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Real embeddings and the Atiyah-Patodi-Singer index theorem for Dirac operators

We present the details of our embedding proof of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary.


Introduction
The index theorem of Atiyah, Patodi and Singer [APS1,(4.3)] for Dirac operators on manifolds with boundary has played important roles in various problems in geometry, topology as well as mathematical physics. Not surprisingly then, there are by now quite a number of proofs of this index theorem other than Atiyah, Patodi and Singer's original proof [APS1]. Among these proofs we mention those of Cheeger [C1, 2] (see also Chou [Ch]),  and Melrose [M]. One common point underlying all these proofs (including the original one) is that they can all be viewed, in one way or another, as certain extensions to manifolds with boundary of the heat kernel proof of the local index theorem for Dirac operators on closed manifolds (cf. [BeGV]). That is, one starts with a Mckean-Singer type formula and then studies the small time asymptotics of the corresponding heat kernels. In particular, one makes use of the explicit formulas for the heat kernel of the Laplace operators on the cylinder ( [APS1], [M]) and/or cone ([BC1], [C1, 2], [Ch]) (being attached the boundary) for the analysis near the boundary. The η-invariant on the boundary, which was first defined in [APS1], appears naturally during the process. Now recall that Atiyah and Singer [AS] also have a K-theoretic proof of their index theorem for elliptic operators on closed manifolds. In such a proof, one transforms the problem, through direct image constructions in K-theory, to a sphere and then applies the Bott periodicity theorem on the sphere to establish the result. 1 It is thus natural to ask whether the strategy of Atiyah-Singer's Ktheoretic ideas can be used to prove the Atiyah-Patodi-Singer index theorem for manifolds with boundary. The purpose of this paper is to present such a proof, of which an announcement of basic ideas has already appeared in [DZ1].
Briefly speaking, we embed the manifold with boundary under consideration into a ball, instead of a sphere, so that it maps the boundary of the original manifold to the boundary sphere of the ball, and reduce the problem to the ball. Now since any vector bundle on the ball is topologically trivial, one obtains the result immediately. This works even when the original manifold has no boundary, giving a proof of the Atiyah-Singer index theorem for Dirac operators. The Bott periodicity theorem is thus not needed.
Observe that in [AS], Atiyah and Singer made heavy use of the techniques of pseudodifferential operators, which is not suitable for treating directly the global elliptic boundary problems. This is the first serious difficulty in extending directly the arguments in [AS] to deal with the Atiyah-Patodi-Singer boundary problems.
On the other hand, Bismut and Lebeau developed in [BL] a general and direct localization procedure which applies to a wide range of localization problems involving Dirac type operators. For example, it has lead to a direct analytic treatment of the index theorem for Dirac operators on closed manifolds along the lines of [AS] (cf. [Z,Remark 2.6]), as well as a localization formula for η-invariants of Dirac operators [BZ] which may be viewed as an odd dimensional analogue of the main result in [BL]. It is these techniques and results that will be used in the present paper, giving an embedding proof of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary [APS1].
In the proof described in [DZ1], we also used in an essential way Cheeger's cone method [C1, 2]. The reason being, in order to apply Bismut-Lebeau's method [BL], we need to transfer the Atiyah-Patodi-Singer boundary problem to an elliptic problem on certain manifolds with cone-like singularity. Now, in the present paper, we will show that how one can avoid the analysis on the cone at all. This is done by considering the Atiyah-Patodi-Singer type boundary value problem for certain non-differetial operators arising naturally from the analysis in [BL]. In this way, one no longer encounters the heat kernel analysis on cylinders and/or cones which are essential for the other proofs of the Atiyah-Patodi-Singer index theorem. We regard this as a major technical simplification with respect to [DZ1].
In a separate paper [DZ3], we will further extend the main result of this paper to the case of families. In particular, we will give a new proof of the family index theorem of 2] and Melrose-Piazza [MP] along the lines of this paper. This paper is organized as follows. In Section 1, we prove an important variation formula for the indices of the Atiyah-Patodi-Singer boundary value problems for Dirac operators on manifolds with boundary. In Section 2, we state a localization formula of Riemann-Roch type for the indices of the Atiyah-Patodi-Singer boundary value problems for Dirac operators on manifolds with boundary. In Section 3, we prove the Riemann-Roch property stated in Section 2. In Section 4, by combining the results in Sections 1, 2 with those of Bismut-Zhang [BZ], we complete our proof of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. There is also an appendix in which we prove a harmonic oscillator property for certain Dirac operators on flat spaces, which plays an essential role in the main text. §1. Dirac type operators on manifolds with boundary: index and its variations In this section, we recall the definition of the Atiyah-Patodi-Singer boundary value problems [APS1] for Dirac type operators on Spin manifolds with boundary. We also prove an important variation formula for the indices of these boundary value problems.
Let X be a compact oriented even dimensional spin manifold with boundary ∂X. We assume that X has been equipped with a fixed spin structure. Then ∂X carries the canonically induced orientation and spin structure.
Let g T X be a metric on T X. Let g T ∂X be its restriction on T ∂X. We assume that g T X is of product structure near the boundary ∂X. That is, there is an open neighborhood U α = [0, α) × ∂X of ∂X in X with α > 0 such that one has the orthogonal splitting on U α , where π α : [0, α) × ∂X → ∂X is the obvious projection onto the second factor. Let ξ be a Hermitian vector bundle over X with Hermitian metric g ξ . Let ∇ ξ be a Hermitian connection on ξ with respect to g ξ . We make the assumption that over the open neighborhood U α of ∂X, one has By taking α > 0 sufficiently small, one can always find g T X , g ξ and ∇ ξ verifying (1.1) and (1.2).
Let S(T X) = S + (T X) ⊕ S − (T X) be the Z 2 -graded Hermitian vector bundle of spinors associated to (T X, g T X ). Let ∇ S(T X) be the Hermitian connection on S(T X) canonically induced from the Levi-Civita connection ∇ T X of g T X . Then ∇ S(T X) preserves the Z 2 -splitting S(T X) = S + (T X) ⊕ S − (T X). We denote by ∇ S ± (T X) the restriction of ∇ S(T X) on S ± (T X). Let ∇ S(T X)⊗ξ (resp. ∇ S ± (T X)⊗ξ ) be the Hermitian connection on S(T X) ⊗ ξ (resp. S ± (T X) ⊗ ξ) obtained from the tensor product of ∇ S(T X) (resp. ∇ S ± (T X) ) and ∇ ξ .
For any e ∈ T X, let c(e) be the Clifford action of e on S(T X). Then c(e) extends to an action on S(T X) ⊗ ξ by acting as identity on ξ. We still denote this extended action by c(e).
Let e 1 , · · · , e dim X be an oriented (local) orthonormal base of T X. We can then define the (total) twisted Dirac operator with coefficient bundle ξ as follows (cf. [BeGV] and [LM]), Definition 1.1. By a Dirac type operator on Γ(S(T X) ⊗ ξ), we mean a first order differential operator D : Γ(S(T X)⊗ξ) → Γ(S(T X)⊗ξ) such that D −D ξ is an odd self-adjoint element of zeroth order, and that for α > 0 sufficiently small, the following identity holds on U α , with B independent of r and its restriction on Γ(S(T X) ⊗ ξ)| ∂X formally selfadjoint. We will also call the restriction D + (resp. D − ) of D to Γ(S + (T X) ⊗ ξ) (resp. Γ(S − (T X) ⊗ ξ)) a Dirac type operator.
When there is no confusion, we will also use B to denote its restriction on (S(T X) ⊗ ξ)| ∂X . Clearly, B preserves the Z 2 -grading of (S(T X) ⊗ ξ)| ∂X = (S + (T X) ⊗ ξ)| ∂X ⊕ (S − (T X) ⊗ ξ)| ∂X . We denote by B ± the restriction of B on (S ± (T X) ⊗ ξ)| ∂X . Now consider the formally self-adjoint first order differential operator B + , which is clearly elliptic, acting on sections of (S + (T X) ⊗ ξ)| ∂X . Then the L 2completion of (S + (T X) ⊗ ξ)| ∂X admits an orthogonal decomposition where E λ is the eigenspace of λ.
Theorem 1.2. The following identity holds, where sf is the notation for the spectral flow of Atiyah-Patodi-Singer [APS2].
Remark 1.3. For a similar variation formula for η-invariants on odd dimensional manifolds with boundary, see Dai-Freed [DF].
Remark 1.4. For an extension of Theorem 1.2 to the case of families, see [DZ2]. §2. A Riemann-Roch theorem under embedding for Dirac operators on manifolds with boundary In this section, we state a Riemann-Roch type formula for indices of Dirac type operators on manifolds with boundary. This formula will be proved in the next section and will play a key role in our proof of the Atiyah-Patodi-Singer index theorem in Section 4.
This section is organized as follows. In a), we describe the basic geometric data. In b), we state the main result of this section, whose proof will be given in the next section. a). The geometric construction of direct images under embedding between manifolds with bounday Let Y be another even dimensional oriented compact spin manifold with boundary ∂Y . Moreover, there is an embedding i : Y ֒→ X such that ∂Y ⊂ ∂X, and that Y intersects transversally with ∂X.
Let g T (∂Y ) be the metric on T (∂Y ) induced from g T (∂X) . Set We can and we will assume that α is small enough so that U ′ α is also a tubular neighborhood of ∂Y .
Clearly, dim N = dim X − dim Y is even. Furthermore, since T X, T Y are oriented and spin, N is also oriented and spin.
Let g T X be a metric on T X such that its restriction on U α is g T Uα . Let g T Y be the restriction of g T X on Y . For simplicity, we can and we will assume that the embedding i : . Let g ξ be a Hermitian metric on ξ such that such that g ξ | Uα = π * α (g ξ | ∂X ) and that ξ + and ξ − are orthogonal to each other with respect to g ξ . Let V ∈ Γ(End odd (ξ)) be a self-adjoint element such that We assume that V is invertible on X \ Y , and that on Y , ker V has locally constant nonzero dimension, so that ker V is a nonzero smooth Z 2 -graded vector subbundle of ξ| Y . Let g ker V be the metric on ker V induced by the metric g ξ | Y . Let P ker V be the orthogonal projection from ξ| Y on ker V . If y ∈ Y , U ∈ T y X, let ∂ U V (y) be the derivative of V with repsect to U in any given smooth trivialization of ξ near y ∈ X. One then verifies that P ker V ∂ U V (y)P ker V does not depend on the trivialization, and only depends on Let µ be a complex vector bundle over Y such that µ| U ′ α = π ′ * α (µ| ∂Y ), equipped with a Hermitian metric g µ such that g µ | U ′ α = π ′ * α (g µ | ∂Y ). We equip S * (N) ⊗ µ the tensor product metric g S * (N )⊗µ . Also, we extend an endomorphism of S * (N) to that of S * (N) ⊗ µ by acting as identity on µ. We now make the fundamental assumption that over the total space of N, we have the identification Let ∇ µ be a Hermitian connection on µ which is of product nature near the boundary. Let ∇ S * (N )⊗µ be the Hermitian connection on S * (N) ⊗ µ obtained from the tensor product of ∇ S * (N ) and ∇ µ .
Let ∇ ξ = ∇ ξ + ⊕ ∇ ξ − be a unitary connection on ξ = ξ + ⊕ ξ − , which preserves the Z 2 -grading of ξ and is of product nature near the boundary. Let ∇ ker V be the unitary connection on ker V given by (2.6) We then make the assumption that under the identification (2.5), we also have the identification of connections (2.7) One easily verifies that there always exists a connection ∇ ξ such that (2.7) holds.
Remark 2.1. By using a well-known construction of Atiyah-Hirzebruch [AH], one verifies easily that given metrics g µ and g N on µ and N, there exist ξ = ξ + ⊕ξ − , g ξ = g ξ + ⊕ g ξ − and V taken as before, such that (2.5) holds (Compare with [BZ,Remark 1.1 Remark 2.2. As an easy but important observation, we note that the restriction of the identifications (2.5), (2.7) on the boundary takes forms of exactly the same nature (Compare with [BZ,Sect. 2b)]). In what follows, whenever such an identification on the boundary will be considered, we will simply use a subscript and/or superscript '∂' to indicate the restriction, when there will be no confusion from the context.

b). A Riemann-Roch theorem under emdedding for Dirac type operators on manifolds with boundary
We continue the discussions in a).
From (2.3), (2.5) and (A.1), one finds The following assumption is essential for this section.
Assumption 2.3. The operator B Y has no zero eigenvalue.
For any T ∈ R, let D T : Γ(S(T X)⊗ξ) → Γ(S(T X)⊗ξ) be the operator defined by where V ∈ End(ξ) extends as an action on S(T X)⊗ξ by 1⊗V , etc. Then by (1.4), its induced boundary operator B T is given by . Let B T,+ the associated boundary operator and P T,+,≥0 the Atiyah-Patodi-Singer projection associated to B T,+ .
We can now state the main result of this section as follows, whose proof will be given in the next section. The purpose of this section is to prove the Riemann-Roch property, Theorem 2.4, for the index of boundary value problems. The proof we described in [DZ1,Sect. 2] relies on Cheeger's cone method. Here, we will give a more direct proof without passing to manifolds with cone-like singularity. We thus avoid the heat kernel analysis on cylinders and/or cones completely.
The methods and techniques developed by Bismut and Lebeau [BL,Sects. 8,9] will play an essential role in this section. In fact, what we will do may be thought of as extensions of the Bismut-Lebeau method to manifolds with boundary.
This section is organized as follows. In a), we construct a natural embedding from the space of sections over Y into the space of sections over X. In b), we decompose the total Dirac operator on X to a sum of four operators according to this embedding and introduce a suitable deformation of the Dirac type operators as well as their associated boundary operators. In c), we prove the elliptic estimates for the deformed operators on the boundary. In d), we prove the Fredholm property of the Atiyah-Patodi-Singer type boundary problem for the deformed operators introduced in b). In e), we complete the proof of Theorem 2.4.
Throughout the rest of the paper, we will make the same assumptions and use the same notation as in Section 2. a). An embedding mapping sections over Y (resp. ∂Y ) to sections over X (resp. ∂X) For any γ ≥ 0, let E γ (resp. E γ ∂X , F γ , F γ ∂Y ) be the set of sections of S(T X)⊗ξ over X (resp. (S(T X)⊗ξ)| ∂X over ∂X, S(T Y )⊗µ over Y , (S(T Y )⊗µ)| ∂Y over ∂Y ) which lie in the γ th Sobolev space.
Following [BL,Sect. 8g)], for any y ∈ Y , Z ∈ N y , let t ∈ R → x t = exp X y (tZ) ∈ X be the geodesic in X with x 0 = y, dxt dt | t=0 = Z. For ε > 0, set B ε = {Z ∈ N : |Z| < ε}. Since X, Y are compact, there exists ε 0 > 0 such that for 0 < ε < ε 0 , the map (y, Z) ∈ N → exp X y (Z) ∈ X is a diffeomorphism from B ε onto a tubular neighborhood U ε of Y in X. From now on, we identify B ε with U ε and use the notation (y, Z) instead of exp X y (Z). In particular, we identify y ∈ Y with (y, 0) ∈ N.
Let dv N be the volume form of the fibers in N. Then dv Y (y)dv N (Z) is a natural volume form on the total space of N. Let k(y, Z) be the smooth positive function on B ε 0 defined by dv X (y, Z) = k(y, Z)dv Y (y)dv N (Z). (3.1) The function k has a positive lower bound on B ε 0 /2 . Also, k(y, 0) = 1. Now for any x = (y, Z) ∈ U ε 0 , we identify (S(T X) ⊗ ξ) x with (S(T X) ⊗ ξ) y by parallel transport with respect to ∇ S(T X)⊗ξ along the geodesic t → (y, tZ). Clearly, this identification preserves the Z 2 -grading of S(T X)⊗ξ.

b). A decomposition of Dirac type operators under consideration and the associated deformation
For any T > 0, let E 0 T (resp. E 0 T,∂X ) denote the image of F 0 (resp. F 0 ∂Y ) under J T (resp. J T,∂ ). Let E 0,⊥ T (resp. E 0,⊥ T,∂X ) be the orthogonal completement of E 0 T (resp. E 0 T,∂X ) in E 0 (resp. E 0 ∂X ). Let p T , p ⊥ T (resp. p T,∂X , p ⊥ T,∂X ) be the orthogonal projections from E 0 (resp. E 0 ∂X ) to E 0 T , E 0,⊥ T (resp. E 0 T,∂X , E 0,⊥ T,∂X ) respectively. Recall that the Dirac operators D T , B T have been defined in (2.11), (2.12). We now decompose D T , B T to We now introduce a deformation of D T (resp. B T ) according to the decomposition (3.6) (resp. (3.7)). One verifies easily that B T (u) is the boundary operator associated to D T (u) in the sense of (1.4).

c). Elliptic estimates for B T (u)
The purpose of this subsection is to show that the operators B T (u) verify the elliptic estimates satisfied by the usual elliptic differential operators, when T is large enough.
In fact, by the geometric assumptions in Section 2a), when restricted to the boundary (see in particular Remark 2.2), as well as Theorem A.3, one can proceed exactly as in [BL,Sect. 8,9] and [BZ] to show that the following estimates for B T,i , 1 ≤ i ≤ 4, hold.
Recall that the construction of J T depends on a parameter ε > 0.
Proposition 3.3. There exist ε > 0 such that (a). as T → +∞, (3.9) (b). there exist C 1 > 0, C 2 > 0 and T 0 > 0 such that for any T ≥ T 0 , any From here, one obtains the following estimates for B T (u), which says that B T (u) is a 'small' perturbation of B T , when T is very large. Thus, it can be regarded as an elliptic estimate for B T (u).
Proposition 3.4. There exist C > 0 and T 0 > 0 such that for any u ∈ [0, 1], T ≥ T 0 and s ∈ E 1 ∂X , the following inequality holds, (3.14) Now one verifies easily that the super commutator [B X , c( ∂ ∂r )V | ∂X ] is of zeroth order. Thus one deduces from (2.12) and the standard estimates for elliptic operators that there exist positive constants A, C 3 , C 4 such that It follows then that there exist constants C 5 > 0, C 6 > 0 such that From (3.14) and (3.16), one gets (3.12). Q.E.D.
Since for any T > 0, B T is a self-adjoint elliptic differential operator, Proposition 3.4 and the standard elliptic method enable one to deduce that when T ≥ max{T 0 , 4C 2 }, each B T (u), for u ∈ [0, 1], is self-adjoint and has discrete eigenvalues with finite multiplicity. Let P T (u) denote the Atiyah-Patodi-Singer projection associated to B T (u). In the next subsection, we will show that the boundary valued problems (D T (u), P T (u)), u ∈ [0, 1], are elliptic when T is large enough.

d). The Fredholm property of the boundary problems (D T (u), P T (u))
We continue the discussion in the previous subsection. In particular, we assume that T ≥ max{T 0 , 4C 2 } so that each B T (u), u ∈ [0, 1], is formally selfadjoint with discrete eigenvalues of finite multiplicity.
Set, for any T ≥ max{T 0 , 4C 2 } and u ∈ [0, 1], be the uniquely determined extension of D T (u). The main result of this subsection can be stated as follows.
Proposition 3.5. There exists T 1 > 0 such that for any u ∈ [0, 1] and T ≥ T 1 , D T,AP S (u) is a Fredholm operator.
Proof. By standard elliptic methods (cf. [BoW,Chap. 20]), in order to get Proposition 3.5, it suffices to prove the following result.
Proposition 3.6. There exist T 1 > 0, C 7 > 0, C 8 > 0 such that for any u ∈ [0, 1], T ≥ T 1 and s ∈ E 1 T (u), one has The rest of this subsection is devoted to a proof of Proposition 3.6.
We decompose X into two parts, the interior and the boundary region: Our proof of Proposition 3.6 consists of three steps, corresponding to the interior, the boundary region and the transition region.
Step 2. The case where s is supported in U 2α/3 : The key observation in this case is that since all the geometric data are of product nature on U α , one can use separation of variables to split the analysis into those along the ∂ ∂r direction and those along the cross section {r} × ∂X's with 0 ≤ r ≤ 2α/3 on which the analysis is the same as on ∂X. In particular, by (1.4), (2.11) and (2.12) one can write on U 2α/3 that Furthermore, by the definition of the embedding J T as well as its restriction on ∂X, and thus on each {r} × ∂X, 0 ≤ r ≤ 2α/3, also, one deduces from (3.26) the following formula on U 2α/3 , One also verifies easily that B T (u) anti-commutes with c( ∂ ∂r ). Thus from (3.27) one gets (D T (u) (3.28) From (3.27), (3.28) and Green's formula (cf. [BoW,Chap. 3]), one deduces easily that for any s ∈ Γ(S(T X) ⊗ ξ) which is supported in U 2α/3 , On the other hand, it is clear that one can apply the analysis in Section 3c) to each {r} × ∂X. Thus by (3.12), (3.16) one deduces that there exist constants C 13 > 0, C 14 > 0 such that when T is large enough, From (3.29), (3.31) and (3.32), one deduces (3.19) easily.
Step 3. The general case: Now by the results in Steps 1 and 2, one can apply the gluing argument in [BL, to complete the proof of Proposition 3.6. Q.E.D.
The proof of Proposition 3.5 is thus also completed. Q.E.D. e). Proof of Theorem 2.4.
We assume that T ≥ T 1 with T 1 determined by Proposition 3.5. We will first show that when T is large enough, the family of Fredholm operators D T,AP S (u), 0 ≤ u ≤ 1, constructed in Proposition 3.5 is a continuous family. For this, one establishes the following result.
The proof of Proposition 3.8 is completed. Q.E.D.
From Proposition 3.5 and Proposition 3.8, we see that when T is large enough, we have a continuous family of Fredholm operators {D T,AP S (u)} 0≤u≤1 . Furthermore, by Proposition 3.8 and Green's formula, the operators D T,AP S (u), 0 ≤ u ≤ 1, are self-adjoint. Now let τ X (resp. τ Y ) be the Z 2 -grading operator of S(T X)⊗ξ (resp. S(T Y )⊗ µ). One verifies directly that that is, J T preserves the Z 2 -gradings of S(T Y ) ⊗ µ and S(T X)⊗ξ.

.3)] for Dirac operators.
This section is organized as follows. In a), we use Theorem 2.4 to refine the main result in [BZ] so that the mod Z term in [BZ,Theorem 2.2] can now be made specific in our situation. In b), we apply the results proved in a) to the case where X is a ball to obtain the Atiyah-Patodi-Singer index theorem for Dirac operators on Y .

a). Real embeddings and η-invariants
Following [BZ,(1.27)], under the geometric assumptions in Section 2a), let γ X be the Chern-Simons current on X defined by Also, if D is a formally self-adjoint Dirac type operator on a closed odd dimensional spin manifold, we define the reduced η-invariant to bē where η(D) is the η-invariant of D in the sense of Atiyah-Patodi-Singer [APS1]. Let D ξ ± +,AP S (resp. D µ +,AP S ) denote the Atiyah-Patodi-Singer boundary problems associated to D ξ ± + (resp. D µ + ). Let B ξ ± + (resp. B µ + ) be the induced Dirac operators on ∂X (resp. ∂Y ) associated to D ξ ± + (resp. D µ + ). Let R T X , R T Y denote the curvature of ∇ T X , ∇ T Y respectively. We can now state the main result of this subsection, which has been announced in [DZ1,Theorem 2.3], as follows.
Theorem 4.1. The following identity holds, Remark 4.2. A weaker mod Z version of Theorem 4.1 has been previously proved in [BZ,Theorem 2.2].
The rest of this subsection is devoted to a proof of Theorem 4.1 by making precise the mod Z contribution in [BZ].
For any T ≥ 0, let D ξ T,+,AP S be the Dirac type operator verifying the Atiyah-Patodi-Singer boundary condition [APS1]. Let B ξ T,+ be the associated boundary operator on ∂X in the sense of (1.4).
We start with the following result which was announced in [DZ1,Prop. 2.1].
Proof. From Theorem 1.2 and a direct counting argument in using the definition of the reduced η-invariant, one sees easily that ind (D ξ T,+,AP S ) +η(B ξ T,+ ) depends smoothly on T . Proposition 4.3 then follows from the local variation formula [BC3, Theorem 2.7] of Bismut and Cheeger. Q.E.D.

Remark 4.4.
To be more precise, in [BC3, Theorem 2.7], Bismut and Cheeger considered the operators of the form B ξ + τ ξ + T V acting on Γ((S + (T X) ⊗ ξ)| Y ), where τ ξ is the Z 2 -grading operator of ξ. However, one sees easily that the map U : is an unitary and verifies that U(B ξ + τ ξ + T V )U −1 = B ξ T,+ . This makes it clear that one can apply the results in [BC3] and [BZ] to the present situation.
We can now proceed as in [BZ]. The key observation is that the geometric assumptions in [BZ,Sect. 1b)] correspond almost exactly to the geometric assumptions on ∂X in the current situation, with the minor diffference that we here use −c( ∂ ∂r )c (Z)τ N * to replace √ −1c (Z) in [BZ,(1.10)]. As a result, we will use here Theorem A.3 in the Appendix to replace [BZ,Theorem 4.5] in obtaining the analogues of the analytic results of [BZ,.
We now examine the arguments in [BZ,Sect. 3e)]. In order to get the required result, we must find out where the mod Z terms in [BZ, Sect. 3e)] arise, and replace them by the exact formulas.
In fact, one finds that this integer term is given by with a 0 > 0 such that B µ + has no non-zero eigenvalues in [−2a 0 , 2a 0 ]. This term must be added in the right hand side of the analogue of [BZ,(3.30)], Secondly, one uses Proposition 4.3 instead of a direct analogue of [BZ,(3.44)].
With the help of these two observations, by proceeding as in [BZ, Sect. 3e)], one finally finds, in our situation, the following refinement of a direct analogue of [BZ,(3.65)] when T 0 > 0 is large enough: We now prove two lemmas which together with (4.6) will give (4.3). The first lemma follows easily from the definitions of the operators under consideration, the definition of the reduced η-invariant as well as the classical Agranovič-Dynin type formula (cf. [BoW,Chap. 21]).
Lemma 4.5. The following identity holds, An equivalent form of the next lemma has been announced in [DZ1,Theorem 2.2].
Lemma 4.6. The following identity holds when T 0 > 0 is sufficiently large, Proof. Let f : X → R be a smooth function such that f ≡ 1 on U α/3 and f ≡ 0 outside of U 2α/3 . Let D T 0 ,−a 0 ,+ be the Dirac type operator defined by (4.9) Let D Y,−a 0 ,+ be the Dirac type operator defined by (4.10) Let D T 0 ,−a 0 ,+,AP S , D Y,−a 0 ,+,AP S be the associated operators verifying the Atiyah-Patodi-Singer boundary condition [APS1].
Since −a 0 is not an eigenvalue of B µ + , one sees that the Assumption 2.3 is verified by the boundary operator associated to D Y,−a 0 ,+ . Thus one can apply Theorem 2.4 to get that when T 0 is sufficiently large, one has ind (D T 0 ,−a 0 ,+,AP S ) = ind (D Y,−a 0 ,+,AP S ) .

b). A proof of the Atiyah-Patodi-Singer index theorem
We first state an easy consequence of a direct analogue in our situation of [BZ,Theorem 1.4].
Lemma 4.7. The following identity holds, det 1/2 R T X /2 sinh (R T X /2) γ X . (4.14) Finally, we are in a position to give a proof of the Atiyah-Patodi-Singer Thorem. Namely, we apply these results to the case where X = D 2n , the 2n dimensional ball with n sufficiently large. That any compact manifold with boundary can be embedded into a large ball in such a fashion is an elementary result from differential topology.

Appendix. Dirac operators and harmonic oscillators
Let E be a real oriented Euclidean vector space of even dimension. Let S(E) = S + (E) ⊕ S − (E) be the Z 2 -graded Hermitian vector space of E-spinors.
If e ∈ E, let e * ∈ E * corresponds to e by the scalar product. Let c(e) denote the Clifford action of e on S(E). Letc(e) denote the corresponding Clifford action of e on S * (E) = S * + (E) ⊕ S * − (E). Let τ be the Z 2 -grading operator of S(E), that is, τ | S ± (E) = ±id S ± (E) . Let τ * be the transpose of τ . Then σ = τ ⊗ τ * is the Z 2 -grading operator on ∧(E * ).
Let Γ(∧(E * )) be the vector space of smooth sections of ∧(E * ) over E.