P_{≤}) be the space of smooth sections ψ of E_{X} such that P_{≤}ψ_{|M} = 0.

Theorem 1.1 (Atiyah-Patodi-Singer) The operator

D_{X} : C^{∞}(X, E_{X}, P_{≤}) → C^{∞}(X, F_{X})

has a finite-dimensional kernel and cokernel and its index is given by

index D_{X} = ∫_{X} Ω(D_{X}) + \frac{η(D) − dim ker D}{2},  (1)

where Ω(D_{X}) is the local index density (defined with the asymptotic expansion of the heat kernel associated to D_{X}) and η(D) is the η-invariant of D defined below.

In the present paper we will only consider η-invariants of generalized Dirac operators. A generalized Dirac operator D on a hermitian or real vector bundle E → M with compatible connection ∇^{E} is a first order elliptic formally selfadjoint differential operator satisfying a Weizenboeck formula of the type D^{2} = (∇^{E})^{∗}∇^{E} + R, where R is a bundle endomorphism of E. We say briefly that D is the Dirac operator associated with the real or complex Dirac bundle E. Examples of generalized Dirac operators are d + δ on E = Λ^{∗}T^{∗}M, the spin Dirac operator on the spinor bundle associated with a spin structure on M, and the odd signature operator on Λ^{ev}T^{∗}M given by (−1)^{k+p}(∗d−d∗)ω for ω ∈ Γ(Λ^{2p}T^{∗}M) if M is oriented and dimM = 4k−1 (some of these examples are in fact real operators). More examples can be obtained by twisting a generalized Dirac operator with an auxiliary real or hermitian vector bundle with compatible connection. A detailed exposition on generalized Dirac operators can be found in Lawson-Michelsohn [30] and Berline-Getzler-Vergne [4]. Readers who are not so familiar with Dirac operators should keep in mind the example of the odd signature operator.

Let M be a closed Riemannian manifold, E → M be a complex or real Dirac bundle and D be the associated generalized Dirac operator on E. The η-invariant η(D) is a global spectral invariant of D in the sense that it is not the integral of a density locally determined by D. It is defined as follows. D has eigenvalues

… λ_{−1} < 0 ≤ λ_{0} ≤ λ_{1} …

which are counted with multiplicity. Its η-function is defined by

η(D)(s) := ∑_{i∈Z} sign(λ_{i})|λ_{i}|^{−s},

where sign(x) is the sign of x if x ≠ 0 and sign(0) = 0. This sum converges for Re(s) > n, where n := dim M. Using the heat-equation method one can show that η(M)(s) has a meromorphic extension to the whole complex plane and is regular at s = 0 (see e.g. Gilkey [25]).

Definition 1.2 The η-invariant of the Dirac operator is defined as the value of its η-function at s = 0

η(D) := η(D)(0).

It is a regularized version of the signature of the quadratic form defined by D, i.e. formally η(D) = ♯{λ_{i} > 0} − ♯{λ_{i} < 0}.

1.3 The gluing problem

The gluing problem for the η-invariant fits well into the class of gluing problems for spectral invariants of elliptic differential operators. Let M be a closed Riemannian manifold, E → M be a real or complex Dirac bundle and D be the generalized Dirac operator on E. Let N ⊂ M be a compact hypersurface cutting M into two components M_{i}, i = 1, 2. We assume that there is a product collar (−ϵ, ϵ)×N → M such that {0} × N → N, carrying a product metric (dr)^{2} + g^{N}, where g^{N} is a Riemannian metric on N which is independent of the normal coordinate. We assume that ∂r points to M_{2}. Thus, N is the right boundary of M_{1} and the left boundary of M_{2}. Concerning the generalized Dirac operator D over (−ϵ, ϵ) × N → M we assume that there is a hermitian vector bundle E_{N} over N with a compatible connection, a parallel automorphism I (i.e.[∇^{E_{N}} , I] = 0), and a generalized Dirac operator D_{N} such that E_{|(−ϵ,ϵ)×N} = pr_{N}^{∗}E_{N} and D = I(∂/∂r + D_{N}). It is easy to see that I satisfies

I^{∗} = −I
I^{2} = −1
ID_{N} + D_{N}I = 0.

The gluing problem for a spectral invariant consists in defining the corresponding spectral invariant for the restrictions D_{i} of D to M_{i} (using a suitable boundary condition) and in computing the invariant of D in terms of the invariants of D_{i} and possibly some invariant depending on the boundary condition and on D_{N}. On evendimensional manifolds the interesting spectral invariant is the index (associated to a Z_{2}-grading of the Dirac bundle E), and the corresponding gluing problem was solved by Atiyah-Patodi-Singer [1] and, more generally, by Booss-Wojciechowski [7]. The gluing problem for the η-invariant is interesting for odd-dimensional manifolds M.

The first step is to define selfadjoint extensions of D_{i} using suitable boundary conditions. In this paper we will consider global boundary conditions of Atiyah-Patodi-Singer type. They are very natural and can be defined in any case (Lesch-Wojciechowski [31]), while there are topological obstructions against the existence of selfadjoint local boundary conditions (e.g. for the odd signature operator). The ηinvariant of Dirac operators with global boundary conditions was first considered by Cheeger [19], p. 612. Cheeger attached cones over N to the boundaries. The choice of an ideal boundary condition there corresponds to the choice of the Lagrangian subspaces in our case (see below).

Let P_{>} denote the positive spectral projection of D_{N}, i.e. the orthogonal projection in L^{2}(N, E_{N}) onto the subspace spanned by all eigenvectors corresponding to positive eigenvalues, let P_{<} be the negative spectral projection and P_{0} = 1−P_{<}−P_{>}. Let V := ker D_{N} = im P_{0}. Note that V is a (”hermitian” in the complex case) symplectic vector space with a symplectic structure given by Φ(x, y) :=< Ix, y >, where < x, y > is the L^{2}-scalar product. In fact, the multiplication by I anticommutes with D_{N} and, thus, acts on ker D_{N}. A Lagrangian subspace L ⊂ V is a subspace satisfying L ⊕ IL = V, Φ(L, L) = 0. Let pr_{L} be the orthogonal projection onto L. Set Π_{L}^{r} = P_{<} + P_{0} − pr_{L}, Π_{L}^{l} = P_{>} + P_{0} − pr_{L}. Here the index l stands for left while r stands for right. The projection Π_{L}^{r(l)} satisfies −IΠ_{L}^{r(l)}I ⊕ Π_{L}^{r(l)} = 1.

Let us choose two Lagrangian subspaces L_{1}, L_{2}. We define essentially selfadjoint operators D_{i} depending on this choice by setting

dom D_{1} := {ψ ∈ C^{∞}(M_{1}, E) , Π_{L_{1}}^{r}ψ_{|N} = 0}
dom D_{2} := {ψ ∈ C^{∞}(M_{2}, E) , Π_{L_{2}}^{l}ψ_{|N} = 0}.

For this kind of boundary conditions one has the usual elliptic regularity (see Atiyah-Patodi-Singer [1], Booss-Wojciechowski [7] and also the calculation in [10]).

The Dirac operators D_{i}, i = 1, 2, have again a pure point spectrum and the η-function η(D_{i}, L_{i})(s) can be defined as above. It is regular in s = 0 (Douglas-Wojciechowski [24]), and we set η(D_{i}, L_{i}) := η(D_{i}, L_{i})(0). We include the Lagrangian subspace L into the notation in order to indicate that η(D, L) depends on L in a very definite way (Lesch-Wojciechowski [31] and Corollary 1.12 of the present paper).

A more precise version of the gluing problem for the η-invariant is to compute

η(D) − η(D_{1}, L_{1}) − η(D_{2}, L_{2}) =: d̃(D_{N}, L_{1}, L_{2}).

For a real number x let [x] be its class in R/Z. Let δD be a deformation of generalized Dirac operators which is trivial in a neighbourhood of the hypersurface N. Then there exists a locally defined density Ω(D, δD) such that

δ[η(D)] = ∫_{M} Ω(D, δD)  (2)
δ[η(D_{i}, L_{i})] = ∫_{M_{i}} Ω(D, δD).  (3)

Hence [d̃(D_{N}, L_{1}, L_{2})] is a deformation invariant with respect to deformations δD in the interior of M_{i}.

We conclude this subsection with some remarks on our sign conventions made for gluing. Note that, above, M_{1} is considered to be on the left-hand side of the boundary N, while M_{2} is on the right-hand side and ∂r points into M_{2}. To make our considerations more symmetric we will sometimes consider two manifolds M_{i} with right boundaries N_{i} and Dirac bundles E_{i} → M_{i} with Dirac operators D_{i}. We assume product structures as above. Let E_{N_{i}} be the corresponding Dirac bundles over N_{i} with Dirac operators D_{N_{i}}. Let N_{1} → N_{2} be an isometry denoted by a and assume an isomorphism of tuples (E_{N_{1}}, D_{N_{1}}, I_{1}) → (E_{N_{2}}, D_{N_{2}}, I_{2}) over a denoted by A. We can glue the manifolds M_{i} using a and obtain M := M_{1} ∪_{a} M_{2}. Moreover we glue the bundles E_{1}, E_{2} using the composition of A with I_{2} and obtain E := E_{1} ∪_{I_{2}∘A} E_{2}. It is easy to check that D_{1} and −D_{2} glue nicely.

We will also use the convention that the symbol M stands for all structures over M. Thus, we write e.g. η(M_{1}, L_{1}) for η(D_{1}, L_{1}). If the left manifold M_{2} is considered as a right manifold with Dirac operator −D_{2}, we write −M_{2} and vice versa. In particular we have

η(M_{2}, L_{2}) = −η(−M_{2}, IL_{2}).  (4)

The bundle E_{N_{2}} is identified with the restriction of E_{2} to N_{2} using I. This is the reason for the appearance of IL_{2} on the r.h.s. of (4). We will always identify V_{i} = ker D_{N_{i}}, i = 1, 2, using A.

1.4 The η-invariant of cylinders

A simple manifold with boundary consisting of two copies of N is the cylinder Z = [−1, 1] × N. Let E_{Z} := pr_{N}^{∗}E_{N}. Then E_{Z} is a real or complex Dirac bundle with the generalized Dirac operator D_{Z} := I(∂/∂r + D_{N}). Of course, Z has a left boundary {−1} × N and a right boundary {1} × N. Hence, we have to fix two Lagrangian subspaces L_{i} ⊂ V, i = 1, 2, in order to define the boundary conditions, where L_{2} is employed at the right boundary {1} × N and L_{1} at the left boundary {−1} × N of Z.

The η-invariant η(Z, L_{1}, L_{2}) was first computed by Lesch-Wojciechowski [31] using separation of variables and the spectral symmetry of D_{N}. In order to state this result we introduce the involution σ_{L} := 1 − 2pr_{L} on V. Note that σ_{L} anticommutes with I, while the unitary σ_{L_{1}}σ_{L_{2}} commutes.

We define the real-valued function m(L_{1}, L_{2}) on pairs of Lagrangian subspaces by

Definition 1.3

m(L_{1}, L_{2}) := −\frac{1}{π} ∑ ; \begin{matrix}e^{ıλ} ∈ spec(−\frac{ı+I}{2ı} σ_{L_{1}}σ_{L_{2}}) \\ λ ∈ (−π, π) \end{matrix} λ

in the complex case and by

Definition 1.4

m(L_{1}, L_{2}) := −\frac{1}{π} ∑ ; \begin{matrix}e^{ıλ} ∈ spec(−σ_{L_{1}}σ_{L_{2}}) \\ λ ∈ (−π, π) \end{matrix} λ

in the real case, where we consider V as a complex vector space with the complex structure given by I.

Theorem 1.5 (Lesch-Wojciechowski) η(Z, L_{1}, L_{2}) = m(L_{1}, L_{2})

Note that η(Z, L_{1}, L_{2}) is independent of the length of Z (which is 1 in [31]). There is a simple relation of the functions m(L_{1}, L_{2}) in the real and the complex case. Consider V to be complex. Assume that V has a real structure V := V^{R} ⊗_{R} C and I respects this real structure, i.e. acts as I^{R} ⊗ 1_{C}. Then V^{R} has an euclidean structure given by < x, y >= Re < x ⊗ 1, y ⊗ 1 > and a symplectic structure Φ^{R}(x, y) =< I^{R}x, y >. We make the following simple observation

Lemma 1.6 If L_{i} = l_{i} ⊗_{R} C are complexifications of real Lagrangian subspaces l_{i}, then m(L_{1}, L_{2}) = m(l_{1}, l_{2}).

Proof: V decomposes as V = V^{+} ⊕ V^{−}, where V^{±} are the ±ı-eigenspaces of I. There is a canonical identification of complex vector spaces W : V^{R} → V^{+} given by W (x) := x⊗1−I^{R}x⊗ı. The extension A⊗1 to V of any operator A ∈ End(V^{R})^{IR} respects the splitting of V and decomposes as A ⊗ 1 = A^{+} ⊕ A^{−}. Moreover, A^{+} = W AW^{−1}. Thus, Lemma 1.6 follows. □

We give now a relation of m to the Maslov index. Let (V, Φ) be a real symplectic vector space with a compatible complex structure (see Definition 2.5). The Maslov index is an integer τ(L_{1}, L_{2}, L_{3}) defined for a triple of Lagrangian subspaces (see Lion-Vergne [32] and Section 2, Definition 2.3). The complex structure I determines a maximal compact subgroup K of the symplectic group G of (V, Φ) as its stabilizer. We have (see Propositions 2.6 and 2.8)

Proposition 1.7 If L_{i} ⊂ V, i = 1, 2, is a pair of Lagrangian subspaces, then

m(L_{1}, L_{2}) = ∫_{K} τ(kL, L_{1}, L_{2}) dk,

where L is an arbitrary Lagrangian subspace.

The function m is the unique function on pairs of Lagrangian subspaces which is invariant under K (with respect to the diagonal action k(L_{1}, L_{2}) = (kL_{1}, kL_{2}), k ∈ K) such that

m(L_{1}, L_{2}) + m(L_{2}, L_{3}) + m(L_{3}, L_{1}) = τ(L_{1}, L_{2}, L_{3}).

1.5 The gluing formula

We start with stating the gluing formula mod Z. Recall the geometric situation described in Subsection 1.3. In particular we consider M_{2} to be on the right-hand side of N. Then

Theorem 1.8 (Gluing formula mod Z) The class

[η(D) − η(D_{1}, L_{1}) − η(D_{2}, L_{2})] = [d̃(D_{N}, L_{1}, L_{2})] ∈ R/Z

only depends on the relative position of the Lagrangian subspaces L_{1}, L_{2} with respect to the automorphism group K of (V, <, >, I) and

[d̃(D_{N}, L_{1}, L_{2})] = [m(L_{1}, L_{2})].

In order to state the real valued version of the gluing formula we modify the situation slightly. We change the notation for the manifolds M_{i} and M denoting them now by M̃_{i} and M̃ respectively. Let M be the manifold obtained by gluing

M̃_{1} ∪_{N} Z ∪_{N} M̃_{2}.

Hence, M is derived from M̃ by stretching the product collar of N such that a copy of Z is embedded into M. There is a natural generalized Dirac operator D over M. Let δD be the infinitesimal deformation of D corresponding to the stretching. It is supported in a tubular neighbourhood of N, which can also be considered to be a subset of Z. Applying the local variation formula first to Z we get

∫_{Z} Ω(D_{Z}, δD_{Z}) = 0

by Theorem 1.5 since η(D_{Z}, L, L) = 0 and hence does not depend on the length of the cylinder. Consequently

δ[η(D)] = ∫_{M} Ω(D, δD) = 0.

Then, varying the length of the cylindrical part of M changes the η-invariant at most by an integer. Alternatively, one could show Ω(D, δD) = 0 by direct computation.

We set M_{1} := M̃_{1} ∪_{N} Z and M_{2} := Z ∪_{N} M̃_{2}. M_{i} can be considered as overlaping subsets of M with the overlap M_{1} ∩ M_{2} = Z. There are generalized Dirac operators D_{i} over M_{i} in a natural way. By the same discussion as above the η-invariants of M_{i} differ from that of M̃_{i} at most by an integer. The gluing problem for this modified geometric situation consists in computing

d(D_{N}, L_{1}, L_{2}) := η(M) − η(M_{1}, L_{1}) − η(M_{2}, L_{2}).

Recall that D is the Dirac operator on M, while D_{Z} lives on Z. The latter is essentially selfadjoint with boundary conditions given by the Lagrangian subspaces L_{2}, L_{1}. Here L_{1} is employed at {1} × N and L_{2} at {−1} × N. Let H := L^{2}(M, E) ⊕ L^{2}(Z, E_{Z}). There we have the Dirac operator D_{+} = D ⊕ D_{Z}. Analogously we define H_{0} := L^{2}(M_{1}, E_{1}) ⊕ L^{2}(M_{2}, E_{2}) and D_{0} := D_{1} ⊕ D_{2}. Again, D_{i} are essentially selfadjoint with the boundary condition given by the L_{i}. The η-invariant is additive under direct sum. Thus, η(D_{+}) = η(M) + η(Z, L_{2}, L_{1}) and η(D_{0}) = η(M_{1}, L_{1}) + η(M_{2}, L_{2}). However η(Z, L_{2}, L_{1}) = m(L_{2}, L_{1}) is known by Theorem 1.5. We want to compare η(D_{+}) and η(D_{0}). This is done by comparing suitable functions of D_{+} and D_{0}. For this, both operators must act on the same Hilbert space, say H. The key idea is the definition of a unitary operator

U : H → H_{0}.

Let D_{−} := U^{∗}D_{0}U. U is defined using the canonical identifications of the parts of M ∪ Z on the one hand and of M_{1} ∪ M_{2} on the other hand. We employ smooth cut-off functions. The unitary U identifies the corresponding boundary pieces such that dom D_{+} = dom D_{−}. Moreover, G := D_{+} − D_{−} is a bounded but non-local operator on H.

Let P_{±} be the positive spectral projections of D_{±}. Then it turns out that P_{+}−P_{−} is compact (see Section 3.3), therefore the relative index I(P_{+}, P_{−}) ∈ Z of these projections is well-defined (see Kato [28], Avron-Seiler-Simon [2], [3], [11] and Section 3.1, Proposition 3.3).

The gluing formula for the η-invariant is given by

Theorem 1.9 (Gluing formula)

η(M)−η(M_{1}, L_{1})−η(M_{2}, L_{2}) = m(L_{1}, L_{2})−2I(P_{+}, P_{−})+dimkerD_{+} −dimkerD_{−}.

The mod Z-gluing formula Theorem 1.8 follows immediately from Theorem 1.9 and the discussion on how stretching the cylinder influences the η-invariant. The proof of Theorem 1.9 is outlined in Subsection 1.6 and completed in Section 3.

Note that dim ker D_{+} = dim kerD_{M} + dim ker D_{Z}, dim ker D_{Z} = dim(L_{1} ∩ L_{2}) and dimker D_{−} = dimker D_{M_{1}} +dimker D_{M_{2}}. In general, the index term as well as the dimensions of the kernels cannot be computed. Hence, we would prefer to have a simpler formula not involving those non-explicit terms. Later in this paper (see Corollary 1.18) we will show that, under a certain regularity condition, the index term and the kernels vanish if one uses a specially adapted boundary condition (given by the limiting values of the extended L^{2}-solutions) and the cylindrical part of M is long enough.

In [20] Cheeger explains the use of a sort of a gluing formula for computing the η-invariant of mapping tori. He employs manifolds with cones attached. In order to untwist the mapping torus, he pinches a cross-section to obtain a manifold with two conical singularities then he deforms this manifold back to a product. The change of the η-invariant during the latter deformation is calculated by his formula for the variation of the η-invariant for manifolds with conical singularities; see also [5]. The pinching process leaves the η-invariant unchanged (mod Z).

The idea of pinching provides an alternative approach to the gluing problem. The η-invariant of a Dirac operator on a manifold with a cone attached differs from the η-invariant of the Dirac operator with APS boundary conditions by at most an integer.

1.6 Outline of the proof of the gluing formula

We compare suitable functions of D_{±}. First we represent the η-function of a generalized Dirac operator using the heat operator

η(D)(s) = \frac{1}{Γ(\frac{s+1}{2})} ∫_{0}^{∞} r^{\frac{s−1}{2}} T r De^{−rD^{2}}dr.

This integral converges absolutely for Re(s) > −2 (see Bismut-Freed [6], Branson-Gilkey [8]). Hence, it also provides the required analytic extension of η(D)(s) up to s = 0.

Define sign(D) := E_{D}(0, ∞) − E_{D}(−∞, 0), where E_{D}() is the family of spectral measures of D. Set

R(s, t) := \frac{1}{Γ(\frac{s+1}{2})} ∫_{0}^{∞} r^{\frac{s−1}{2}} De^{−(t+r)D^{2}}dr,

where the integral converges in the trace norm for Re(t) > 0 and Re(s) > −1 or Re(t) ≥ 0 and Re(s) > n. Then

R(s, t) = e^{−tD^{2}}sign(D)|D + E_{D}{0}|^{−s}.

For Re(s) > n we have

η(D)(s) = T rR(s, 0).

The idea of introducing the new parameter t is that we get an holomorphic extension of T r R(s, t) up to the interesting point (s, t) = (0, 0). Thus, we can replace analytic continuation by the limit (0, t) →^{t→0} (0, 0) in the discussion below.

Let R_{i}(s, t) be defined as above for D_{i} with i = ±. Set Δ(s, t) := T r(R_{+}(s, t) − R_{−}(s, t)) and A(t) := R_{+}(0, t) − R_{−}(0, t). The function Δ(s, t) is holomorphic for Re(s) > −1/2, Re(t) ≥ 0 and

lim_{t→0} Δ(s, t) = η(D_{+})(s) − η(D_{−})(s)

exists uniformly in s on a compact set for Re(s) ≥ −1/2. This follows from the estimate

|T r(D_{+}e^{−tD_{+}^{2}}) − T r(D_{−}e^{−tD_{−}^{2}})| ≤ Ce^{−ct}e^{− \frac{c}{t}} ∀t > 0

with C < ∞, c > 0, which is proved by the finite propagation speed method. We want to compute

δ := lim_{t→0} Δ(0, t) = η(D_{+}) − η(D_{−}).  (5)

Formally, δ is the trace of the strong limit

s − lim_{t→0} A(t) = sign(D_{+}) − sign(D_{−}) = 2P_{+} − 2P_{−} + E_{D_{+}}{0} − E_{D_{−}}{0}.

Unfortunately the limit on the l.h.s of this equation does not exist in the trace norm and the r.h.s is not of trace class.

The first idea is that one can add a suitable commutator K to P_{+} − P_{−} (see Lemma 3.2) such that P_{+} − P_{−} + K is of trace class and such that

T r(P_{+} − P_{−} + K) = −I(P_{+}, P_{−}).

The second idea is to get control over the convergence of A(t) in the trace norm as follows. We can extend K to a holomorphic family of commutators K(t) that are of trace class for Re(t) > 0 (see Subsection 3.3, Definition 3.4). We find another holomorphic family of trace class operators B(t) with T r B(t) = 0 and commutators K̃(t) such that the limits B := s−lim_{t→0} B(t), K̃ := s−lim_{t→0} K̃(t) exist. Moreover, B + 2K̃ is of trace class and T r(B + 2K̃) = 0. These families are constructed such that

A(t) + 2K(t) − B(t) − 2K̃(t) →^{t→0} 2P_{+} − 2P_{−} + 2K − B − 2K̃ + E_{D_{+}}{0} − E_{D_{−}}{0}

in the trace norm. It follows that

δ = lim_{t→0} T r(A(t) + 2K(t) − B(t) − 2K̃(t))
= T r(2P_{+} − 2P_{−} + 2K − B − 2K̃) + T r(E_{D_{+}}{0} − E_{D_{−}}{0})
= −2I(P_{+}, P_{−}) + dim ker D_{+} − dim ker D_{−}.

The gluing formula follows immediately.

The families B(t), K̃(t) will be constructed analogously to A(t), K(t). Note that M ∪Z and also M_{1} ∪M_{2} contain two copies of the cylinder Z. Thus, we can consider the corresponding Hilbert space h := L^{2}(Z ∪ Z, E_{Z} ∪ E_{Z}) as a subspace of H. On h we consider the unbounded operators D̃_{+} := D_{Z} ⊕ D_{Z} and D̃_{−} := U^{∗}(D_{Z} ⊕ D_{Z})U. We extend these operators by zero in order to obtain operators on H, and we define R̃(s, t)_{±} and K̃(t) by the same procedure as for D_{±}. Then B(t) := R̃_{+}(0, t)−R̃_{−}(0, t). The main point behind this construction is that D_{±} and D̃_{±} coincide locally on the set where the cut-off functions used for defining U are not constant. Note that D_{±} and D̃_{±} should not be compared in a neighbourhood of the boundary of the copies of Z.

1.7 Applications

Let (M_{i}, N_{i}), i = 1, 2, 3, be a triple of compact Riemannian manifolds with boundary, and E_{i} → M_{i} be real or complex Dirac bundles with Dirac operators D_{i} over M_{i}. All manifolds are considered as left manifolds. We assume that N_{i} are pairwise isometric (we denote the isometry by a) and (E_{i|N_{i}}, I_{i}, D_{N_{i}}) are isomorphic (with the isomorphism A over the given isometry a) as explained in Subsection 1.3 such that we can glue arbitrary pairs, thus obtaining M_{12} := M_{1}∪_{N} −M_{2}, M_{23} := M_{2}∪_{N} −M_{3} and M_{31} := M_{3} ∪_{N} −M_{1} together with the corresponding Dirac operators D_{12}, D_{23}, D_{31}. From the gluing formula Theorem 1.9 and (see the proof of Lemma 2.9)

τ_{I}(L_{1}, L_{2}, L_{3}) := m(L_{1}, IL_{2}) + m(L_{2}, IL_{3}) + m(L_{3}, IL_{1}) ∈ Z  (6)

we obtain

Corollary 1.10 (Cocycle) [η(M_{12}) + η(M_{23}) + η(M_{31})] = 0 ∈ R/Z.

Later we will study the integer-valued cocycle

η(M_{12}) + η(M_{23}) + η(M_{31})

in the adiabatic limit. In the special case of the odd signature operator this cocycle can be computed from the non-additivity theorem for the signature of Wall [39]. For the odd signature operator on oriented manifolds (M, N) of dimension n = 4k − 1 we consider the real symplectic vector space V := H^{2k−1}(N, R) with the symplectic structure given by the intersection form. Let l_{i} := im(H^{2k−1}(M_{i}, R) → H^{2k−1}(N, R)), i = 1, 2, 3. Then the l_{i} are Lagrangian subspaces. In the signature operator case we have (see Lemma 2.12)

η(M_{12}) + η(M_{23}) + η(M_{31}) = τ(l_{1}, l_{2}, l_{3}),  (7)

where τ(l_{1}, l_{2}, l_{3}) is the Maslov index of l_{1}, l_{2}, l_{3}. Note that the identifications of the bundles of differential forms are different from those used in the case of general Dirac operators. This is the reason for the appearance of the Maslov index instead of the I-twisted Maslov index (6) in (7).

Let (M̃_{i}, N_{i}), i = 1, 2, be a pair of compact Riemannian manifolds with boundary, and E_{i} → M̃_{i} be real or complex Dirac bundles with the Dirac operator D_{i} being again isomorphic over N_{i}. If M̃_{1} ⊃ N_{1} →^{f} N_{2} ⊂ M̃_{2} is another isometry and f is covered by some isomorphism F : E_{N_{1}} → E_{N_{2}} which intertwines (I_{1}, D_{N_{1}}) with (I_{2}, D_{N_{2}}) we can also use (f, F) instead of (a, A) for gluing. We denote the resulting manifold by M^{f} = M̃_{1} ∪ Z ∪_{f} −M̃_{2}. It carries the Dirac bundle E^{F} → M^{f} and the Dirac operator D^{F}. We use boundary conditions defined by the Lagrangian subspace L at N_{1} and IL at N_{2}.

Let P_{±}^{F} be the positive spectral projections for D_{±}^{F} (of course D_{−} and D_{−}^{F} are unitary equivalent).

Corollary 1.11 (Cutting and pasting formula)

η(M^{f}) − η(M) = m(F (L), L) + dim(L ∩ F (L)) − \frac{1}{2}dim ker D_{N}
+ dim ker D^{F} − dim ker D − 2I(P_{+}^{F} , P_{−}^{F}) + 2I(P_{+}, P_{−}),

where F (L) is the image of L under the action of F on ker D_{N}. In particular,

[η(M^{f}) − η(M)] = [m(F (L), L)] ∈ R/Z.

One can simplify

−I(P_{+}^{F} , P_{−}^{F}) + I(P_{+}, P_{−}) = I(UP_{+}U^{∗}, U^{F} P_{+}^{F} (U^{F})^{∗}).

One can also reproduce and refine the result of Lesch-Wojciechowski about the dependence of the η-invariant on the boundary condition. Let (M̃_{0}, N) be a compact Riemannian manifold with boundary, and E_{0} → M̃_{0} be a real or complex Dirac bundle with the Dirac operator D. If we are given two Lagrangian subspaces L_{i}, i = 1, 2, we can take (M̃_{1}, N) := (M̃_{0}, N) with the boundary condition L_{1} and (M̃_{2}, N) := (M̃_{0}, N) with the boundary condition L_{2}. By gluing we obtain M = M̃_{1} ∪ Z ∪_{N} −M̃_{2} and η(M) = 0 for symmetry reasons. The gluing formula gives

Corollary 1.12 (Dependence of η on the Lagrangian subspace)

η(M_{0}, L_{1}) − η(M_{0}, L_{2}) = m(IL_{2}, L_{1}) + 2I(P_{+}, P_{−})
− dim ker D_{M} − dim(L_{1} ∩ IL_{2}) + dim ker D_{M_{1}} + dim ker D_{M_{2}}.

Taking this modulo Z we obtain Theorem 3.1. of Lesch-Wojciechowski [31]. Note that m(L_{2}, IL_{2}) + m(IL_{2}, L_{1}) + m(L_{1}, L_{2}) = τ(L_{2}, IL_{2}, L_{1}), m(L_{2}, IL_{2}) = 0 and, hence,

m(IL_{2}, L_{1}) = m(L_{2}, L_{1}) mod Z.

1.8 The adiabatic limit

In this subsection we discuss the vanishing of the index term I(P_{+}, P_{−}) and of the kernels of D_{±} in the adiabatic limit. Recall the modification of the geometric situation made in Subsection 1.5. There we have defined M := M̃_{1} ∪_{N} Z ∪_{N} M̃_{2}, M_{1} := M̃_{1} ∪_{N} Z and M_{2} := Z ∪_{N} M̃_{2}. For r ≥ 0 we could glue a cylinder Z_{r} := [−r, r] × N into M̃ and obtain M_{r} := M̃_{1} ∪_{N} Z_{r} ∪_{N} M̃_{2}. Let M_{1,r} = M̃_{1} ∪_{N} Z_{r} and M_{2,r} = Z_{r} ∪_{N} M̃_{2}. There are generalized Dirac operators D_{r}, D_{i,r} over M_{r}, M_{i,r} in a natural way. We will often omit the parameter r to make the notation more compact. The gluing formula Theorem 1.9 is valid for any r ≥ 1. We will study the index term I(P_{+}, P_{−}) and dim ker D_{±} for large r. Note that both terms depend very sensitively on the data D_{i}, L_{i}, i = 1, 2, as e.g. the dimension of kernels or eigenspaces in general. Thus, there is no hope to compute I(P_{+}, P_{−}) in general. Our way out is to make a special choice of the Lagrangian subspaces L_{i} depending on D_{i} (and not on r) such that, under certain regularity conditions, I(P_{+}, P_{−}) = 0 and ker D_{±} = {0} for all r ≥ r_{0}.

We will also need more general boundary conditions. Recall that V := ker D_{N}. To a general subspace L ⊂ V we associate the boundary conditions B_{L} := {(P_{<} + pr_{L^{⊥}})ϕ_{|N} = 0} at right boundaries and B_{L} := {(P_{>} + pr_{L^{⊥}})ϕ_{|N} = 0} at left boundaries. Of course, B_{L} gives rise to a selfadjoint extension of the corresponding Dirac operator iff L is Lagrangian.

Let B_{i} be the closure of D_{i} subject to the boundary condition B_{V}. Then ker B_{i} is finite dimensional. Let for i = 1, 2

Definition 1.13

L_{M_{i}} := {pr_{V} ϕ_{|N} | ϕ ∈ ker B_{i}}.  (8)

In other words, L_{M_{i}} is the space of the limiting values of the extended L^{2}-solutions of the Dirac operator if one glues a complete half-cylinder on M_{i}. The following fact was proved by Yoshida [41] in a special case (with a different method) and also by Nicolaescu [37].

Proposition 1.14 The L_{M_{i}} ⊂ V, i = 1, 2 are Lagrangian subspaces.

See Proposition 4.1 for a proof. Of course, the position of the L_{M_{i}} inside V is highly sensitive with respect to D_{i}.

Next we discuss the regularity assumptions. Note that B_{i}^{∗} is the closure of D_{i} subject to the boundary conditions B_{{0}}.

Assumption 1.15 ker B_{i}^{∗} = {0}.

Assumption 1.16 L_{M_{1}} ∩ L_{M_{2}} = {0}.

Set L_{1} := L_{M_{1}} and L_{2} := L_{M_{2}} and define the selfadjoint extensions D_{i} and D_{Z} as above using the Lagrangian subspaces L_{i}. We employ L_{1} at {−r} × N and L_{2} at {r} × N. Then the assumptions will imply

• ker D_{r} = {0} for r ≥ r_{0},

• ker D_{Z} = 0,

• ker D_{i} = 0, i = 1, 2.

In fact, ker D_{Z} ≅ L_{1} ∩ L_{2} and with Assumption 1.15 also ker D_{i} ≅ L_{1} ∩ L_{2}. The claim for D_{r} follows from the discussion of the small eigenvalues when r becomes large. Thus, for r ≥ r_{0}, also ker D_{+} = ker D_{−} = {0}.

Unfortunately the regularity condition excludes a lot of interesting cases. For the odd signature operator this condition is valid iff the image of H^{∗}(M_{i}, N) in H^{∗}(M) vanishes. This follows from the discussion of the limiting L^{2}-solutions given in Atiyah-Patodi-Singer [1]. On the other hand, in Yoshida [41] it is shown by example that regularity holds in other cases.

Theorem 1.17 [Vanishing of the relative index term] Assume 1.15, 1.16. Then there is an r_{0} ≥ 1 such that for all r ≥ r_{0}

I(P_{+}, P_{−}) = 0.

This vanishing is stable against small perturbations of the L_{i}. We prove Theorem 1.17 in Section 4 by first interpreting the relative index as the spectral flow of the family D(u) := D_{−} + uG, u ∈ [0, 1] (see [11]) and then proving a splitting formula showing that this spectral flow vanishes indeed if r ≥ r_{0}. The argument is a variation of Yoshida’s proof of the splitting formula [41].

Using Theorem 1.17 we can restate our gluing formula for the η-invariant as follows

Corollary 1.18 (Simplified gluing formula) Assume 1.15, 1.16. Then there is an r_{0} ≥ 1 such that for all r ≥ r_{0}

η(M) = η(M_{1}, L_{2}) + η(M_{2}, L_{1}) + m(L_{2}, L_{1}).

Note that the L_{i} are not arbitrary.

1.9 Applications in the adiabatic limit

As in Subsection 1.7 we consider here all manifolds (M_{i}, N_{i}) with boundary to be left manifolds. Moreover, we assume isometries (called a in Subsection 1.3) of the N_{i} and identifications (denoted by A) of the bundles E_{N_{i}}. We will denote L_{M_{i}} by L_{i}. Assumption 1.16 has to be modified to

Assumption 1.19 L_{1} ∩ IL_{2} = {0}.

If M and D are obtained by gluing M_{1} and −M_{2} using the given identifications under the regularity conditions 1.15 and 1.19, the gluing formula for the η-invariant reads

η(M) = η(M_{1}, IL_{2}) − η(M_{2}, IL_{1}) + m(IL_{2}, L_{1})  (9)

if r ≥ r_{0}.

Let (M̃_{i}, N_{i}), i = 1, 2, 3, be a triple of compact Riemannian manifolds with boundary, and E_{i} → M̃_{i} be real or complex Dirac bundles with Dirac operators D_{i} over M_{i}. We assume product structures and that N_{i} and (E_{N_{i}}, I_{i}, D_{N_{i}}) are pairwise isomorphic such that we can glue arbitrary pairs obtaining M_{12} := M̃_{1}∪_{N} Z_{r}∪_{N} −M̃_{2}, M_{23} := M̃_{2} ∪_{N} Z_{r} ∪_{N} −M̃_{3} and M_{31} := M̃_{3} ∪_{N} Z_{r} ∪_{N} −M̃_{1}, together with the corresponding Dirac operators D_{12}, D_{23} and D_{31}. Moreover, recall that L_{i}, i = 1, 2, 3, are given by

L_{i} := {pr_{V} ϕ_{|N_{i}} | ϕ ∈ ker B_{i}},  (10)

where B_{i} is the closure of D_{i} with respect to the boundary condition B_{V}.

Recall the I-twisted Maslov index of the triple of Lagrangian subspaces (L_{1}, L_{2}, L_{3})

τ_{I}(L_{1}, L_{2}, L_{3}) := m(L_{1}, IL_{2}) + m(L_{2}, IL_{3}) + m(L_{3}, IL_{1}).  (11)

Corollary 1.20 (Cocycle) Assume the regularity condition 1.15, i.e. ker B_{i}^{∗} = {0} for i = 1, 2, 3, and 1.19, i.e. L_{i} ∩ IL_{j} = {0} for i ≠ j, i, j = 1, 2, 3. Then there is an r_{0} > 1 such that for all r ≥ r_{0}

η(M_{12}) + η(M_{23}) + η(M_{31}) = τ_{I}(L_{1}, L_{2}, L_{3}) ∈ 2Z.

Proof: Recall that we consider all M_{i} to be left manifolds. We apply equation (9). Let r_{0} be large enough. Then

η(M_{12}) + η(M_{23}) + η(M_{31}) = η(M_{1}, IL_{2}) − η(M_{2}, IL_{1}) − m(L_{1}, IL_{2}) (12)
+ η(M_{2}, IL_{3}) − η(M_{3}, IL_{2}) − m(L_{2}, IL_{3})
+ η(M_{3}, IL_{1}) − η(M_{1}, IL_{3}) − m(L_{3}, IL_{1}).

We also have

η(M_{1}, IL_{2}) − η(M_{1}, IL_{3}) = m(L_{1}, IL_{2}) − m(L_{1}, IL_{3})  (13)
η(M_{2}, IL_{3}) − η(M_{2}, IL_{1}) = m(L_{2}, IL_{3}) − m(L_{2}, IL_{1})
η(M_{3}, IL_{1}) − η(M_{3}, IL_{2}) = m(L_{3}, IL_{1}) − m(L_{3}, IL_{2}).

We explain e.g. (13). Note that there is a path of Lagrangian subspaces L(t) from IL_{3} to IL_{2} inside of the affine set Λ_{L_{1}} := {L ∈ V |L ∩ L_{1} = {0}}. Consider D_{Z}(t) on Z_{r} subject to the boundary conditions B_{L_{1}} at {−r} × N and B_{L(t)} at {r} × N. Then, by the method of Lesch-Wojciechowski [31],

\frac{∂}{∂t}η(D_{Z}(t)) = \frac{∂}{∂t}η(D_{1}, L(t)).

Both families of η-invariants are smooth by the definition of the path L(t). (13) follows from Theorem 1.5. Inserting this into (12) we get the claim. It will be shown in Lemma 2.9 that the I-twisted Maslov index is even. □

Via the index theorem of Atiyah-Patodi-Singer [1] (see Theorem 1.1) Corollary 1.20 provides a generalization of Wall’s non-additivity of the index for the signature operator [39] (which is discussed extensively in Section 2) to general Atiyah-Patodi-Singer boundary value problems. We recall the geometric situation described in Subsection 1.2 adapted to our current problem. Assume that X_{±} are n + 1-dimensional Riemannian manifolds with Z_{2}-graded Dirac bundles F_{±} = F_{±}^{+} ⊕ F_{±}^{−} and Dirac operators D_{±}. Moreover, assume that ∂X_{+} = M_{12}, ∂X_{−} = M_{23} (metrically) and that the metric and the bundles respect a product structure near the boundary. Finally, we assume that

D_{+}^{+} = σ(\frac{∂}{∂s} + D_{12})
D_{−}^{+} = σ(\frac{∂}{∂s} + D_{23})

near the boundary, where D_{±}^{+} : Γ(F_{±}^{+}) → Γ(F_{±}^{−}) is ’half’ of D_{±} with respect to the grading of F_{±}. Hence, we assume that

F_{+|M_{12}}^{+} = E_{12}
F_{−|M_{23}}^{+} = E_{23}.

Recall that e.g. M_{12} = M̃_{1} ∪_{N} Z_{r} ∪_{N} −M̃_{2}. One can now glue X_{+} ∪_{M̃_{2}∪_{N} [0,r]×N} X_{−}, obtaining the Riemannian manifold X and a Dirac bundle F → X with the Dirac operator D (here one has to smooth the corner in some non-unique way). X has the boundary ∂X = M̃_{1} ∪_{N} Z_{r} ∪_{N} −M̃_{3}. Consider the Dirac operators as maps

D_{+}^{+} : {ψ ∈ C^{∞}(X_{+}, F_{+}^{+})|E_{D_{12}}(−∞, 0]ψ_{|M_{12}} = 0} → C^{∞}(X_{+}, F_{+}^{−})
D_{−}^{+} : {ψ ∈ C^{∞}(X_{−}, F_{−}^{+})|E_{D_{23}}(−∞, 0]ψ_{|M_{23}} = 0} → C^{∞}(X_{−}, F_{−}^{−})
D^{+} : {ψ ∈ C^{∞}(X, F^{+})|E_{D_{13}}(−∞, 0]ψ_{|M_{13}} = 0} → C^{∞}(X, F^{−}),

where E_{D}(−∞, 0] denotes the spectral projection onto the non-positive subspace of D. Then the index of these maps is well-defined and by the theorem of Atiyah-Patodi-Singer

index(D_{+}^{+}) = ∫_{X_{+}} Ω(D_{+}^{+}) + \frac{η(D_{12}) − dim ker D_{12}}{2}
index(D_{−}^{+}) = ∫_{X_{−}} Ω(D_{−}^{+}) + \frac{η(D_{23}) − dim ker D_{23}}{2}
index(D^{+}) = ∫_{X} Ω(D^{+}) + \frac{η(D_{13}) − dim ker D_{13}}{2} .

Applying Corollary 1.20 we obtain

Corollary 1.21 Assume the regularity condition 1.15, i.e. ker B_{i} = {0} for i = 1, 2, 3, as well as 1.19, i.e. L_{i} ∩ IL_{j} = {0} for i ≠ j, i, j = 1, 2, 3. Then there is an r_{0} > 1 such that for all r ≥ r_{0}

index(D_{+}^{+}) + index(D_{−}^{+}) − index(D^{+}) = \frac{1}{2}τ_{I}(L_{1}, L_{2}, L_{3}).

In fact, the integrals of the index densities cancel out each other. Using the product structure one can smooth the objects near the corner such that the index density vanishes in that region.

Let (M̃_{i}, N_{i}), i = 1, 2, be a pair of compact Riemannian manifolds with boundary, and E_{i} → M̃_{i} be real or complex Dirac bundles with Dirac operators D_{i}. We assume product structures near the boundary. Moreover, we assume that a : N_{1} → N_{2} is an isometry and there is an isomorphism A : E_{N_{1}} → E_{N_{2}} of bundles intertwining with I_{i} and D_{N_{i}} living over a.

If M̃_{1} ⊃ N_{1} →^{f} N_{2} ⊂ M̃_{2} is another isometry and f is covered by some isomorphism F : E_{N_{1}} → E_{N_{2}} of bundles intertwining with I_{i} and D_{N_{i}}, we can also use (f, F) for gluing. We denote the resulting manifold by M^{f} = M̃_{1} ∪ Z_{r} ∪_{f} −M̃_{2}. It carries the Dirac bundle E^{F} → M^{f} and the Dirac operator D^{F}. Again this construction contains a parameter r being the half of the length of the cylinder glued in. Recall that we identify ker D_{N_{i}} using A.

Corollary 1.22 Assume 1.15 for i = 1, 2 and

L_{1} ∩ IL_{2} = F (L_{1}) ∩ IL_{2} = {0}.

Then there is an r_{0} > 1 such that for all r ≥ r_{0}

η(D_{r}^{F}) − η(D_{r}) = m(F (L_{1}), IL_{2}) − m(L_{1}, IL_{2}).

Proof: Both D_{i}, i = 1, 2, are considered to live on the l.h.s. of the boundary N_{i}. We apply equation (9).

η(M^{f}) − η(M) = η(M_{1}, F^{−1}(IL_{2})) − η(M_{2}, IF (L_{1})) − m(F (L_{1}), IL_{2})
− η(M_{1}, IL_{2}) + η(M_{2}, IL_{1}) + m(L_{1}, IL_{2}).

Moreover, similarly as for equation (13)

η(M_{1}, IF^{−1}(L_{2})) − η(M_{1}, IL_{2}) = m(F (L_{1}), IL_{2}) − m(L_{1}, IL_{2})
η(M_{2}, IF (L_{1})) − η(M_{2}, IL_{1}) = m(IL_{2}, F (L_{1})) − m(IL_{2}, L_{1}).

The claim now follows. □

1.10 A Splitting formula for the spectral flow

Another application of Theorem 1.17 provides a splitting formula for the spectral flow. Let (M̃_{i}, N_{i}) be a pair of compact Riemannian manifolds with boundary and E_{i} → M̃_{i} be Dirac bundles. We assume a product structure of the metrics and the Dirac bundles near the boundary N_{i}. This provides E_{N_{i}} → N_{i} and I_{i} ∈ Γ(End(E_{N_{i}})) as explained in Subsection 1.3. Moreover, we assume that N_{i} are isometric with an isometry denoted by a and that there is an isomorphism A : (E_{N_{1}}, I_{1}) → (E_{N_{2}}, I_{2}). Let D_{i}(t), i = 1, 2, t ∈ [0, 1], be a family of Dirac operators on E_{i} such that D_{i}(t) have a constant principal symbol. Equivalently, D_{i}(t) − D_{i}(s), s, t ∈ [0, 1], is a bundle endomorphism. Such a family, for example, comes from a family of connections on the Clifford bundles E_{i}. We assume that the Dirac operators D_{i}(t) have a product structure near N for all t ∈ [0, 1]. We obtain a family D_{N_{i}}(t) on E_{N_{i}}. Moreover, we assume that (E_{N_{i}}, I_{i}, D_{N_{i}}(t)) are isomorphic for i = 1, 2 and all t via the identification A. Then we can glue the operators for each t ∈ [0, 1] and obtain a family D(t) of Dirac operators on M (recall that we glue in a cylinder of length r). Let V (t) := ker D_{N_{1}}(t). Note that the dimension of V (t) may be discontinuous. Let L_{i}(t) ⊂ V (t), i = 1, 2, be families of Lagrangian subspaces and define families of selfadjoint extensions D_{i}(t), i = 1, 2, and D_{Z}(t) as above using the boundary conditions given by L_{i}(t). Recall the construction of D_{±} in Subsection 1.5. Thus, we can also form the families D_{±}(t) and make the following assumptions.

Assumption 1.23 D_{±}(t) are ’continuous’ such that the spectral flow is well-defined.

Assumption 1.24 ker D_{+}(t) = ker D_{−}(t) = {0} for t = 0, 1.

Assumption 1.23 is formulated informally, but it is, what we really need. In the points t where dim V (t) is continuous, it is enough to have a continuous family of Lagrangian subspaces L_{i}(t) (the union of all V (t), t ∈ [0, 1], has the structure of a continuous bundle of symplectic vector spaces in those points as explained in Yoshida [41]). The continuity condition for L_{i}(t) at points where dim V (t) jumps has to take into account which vectors of V (t) come from or move into the positive or negative spectral subspace of D_{N}(t) (see Nicolaescu [37] for a detailed account).

Assuming 1.23 and 1.24 it can be shown (compare [11] or Proposition 3.3) that P_{±}(1) − P_{±}(0) is compact and that

sf{D_{±}(t)} = I(P_{±}(1), P_{±}(0)),

where P_{±}(t) are the positive spectral projections of D_{±}(t). By the algebraic properties of the relative index

0 = I(P_{+}(1), P_{+}(0)) + I(P_{+}(0), P_{−}(0)) + I(P_{−}(0), P_{−}(1)) + I(P_{−}(1), P_{+}(1))

and, thus,

sf{D_{+}(t)} − sf{D_{−}(t)} = −I(P_{+}(0), P_{−}(0)) + I(P_{+}(1), P_{−}(1)).  (14)

We obtain

Corollary 1.25 (Splitting Formula for the Spectral Flow) Assume 1.15,1.19 (1.24 is implied by 1.15 and 1.19 if r ≥ r_{0}) for D_{i}(0) and D_{i}(1), i = 1, 2, and 1.23. If L_{i}(0) and L_{i}(1) are given by (10), then there is an r_{0} ≥ 1 such that for all r ≥ r_{0}

sf{D(t)} = sf{D_{1}(t)} + sf{D_{2}(t)} − sf{D_{Z}(t)}.

The spectral flow of {D_{Z}(t)} is related to a symplectic invariant of the family of pairs of Lagrangian subspaces {L_{1}(t), IL_{2}(t)} as explained in Yoshida [41] (see also [18]). Such a splitting formula was proved also by Cappell-Lee-Miller in [17] as cited in Kirk-Klassen [29]. Moreover there is a very interesting paper by Nicolaescu [37], which the author received after this work was finished. There, a more general splitting theorem is proved without the assumption of a constant principal symbol and regularity. Moreover, one finds also a nice discussion of the Maslov index for families of pairs of Lagrangian subspaces in a infinite-dimensional framework.

2 The Maslov index

2.1 Cochains

We recall the definition of the complex of (measurable) cochains C_{(λ)}^{∗}(X) associated with a (measurable) space X. If there is a group G acting on X, we consider the cohomology of the complex of G-invariant cochains.

Let X be a measurable space. Set

C^{q}(X) := \{bounded measurable functions on \underbrace{X × … × X}_{q+1}\}.

Without further notice we will consider real valued functions. The differential

d : C^{q}(X) → C^{q+1}(X)

is given by the formula

df(x_{0}, …, x_{q+1}) := ∑_{i=0}^{q+1}(−1)^{i}f(x_{0}, …, x̂_{i}, …, x_{q+1})

for f ∈ C^{q}(X). Let C_{λ}^{∗}(X) ⊂ C^{∗}(X) be the subcomplex of completely antisymmetric functions.

Lemma 2.1 The complexes (C^{∗}(X), d) and (C_{λ}^{∗}(X), d) are acyclic.

Proof: Fix a point o ∈ X. Then there is a contraction

s_{o} : C_{(λ)}^{q}(X) → C_{(λ)}^{q−1}(X)

given by

(s_{o}f)(x_{0}, …, x_{q−1}) := f(o, x_{0}, …, x_{q−1}).

In fact

ds_{o} + s_{o}d = id.  (15)

See also Spanier [38]. □

Let G be a group acting on X (preserving the measurable structure). Then G acts on

\underbrace{X × … × X}_{q+1}

diagonally. Hence, it acts on the cochains C_{(λ)}^{∗}(X) by

gf(x_{0}, …, x_{q}) := f(g^{−1}x_{0}, …, g^{−1}x_{q}),

where g ∈ G and gf, f ∈ C_{(λ)}^{q}(X). Let C_{(λ)}^{∗}(X)^{G} ⊂ C_{(λ)}^{∗}(X) denote the G-invariant functions.

The complex (C_{(λ)}^{∗}(X)^{G}, d) need not be acyclic in general. If e.g. G is a countable group and X = G, then

H^{∗}(C^{.}(G)^{G}) = H^{∗}(G, R)

(compare Brown [9]).

Let now G be a compact topological group with normalized Haar measure dg. Assume that

G × X → X

is measurable. Then there is a projection

m : C_{(λ)}^{∗}(X) → C_{(λ)}^{∗}(X)^{G}

given by

mf := ∫_{G} gf dg.

It is easy to check that

dm = md.  (16)

It follows

Lemma 2.2 If G is a compact group acting on X such that G × X → X is measurable, then

H^{∗}(C_{(λ)}(X)^{G}) = 0.

In fact, for finite groups H^{∗}(G, Z) is a torsion group and thus H^{∗}(G, R) = 0.

2.2 The Maslov index

We recall the definition of the Maslov index τ(l_{1}, l_{2}, l_{3}) associated to a triple (l_{1}, l_{2}, l_{3}) of Lagrangian subspaces of a real symplectic vector space (V, Φ). τ gives a 2-cocycle over Λ, the space of Lagrangian subspaces of V (see Lion-Vergne [32] for most of that material). We will find 1-cochains μ such that dμ = τ.

Let (V, Φ) be a symplectic vector space over R and G be the group of linear symplectic automorphisms G = Sp(V). Let Λ be the space of Lagrangian subspaces of V. Λ is a compact manifold. There is an obvious G-action on Λ.

A triple l_{1}, l_{2}, l_{3} ∈ Λ determines a quadratic form Q on l_{1} ⊕ l_{2} ⊕ l_{3}

Q(x_{1}, x_{2}, x_{3}) = Φ(x_{1}, x_{2}) + Φ(x_{2}, x_{3}) + Φ(x_{3}, x_{1}),

where x_{i} ∈ l_{i}, i = 1, 2, 3.

Definition 2.3 The Maslov index of l_{1}, l_{2}, l_{3} is given by the integer

τ(l_{1}, l_{2}, l_{3}) := sign Q.

Here sign Q := p − q, where p is the number of 1’s while q is the number of −1’s on the diagonal of a matrix representing Q with respect to a suitable diagonalizing basis.

The following is proved in Lion-Vergne [32].

Theorem 2.4 (Vergne) The Maslov index τ satisfies

1. τ ∈ C_{λ}^{2}(Λ)^{G} and

2. dτ = 0.

Note that there is no G-invariant μ ∈ C_{λ}^{1}(Λ) with dμ = τ. In fact,

(g_{1}, g_{2}) → c_{l}(g_{1}, g_{2}) := τ(l, g_{1}l, g_{1}g_{2}l)

is a non-trivial (over R) group cochain and, thus 0 ≠ [τ] ∈ H^{2}(C_{λ}^{.}(Λ)^{G}).

If K ⊂ G is a maximal compact subgroup, we can find of course a K-invariant cochain μ with dμ = τ. Such a cochain is then naturally associated to a compatible complex structure on V.

Definition 2.5 A complex structure I in V is called compatible if < ., . >:= Φ(., I.) is a positive-definite scalar product.

Let us fix a compatible complex structure I on V. Then K := Stab_{G}(I) = Gl(V, I)∩ Sp(V) ≅ U(V) is a maximal compact subgroup of G. Note that K acts transitively on Λ.

Proposition 2.6 There is a unique μ ∈ C_{λ}^{1}(Λ)^{K} with dμ = τ.

Proof: Fix some l ∈ Λ and set μ := m s_{l}(τ), where m is the averaging with respect to K. Thus, by the G-invariance of τ

μ(l_{1}, l_{2}) = ∫_{K} τ(kl, l_{1}, l_{2}) dk.

By (15), (16) and Theorem 2.4(2) we have d m s_{l}(τ) = m d s_{l}(τ) = m(τ) = τ. Obviously, μ is K-invariant.

We have to show uniqueness. Let μ_{i} ∈ C_{λ}^{1}(Λ)^{K}, i = 1, 2, with dμ_{1} = dμ_{2} = τ. Then d(μ_{1} − μ_{2}) = 0 and there is a ξ ∈ C_{λ}^{0}(Λ)^{K} with dξ = μ_{1} − μ_{2} by Lemma 2.2. However K acts transitively on Λ and thus ξ is constant. Hence, dξ = 0. □

We compute μ in the two-dimensional case V := R^{2}. Let V := RP ⊕RQ, where (P, Q) is a basis, with the symplectic structure given by Φ(P, Q) = 1, Φ(P, P) = Φ(Q, Q) = 0. Fix the complex structure I by I(P) = Q, I(Q) = −P. Then < P, P >=< Q, Q >= 1, < P, Q >= 0. We have G = Aut(V, Φ) = SL(2, R) and K = Stab_{G}(I) = SO(2). Any one-dimensional subspace l parametrized by the angle α(l) between RP and l, with α ∈ [0, π), is a Lagrangian subspace.

Lemma 2.7

μ(l_{1}, l_{2}) = \{ \begin{matrix}1 − \frac{2(α(l_{2})−α(l_{1}))}{π} & : α(l_{2}) > α(l_{1}) \\ −1 − \frac{2(α(l_{2})−α(l_{1}))}{π} & : α(l_{2}) < α(l_{1}) \\ 0 & : α(l_{1}) = α(l_{2}) \end{matrix}

Proof: This is an easy computation if one uses that τ(l_{1}, l_{2}, l_{3}) = 0 if any two of the subspaces coincide and that τ(l_{1}, l_{2}, l_{3}) = 1 if α(l_{1}) < α(l_{2}) < α(l_{3}). □

2.3 The equality μ = m

Let V be a real vector space with an euclidean structure <, > and a compatible complex structure I, i.e. I = −I^{∗} and I^{2} = −1. I defines a symplectic structure Φ(x, y) :=< Ix, y >. Let Λ := Λ(V) be the space of Lagrangian subspaces of V and K be the automorphism group of (V, <, >, I). For l ∈ Λ we define the involution σ_{l} := pr_{l} − (1 − pr_{l}) that acts as the identity on l and by −1 on the orthogonal complement l^{⊥}. The following properties are easy to verify:

• σ_{l}^{2} = 1, σ_{l}^{∗} = σ_{l}

• σ_{l}I + Iσ_{l} = 0

• σ_{ul} = uσ_{l}u^{∗}, u ∈ K.

For a pair l_{1}, l_{2} ∈ Λ we form M(l_{1}, l_{2}) := σ_{l_{1}}σ_{l_{2}}. Then

• M(l_{1}, l_{2}) = M(l_{2}, l_{1})^{−1}

• M(l_{1}, l_{2}) ∈ K

• M(l_{1}, l_{2})M(l_{2}, l_{3})M(l_{3}, l_{1}) = 1

• M(ul_{1}, ul_{2}) = uM(l_{1}, l_{2})u^{∗}

• M(Al, l) = AĀ^{−1} for A ∈ Sp(V) with Ā := σ_{l}Aσ_{l}.

Recall that from Definition 1.4.

m(l_{1}, l_{2}) := −\frac{1}{π} ∑ ; \begin{matrix}e^{ıλ} ∈ spec(−M(l_{1}, l_{2})) \\ λ ∈ (−π, π) \end{matrix} λ.

and Proposition 2.6.

μ(l_{1}, l_{2}) = ∫_{K} τ(kl, l_{1}, l_{2})dk.  (17)

Proposition 2.8 m(l_{1}, l_{2}) = μ(l_{1}, l_{2}).

Proof:

If l_{1}, l_{2}, l_{3} ∈ Λ are pairwise transverse, then (see Guillemin-Sternberg [27], 2.2.26)

m(l_{1}, l_{2}) + m(l_{2}, l_{3}) + m(l_{3}, l_{1}) = τ(l_{1}, l_{2}, l_{3}).  (18)

By the properties of M(l_{1}, l_{2}) listed above m(ul_{1}, ul_{2}) = m(l_{1}, l_{2}) for all u ∈ K and m(l_{1}, l_{2}) = −m(l_{2}, l_{1}). Let l_{1}, l_{2} ∈ Λ be transverse, and define

K̇ := {u ∈ K|ul ∩ l_{1} = {0}, ul ∩ l_{2} = {0}}.

Then K \ K̇ has measure zero. Inserting (18) into (17) yields

μ(l_{1}, l_{2}) = ∫K̇ τ(ul, l_{1}, l_{2}) du
= ∫K̇(m(ul, l_{1}) + m(l_{2}, ul))du + m(l_{1}, l_{2})
= m(l_{1}, l_{2}).

Here the integral vanishes for symmetry reasons. It requires a little bit more effort to check the equality in the case if l_{1}, l_{2} are not transverse. This can be done by studying the jumps of m(l_{1}, l(t)) and μ(l_{1}, l(t)) if l(t) is a family with l(0) = l_{2} that goes transversely through {l ∈ Λ | l ∩ l_{1} ≠ {0}}. Then

2m(l_{1}, l_{2}) = m(l_{1}, l(+0)) + m(l_{1}, l(−0))
2μ(l_{1}, l_{2}) = μ(l_{1}, l(+0)) + μ(l_{1}, l(−0)).

We omit the details. □.

2.4 The twisted Maslov index

Recall the definition of the twisted Maslov index (11)

τ_{I}(L_{1}, L_{2}, L_{3}) = m(L_{1}, IL_{2}) + m(L_{2}, IL_{3}) + m(L_{3}, IL_{1}).

Lemma 2.9 If L_{i} ∩ IL_{j} = {0} for j, i = 1, 2, 3, i ≠ j, then τ_{I}(L_{1}, L_{2}, L_{3}) ∈ 2Z.

Proof: Let L be any Lagrangian subspace. By the antisymmetry of m

m(L, L) = 0.

Since I ∈ G and I^{2} = −1 we have m(L, IL) = m(IL, L) = −m(L, IL). Hence,

m(L, IL) = 0.

A simple computation shows

τ_{I}(L_{1}, L_{2}, L_{3}) = τ(L_{1}, IL_{2}, L_{2}) + τ(L_{2}, IL_{3}, L_{3})
+ τ(L_{3}, IL_{1}, L_{1}) + τ(L_{1}, L_{2}, L_{3}).

By Prop. 1.9.3 of Lion/Vergne [32] for any triple of Lagrangian subspaces L_{1}, L_{2}, L_{3}

τ(L_{1}, L_{2}, L_{3}) = n + dim(L_{1} ∩ L_{2}) + dim(L_{2} ∩ L_{3}) + dim(L_{3} ∩ L_{1}) (mod 2),

where n = dim(V)/2. We obtain

τ_{I}(L_{1}, L_{2}, L_{3}) = 2(dim(L_{1} ∩ L_{2}) + dim(L_{2} ∩ L_{3}) + dim(L_{3} ∩ L_{1})) + 4n = 0 (mod 2).

□

2.5 The signature defect

We recall the theorem of Atiyah-Patodi-Singer [1] on the signature of a compact manifold with boundary and the result of Wall [39] on the signature defect.

Let (X, M) be a compact oriented Riemannian manifold with boundary of dimension dim X = 4k. Assume that the metric is product near the boundary. Let η(M) be the η-invariant associated with the odd signature operator introduced by Atiyah-Patodi-Singer [1]. Then

Theorem 2.10 (APS)

sign(X) = ∫_{X} L − η(M).

Here L is the Hirzebruch-L polynomial in the Pontrjagin forms of X.

Let (X_{+}, M_{1}, −M_{2}, N), (X_{−}, M_{2}, −M_{3}, N) be oriented manifold 4-tuples (the sign indicates the orientation) with M_{1}∩M_{2} = N, M_{2}∩M_{3} = N, M_{1}∪_{N} −M_{2} = ∂X_{+}, M_{2} ∪_{N} −M_{3} = ∂X_{−}. Give N the orientation as the boundary of M_{2}. We form X := X_{+} ∪_{M_{2}} X_{−}. Let V := H_{2k−1}(N, R). V is a symplectic vector space, where the symplectic structure Φ is given by the intersection number. Let

l_{i} := ker(V → H_{2k−1}(M_{i})), i = 1, 2, 3.

It can be shown that the l_{i} are Lagrangian subspaces. The result of Wall [39] is Theorem 2.11

sign(X) = sign(X_{+}) + sign(X_{−}) + τ(l_{1}, l_{2}, l_{3})  (19)

By Theorem 2.10 we have

sign(X_{+}) = ∫_{X_{+}} L − η(M_{1} ∪_{N} −M_{2})
sign(X_{−}) = ∫_{X_{−}} L − η(M_{2} ∪_{N} −M_{3})
sign(X) = ∫_{X} L − η(M_{1} ∪_{N} −M_{3}).

Inserting these equations into (19) we obtain

η(M_{1} ∪_{N} −M_{2}) + η(M_{2} ∪_{N} −M_{3}) + η(M_{3} ∪_{N} −M_{1}) = τ(l_{1}, l_{2}, l_{3}).  (20)

Lemma 2.12 Equation (20) holds for any three pairs (M_{i}, N), i = 1, 2, 3, of compact, oriented Riemannian manifolds of dimension 4k −1 with a boundary isometric to N and a product metric near the boundary.

Proof: We form the compact oriented Riemannian manifolds

Y_{1} := M_{1} ∪_{N} −M_{2}, Y_{2} := M_{2} ∪_{N} −M_{3}.

Since the oriented cobordism vanishes rationally

Ω_{4k−1}^{SO} ⊗ Q = {0}

in dimension 4k − 1, there are oriented manifolds with boundary

(X_{+}, nY_{1}), (X_{−}, mY_{2}), n, m ∈ N.

In fact, we can assume n = m. Applying now Theorem 2.10 to the 4-tuples

(X_{+}, nM_{1}, −nM_{2}, nN), (X_{−}, nM_{2}, −nM_{3}, nN),

we see that n-times the equation (20) holds. The Lemma follows. □

3 The proof of the gluing formula

3.1 The relative index of projections

Let H be a separable Hilbert space. The space of bounded operators L(H) contains the ideals K of compact operators and L^{1}(H) ⊂ K of trace class operators. Moreover, there is the family L^{1}(H) ⊂ L^{N}(H) ⊂ K of Schatten classes for N ≥ 1 (see Connes [23], Appendix). The Nth Schatten class consists of all operators A ∈ L(H) such that |A|^{N} is of trace class. A projection P ∈ L(H) is a selfadjoint idempotent, i.e. P^{∗} = P and P^{2} = P.

Let P, Q be infinite dimensional projections on H such that P − Q ∈ K. Then the operator P Q : Im(Q) → Im(P) is Fredholm, and its index is the relative index of P and Q denoted by I(P, Q). We consider the space h := Im(P) ⊕ Im(Q) and the operator

A := ( \begin{matrix}0 & P Q \\ W & 0 \end{matrix} ) ∈ L(h),

where W : Im(P) → Im(Q) is some isometric isomorphism which exists by the assumption on the dimensions of P, Q. Let

B := ( \begin{matrix}0 & W^{∗} \\ QP & 0 \end{matrix} ) ∈ L(h).

Then

S := 1 − BA = ( \begin{matrix}0 & 0 \\ 0 & Q(Q − P)Q \end{matrix} )

and

T := 1 − AB = ( \begin{matrix}P (P − Q)P & 0 \\ 0 & 0 \end{matrix} )

are compact operators. Thus, A is Fredholm with parametrix B and

index(A) = I(P, Q).

Assume now that S, T ∈ L^{N}(H), i.e. they are in the Nth Schatten class for some N ≥ 1. Then we have the following Proposition (proved e.g. in Connes [23])

Proposition 3.1 I(P, Q) := T r S^{N} − T r T^{N}.

In terms of the projections P, Q this formula is equivalent to

I(P, Q) = T r ((Q − QP Q)^{N} − (P − P QP)^{N} ) .

The following Lemma provides us with the commutator K announced in the outline of the proof in Subsection 1.6.

Lemma 3.2 If P, Q are projections on H, then

(Q − QP Q)^{N} − (P − P QP)^{N} = Q − P + K,

where K is a commutator given by

K = [(QP − P Q), \frac{1}{2} ∑_{i=0}^{N−1} ( \begin{matrix}N \\ i \end{matrix} ) (−1)^{N−i} (Q(P Q)^{N−i−1} + P (QP)^{N−i−1})] .

Proof: A simple computation shows that for l ≥ 0

[[Q, P], (Q(P Q)^{l} + P (QP)^{l})] = 2(Q(P Q)^{l+1} − P (QP)^{l+1}).  (21)

Now we expand the N’th powers (Q − QP Q)^{N} and (P − P QP)^{N} and use the fact that Q commutes with QP Q and P commutes with P QP. Inserting (21) into these expansions and collecting all commutator terms proves the Lemma. □

3.2 Construction of the isometry U

Recall the modified geometric situation introduced in Section 1.5 and the Hilbert spaces H := L^{2}(M, E) ⊕ L^{2}(Z, E_{Z}) and H_{0} := L^{2}(M_{1}, E_{1}) ⊕ L^{2}(M_{2}, E_{2}). We define an isometry U : H → H_{0}.

We start with choosing a function χ ∈ C^{∞}[−1, 1] with χ = 1 on [−1, −1/2], χ(r) ∈ [0, 1] if r ∈ [−1/2, 1/2] and χ = 0 on [1/2, 1] such that γ := \sqrt{1 − χ^{2}} is also smooth. χ gives rise to a function on Z depending only on the radial variable. We extend χ by 1 to the left on M_{1} and by 0 to the right on M_{2}. The corresponding extension of γ is γ := \sqrt{1 − χ^{2}}. Thus, we obtain functions on M, M_{1}, M_{2} also denoted by χ, γ.

We define operators a, b, c, d : H → H_{0} as follows (compare [12])

• a is the multiplication by χ on M followed by the transfer to M_{1}.

• b is the multiplication by χ on Z followed by the transfer to M_{2}.

• c is the multiplication by γ on M followed by the transfer to M_{2}.

• d is the multiplication by γ on Z followed by the transfer to M_{1}.

Here we use the canonical identifications of the different parts of M, M_{1}, M_{2} and Z and of the Dirac bundles over these parts.

Note that a^{∗}a is the multiplication by χ^{2} on M. Analogously, c^{∗}c is the multiplication by γ^{2} on M and similarly for b, d. Thus, a^{∗}a + b^{∗}b + c^{∗}c + d^{∗}d = 1_{H}. We define

U := a + b + c − d.

Then

U^{∗}U = a^{∗}a + b^{∗}b + c^{∗}c + d^{∗}d + a^{∗}(b + c − d)
+ b^{∗}(a + c − d) + c^{∗}(a + b − d) − d^{∗}(a + b + c)
= 1_{H} − a^{∗}d + b^{∗}c + c^{∗}b − d^{∗}a
= 1_{H},

where we have employed that a^{∗}d = c^{∗}b, b^{∗}c = d^{∗}a, a^{∗}b = 0, a^{∗}c = 0, b^{∗}a = 0, b^{∗}d = 0, c^{∗}a = 0, c^{∗}d = 0, d^{∗}b = 0 and d^{∗}c = 0. Analogously UU^{∗} = 1_{H_{0}}.

3.3 The difference of the positive spectral projections

We consider now the Dirac operators D_{+} on H and D_{0} on H_{0}. Their domains are specified by boundary conditions defined by Lagrangian subspaces L_{1} at {1}×N and L_{2} at {−1} × N. Since the transfer used above identifies the boundary components, where we have put the same condition, and U is defined by smooth cut-off functions, we have U^{∗} dom D_{0} = dom D_{+}. Let D_{−} := U^{∗}D_{0}U. Then we can consider D_{−} as a bounded perturbation of D_{+}. The operator G := D_{+} − D_{−} is the sum of non-local bundle endomorphisms over the flip interchanging the cylindrical parts of M ∪ Z. It is supported on (−1/2, 1/2) × N ∪ (−1/2, 1/2) × N in the cylindrical parts, and it is the sum of products of functions depending only on the r-coordinate with the composition of the flip and the multiplication by I.

We define Sobolev spaces H^{k}, k ∈ Z, using the operator D_{+}. We set H^{k} = dom D_{+}^{k} with the norm ∥ψ∥_{k} := ∥(1 + |D_{+}|^{k})ψ∥. Using D_{−} yields equivalent spaces. Let L^{N}(H) denote the N-th Schatten class of H. Let P_{±} be the positive spectral projections of D_{±}.

Proposition 3.3 The difference P_{+} −P_{−} is continuous from H to H^{1/2} and, hence, (P_{+} − P_{−}) ∈ L^{N}(H) for N > 2n.

Proof: We use the technique of [11], Sec.3. We represent P_{±} by strongly convergent integrals

P_{±} = 1/2 + lim_{S→∞} \frac{1}{2π} ∫_{−S}^{S}(D_{±} − ϵ + ıλ)^{−1}dλ,

where ϵ > 0 is smaller than the smallest positive eigenvalue of D_{±}. Then

P_{+} − P_{−} = \frac{1}{2π} ∫_{−∞}^{∞} [(D_{+} − ϵ + ıλ)^{−1} − (D_{−} − ϵ + ıλ)^{−1}] dλ

and this integral converges in the norm of L(H, H^{1/2}). In fact, using the resolvent identity we can estimate the norm of the integrand by

∥(D_{+} − ϵ + ıλ)^{−1}G(D_{−} − ϵ + ıλ)^{−1}∥_{L(H,H^{1/2})} ≤ C(1 + |λ|)^{−3/2}.

The Proposition now follows. □.

Applying now Proposition 3.1 we see that the relative index of P_{+} and P_{−} is well defined and can be computed by the formula given in Lemma 3.2. We have

I(P_{−}, P_{+}) = T r(P_{+} − P_{−} + K),

where

K := [[P_{+}, P_{−}], Q]
Q := \frac{1}{2} ∑_{i=0}^{N−1} ( \begin{matrix}N \\ i \end{matrix} ) (−1)^{N−i} (P_{+}(P_{−}P_{+})^{N−i−1} + P_{−}(P_{+}P_{−})^{N−i−1}) .

For Re(t) ≥ 0 we give now the extension of K to a holomorphic family of commutators K(t) announced in Section 1.6. We introduce the regularized ’projections’ P_{±}(t) = e^{−tD_{±}^{2}}P_{±}.

Definition 3.4 K(t) := [[P_{+}(t), P_{−}(t)], Q].

Then K(t) is of trace class and obviously T rK(t) = 0 if Re(t) > 0.

3.4 Finite propagation speed and comparison of heat kernels

In this section we compare the heat kernels of two Dirac operators in places where they coincide locally. We explain the general method (introduced by Cheeger-Gromov-Taylor [21] and employed for comparison in [10]), that will be frequently applied in the sequel.

For i = 0, 1 let (M_{i}, g_{i}) be Riemannian manifolds of dimension n and E_{i} → M_{i} be Dirac bundles with Dirac operators D_{i}. We assume that M_{i}, i = 0, 1, decompose as K_{i} ∪ U_{i} with U_{i} open and precompact, and there exists an isometry between U_{0} and U_{1} covered by a bundle isomorphism E_{0|U_{0}} → E_{1|U_{1}} intertwining with the Dirac operators. Let

H = L^{2}(K_{0}, E_{0}) ⊕ L^{2}(K_{1}, E_{1}) ⊕ L^{2}(U_{0}, E_{0}).

Identifying the sections over U_{0} with those over U_{1} there are natural embeddings L^{2}(M_{i}, E_{i}) → H. Let P_{i} be the projections onto these subspaces. We extend the Dirac operators to H by zero where they have not been defined before. Let H_{i} := D_{i}^{2}. Let W_{i}(t, x, y), t ≥ 0, be the heat kernel of D, i.e. the integral kernel of e^{−tH_{i}}. If M_{i} has a non-empty boundary, we assume a selfadjoint elliptic boundary condition. We represent the heat kernels by the Fourier transform in terms of the wave operator.

e^{−tH} = \frac{1}{\sqrt{4πt}} ∫_{−∞}^{∞} e^{− \frac{λ^{2}}{4t}} e^{ıλD}dλ ,

where the integral converges in the operator norm for t > 0. The operators D_{i}^{l}e^{−tH_{i}}, l ≥ 0, are smoothing and we represent the kernels D_{i}^{l}W_{i} by

D_{i}^{l}W_{i}(t, x, y) =< δ(x), D_{i}^{l}e^{−tH_{i}}δ(y) >∈ E_{i,x} ⊗ E_{i,y}^{∗}.

The following lemma states the finite propagation speed property of the wave operator (Chernoff [22]).

Lemma 3.5 (Finite propagation speed) The operators e^{ıλH_{i}} extend to all Sobolev spaces H^{k}(M_{i}, E_{i}) and, for ψ ∈ H^{k}(M_{i}, E_{i}), supp e^{ıλH_{i}}ψ is contained in a |λ|- neighbourhood of supp ψ.

Let x, y ∈ U_{i} such that max(dist(x, K_{1}), dist(y, K_{1})) =: u > 0. We can choose a family of smooth cut-off functions χ_{u}(r) such that χ_{u}(r) = 1 if |r| ≥ u/2 and χ_{u}(r)=0 if |r| ≤ u/4 such that the C^{k}-norms of χ_{u} depend polynomially on u^{−1}. Then

D_{0}^{l}W_{0}(t, x, y) − D_{1}^{l}W_{1}(t, x, y)  (22)
= < δ(x), \frac{1}{\sqrt{4πt}} ∫_{R} e^{− \frac{λ^{2}}{4t}} [D_{0}^{l}e^{ıλD_{0}} − D_{1}^{l}e^{ıλD_{1}}] dλδ(y) > .  (23)

We can exclude [−u, u] from the integration since by Lemma 3.5 the integrand vanishes in that interval, if it is applied to the delta distributions. Thus,

D_{0}^{l}W_{0}(t, x, y) − D_{1}^{l}W_{1}(t, x, y)  (24)
= < δ(x), \frac{1}{\sqrt{4πt}} ∫_{R} χ_{u}(λ)e^{− \frac{λ^{2}}{4t}} [D_{0}^{l}e^{ıλD_{0}} − D_{1}^{l}e^{ıλD_{1}}] dλδ(y) > .

Using partial integration in order to regularize the difference, (24) can be estimated by

Ce^{− \frac{cu^{2}}{t}} ∥δ(x)∥_{H^{−k}(B_{ϵ}(x),E)^{∗}⊗E_{x}} ∥δ(y)∥_{H^{−k}(B_{ϵ}(y),E)^{∗}⊗E_{y}}.

Here B_{ϵ}(x) is the ball of radius ϵ > 0 centered at x and c > 0. If k > n/2, the norms of the δ-distributions are bounded.

Lemma 3.6 (Estimate of the difference of heat kernels)

∥D_{0}^{l}W_{0}(t, x, y) − D_{1}^{l}W_{1}(t, x, y)∥ < Ce^{− \frac{cu^{2}}{t}} .  (25)

By essentially the same technique we also have the off-diagonal estimate

Lemma 3.7 (Off-diagonal estimate of a heat kernel)

∥D_{i}^{l}W_{i}(t, x, y)∥ < Ce^{− \frac{c dist(x,y)^{2}}{t}}  (26)

for all x, y ∈ M_{i}.

The finite propagation speed property fails to be true along the boundary ∂M_{i}, where a global boundary condition is posed. Let x ∈ M_{0} and consider the distribution ϕ_{λ} := e^{ıλD_{0}}δ(x). As long as |λ| < dist(∂M_{0}, x) we have supp(ϕ_{λ}) ⊂ B(x, |λ|). If supp(ϕ_{λ}) hits ∂M_{0}, then it instantaneously spreads out along ∂M_{0}. Thus, supp(ϕ_{λ}) ⊂ B(x, |λ|) ∪ {y ∈ M_{0} | dist(y, ∂M_{0}) ≤ |λ| − dist(x, ∂M_{0})}. The same holds on M_{1}. Consider now the integrand of (22). It vanishes as long as the supports of e^{ıλD_{i}}δ(z), i = 0, 1, z = x, y are contained in the set where D_{0} and D_{1} coincide. Let us redefine the distance of points of U_{i} from K_{i}. For x ∈ U_{i} let

d̃ist(x, K_{i}) := min(dist(x, K_{i}), dist(x, ∂M_{i}) + dist(∂M_{i}, K_{i})).

With this definition of the distance the estimate (25) holds. A similar modification works for (26).

3.5 Comparison away from the cylinder

Recall the definition of R(s, t)_{±}:

R(s, t)_{±} = e^{−tD_{±}^{2}}sign(D_{±})|D_{±} + E_{D_{±}}{0}|^{−s}.

We use the finite propagation speed method in order to compare these functions away from the set (−5/8, 5/8) × N ∪ (−5/8, 5/8) × N ⊂ M ∪ Z, where D_{+} and D_{−} differ. Let v, l, r be smooth cut-off functions with l + v + r = 1 and such that • supp v ⊂ [−6/8, 6/8] × N ∪ [−6/8, 6/8] × N

• supp l ⊂ M̃_{1} ∪_{N} Z ∪ Z

• l = 0 on [−5/8, 1] × N ∪ [−5/8, 1] × N

• supp r ⊂ Z ∪ Z ∪_{N} M̃_{2}

• r = 0 on [−1, 5/8] × N ∪ [−1, 5/8] × N

holds. Let W_{±}(t, x, y) be the integral kernels of e^{−tD_{±}^{2}} and M_{+} := M ∪ Z. The kernels W_{±}(t, x, y) are smooth on (t, x, y) ∈ (0, ∞) × M_{+} × M_{+} up to the boundary of M_{+}. They diverge on the diagonal if t → 0. However, these divergences cancel each other if one forms differences. This can by quantified by the finite propagation speed method which provides the following estimates.

Corollary 3.8 There are constants c > 0 and C < ∞ such that for all t > 0, (x, y) ∈ M_{+} × M_{+}

|l(x)(W_{+}(t, x, y) − W_{−}(t, x, y))| ≤ Ce^{− \frac{c}{t}}
|l(x)(D_{+}W_{+}(t, x, y) − D_{−}W_{−}(t, x, y))| ≤ Ce^{− \frac{c}{t}} e^{−ct}
|(W_{+}(t, x, y) − W_{−}(t, x, y))l(x)| ≤ Ce^{− \frac{c}{t}}
|(D_{+}W_{+}(t, x, y) − D_{−}W_{−}(t, x, y))l(x)| ≤ Ce^{− \frac{c}{t}} e^{−ct}
|r(x)(W_{+}(t, x, y) − W_{−}(t, x, y))| ≤ Ce^{− \frac{c}{t}}
|r(x)(D_{+}W_{+}(t, x, y) − D_{−}W_{−}(t, x, y))| ≤ Ce^{− \frac{c}{t}} e^{−ct}
|(W_{+}(t, x, y) − W_{−}(t, x, y))r(x)| ≤ Ce^{− \frac{c}{t}}
|(D_{+}W_{+}(t, x, y) − D_{−}W_{−}(t, x, y))r(x)| ≤ Ce^{− \frac{c}{t}} e^{−ct}.

Similar estimates hold for higher derivatives in the (x, y)-coordinates.

Lemma 3.9 The integral kernels of the restricted differences

l(R(s, t)_{+} − R(s, t)_{−})
r(R(s, t)_{+} − R(s, t)_{−})
(R(s, t)_{+} − R(s, t)_{−})l
(R(s, t)_{+} − R(s, t)_{−})r

are smooth and uniformly bounded on (s, t, x, y) ∈ (−1/3, C] × [0, ∞] × M_{+} × M_{+}. In particular, the limits

lim_{t→0} l(R(0, t)_{+} − R(0, t)_{−}) = l(R(0, 0)_{+} − R(0, 0)_{−})
lim_{t→0} r(R(0, t)_{+} − R(0, t)_{−}) = r(R(0, 0)_{+} − R(0, 0)_{−})
lim_{t→0}(R(0, t)_{+} − R(0, t)_{−})l = (R(0, 0)_{+} − R(0, 0)_{−})l
lim_{t→0}(R(0, t)_{+} − R(0, t)_{−})r = (R(0, 0)_{+} − R(0, 0)_{−})r

exist in the trace norm.

Proof: The operator valued function R(s, t) has the following integral representation:

R(s, t) = \frac{1}{Γ(\frac{s+1}{2})} ∫_{0}^{∞} u^{\frac{s−1}{2}} De^{−(u+t)D^{2}}du.  (27)

The integral kernel of difference l(R(s, t)_{+} − R(s, t)_{−}) is given by the integral

\frac{1}{Γ(\frac{s+1}{2})} ∫_{0}^{∞} u^{\frac{s−1}{2}} l(x)(D_{+}W_{+}(t + u, x, y) − D_{−}W_{−}(t + u, x, y))du.

By Corollary 3.8 this integral absolutely converges in C^{k}, k ≥ 0, in the region (s, t, x, y) ∈ (−1/3, C] × [0, ∞] × M_{+} × M_{+}. The claim for the kernel of l(R(s, t)_{+} − R(s, t)_{−}) follows. The other differences are treated similarly. □

It remains to consider the behaviour of

v(R(0, t)_{+} − R(0, t)_{−})v

as t → 0. Unfortunately it does not converge in the trace norm. This is exactly the reason for introducing other families R̃_{±}(s, t) to compare with.

3.6 Comparison on the cylinder

In this section we carry out the construction of the operators B(t), K̃(t) used for comparison on the cylinder. Note that M ∪ Z contains two copies of the cylinder Z, namely Z itself and the middle part of M. Thus, we have a canonical embedding

L^{2}(Z, E_{Z}) ⊕ L^{2}(Z, E_{Z}) → H,

where the first summand is the subspace of L^{2}(M, E). Moreover, M_{1} ∪ M_{2} contains two copies of the cylinder Z (recall the modified geometry in Subsection 1.5) and we have the embedding

L^{2}(Z, E_{Z}) ⊕ L^{2}(Z, E_{Z}) → H_{0}.

Then U restricts to a unitary isomorphism of these subspaces. Let D̃_{+} be the sum of two copies of D_{Z} with boundary conditions given by the Lagrangian subspaces L_{1}, L_{2} as before and let D̃_{0} be the sum of two copies of D_{Z} now considered on H_{0}. Define D̃_{−} := U^{∗}D̃_{0}U. We extend these operators by zero to the complement of the cylinders. Repeating all the constructions above for D̃_{±} we obtain R̃(s, t)_{±}, P̃_{±}, P̃_{±}(t) and K̃(t). Let W̃_{±}(t, x, y) be the integral kernels of e^{−tD̃_{±}^{2}}. They are smooth on (t, x, y) ∈ (0, ∞) × [Z ∪ Z] × [Z ∪ Z]. By the finite propagation speed method we have Corollary 3.10 There are constants c > 0 and C < ∞ such that for all t > 0, (x, y) ∈ M_{+} × M_{+}

|l(x)(W̃_{+}(t, x, y) − W̃_{−}(t, x, y))| ≤ Ce^{− \frac{c}{t}}
|l(x)(D̃_{+}W̃_{+}(t, x, y) − D̃_{−}W̃_{−}(t, x, y))| ≤ Ce^{− \frac{c}{t}} e^{−ct}
|(W̃_{+}(t, x, y) − W̃_{−}(t, x, y))l(x)| ≤ Ce^{− \frac{c}{t}}
|(D̃_{+}W̃_{+}(t, x, y) − D̃_{−}W̃_{−}(t, x, y))l(x)| ≤ Ce^{− \frac{c}{t}} e^{−ct}
|r(x)(W̃_{+}(t, x, y) − W̃_{−}(t, x, y))| ≤ Ce^{− \frac{c}{t}}
|r(x)(D̃_{+}W̃_{+}(t, x, y) − D̃_{−}W̃_{−}(t, x, y))| ≤ Ce^{− \frac{c}{t}} e^{−ct}
|(W̃_{+}(t, x, y) − W̃_{−}(t, x, y))r(x)| ≤ Ce^{− \frac{c}{t}}
|(D̃_{+}W̃_{+}(t, x, y) − D̃_{−}W̃_{−}(t, x, y))r(x)| ≤ Ce^{− \frac{c}{t}} e^{−ct}.

Similar estimates hold for higher derivatives in the (x, y)-coordinates.

Lemma 3.11 The integral kernels of the restricted differences

l(R̃(s, t)_{+} − R̃(s, t)_{−})
r(R̃(s, t)_{+} − R̃(s, t)_{−})
(R̃(s, t)_{+} − R̃(s, t)_{−})l
(R̃(s, t)_{+} − R̃(s, t)_{−})r

are smooth and uniformly bounded on (s, t, x, y) ∈ (−1/3, C] × [0, ∞] × [Z ∪ Z] × [Z ∪ Z]. In particular the limits

lim_{t→0} l(R̃(0, t)_{+} − R̃(0, t)_{−}) = l(R̃(0, 0)_{+} − R̃(0, 0)_{−})
lim_{t→0} r(R̃(0, t)_{+} − R̃(0, t)_{−}) = r(R̃(0, 0)_{+} − R̃(0, 0)_{−})
lim_{t→0}(R̃(0, t)_{+} − R̃(0, t)_{−})l = (R̃(0, 0)_{+} − R̃(0, 0)_{−})l
lim_{t→0}(R̃(0, t)_{+} − R̃(0, t)_{−})r = (R̃(0, 0)_{+} − R̃(0, 0)_{−})r

exist in the trace norm.

Lemma 3.12

T r(R̃(s, t)_{+} − R̃(s, t)_{−}) = 0
T r(P̃_{+} − P̃_{−} + K̃) = 0.

Proof: The first equation is obvious since R̃(s, t)_{+} and R̃(s, t)_{−} are unitary equivalent. The second equation is true since one can interchange the roles of D̃_{±} by applying U. □

Note that the domains of D_{±} and D̃_{±} are not compatible, but D_{±} and D̃_{±} coincide on ([−7/8, 7/8] × N) ∪ ([−7/8, 7/8] × N) locally.

Lemma 3.13 The integral kernels of v(R(s, t)_{±} − R̃(s, t)_{±})v are uniformly bounded and smooth on (s, t, x, y) ∈ [−1/3, C] × [0, ∞] × M_{+} × M_{+}. In particular, the limits

lim_{t→0} v(R(0, t)_{±} − R̃(0, t)_{±})v = v(R(0, 0)_{±} − R̃(0, 0)_{±})v

exist in the trace norm.

Proof: We employ again the integral representation (27). The finite propagation speed method provides constants C < ∞, c > 0 such that for all t > 0 and (x, y) ∈ M_{+} × M_{+}

|v(x)(D_{±}W_{±}(t, x, y) − D̃_{±}W̃_{±}(t, x, y))v(y)| ≤ Ce^{−ct}e^{− \frac{c}{t}}

holds. A similar estimate holds for higher derivatives in the (x, y)-coordinates. □ Define B(t) = R̃(s, t)_{+} − R̃(s, t)_{−}. Then

s − lim_{t→0} B(t) = 2P̃_{+} − 2P̃_{−} + ED̃_{+}{0} − ED̃_{−}{0}.

3.7 The proof of the gluing formula

In the preceding sections we have constructed families of trace class operators A(t), K(t), B(t) and K̃(t). We already know that

lim_{t→0} T rA(t) = lim_{t→0} T r(A(t) + 2K(t)) = δ

(see equation (5)). Moreover, we know that

A(0) + 2K(0) = 2(P_{+} − P_{−}) + 2K + E_{D_{+}}{0} − E_{D_{−}}{0}

is of trace class. Since the limit

lim_{t→0}(A(t) + 2K(t))

does not exist in the trace norm, we use the family B(t) + 2K(t) for comparison. In fact,

T r(A(t) + 2K(t)) = T r(A(t) + 2K(t) − B(t) − 2K̃(t))

and

T r(A(0) + 2K(0)) = T r(A(0) + 2K(0) − B(0) − 2K̃(0))

by Lemma 3.12.

Theorem 3.14 The limit

lim_{t→0}(A(t) + 2K(t) − B(t) − 2K̃(t)) = A(0) + 2K(0) − B(0) − 2K̃(0)

exists in the trace norm. Hence,

δ = T r(A(0) + 2K(0)) = −2I(P_{+}, P_{−}) + dim kerD_{+} − dim kerD_{−}.

Proof: It is sufficient to show that the following families converge in the trace norm.

l(A(t) + 2K(t) − B(t) − 2K̃(t))
r(A(t) + 2K(t) − B(t) − 2K̃(t))
v(A(t) + 2K(t) − B(t) − 2K̃(t))(l + r)
v(A(t) + 2K(t) − B(t) − 2K̃(t))v.

According to the Lemmas 3.9, 3.11 and 3.13 it is enough to show the convergence in the trace norm of the terms lK(t), lK̃(t), rK(t), rK̃(t), vK(t)(l + r), vK̃(t)(l + r), v(K(t) − K̃(t))v. The arguments for all these terms are very similar. Thus, we consider only the first and the last one. Let us start with

lK(t) = l [[P_{+}(t), P_{−}(t)], Q]
= l[P_{+}(t), P_{−}(t)]Q − lQ[P_{+}(t), P_{−}(t)].

Let l_{1} be a smooth cut-off function with ll_{1} = l and l_{1} = 0 in a 1/100-neighbourhood of ([−6/8, 6/8]×N)∪([−6/8, 6/8]×N). Inserting l_{1} between Q and the commutator provides

lK(t) = l[P_{+}(t), P_{−}(t)]Q − lQl_{1}[P_{+}(t), P_{−}(t)] − lQ(1 − l_{1})[P_{+}(t), P_{−}(t)].

It is enough to show that

l[P_{+}(t), P_{−}(t)] →^{t→0} l[P_{+}, P_{−}]  (28)
l_{1}[P_{+}(t), P_{−}(t)] →^{t→0} l[P_{+}, P_{−}]  (29)
lQ(1 − l_{1})[P_{+}(t), P_{−}(t)] →^{t→0} [P_{+}, P_{−}]  (30)

exist in the trace norm. First we consider the term (28). For this, we need the following Lemmas.

Lemma 3.15 The integral kernel of l(P_{+}(t) − P_{−}(t)) is uniformly bounded and smooth on (t, x, y) ∈ [0, ∞) × M_{+} × M_{+}.

Proof: We represent P_{±}(t) as follows.

2P_{±}(t) = R(0, t)_{±} + e^{−tD_{±}^{2}} − e^{−tD_{±}^{2}}E_{D_{±}}{0}

Hence, the claim follows immediately from Corollary 3.8 and Lemma 3.9. □

The same is true for l replaced by l_{1}. Applying the finite propagation speed method in order to obtain an off-diagonal estimate we obtain

Lemma 3.16 The integral kernels of

lP_{±}(t)(1 − l_{1}), (1 − l_{1})P_{±}(t)l

are uniformly bounded and smooth on (t, x, y) ∈ [0, ∞) × M_{+} × M_{+}.

We have

l[P_{+}(t), P_{−}(t)] = lP_{+}(t)(P_{−}(t) − P_{+}(t)) − l(P_{−}(t) − P_{+}(t))P_{+}(t). (31)

We discuss the first term of the r.h.s. of (31)

lP_{+}(t)(P_{−}(t)−P_{+}(t)) = lP_{+}(t)l_{1}(P_{−}(t)−P_{+}(t))+lP_{+}(t)(1−l_{1})(P_{−}(t)−P_{+}(t)). (32)

Let us again consider the first term in more detail. By Lemma 3.15 one can find a compact operator T and a family S(t) converging in the trace norm as t → 0 such that l_{1}(P_{−}(t) − P_{+}(t)) = T S(t). Since lP_{+}(t) →^{t→0} lP_{+} strongly, the family P_{+}(t)T converges uniformly. Hence, P_{+}(t)T S(t) = lP_{+}(t)l_{1}(P_{−}(t) − P_{+}(t)) converges in the trace norm. The second term in (32) is handled in a similar way using Lemma 3.16. Analogously, we show the convergence in the trace norm of the second term in (31). The same method applies to (29). The term (30) converges in the trace norm since

• lQ(1 − l_{1}) has a bounded smooth kernel. This can also be shown with the finite propagation speed method without appealing for the pseudodifferential calculus and using the fact that l(1 − l_{1}) = 0.

• [P_{+}(t), P_{−}(t)] converges strongly as t → 0.

Hence, we have proved the convergence in the trace norm of lK(t) as t → 0.

Now we show how to deal with v(K(t) − K̃(t))v. Let v_{1} be a smooth cut-off function with v_{1}v = v and supp v_{1} ⊂ ([−6/8, 6/8] × N) ∪ ([−6/8, 6/8] × N). Set

Q̃ := \frac{1}{2} ∑_{i=0}^{N−1} ( \begin{matrix}N \\ i \end{matrix} ) (−1)^{N−i} (P̃_{+}(P̃_{−}P̃_{+})^{N−i−1} + P̃_{−}(P̃_{+}P̃_{−})^{N−i−1}) .

We have to consider

v(K(t) − K̃(t))v
= v [[P_{+}(t), P_{−}(t)], Q] v − v [[P̃_{+}(t), P̃_{−}(t)], Q̃] v
= v [[P_{+}(t), P_{−}(t)], (Q − Q̃)] v  (33)
+ v [[P_{+}(t), P_{−}(t)] − [P̃_{+}(t), P̃_{−}(t)], Q̃] v.  (34)

We write the term (33) as

v [[P_{+}(t), P_{−}(t)], (Q − Q̃)] v
= v[P_{+}(t), P_{−}(t)](Q − Q̃)v − v(Q − Q̃)[P_{+}(t), P_{−}(t)]v
= v[P_{+}(t), P_{−}(t)]v_{1}(Q − Q̃)v − v(Q − Q̃)v_{1}[P_{+}(t), P_{−}(t)]v
+ v[P_{+}(t), P_{−}(t)](1 − v_{1})(Q − Q̃)v − v(Q − Q̃)(1 − v_{1})[P_{+}(t), P_{−}(t)]v.

All these terms converge in the trace norm as t → 0. To see this we use the following facts:

• v_{1}(Q − Q̃)v has a smooth bounded integral kernel

• v[P_{+}(t), P_{−}(t)]v_{1} converges strongly

• v[P_{+}(t), P_{−}(t)](1 − v_{1}) converges in the trace norm due to the off-diagonal estimates.

The term (34) is handled in a similar way. Thus, we have proved the theorem. □

This also finishes the proof of the gluing formula for the η-invariant Theorem 1.9:

η(M)−η(M_{1}, L_{1})−η(M_{2}, L_{2}) = m(L_{1}, L_{2})−2I(P_{+}, P_{−})+dimker D_{+} −dim ker D_{−}.

Probably this formula could also be proved using a variation formula. In fact D_{+}−D_{−} = G is a compactly supported bundle endomorphism (non-local!). It seems not to be too complicated to show that the variation of the η-invariant for the family D_{t} = D_{+} − tG vanishes. Then one has only to count the jumps of the η-invariant of D_{t}. The resulting integer is exactly −2I(P_{+}, P_{−}) + dim ker D_{+} − dim ker D_{−}. An related application of the variation formula was discussed by Cheeger [20].

4 Vanishing of the relative index term

4.1 The space of limiting values is Lagrangian

We start with proving Proposition 1.14. Let (M, N) be a compact odd-dimensional Riemannian manifold with right boundary N, and E → M be a Dirac bundle with Dirac operator D. We assume a product structure for the Riemannian metric and for E near N as explained in the Subsection 1.3. Moreover, let B be the closure of D with boundary condition B_{{V }} (Subsection 1.8), i.e. for ϕ ∈ dom B we require that P_{−}ϕ_{|N} = 0, where P_{−} is the negative spectral projection of D_{N}. Recall the definition of

L := L_{M} := {pr_{V} ϕ_{|N} | ϕ ∈ ker B}.

L is a subspace of the symplectic vector space V := ker D_{N} with the symplectic structure given by Φ(u, v) =< Iu, v >_{L^{2}}.

Proposition 4.1 L is a Lagrangian subspace of V, i.e. L ⊥ IL and L ⊕ IL = V.

Proof: We show that the restriction of Φ to L vanishes. Let u, v ∈ L and ϕ, ψ ∈ ker B with pr_{V} ϕ_{|N} = u, pr_{V} ψ_{|N} = v. Then, because of IP_{−}I = −P_{+}, by the partial integration formula for the Dirac operator

0 = < Bϕ, ψ >_{L^{2}} − < ϕ, Bψ >_{L^{2}}
= < Iu, v >_{L^{2}(N,E_{N})}= Φ(u, v).

Since this is true for all pairs u, v ∈ L, the claim follows. Note that B^{∗} is the closure of D with the initial domain given by the boundary condition B_{{0}}. Thus, a vector ϕ ∈ ker B satisfying pr_{V} ϕ_{|N} = 0 belongs already to ker B^{∗}. It follows

dim L = dim ker B − dim ker B^{∗} = index B.

It remains to show that index B = dim V/2. We consider M̄ := [−1, 0] × N and let Ē → M̄ be the Dirac bundle induced from that of M over the product collar with Dirac operator D̄. We define B̄ to be the closure of D̄ subject to the boundary condition B_{V} at {0} × N and BL̄ at {−1} × N, where L̄ is an arbitrary Lagrangian subspace of V. By separation of variables one can compute ker B̄ ≅ L̄ and coker B̄ = {0}. Hence, index B̄ = dim L̄ = dim V/2. We can apply a variant of the relative index theorem of Gromov-Lawson [26] in order to infer index B = dim V/2. In fact, B̄ and B coincide on the cocompact set (−ϵ, 0] × N and are both Fredholm. Hence, the difference of their indices is the difference of integrals of index densities (the left boundary of M̄ does not contribute since we have chosen a selfadjoint boundary condition). Since the dimension of M is odd, this integral vanishes. This proves the claim and finishes also the proof of the Proposition. □

4.2 Adjusting U in the adiabatic limit

Recall that M, M_{i}, D, D_{i} depend on a parameter r > 1 that is half of the length of the cylinder glued in at the the hypersurface N as explained in Subsection 1.8. We will now make a more convenient r-dependent choice of the unitary U. Let F_{s} : [−s, s] × (N ∪ N) → [−1, 1] × (N ∪ N) be the stretching map and F^{∗} be the induced pull back of sections of the Dirac bundle. Let U_{1} denote the isometry U defined above on the cylinder of length 2. For r ≥ 1 we define

• U_{r} := U_{1} outside [−r/2, r/2] × (N ∪ N)

• U_{r} := F_{r}^{∗}U_{1}F_{r}^{−1∗} on [−r/2, r/2] × (N ∪ N).

In what follows we will again omit the index r assuming this new choice of U. Then also G becomes r-dependent and supported in [−r/2, r/2] × (N ∪ N). Since G involves derivatives of cut-off functions there is a C < ∞ such that

∥G∥ ≤ \frac{C}{r} .  (35)

Since in Theorem 1.9 the relative index term I(P_{+}, P_{−}) (note that P_{±} are the positive spectral projections of D_{±}) is the only one depending on the unitary U, it turns out that I(P_{+}, P_{−}) is in fact independent of the choice of U.

4.3 The operator B(u)

Recall the definition of D(u) := D_{−} + uG, u ∈ [0, 1], with G := D_{+} − D_{−}. As explained below the statement of Theorem 1.17 we have to show the vanishing of the spectral flow of this family of Dirac operators with constant principal symbol if r is large. These operators live on M ∪ Z. Note that M ∪ Z contains cylindrical parts Z ∪Z, Z^{+} := [r/2, r]×(N ∪N) and Z^{−} := [−r, −r/2]×(N ∪N), where the operator D(u) has a product structure for all u. We refer to the component of Z ∪ Z which is not glued to M̃_{i} as the second component, while the first component is a part of M. On Z ∪ Z we define another family B(u) as an extension of (D_{Z} ⊕ D_{Z}) + (u − 1)G subject to the following boundary conditions:

• We require B_{L_{1}} at the left boundary of the second component {−r} × N and B_{L_{2}} at the right boundary of the second component {r} × N.

• On the first component we require that ψ ∈ dom B(u) satisfies ψ ∈ (L_{1} ⊕ N_{+}) at {−r} × N and ψ ∈ (L_{2} ⊕ N_{−}) at {r} × N, where N_{±} are the positive and negative spectral subspaces of D_{N}.

On the first component the boundary condition is non-elliptic. However B(u) decomposes into a sum of one-dimensional operators labeled by spec(D_{N}). These one-dimensional operators are essentially selfadjoint and elliptic. This gives the selfadjoint extension of B(u). Note that D(u) and B(u) coincide locally over Z^{±}. We will compare the spectral flow of B(u) with that of D(u) for large r. Since this concerns the small eigenvalues, actually only a part of B(u) is interesting. The next Lemma takes the special structure of G into account.

Lemma 4.2 Let h_{0} := L^{2}([−r, r], V ⊕ V) ⊂ L^{2}(Z ∪ Z, E_{Z} ∪ E_{Z}) be the subspace of ψ with ψ(s) ∈ V ⊕ V for all s ∈ [−r, r]. Then B(u) leaves h_{0} invariant.

Proof: This holds for B(1) = I(∂/∂r +(D_{N} ⊕D_{N})). However G consists of compositions of multiplication operators ρI with functions ρ only depending on the r-variable with transportation using the flip interchanging the two copies of Z. Hence, G also leaves h_{0} invariant. □

We define B_{0}(u) to be the restriction of B(u) to h_{0}. Let

d := inf(|spec(D_{N})| \ {0}).

Note that |spec(B(1)_{|h_{0}^{⊥}})| ≥ d. There is an r_{0} such that if r ≥ r_{0} ≥ 1, then by (35) ∥G∥ ≤ d/2 and also |spec(B(u)_{|h_{0}^{⊥}})| ≥ d/2 for all u ∈ [0, 1]. In this case all eigenvectors of B(u) corresponding to eigenvalues λ with |λ| ≤ d/2 are in fact eigenvectors of B_{0}(u). Hence, for r > r_{0} we have sf{B(u)} = sf{B_{0}(u)}.

4.4 The structure of the eigenvectors on the cylinder

We identify L^{2}(Z ∪ Z, E_{Z} ∪ E_{Z}) with L^{2}([−r, r], L^{2}(N, E_{N}) ⊕ L^{2}(N, E_{N})). Let N_{±} ⊂ L^{2}(N, E_{Z}) be the positive and negative spectral subspaces of D_{N}, respectively. If ψ is an eigenvector of D(u) or B(u) corresponding to the eigenvalue λ, then we can split it over Z ∪ Z as

ψ = ψ_{0} + ψ_{+} + ψ_{−},

where ψ_{0}(s) ∈ (V ⊕ V), ψ_{+}(s) ∈ N_{+} ⊕ N_{+} and ψ_{−}(s) ∈ (N_{−} ⊕ N_{−}).

We give now expressions for the eigenfunctions of I(∂/∂r + D_{N}). Let {h_{μ}} and {Ih_{μ}} be orthonormal bases of N_{+} and N_{−} such that D_{N}h_{μ} = μh_{μ} with μ > 0 and D_{N}Ih_{μ} = −μIh_{μ}. If |λ| ≤ d/2, we can define α_{μ} ∈ (−π/2, π/2) by sinα_{μ} = λ/μ. Let

H_{μ}^{+}(s) := [(1 − cosα_{μ})h_{μ} + sinα_{μ}Ih_{μ}]e^{sμcosα_{μ}}  (36)
H_{μ}^{−}(s) := [sinα_{μ}h_{μ} + (1 − cosα_{μ})Ih_{μ}]e^{−sμcosα_{μ}}.  (37)

Then one can check I(∂/∂r + D_{N})H_{μ}^{±} = λH_{μ}^{±}.

4.5 Definition of the map K

We start now with the comparison of the spectral flow of the families D(u) and B(u) by defining a linear, almost unitary map K mapping eigenvectors of B(u) corresponding to small eigenvalues to approximate eigenvectors of D(u) of the same eigenvalue. The approximation becomes better when r becomes larger and |λ| smaller. Starting from now on we define D(u) and B(u) using the Lagrangian subspaces L_{1} := L_{M_{1}} and L_{2} := L_{M_{2}} given by the limiting values. Note that M_{2} is considered as a right manifold. L_{1} is employed at {−r} × (N ∪ N), while L_{2} at {r} × (N ∪ N).

We fix linear maps x_{i} : L_{i} → ker B̃_{i} (B̃_{i} is the extension of D̃_{i} subject to the boundary condition B_{V} living on M̃_{i}, i.e. without the cylinder glued on (see Subsection 1.8), such that pr_{V} x_{1}(v)(−r) = v, v ∈ L_{1} and pr_{V} x_{2}(u)(r) = u, u ∈ L_{2}. Note that every element ϕ ∈ ker B̃_{1} has an obvious extension to the first component of Z^{−} of the form ϕ = ϕ_{0} + ϕ_{+}, where

∥ϕ_{+}(s)∥_{N} ≤ Ce^{c(−r−s)}∥ϕ_{0}(−r)∥_{N}  (38)

and ∥.∥_{N} is the norm on L^{2}(N, E_{N}). Every χ ∈ ker B̃_{2} has an extension χ_{0} + χ_{−} to the first component of Z_{r}^{+} with

∥χ_{−}(s)∥_{N} ≤ Ce^{c(s−r)}∥χ_{0}(r)∥_{N}.  (39)

We choose smooth cut-off functions α_{±} with

• α_{−}(s) = 1 for s < −3r/4 − 1,

• α_{−}(s) = 0 for s > −3r/4 + 1,

• α_{+}(s) = 1 for s > 3r/4 + 1,

• α_{+}(s) = 0 for s < 3r/4 − 1 such that the C^{1}-norm is uniformly bounded with respect to r. Let ψ be an eigenvector of B(u) corresponding to an eigenvalue λ with |λ| ≤ d/2. Then we have ψ = ψ_{0} + ψ_{−} + ψ_{+} on Z ∪ Z. Since λ is small, actually ψ = ψ_{0}. Moreover, ψ_{0}(−r) ∈ L_{1} and ψ_{0}(r) ∈ L_{2} on both components.

Let ϕ := x_{1}(ψ_{0}(−r)) and χ := x_{2}(ψ_{0}(r)) (we take the boundary values on the first component). We define K(ψ) to be ψ on the second component of Z ∪ Z and by

• ϕ on M̃_{1}

• ψ + ϕ_{+} on [−r, −3r/4 − 1] × N

• ψ + α_{−}ϕ_{+} on [−3r/4 − 1, −3r/4 + 1] × N

• ψ on [−3r/4 + 1, 3r/4 − 1] × N

• ψ + α_{+}χ_{−} on [3r/4 − 1, 3r/4 + 1] × N

• ψ + χ_{−} on [3r/4 + 1, r] × N

• χ on M̃_{2}. Then K(ψ) is continuous, piecewise smooth and satisfies the boundary conditions for D(u). We extend K linearly to linear combinations of eigenvectors. Lemma 4.3 (K maps eigenvectors to approximate eigenvectors) There are constants C < 0, c > 0 and r_{0} ≥ 2 such that the following holds for all r ≥ r_{0}, u ∈ [0, 1]: Let ψ be a normed eigenvector of B(u) to the eigenvalue λ with |λ| ≤ d/2. Then

∥D(u)K(ψ) − λK(ψ)∥ ≤ C(\frac{|λ|}{\sqrt{r}} + e^{−cr}).

If ψ_{1}, ψ_{2} are two such eigenvectors, then

| < ψ_{1}, ψ_{2} > − < K(ψ_{1}), K(ψ_{2}) > | ≤ \frac{C}{\sqrt{r}}.

Proof: We compute D(u)K(ψ) − λK(ψ) obtaining 0 on the second component, and on the first component the result is

• −λϕ on M̃_{1}

• −λϕ_{+} on [−r, −3r/4 − 1] × N

• grad α_{−}ϕ_{+} − λα_{−}ϕ_{+} on [−3r/4 − 1, −3r/4 + 1] × N

• 0 on [−3r/4 + 1, 3r/4 − 1] × N

• grad α_{+}χ_{−} − λα_{+}χ_{−} on [3r/4 − 1, 3r/4 + 1] × N • −λχ_{−} on [3r/4 + 1, r] × N

• −λχ on M̃_{2}.

Note that ψ_{0} is oscillating. Hence,

1 = ∥ψ∥_{L^{2}} ≥ ∥ψ_{0}∥_{L^{2}(Z^{±},E_{Z})} = \sqrt{r/2}∥ψ_{0}(±r)∥_{N}.  (40)

Applying this and (38), (39) we obtain

• ∥λϕ∥_{L^{2}} ≤ C \frac{|λ|}{\sqrt{r}} on M̃_{1}

• ∥λϕ_{+}∥_{L^{2}} ≤ C \frac{|λ|}{\sqrt{r}} on [−r, −3r/4 − 1] × N

• ∥grad α_{−}ϕ_{+} − λα_{−}ϕ_{+}∥_{L^{2}} ≤ C \frac{1}{\sqrt{r}}e^{−cr} on [−3r/4 − 1, −3r/4 + 1] × N

• ∥grad α_{+}χ_{−} − λα_{+}χ_{−}∥_{L^{2}} ≤ C \frac{1}{\sqrt{r}}e^{−cr} on [3r/4 − 1, 3r/4 + 1] × N

• ∥λχ_{−}∥_{L^{2}} ≤ C \frac{|λ|}{\sqrt{r}} on [3r/4 + 1, r] × N

• ∥λχ∥_{L^{2}} ≤ C \frac{|λ|}{\sqrt{r}} on M̃_{2}.

Summing up the terms we obtain the first inequality of the Lemma. In order to see the second inequality note that most of the mass of ψ_{i} is concentrated in (ψ_{i})_{0} on the cylinder. □

Thus for large r we know by the variational principle, that for every small eigenvalue λ of multiplicity m of B(u), there are eigenvalues of D(u) of total multiplicity at least m in a neighbourhood of λ of size C(\frac{|λ|}{\sqrt{r}} + e^{−cr}). In order to check that we cover all small eigenvalues with the right multiplicity we will construct a map A from the eigenvectors of D(u) to the approximate eigenvectors of B(u).

4.6 The small eigenvectors of D(u)

We first study the structure of the eigenvectors of D(u) corresponding to small eigenvalues.

Lemma 4.4 For every J_{0} there are constants C < ∞, c > 0 and r_{0} > 1 such that for all r ≥ r_{0} the following holds: Let ψ be a normed eigenvector of I(∂/∂r + D_{N}) on [0, r] × N corresponding to the eigenvalue λ with |λ| ≤ c and assume

∥ψ(0)∥_{N} ≤ J ≤ J_{0}.  (41)

Then

∥ψ_{−}(0)∥_{N} ≤ C(J|λ| + e^{−cr}).

Proof: Since

1 = ∥ψ∥_{L^{2}}^{2} ≥ ∫_{0}^{r} ∥ψ(s)_{+}∥_{N}^{2} ds + ∫_{0}^{r} ∥ψ_{−}(s)∥_{N}^{2} ds,

there is a point v ∈ [3r/4, r] with

∥ψ_{−}(v)∥_{N}^{2} ≤ \frac{4}{r}.  (42)

Without loss of generality we can assume that ψ = c^{+}H_{μ}^{+} + c^{−}H_{μ}^{−} (see (36), (37)). Writing (41) and (42) out provides

|(1 − cosα_{μ})c^{+} + sinα_{μ}c^{−}| ≤ J  (43)
|sinα_{μ}c^{+} + (1 − cosα_{μ})c^{−}| ≤ J  (44)
|sinα_{μ}e^{vμcosα_{μ}}c^{+} + (1 − cosα_{μ})e^{−vμcosα_{μ}}c^{−}| ≤ \frac{2}{\sqrt{r}}.  (45)

From (43) we derive

|c^{−}| ≤ \frac{1}{|sinα_{μ}|}(J + (1 − cosα_{μ})|c^{+}|).  (46)

Inserting (46) into (45) yields

|c^{+}|(|sinα_{μ}|e^{vμcosα_{μ}} − \frac{(1 − cosα_{μ})^{2}e^{−vμcosα_{μ}}}{|sinα_{μ}|}) ≤ \frac{2}{\sqrt{r}} + \frac{J(1 − cosα_{μ})e^{−vμcosα_{μ}}}{|sinα_{μ}|} . (47)

If c > 0 is small enough, the factor at |c^{+}| is positive. Choosing c > 0 even smaller (depending on J_{0}) we can simplify (47) to

|c^{+}| ≤ \frac{8e^{−vμcosα_{μ}}}{\sqrt{r}|sinα_{μ}|}.  (48)

Inserting (48) into (46) we obtain

|c^{−}| ≤ \frac{J}{|sinα_{μ}|} + \frac{8(1 − cosα_{μ})e^{−vμcosα_{μ}}}{\sqrt{r}(sinα_{μ})^{2}}  (49)
≤ \frac{J}{|sinα_{μ}|} + \frac{100e^{−vμcosα_{μ}}}{\sqrt{r}} .  (50)

From (44), (48) and (49) we get

∥ψ_{−}(0)∥_{N} ≤ \frac{8e^{−vμcosα_{μ}}}{\sqrt{r}} + \frac{J(1 − cosα_{μ})}{|sinα_{μ}|} + \frac{100(1 − cosα_{μ})e^{−vμcosα_{μ}}}{\sqrt{r}}
≤ C(J|λ| + e^{−cr})

since \frac{1−cosα_{μ}}{|sinα_{μ}|} ≤ C|λ| and v ≥ 3r/4. Here C and c are independent of μ. □

Lemma 4.5 There are constants C > ∞, c > 0 and r_{0} > 1 such that the following holds for all r ≥ r_{0} and u ∈ [0, 1]: Let ψ be a normed eigenvector of D(u) corresponding to the eigenvalue λ with |λ| ≤ c. Then, on the first component

∥ψ_{−}(−r)∥_{N} ≤ C(|λ| + e^{−cr})  (51)
∥ψ_{+}(r)∥_{N} ≤ C(|λ| + e^{−cr}).  (52)

Proof: In order to get the first estimate we apply Lemma 4.4 to ψ on the first component of Z^{−}, noting that apriori there is a J_{0} < ∞ with ∥ψ(−r)∥_{N} ≤ J_{0}∥ψ∥_{L^{2}} ≤ J_{0}. The second estimate is obtained by a reflection-symmetric argument. □

Lemma 4.6 Assume 1.15. Then there are constants C < ∞, c > 0 and r_{0} > 1 such that for all r ≥ r_{0} and u ∈ [0, 1] the following holds: Let ψ be a normed eigenvector of D(u) corresponding to an eigenvalue λ with |λ| < c. Then

∥ψ_{|M̃_{i}}∥_{L^{2}} ≤ C(|λ| + \frac{1}{\sqrt{r}})

for i = 1, 2.

Proof: We consider the assertion for M̃_{1}. Let F be the closure of D̃_{1} with the boundary condition B_{L_{1}^{⊥}}. Then F is selfadjoint and inf |spec(F)| ≥ 2c > 0 for some c > 0 by the regularity assumption 1.15. Hence, for any ϕ ∈ dom F if |λ| ≤ c we have

∥(F − λ)ϕ∥_{L^{2}} ≥ c∥ϕ∥_{L^{2}}.  (53)

Let γ be a smooth cut-off function on M̃_{1} with γ = 0 on {−r} × N and γ = 1 outside of the collar (−r − ϵ, −r] × N.

Let now ψ be a normed eigenvector of D(u) corresponding to the eigenvalue λ with |λ| ≤ c and ψ = ψ_{0} + ψ_{−} + ψ_{+} near {−r} × N of the first component. We define ϕ ∈ dom F to be

• ψ on M_{1} \ (−r − ϵ, −r] × N

• ψ_{+} + γ(ψ_{0} + ψ_{−}) on (−r − ϵ, −r] × N.

Since ψ_{0} is oscillating, we have ψ_{0}(s) ≤ \frac{C}{\sqrt{r}} for all s ∈ (−ϵ, 0] × N. By Lemma 4.5 we have also

∥ψ_{−}(s)∥_{N} ≤ C(|λ| + e^{−cr})

for s ∈ (−ϵ, 0] × N. This shows

∥ψ_{|M̃_{1}} − ϕ∥_{L^{2}} ≤ C(|λ| + \frac{1}{\sqrt{r}}).  (54)

On s ∈ (−r − ϵ, −r] × N we compute

(F − λ)ϕ = [λ(1 − γ) + grad γ]ψ_{−}
+ λ(γ − 1)ψ_{+} + grad γψ_{0}.

We give the L^{2}-norm estimates of the various terms using Lemma 4.5.

∥[λ(1 − γ) + grad γ]ψ_{−}∥_{L^{2}} ≤ C(|λ| + e^{−cr})  (55)
∥λ(γ − 1)ψ_{+}∥_{L^{2}} ≤ C|λ|  (56)
∥grad γψ_{0}∥_{L^{2}} ≤ \frac{C}{\sqrt{r}}.

Hence,

∥(F − λ)ϕ∥_{L^{2}} ≤ C(|λ| + \frac{1}{\sqrt{r}}).

Now (53) also implies ∥ϕ∥_{L^{2}} ≤ C(|λ| + \frac{1}{\sqrt{r}}) and the claim follows from (54). □

Lemma 4.7 Assume 1.15. Then there are constants C < ∞, c > 0 and r_{0} > 1 such that for all r ≥ r_{0} and u ∈ [0, 1] the following is holds: Let ψ be a normed eigenvector of D(u) corresponding to an eigenvalue λ with |λ| < c. For the boundary values on the first component it holds

∥ψ(−r)∥_{N} ≤ C(|λ| + \frac{1}{\sqrt{r}})  (57)
∥ψ(r)∥_{N} ≤ C(|λ| + \frac{1}{\sqrt{r}})  (58)
∥ψ_{−}(−r)∥_{N} ≤ C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr})  (59)
∥ψ_{+}(r)∥_{N} ≤ C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr}).  (60)

Proof: The first two estimates (57) and (58) follow from Lemma 4.6 since there is an apriori estimate of the C^{1}-norm. In order to get (59) and (60) insert (57) and (58) into Lemma 4.4, i.e. use J = C(|λ| + \frac{1}{\sqrt{r}}). □

Lemma 4.8 Assume 1.15. Then there are constants C < ∞, c > 0 and r_{0} > 1 such that for all r ≥ r_{0} and u ∈ [0, 1] the following holds: Let ψ be a normed eigenvector of D(u) corresponding to an eigenvalue λ with |λ| < c. Then

∥ψ_{|M̃_{i}}∥_{L^{2}} ≤ C(λ^{2} + \frac{1}{\sqrt{r}})

for i = 1, 2.

Proof: We proceed as in the proof of Lemma 4.6, but we insert the better estimates (59) for ψ_{−} into (55) and (57) for ψ_{+} into (56). □

Lemma 4.9 Assume 1.15. Then there are constants C < ∞, c ≥ 0 and r_{0} > 1 such that for all r ≥ r_{0} and u ∈ [0, 1] the following holds: Let ψ be a normed eigenvector of D(u) corresponding to the eigenvalue λ with |λ| ≤ c. Then, on the first component, • ∥pr_{L_{1}^{⊥}}ψ_{0}(−r)∥_{N} ≤ C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr}) and

• ∥pr_{L_{2}^{⊥}}ψ_{0}(r)∥_{N} ≤ C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr})

holds.

Proof: We consider the first estimate. Let F_{L} be the closure of D̃_{1} with the boundary condition B_{L}, where L is Lagrangian. By the method of Lesch-Wojciechowski [31] it can be shown that the spectrum of F_{L} depends continuously on L. Since the parameter space of all Lagrangian subspaces of V is compact, we have

inf_{x∈V} sup_{x∈L⊂V} inf(|spec(F_{L})| \ {0}) = 2c_{1} > 0.  (61)

Let ψ be a normed eigenvector of D(u) corresponding to the eigenvalue λ with |λ| ≤ min(c, c_{1}) (the c from Lemma 4.8) and ψ = ψ_{0} + ψ_{−} + ψ_{+} its decomposition on Z ∪ Z. We define ϕ on M̃_{1} by

• ψ on M̃_{1} \ (−r − ϵ, −r] × N

• ψ_{0} + ψ_{+} + γψ_{−} on (−r − ϵ, −r] × N.

Let L ⊂ V realize the maximum of inf|(spec(F_{L}) \ {0}|) subject to ψ_{0}(−r) ∈ L. Then ϕ ∈ dom F_{L}. We compute

(F_{L} − λ)ϕ = [λ(1 − γ) + grad γ]ψ_{−} + λ(γ − 1)ψ_{+}.

Applying now Lemma 4.7 to ψ_{+} and to ψ_{−} yields

∥(F_{L} − λ)ϕ∥_{L^{2}} ≤ C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr}).

Let ϕ = ϕ_{n} + ϕ_{o} with F_{L}ϕ_{n} = 0 and ϕ_{o} ⊥ ker F_{L}. Then ∥(F_{L} − λ)ϕ_{o}∥_{L^{2}} ≤ C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr}) and, hence, ∥ϕ_{o}∥_{L^{2}} ≤ C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr}). Since the C^{1}-norm of the ϕ is apriori bounded, we have ϕ_{o}(−r) ≤ C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr}) too. Due to pr_{V} ϕ_{n}(−r) ∈ L ∩ L_{1} ⊂ L_{1} we have pr_{L_{1}^{⊥}}ψ_{0}(−r) = pr_{L_{1}^{⊥}}ϕ_{o}(−r). The Lemma follows. □

4.7 The construction of the map A

Now we construct a linear, almost unitary map A mapping eigenvectors of D(u) corresponding to small eigenvalues λ to approximate eigenvectors of B(u). The approximation becomes better when r increases and |λ| becomes small.

Choose smooth cut-off functions ρ_{±} with

• ρ_{−} = 0 near {−r} • ρ_{−} = 1 on [−r + 1, r]

• ρ_{+} = 0 near {r}

• ρ_{+} = 1 on [−r, r − 1]

such that the C^{1}-norms are uniformly bounded with respect to r. Let ψ be an eigenvector of D(u) corresponding to λ and let ψ = ψ_{0} + ψ_{−} + ψ_{+} be the decomposition on Z ∪ Z. We define A(ψ) to be ψ on the second component, and on the first component to be

• pr_{L_{1}}ψ_{0} + ψ_{+} + ρ_{−}(pr_{L_{1}^{⊥}}ψ_{0} + ψ_{−}) on [−r, −r + 1] × N

• ψ on [−r + 1, r − 1] × (N ∪ N)

• pr_{L_{2}}ψ_{0} + ψ_{−} + ρ_{+}(pr_{L_{2}^{⊥}}ψ_{0} + ψ_{+}) on [r − 1, r] × N.

Then A(ψ) ∈ dom B(u). A will be extended linearly to linear combinations of eigenvectors.

Lemma 4.10 (A maps eigenvectors to approximate eigenvectors) There are constants C < ∞, c > 0 and r_{0} > 1 such that for all r ≥ r_{0}, u ∈ [0, 1], the following holds: Let ψ be a normed eigenvector of D(u) corresponding to an eigenvalue λ with |λ| < c. Then

∥B(u)A(ψ) − λA(ψ)∥_{L^{2}} ≤ C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr}).

If ψ_{1}, ψ_{2} are two such eigenvectors, then

| < ψ_{1}, ψ_{2} > − < A(ψ_{1}), A(ψ_{2}) > | ≤ C(\frac{1}{\sqrt{r}} + λ^{2}).  (62)

Proof: We compute B(u)A(ψ) − λA(ψ) and obtain 0 on the second component,

([λ(1 − ρ_{−}) + grad ρ_{−}]pr_{L_{1}^{⊥}} + λ(ρ_{−} − 1)pr_{L_{1}})ψ_{0}
+ (grad ρ_{−} + λ(1 − ρ_{−}))ψ_{−} + λ(ρ_{−} − 1)ψ_{+}

on [−r, −r + 1] × N, 0 on [−r + 1, r − 1] × N, and

([λ(1 − ρ_{+}) + grad ρ_{+}]pr_{L_{2}^{⊥}} + λ(ρ_{+} − 1)pr_{L_{2}})ψ_{0}
+ (grad ρ_{+} + λ(1 − ρ_{+}))ψ_{+} + λ(ρ_{+} − 1)ψ_{−}

on [r − 1, r] × N. Lemma 4.9 and 4.7 provide

∥B(u)A(ψ) − λA(ψ)∥_{L^{2}} ≤ C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr}).

Using Lemma 4.7 and 4.8 we further obtain (62). □

Thus, for large r we know by the variational principle, that for every small eigenvalue λ of multiplicity m of D(u), there are eigenvalues of B(u) of total multiplicity at least m in a neighbourhood of λ of size C(λ^{2} + \frac{|λ|}{\sqrt{r}} + e^{−cr}).

4.8 Comparison of the spectral flows of D(u) and B(u)

We fix now some large r_{0}. The small eigenvalues λ of B(u) with |λ| ≤ d/2 are eigenvalues of B_{0}(u). For r ≥ r_{0} and an eigenvector ψ of B_{0}(u) let ψ_{r}(s) := ψ(\frac{r_{0}s}{r}). Then ψ_{r} is an eigenvector of B_{0}(u)_{r} (it is more convenient not to omit the r in this Subsection) to the eigenvalue \frac{r_{0}λ}{r}. Here we employ the r-dependent choice of U.

Lemma 4.11 Assume 1.15 and 1.16. There is an r_{1} ≥ r_{0} such that, for all r ≥ r_{1}, we have equality of the spectral flows

sf{B(u)_{r}} = sf{D(u)}.

Proof: We label the eigenvalues of B_{0}(1)_{r_{0}} as follows (note that ker B_{0}(1)_{r_{0}} = {0} by Assumption 1.16, i.e. L_{1} ∩ L_{2} = {0})

… ≤ λ_{−2} ≤ λ_{−1} < 0 < λ_{0} ≤ λ_{1} ≤ ….

Then we extend this labeling by continuity to all u ∈ [0, 1] such that the order is maintained, i.e. λ_{i}(u) ≤ λ_{i+1}(u). By rescaling this induces a labeling of the small eigenvalues (i.e. with |λ| ≤ d/2) of B(u)_{r} for all r ≥ r_{0}.

There exists a small c_{1} > 0 such that for r = r_{0} and every u ∈ [0, 1] there is an N(u) ∈ Z such that

|λ_{N(u)}(v) − λ_{N(u)+1}(v)| ≥ c_{1}

with v ∈ [u^{−}, u^{+}], where [u^{−}, u^{+}] is a small neighbourhood of u. We choose finitely many u_{i} such that ∪_{i}[u_{i}^{−}, u_{i}^{+}] = [0, 1].

Now we choose the r_{1} so large that the following holds for all r ≥ r_{1}:

• λ_{N(u_{i})}(v) ≤ c for all i and v ∈ [u_{i}^{−}, u_{i}^{+}], where c is the smallest c of the preceding Lemmas.

•

\frac{r_{0}c_{1}}{r} ≥ 100C(λ_{N(u_{i})}(v)^{2} + \frac{|λ_{N(u_{i})}(v)|}{\sqrt{r}} + e^{−cr})  (63)

for all v ∈ [u_{i}^{−}, u_{i}^{+}] and i, where C is the maximum of the C’s given in the preceding Lemmas and c is again the minimum of the c’s given there. Note that λ_{N(u_{i})}(v) ∼ \frac{1}{r}. Hence, the inequality above can be fulfilled if r_{1} is large enough.

• Let q := inf|spec(B_{0}(1)_{r_{0}})| > 0 (by 1.16!). Then

\frac{qr_{0}}{r} − 100C((\frac{qr_{0}}{r})^{2} + \frac{qr_{0}}{r^{3/2}} + e^{−cr}) > 0.  (64)

To every eigenvalue λ of D(u) with |λ| ≤ c, there corresponds an eigenvalue of B(u)_{r} within a neighbourhood of λ of size C(λ^{2} + \frac{λ}{\sqrt{r}} + e^{cr}) and vice versa. Here all eigenvalues are counted with multiplicity. We will use this in order to construct an induced labeling of the small eigenvalues of D(u). Let us consider some i. For v ∈ [u_{i}^{−}, u_{i}^{+}] define ρ_{N(u_{i})}(v) to be the largest eigenvalue of D(v) smaller than

λ_{N(u_{i})} + C(λ_{N(u_{i})}^{2} + \frac{|λ_{N(u_{i})}|}{\sqrt{r}} + e^{−cr}).

It is easy to see that ρ_{N(u_{i})}(v) is continuous since no other branch of eigenvalues of D(v) may enter the neighbourhood of λ_{N(u_{i})} from above because we have chosen N(u_{i}) such that there is a ’large’ gap according to (63). Now we can label all the other eigenvalues of D(v) using their order. Then all branches ρ_{i}(v) become continuous. Moreover, one can check that the labeling is compatible over the intersections [u_{i}^{−}, u_{i}^{+}]∩[u_{j}^{−}, u_{j}^{+}]. In fact, there are only finitely many eigenvalues between λ_{N(u_{i})}(v) and λ_{N(u_{j})}(v) and the same is true for ρ_{N(u_{i})}(v) and ρ_{N(u_{j})}(v). Furthermore we have for all u ∈ [0, 1], i ∈ Z,

|λ_{i}(u) − ρ_{i}(u)| ≤ C(λ_{i}(u)^{2} + \frac{|λ_{i}(u)|}{\sqrt{r}} + e^{−cr})  (65)

if |λ_{i}(u)| ≤ c. Due to (64) the choice of r_{1} assures that

… ≤ ρ_{−2}(1) ≤ ρ_{−1}(1) < 0 < ρ_{0}(1) ≤ ρ_{1}(1) ≤ ….

Let f := −sf{B(u)_{r}} be the spectral flow of {B(u)_{r}}. Then

… ≤ λ_{−2+f}(0) ≤ λ_{−1+f}(0) < 0 < λ_{f}(0) ≤ λ_{1+f}(0) ≤ ….

Hence, again by (64) and (65)

… ≤ ρ_{−2+f}(0) ≤ ρ_{−1+f}(0) < 0 < ρ_{f}(0) ≤ ρ_{1+f}(0) ≤ ….

Thus sf{D(u)} = sf{B(u)_{r}}. □

The following Lemma finishes the proof of our main Theorem 1.17.

Lemma 4.12 sf{B(u)_{r}} = 0

Proof: Let V be the restriction of the unitary U to h_{0}. Then, by the definition of G, we have V B_{0}(u)_{r}V^{∗} = B_{0}(1 − u)_{r} and the claim follows since sf{B(u)_{r}} = sf{B_{0}(u)_{r}}. □

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