Commun. Math. Phys. 201, 423 – 444 (1999)

Communications in

Mathematical Physics © Springer-Verlag 1999

The ζ -Determinant and the Additivity of the η -Invariant on the Smooth, Self-Adjoint Grassmannian Krzysztof P. Wojciechowski Department of Mathematics, IUPUI (Indiana/Purdue), Indianapolis, IN 46202-3216, USA. E-mail: [email protected] Received: 28 March 1996 / Accepted: 2 September 1998

Abstract: In this paper we discuss the existence of the ζ-determinant of a Dirac operator D on a compact manifold with boundary. We show that the determinant is well defined in the case of the operator D with a domain determined by a boundary condition from the smooth, self-adjoint Grassmannian Gr∗∞ (D) discussed in the papers [5, 13, 29]. We prove a generalization of a pasting formula for the η-invariant (see [34]). The results of the paper are used in the recent proof of the projective equality of the ζ-determinant and Quillen determinant on Gr∗∞ (D) (see [30, 31]). Introduction Recent studies in Quantum Field Theory and Topology have stressed the importance of the correct definition of the renormalized determinant of the Dirac operator over a manifold with boundary. The renormalization successfully used in the case of a closed manifold is the ζ-renormalization introduced by Ray and Singer in [27] (see also [32]). The ζ-determinant of the Dirac operator D on a closed manifold is given by the formula: det ζ D = e

iπ 2 (ηD (0)−ζD 2 (0))

·e−1/2·(d/ds(ζD2 (s))|s=0 ) ,

(0.1)

where ζD2 (s) and ηD (s) are functions constructed from the eigenvalues of the operator D. Now let us assume that D : C ∞ (M ; S) → C ∞ (M ; S) is a compatible Dirac operator acting on sections of a bundle of Clifford modules S over a compact Riemannian manifold M with boundary Y . In this paper we concentrate on the case of an odd-dimensional manifold M , and from now on we assume that n = dim M = 2m + 1. Let us point out, however, that our results are true for Dirac operators on an evendimensional manifold as well. The necessary modifications due to the different algebraic structure of the spinors in the odd and even case can be found in [7], where we discuss

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the applications of our results in the even-dimensional situation (see also [3] for an introductory discussion of applications of the ζ-determinant of elliptic boundary problems in Quantum Chromodynamics). In the present paper we discuss only the Product Case. Namely we assume that the Riemannian metric on M and the Hermitian structure on S are products in a certain collar neighborhood of the boundary. Let us fix N = [0, 1] × Y the collar. Then in N the operator D has the form D = G(∂u + B),

(0.2)

where G : S|Y → S|Y is a unitary bundle isomorphism (Clifford multiplication by the unit normal vector) and B : C ∞ (Y ; S|Y ) → C ∞ (Y ; S|Y ) is the corresponding Dirac operator on Y , which is an elliptic self-adjoint operator of first order. Furthermore, G and B do not depend on the normal coordinate u and they satisfy the identities G2 = −Id and GB = −BG.

(0.3)

Since Y has dimension 2m the bundle L S|Y decomposes into its positive and negative chirality components S|Y = S + S − and we have a corresponding splitting of the operator B into B ± : C ∞ (Y ; S ± ) → C ∞ (Y ; S ∓ ), where (B + )∗ = B − . Equation (0.2) can be rewritten in the following form     i 0 0 B− . ∂u + 0 −i B+ 0 In order to obtain a nice unbounded Fredholm operator we have to impose a boundary condition on the operator D. Let 5≥ denote the spectral projection of B onto the subspace of L2 (Y ; S|Y ) spanned by the eigenvectors corresponding to the nonnegative eigenvalues of B. It is well known that 5≥ is an elliptic boundary condition for the operator D (see [1, 6]). The meaning of the ellipticity is as follows. We introduce the unbounded operator D5≥ equal to the operator D with domain dom D5≥ = {s ∈ H 1 (M ; S) ; 5≥ (s|Y ) = 0}, where H 1 denotes the first Sobolev space. Then the operator D5≥ = D : dom(D5≥ ) → L2 (M ; S) is a Fredholm operator with kernel and cokernel consisting only of smooth sections. The orthogonal projection 5≥ is a pseudodifferential operator of order 0 (see [6]) . In fact we can take any pseudodifferential operator R of order 0 with principal symbol equal to the principal symbol of 5≥ and obtain an operator DR which satisfies the aforementioned properties. Let us point out, however, that only the projection onto the kernel of the operator R is used in the construction of the operator DR . Therefore we can restrict ourselves to the study of the Grassmannian Gr(D) of all pseudodifferential projections which differ from 5≥ by an operator of order −1. The space Gr(D) has infinitely many connected components and two boundary conditions P1 and P2 belong to the same connected component if and only if index DP1 = index DP2 . We are interested however in self-adjoint realizations of the operator D. The involution G : S|Y → S|Y equips L2 (Y ; S|Y ) with a symplectic structure, and Green’s formula (see [6])

ζ-Determinant on the Smooth, Self-Adjoint Grassmannian

425

Z (Ds1 , s2 ) − (s1 , Ds2 ) = −

Y

< G(s1 |Y ); s2 |Y > dy,

(0.4)

shows that the boundary condition R provides a self-adjoint realization DR of the operator D if and only if ker R is a Lagrangian subspace of L2 (Y ; S|Y ) (see [5, 6, 13]). It is therefore reasonable to restrict ourselves to those elements of Gr(D) which are Lagrangian subspaces of L2 (Y ; S|Y ). More precisely we introduce Gr∗ (D), the Grassmannian of orthogonal, pseudodifferential projections P such that P −5≥ is an operator of order -1 and − GP G = Id − P.

(0.5)

The space Gr∗ (D) is contained in the connected component of Gr(D) parameterizing projections P with index DP = 0. In this paper we discuss only the Smooth, Self-adjoint Grassmannian, a dense subset of the space Gr∗ (D), defined by ∗ (D) = {P ∈ Gr∗ (D) ; P − 5≥ has a smooth kernel}. Gr∞

(0.6)

∗ (D) if and only if ker B The spectral projection 5≥ is an element of Gr∞

= Remark 0.1. {0}. However, it is well-known that P (D) the (orthogonal) Calderon projection is an element of Gr∗ (D) (see for instance [5]), and it has been recently observed by Simon Scott (see [28], Proposition 2.2.) that P (D) − 5≥ is a smoothing operator, and ∗ (D). The finite-dimensional perturbations of 5≥ hence that P (D) is an element of Gr∞ discussed below (see also [13], [21] and [34]) provide further examples of boundary ∗ (D). The latter were introduced by Jeff Cheeger, who called them conditions from Gr∞ Ideal Boundary Conditions (see [10, 11]). For any P ∈ Gr∗ (D) the operator DP has a discrete spectrum nicely distributed along the real line (see [5, 13]). Therefore one might expect that det ζ (DP ) is well defined. To see that, we have to study the asymptotic expansion of the heat kernels involved in the construction of the determinant, or equivalently the expansion of the operator (DP − λ)−1 . The existence of a nice asymptotic expansion of the trace of the heat kernels used in the constructions of ηDP (s) and ζDP2 (s) was established in a recent work of Gerd Grubb [18] . She used the machinery developed in her earlier work and her joint work with Bob Seeley (see [15, 16, 17]). However, at the moment, the problem of explicit computation of the coefficents in the expansion is open. From this point of view the existence of the invariants used to define det ζ depends on the vanishing of particular coefficients in the corresponding expansions. We choose a different route. It follows from our earlier work on Grassmannians (see ∗ (D) is a path connected space. As a consequence [5, 6], and [13] Appendix B) that Gr∞ we can perform a Unitary Twist and replace the operator DP by a unitarily equivalent ∗ (D) denotes an appropriate finite-dimensional operator (D + R)5σ , where 5σ ∈ Gr∞ modification of 5≥ defined below in Sect. 1. The operator D5σ has a well-defined ζdeterminant and the correction term R lives in the collar N . The operator R is no longer a differential operator, but for any 0 ≤ u ≤ 1, Ru = R|{u}×Y is a pseudodifferential operator. If we assume that P − 5≥ has a smooth kernel then the operator Ru has a smooth kernel for each 0 ≤ u ≤ 1 and this is all that one needs in order to study the correction terms appearing in the corresponding heat kernels. This is the reason why we ∗ (D). The results of the paper should hold also in the restrict attention to the space Gr∞ ∗ case of P ∈ Gr (D). The main result of this paper is the following theorem.

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∗ Theorem 0.2. For any projection P ∈ Gr∞ (D), ηDP (s) and ζDP2 (s) are holomorphic functions of s in a neighborhood of s = 0. ∗ (D). Corollary 0.3. The ζ-determinant is a well-defined smooth function on Gr∞

Remark 0.4. (1) The result stated above implies the existence of the Quillen ζ-function metric for families of elliptic boundary value problems. This metric was studied before by Piazza [25] in the context of b-calculus developed by Melrose and his collaborators. (2) In fact, we are able to obtain complete asymptotic expansions of the heat kernels for the operator DP . The reason is that Duhamel’s Principle allows one to study the interior contribution and the boundary contribution separately and identify the singularities caused by the boundary contribution. This procedure was used before in [13] (Sect. 4 and Appendix A) and [20] (Sect. 1). As the asymptotic expansion has been already studied (see [16, 17, 18]) we leave the details to the reader and in this paper we concentrate instead on the analysis of the ζ-determinant. (3) A more difficult problem than the existence of the asymptotic expansion is to show that the invariants used in the construction of the determinant are well defined. For instance Grubb and Seeley showed the regularity of the η-function only for finitedimensional perturbations of the Atiyah–Patodi–Singer boundary condition (see [16]). A similar result was also obtained by Dai and Freed (see [12]). The η-invariant of a more general class of boundary problems was also studied recently by Br¨uning and Lesch (see [8]). There, however, the authors had to deal with the residuum of the η-function at s = 0, which is not present in our situation. (4) The results of this paper were announced in a talk the author gave at the Annual Meeting of the AMS in San Francisco in January 1995. The delay in publication was partly due to work on the applications, which were the motivation for the present work (see [3, 4, 7, 29, 30], and [31]). We also want to single out one particular result, which is related to the discussion of the dependence of spectral invariants on the symbol of the operator given in [35]. ∗ (D), i.e. Proposition 0.5. The value of the ζ-function at s = 0 is constant on Gr∞

ζDP2 (0) = ζDP2 (0), 1

(0.7)

2

∗ (D). for any P1 , P2 ∈ Gr∞

The results of this paper allow us to study the ζ-determinant as a function on ∗ (D). In particular, we are interested in the relation of the ζ-determinant and the Gr∞ Quillen determinant defined as a canonical section of the determinant line bundle over the Grassmannian. It was observed by Scott [28] that when restricted to the self-adjoint Grassmannian the determinant line bundle over Gr(D) becomes trivial. Moreover, it has ∗ (D). The Quillen determinant expressed in this trivia natural trivialization over Gr∞ alization becomes a function. We refer to the determinant obtained in this way as the canonical determinant and we denote it by det C DP (see [28] for details). In recent work of the author and Simon Scott the relation between det ζ DP and det C DP is studied. In fact, it has been shown that, up to a natural multiplicative constant, the two determinants are equal. Proposition 0.5 and Proposition 4.7 are used in an essential way in the proof of this result. We refer the reader to [29, 4, 30, 31] for details. In this paper we discuss another application of Theorem 0.2, the extension of the ∗ (D). This formula additivity formula for the η-invariant to boundary conditions from Gr∞

ζ-Determinant on the Smooth, Self-Adjoint Grassmannian

427

has been previously known only for finite-dimensional perturbations of the Atiyah– Patodi–Singer condition (see [34]). Let us point out that the additivity formula for the η-invariant stated in Theorem 4.1 and Proposition 0.5 implies the additivity of the phase of the ζ-determinant under the pasting of two manifolds with the same boundary. This extends the result of Dai and Freed (see [12]). In the first two sections of the paper we study the η-function of DP . We obtain the following result as a conclusion of our computations. ∗ (D) the function ηDP (s) is a holomorphic function of Theorem 0.6. For any P ∈ Gr∞ s in the half-plane Re(s) > −1.

Section 3 contains a discussion of ζDP2 (s) and d/ds(ζDP2 (s))|s=0 . In Sect. 4 we discuss the additivity formula for the η-invariant. In the Appendix we present proofs of two technical results used in Sect. 2 and Sect. 3. 1. Boundary Contribution to the η-Function. Unitary Twist and Duhamel’s Principle Let us assume for a moment that the manifold M does not have a boundary. The Dirac operator D is then a self-adjoint elliptic operator with a discrete spectrum {λk }k∈Z . We define the η-function of D as follows: X sign(λk )|λk |−s . (1.1) ηD (s) = λk6 =0

The function ηD (s) is a holomorphic function of s for Re(s) > dim (M ) and it has a meromorphic extension to C with isolated simple poles on the real axis. The point s = 0 is not a pole and ηD = ηD (0) the η-invariant of the operator D is an important invariant, which has found numerous applications in geometry, topology and physics. In the case of a compatible Dirac operator D the η-function is actually a holomorphic function of s for Re(s) > −2. This was shown by Bismut and Freed [2] , who used the heat kernel representation of the η-function Z ∞ s−1 2 1 t 2 ·Tr De−tD dt, (1.2) ηD (s) = s+1 0( 2 ) 0 which in particular allows us to express the η-invariant as Z ∞ 2 1 1 √ ·Tr De−tD dt. ηD (0) = √ π 0 t

(1.3)

It follows from (1.2) that the estimate

√ 2 |Tr De−tD | < c t

implies the regularity of the η-function. In fact, Bismut and Freed proved a sharper 2 result: Let E(t; x, y) denote the kernel of the operator De−tD , then there exists a positive constant c such that for any x ∈ M and for any 0 < t < 1, √ (1.4) |Tr E(t; x, x)| < c t. We argue along the same lines and prove the following proposition, which implies Theorem 0.6.

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∗ Proposition 1.1. For any P ∈ Gr∞ (D) there exists a positive constant c > 0 such that for any 0 < t < 1 the following estimate holds:

|Tr DP e−tDP | < c. 2

(1.5)

The proof of Proposition 1.1 occupies Sect. 1 and Sect. 2 of the paper. Proposition 1.1 is a statement on the small time asymptotics, which by Duhamel’s Principle allows us to replace the kernel of the operator by a suitable parametrix built from the heat kernel ˜ , the closed double of the manifold M , and the heat kernel of the of the operator on M operator G(∂u + B) subject to the boundary condition P on a cylinder [0, ∞) × Y . However, we need to start with a concrete representation of the heat kernel on the cylinder. Such a representation is well-known for the original Atiyah–Patodi–Singer condition 5≥ (see [1], or [6] Sect. 22). In general the projection 5≥ is not an element of ∗ (D). Nevertheless, one can find easily a finite-dimensional modthe Grassmannian Gr∞ ification of 5≥ which belongs to this Grassmannian and then use the explicit formulas for the heat kernel on the cylinder. We obtain our modification of the Atiyah–Patodi–Singer condition in the following way. The involution G (see (0.3)) restricted to ker(B) defines a symplectic structure on this subspace of L2 (Y ; S|Y ) and the Cobordism Theorem for Dirac Operators (see for instance [6], Corollary 21.16) implies dim ker(B + ) = dim ker(B − ). The last equality shows the existence of Lagrangian subspaces of ker(B). We choose such a subspace W and let σ : L2 (Y ; S|Y ) → L2 (Y ; S|Y ) denote the orthogonal projection of L2 (Y ; S|Y ) onto W . Let 5> denote the orthogonal projection of L2 (Y ; S|Y ) onto the subspace spanned by eigenvectors of B corresponding to the positive eigenvalues. Then ∗ (D), 5σ = 5> + σ ∈ Gr∞

(1.6)

∗ (D), which is a finite-dimensional perturbation of the Atiyah– gives an element of Gr∞ Patodi–Singer condition. The operator Dσ = D5σ is a self-adjoint operator and the properties of its η-function were studied in [13] (see Sect. 4 and Appendix A). It follows that ηDσ (s) is a holomorphic function for Re(s) > −2. To make a connection with the operator DP we need the following result, which is an easy consequence of the topological structure of the Grassmannians studied in [5, 6, 13] (Appendix B). ∗ (D) there exists a smooth path {gu }0≤u≤1 of unitary Lemma 1.2. For any P ∈ Gr∞ 2 operators on L (Y ; S|Y ) which satisfies

Ggu = gu G and gu − Id has a smooth kernel, ∗ (D) connects P0 = P with such that g1 = Id and the path {Pu = gu 5σ gu−1 } ⊂ Gr∞ P1 = 5σ .

We can always assume that the path {gu } is constant on subintervals [0, 1/4] and [3/4, 1]. We introduce U a unitary operator on L2 (M ; S) using the formula ( Id on M \ N . (1.7) U := gu on N The following lemma introduces the Unitary Twist, which allows us to replace the operator DP by a modified operator D + R subject to the boundary condition 5σ . This makes possible an explicit construction of the heat kernels on a cylinder.

ζ-Determinant on the Smooth, Self-Adjoint Grassmannian

429

Lemma 1.3. The operators DP and DU,σ = (U −1 DU )5σ are unitarily equivalent operators. Proof. Let {fk ; µk }k∈Z denote a spectral resolution of the operator DP . This means that for each k we have Dfk = µk fk and P (fk |Y ) = 0 . This implies U −1 DU (U −1 fk ) = µk (U −1 fk ) and 5σ ((U −1 fk )|Y ) = g0−1 P (fk |Y ) = 0, hence {U −1 fk ; µk } is a spectral resolution of (U −1 DU )5σ .



In the collar N , we have formulas U −1 DU = D + GU −1

∂U + GU −1 [B, U ] , ∂u

and

∂U ∂2U ∂u − U −1 2 + U −1 [B 2 , U ] , ∂u ∂u which restricted to the collar [0, 1/4] × Y give U −1 D2 U = D2 − 2U −1

U −1 DU = D + GU −1 [B, U ] and U −1 D2 U = D2 + U −1 [B 2 , U ].

(1.8)

It follows from Lemma 1.3 that we can study the operator DU,σ instead of the operator DP . We use the representation (1.8) in the construction of the parametrix of 2 the kernel of the operator DU,σ e−tDU,σ . This parametrix is built from the heat kernel ˜ and the heat kernel on the cylinder. The bundle S and the on the double manifold M ˜ (see [13]; see [6] for a operator D extend to the corresponding objects S˜ and D˜ on M detailed discussion of the glueing constructions). There is also the obvious double U˜ of the unitary transformation U . We introduce the operator ˜ ; S) ˜ → C ∞ (M ˜ ; S), ˜ U˜ −1 D˜ U˜ : C ∞ (M ˜ Therefore the estimate (1.4) holds for the kernel which is unitarily equivalent to D. E˜U (t; x, y) of the operator ˜ −1 ˜ ˜ 2 ˜ −tD˜ 2 U˜ . U˜ −1 D˜ U˜ e−t(U DU ) = U˜ −1 De

It follows from Duhamel’s Principle that on M \ N up to exponentially small error 2 ˜ −tD˜ 2 U˜ is equal to the kernel of DU,σ e−tDU,σ for 0 < t < 1 (in t), the kernel of U˜ −1 De (see [13, 20]; see [6] for a detailed discussion of the variant of Duhamel’s Principle we need in this paper). More precisely, we have the following Lemma, which takes care of the situation in the interior of M Lemma 1.4. Let EU,σ (t; x, y) denote the kernel of the operator DU,σ e−tDU,σ , 2

then there exist positive constants c1 , c2 such that for any x ∈ M1/8 = M \ [0, 1/8] × Y and any 0 < t < 1 the following estimate holds c

2 kEU,σ (t; x, x) − E˜U (t; x, x)k ≤ c1 e− t .

(1.9) 2 −tDU,σ

Hence the estimate (1.4) holds for the kernel of the operator DU,σ e

in M1/8 .

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K. P. Wojciechowski

Now, we study the heat kernel in the collar neighborhood of Y . Once again we apply 2 Duhamel’s Principle to replace the kernel EU,σ (t; x, y) of the operator DU,σ e−tDU,σ by the corresponding kernel on [0, +∞) × Y . It follows from Eq. (1.8) that up to an exponentially small error we can use the kernel of the operator (G(∂u + B) + K1 )e−t(−∂u +B 2

where

2

+K2 )σ

,

(1.10)

K1 = GU −1 [B, U ] and K2 = U −1 [B 2 , U ] .

Let us observe that K1 anticommutes and K2 commutes with the involution G. The symbol exp(−t(−∂u2 + B 2 + K2 )σ ) in (1.10) denotes the following operator. We consider the operator G(∂u + B)5σ on the infinite cylinder [0, +∞) × Y and its square, which we denote by (−∂u2 + B 2 )σ . The operator (−∂u2 + B 2 )σ is an unbounded self-adjoint operator in L2 ([0, +∞) × Y ; S) and the kernel of the operator exp(−t(−∂u2 + B 2 )σ ) is given by explicit formulas (see [1, 6]). We add the bounded operator K2 and obtain the operator (−∂u2 + B 2 + K2 )σ . It follows from standard theory (see for instance [26]) that the semigroup exp(−t(−∂u2 + B 2 + K2 )σ ) is well defined. We study the trace of the 2 2 kernel of (G(∂u + B) + K1 )e−t(−∂u +B +K2 )σ in the next section. 2. Boundary Contribution to the η-Function. Heat Kernel on the Cylinder In this section we continue the proof of Proposition 1.1. We have to show that the 2 boundary contribution to Tr DP e−tDP is bounded for t sufficiently small. Let e(t) denote the operator exp(−t(−∂u2 +B 2 +K2 )σ ) and e1 (t) denote the operator exp(−t(−∂u2 +B 2 )σ ). We have the formula e(t) = e1 (t) +

∞ X

{e1 ∗ K2 e1 ∗ K2 e1 ∗ ... ∗ K2 e1 }(t),

(2.1)

n=1

where the term K2 e1 appears n-times in the curly bracket under the summation sign and ∗ denotes convolution (see for instance [6]; Sect. 22C). It follows from the explicit formulas giving the kernel of the operator e1 (t) (see (2.5) and Appendix formula (A.1)) that Z tr G(∂u + B)e1 (t; (u, y), (v, z))|y=z dy = 0 , y, z ∈ Y. u=v Y

We want to show that there exists a positive constant C such that for any 0 ≤ u ≤ 1/8, Z tr (G(∂u + B) + K1 )e(t; (u, y), (v, z))|y=z dy| < C, y, z ∈ Y. (2.2) | u=v Y

It follows from Formula (2.1) that we have to study the trace Z tr (G(∂u + B) + K1 ) Y

{e1 +

∞ X n=1

{e1 ∗ K2 e1 ∗ K2 e1 ∗ ... ∗ K2 e1 }}(t; (u, y), (v, z))|y=z dy. u=v

The involution G commutes with the operators e1 and K2 and anticommutes with B and K1 , which gives us

ζ-Determinant on the Smooth, Self-Adjoint Grassmannian

Z Y

tr GB{e1 (t) +

n=1

andZ Y

∞ X

Tr K1 {e1 (t) +

∞ X n=1

431

{e1 ∗ K2 e1 ∗ K2 e1 ∗ ... ∗ K2 e1 }}(t; (u, y), (v, z))|y=z dy = 0, u=v

{e1 ∗ K2 e1 ∗ K2 e1 ∗ ... ∗ K2 e1 }}(t; (u, y), (v, z))|y=z dy = 0. u=v

Therefore we have the equality Z

Z Y

tr G(∂u + B) + K1 )e(t; (u, y), (v, z))|y=z dy = u=v

+ Z

∞ X n=1

= Y

Y

tr G∂u {e1 (t) +

{e1 ∗ K2 e1 ∗ K2 e1 ∗ ... ∗ K2 e1 }}(t; (u, y), (v, z))|y=z dy u=v

∞ X tr G∂u { {e1 ∗ K2 e1 ∗ K2 e1 ∗ ... ∗ K2 e1 }}(t; (u, y), (v, z))|y=z dy. u=v n=1

(2.3)

The last equality in (2.3) follows from the fact that Z Y

tr G(∂u e1 )(t; (u, y), (v, z))|y=z dy = 0, u=v

(see formula (A.1)). We have to study the right side of (2.3). The crucial point here is to estimate the first term Z Y

tr G(∂u e1 ) ∗ K2 e1 (t; (u, y), (v, z))|y=z dy. u=v

We estimate the trace in the Y -direction of the operator Z G(∂u e1 ) ∗ K2 e1 (t) =

t

G(∂u e1 (s))K2 e1 (t − s)ds.

0

Our result essentially follows from the fact that E1 (t − s; (u, y), (v, z)), the kernel of the operator e1 (t − s), and F(s; (u, y), (v, z)), the kernel of the operator ∂u e1 (s), have nice “diagonal” representations on the cylinder. We can choose a spectral resolution {ϕn ; µn }n∈Z\{0} of the tangential operator B, such that ϕn corresponds to a positive eigenvalue or is an element of Ran σ and Gϕn = ϕ−n . This means that one has Bϕn = µn ϕn and 5σ ϕn = 0,

(2.4)

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K. P. Wojciechowski

for µn ≥ 0, and B(Gϕn ) = −µn (Gϕn ) and 5σ (Gϕn ) = Gϕn . Now we can represent our kernels in the following way. X gn (t; u, v)ϕn (y)⊗ϕ∗n (z), E1 (t; (u, y), (v, z)) =

(2.5)

n∈Z\{0}

and X

F(t; (u, y), (v, z)) =

hn (t; u, v)ϕn (y)⊗ϕ∗n (z),

n∈Z\{0}

where gn (t; u, v) and hn (t; u, v) are given by explicit formulas (see (A.1)). We have X (G(∂u e1 (s))K2 e1 (t − s)(ϕn ); ϕn )|u=u0 , T rY G(∂u e1 (s))K2 e1 (t − s)|u=u0 = n∈Z\{0}

and (∂u e1 (s))K2 e1 (t − s)(ϕn )(y)|u=u0 = X Z ∞ dv·gm (s; u0 , v)hn (t − s; v, u0 )(ϕm ; K2 ϕn )ϕm (y). m∈Z\{0}

0

This gives us the following expressions T rY G(∂u e1 (s))K2 e1 (t − s)|u=u0 = X Z ∞ h−n (s; u0 , v)gn (t − s; v, u0 )dv·(ϕn ; K2 ϕn ), m∈Z\{0}

and T rY e1 (s)K2 e1 (t − s)|u=u0 =

(2.6)

0

X m∈Z\{0}

Z

∞

gn (s; u0 , v)gn (t − s; v, u0 )dv·(ϕn ; K2 ϕn ).

0

The existence of the η-invariant for the operator DP is now a consequence of the first part of the following theorem. The second part of the theorem is used below in Sect. 3, where we deal with the ζ-function and its derivative. Theorem 2.1. There exists a positive constant c > 0 such that for any n 6 = 0 and for any 0 < t < 1 we have the following estimates Z ∞ c , (2.7) h−n (s; u0 , v)gn (t − s; v, u0 )dv| < √ | s(t − s) 0 and Z ∞ c | (2.8) gn (s; u0 , v)gn (t − s; v, u0 )dv| < √ . t 0 The proof of Theorem 2.1 is completely elementary and consists of long and tedious computations. We present the proof in the Appendix at the end of the paper. Theorem 2.1 has the following immediate corollary.

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433

Corollary 2.2. Let γ(u) denote a non-increasing smooth function equal to 1 for u ≤ 1/8, and equal to 0 for u ≥ 1/4. Then there exists a positive constant c such that |Tr γ(u){G(∂u e1 ) ∗ K2 e1 }(t)| ≤ c·T r|K2 | and

√ |Tr γ(u){e1 ∗ K2 e1 }(t)| ≤ c t·T r|K2 |.

(2.9)

Proof. We prove the first estimate in (2.9), the proof of the second is completely analogous. |Tr γ(u){G(∂u e1 ) ∗ K2 e1 }(t)| Z X Z t Z ∞ ds γ(u)du ≤| n∈Z\{0}

0

0

X Z

≤ c1 ·

n∈Z\{0}

< c2 ·(

X

t

∞

ds|

0

Z

0 t

√

0 t/2

0

h−n (s; u0 , v)gn (t − s; v, u0 )dv|·|(ϕn ; K2 ϕn )|

0

|(ϕn ; K2 ϕn )|)· Z

∞

γ(u)du Z

1 < c3 ·T r|K2 |· p · t/2

h−n (s; u0 , v)gn (t − s; v, u0 )dv·(ϕn ; K2 ϕn )|

0

Z

n∈Z\{0}

∞

ds ≤ c3 ·T r|K2 |· s(t − s)

ds √ < c4 ·T r|K2 |. s

Z

t/2

√

0

ds s(t − s)



Proof of Proposition 1.1. We have |T r γ(u){G(∂u + B) + K1 )e}(t)| ∞ X = |Tr γ(u)G∂u { {e1 ∗ K2 e1 ∗ K2 e1 ∗ ... ∗ K2 e1 }}(t) n=1

≤ |Tr γ(u){G(∂u e1 ) ∗ K2 e1 }(t)| ∞ X γ(u){G(∂u e1 ) ∗ K2 e1 ∗ K2 e1 ∗ ... ∗ K2 e1 }(t)| + |Tr n=2

≤ c·T r|K2 | +

∞ X

Z |T r

t

Z ds1

Z ds2 ...

0

0

n=2

s1

sn−1

dsn ·

0

γ(u)(G∂u e1 )(sn )◦(K2 e1 )(sn−1 − sn )◦...◦(K2 e1 )(t − s1 )| Z s1 Z sn−1 ∞ Z t X ds1 ds2 .. dsn ≤ c·T r|K2 | + n=2

0

0

0

{T r|γ(u)(G∂u e1 )(sn )◦(K2 e1 )(sn−1 − sn )|· k(K2 e1 )(sn−2 − sn−1 k..k(K2 e1 )(t − s1 )k} Z sn−1 Z s1 ∞ Z t X ds1 ds2 ... dsn · ≤ c·T r|K2 | + n=2

0

0

0

T r|γ(u)(G∂u e1 )(sn )◦(K2 e1 )(sn−1 − sn )|·kK2 kn−1 Z sn−2 Z t Z s1 ∞ X n−1 ≤ c·T r|K2 |{1 + c· kK2 k ds1 ds2 ... dsn−1 } n=2

0

0

0

434

K. P. Wojciechowski

= c·T r|K2 |{1 + c·kK2 k·

∞ X (kK2 kt)n−2

(n − 2)!

n=2

} ≤ c1 ·T r|K2 |·ec2 tkK2 k

for some positive constants c1 and c2 . This ends the proof of Proposition 1.1.



3. The Modulus of the ζ-Determinant on the Grassmannian In this section we study the spectral invariants of the operator DP2 used in the construction of the ζ-determinant, namely ζDP2 (0) and d/ds(ζDP2 (s))|s=0 (see (0.1)). Let us review briefly the situation in the case of a closed manifold M . We follow here the presentation in [32] and the necessary technicalities can be found in [14]. We assume that D is an invertible operator. Otherwise det ζ D = 0. We have Z ∞ 2 1 ts−1 Tr e−tD dt (3.1) ζD2 (s) = Tr (D2 )−s = 0(s) 0 which is a well defined holomorphic function of s for Re(s) > n2 , where n = dim M , and has a meromorphic extension to the whole complex plane with only simple poles. The poles and residues are determined by the small time asymptotics of the heat kernel. Let E(t; x, y) denote the kernel of the operator exp(−tD2 ). The pointwise trace Tr E(t; x, x) has an asymptotic expansion as t → 0, tr E(t; x, x) = t−n/2

N X

tk/2 ak (D2 ; x) + o(t

N −n 2

),

(3.2)

k=0

where ak (D2 ; x) are computed from the coefficients of the operator D2 at the point x (see [14]). It follows that the meromorphic extension of ζD2 (s) has poles at the points sk = n−k 2 with residues given by Ress=sk ζD2 (s) = where ak (D2 ) denotes the integral

1

ak (D 0( n−k 2 )

2

),

(3.3)

Z

ak (D ) = 2

M

ak (D2 ; x)dx.

In particular, there are no poles at non-positive integers and ζD2 (0) is given by ζD2 (0) = an (D2 ). R∞

(3.4)

The functions 0(s) and 0 ts−1 Tr e−tD dt have the following asymptotic expansion in a neighborhood of s = 0: Z ∞ 2 1 an (D2 ) + b + sf (s) and 0(s) = + γ + sh(s), (3.5) ts−1 Tr e−tD dt = s s 0 2

where f (s) and h(s) are holomorphic functions of s and γ denotes Euler’s constant. R∞ 2 The number b denotes the regularized value of the integral 0 t−1 Tr e−tD dt. Now we differentiate

ζ-Determinant on the Smooth, Self-Adjoint Grassmannian

435

− ln det ζ (D2 ) = d/ds(ζD2 )|s=0 ) + b + sf (s) d an (D { 1s }|s=0 = b − γ·an (D2 ), = ds + γ + sh(s) s 2

(3.6)

and obtain the formula for the derivative of ζD2 (s) at s = 0. We want to study the corresponding invariants on a manifold with boundary for the ∗ (D). We show that despite the additional poles, caused operator DP , where P ∈ Gr∞ by the boundary contribution, at least in a neighborhood of s = 0 the situation is not different from the case of a closed manifold. First we have the following result which holds in the case of the operator Dσ . R∞ 2 Proposition 3.1. The function 0(s)ζDσ (s) = 0 ts−1 Tr e−tDσ dt has a simple pole at s = 0. Hence ζDσ2 (0) and, according to formula (3.6), ln det ζ (Dσ )2 = −d/ds(ζDσ2 (s))|s=0 are well defined. The proof of Proposition 3.1 consists of a straightforward computation of the boundary contribution and is included in the Appendix. Now the fact that ζDP2 (0) and d/ds(ζDP2 (s))|s=0 are well defined is an immediate consequence of the next theorem. ∗ (D) there exists a constant c > 0 such that the Theorem 3.2. For any P ∈ Gr∞ following estimate holds for any 0 < t < 1: √ 2 2 (3.7) |Tr e−tDP − Tr e−tDσ | < c t·T r|K2 |etkK2 k .

Proof. We essentially repeat the proof of Proposition 1.1. We replace the operator DP by the operator DU,σ and use Duhamel’s Principle to obtain |Tr e−tDP − Tr e−tD5s | = ∞ X {e1 ∗ K2 e1 ∗ K2 e1 ∗ ... ∗ K2 e1 }}(t)| + O(e−c/t ). (3.8) |Tr 2

2

n=1

Now we use the second part of Theorem 2.1 in order to estimate the sum on the right side of (3.8) in exactly the same way as in the proof of Proposition 1.1.  Theorem 3.2 shows that the difference ζDP2 (s)−ζDσ2 (s) is a holomorphic function of s for Re(s) > − 21 . Therefore ζDP2 (s) is a holomorphic function of s in a neighborhood of s = 0. The proof of Theorem 0.2 is now complete. Proof of Proposition 0.5. The proposition is an easy corollary of Theorem 3.2. It follows from (3.1) and (3.5) that we have the equality Z ∞ 2 2 1 2 2 ts−1 Tr (e−tDP − e−tDσ )dt =, ζDP (0) − ζDσ (0) = lim s→0 0(s) 0 Z 1 2 2 ts−1 T r(e−tDP − e−tDσ )dt. lim s s→0

0

Now we apply Theorem 3.2 and obtain

436

K. P. Wojciechowski

Z |ζDP2 (0) − ζDσ2 (0)| < lim s s→0

1

ts−1 |Tr e−tDP − Tr e−tDσ |dt 2

2

0

Z

1

< c·lim s s→0

ts−1/2 dt = 0.

0



This ends the Proof of Proposition 0.5.

∗ (D) 4. The Additivity of the η-Invariant on the Grassmannian Gr∞

Now we assume that D : C ∞ (X; S) → C ∞ (X; S) is a compatible Dirac operator acting on sections of a bundle of Clifford modules S over a closed partitioned odd-dimensional manifold X. In this section we study the decomposition of the η-invariant ηD = ηD (0) of the operator D into contributions coming from the different parts of the manifold. So, assume that we have a decomposition of X as M1 ∪ M2 , where M1 and M2 are compact manifolds with boundary such that M1 ∩ M2 = Y = ∂M1 = ∂M2 .

(4.1)

We also assume that the Riemannian metric on X and the Hermitian product on S are products in the bicollar neighborhood N˜ = [−1, 1] × Y of Y , where M1 ∩ N˜ = [−1, 0]×Y . The operator D is given by the formula (0.2) in N˜ . Let Di = D|Mi (i = 1, 2) ∗ (D). Let η(P1 , P2 ) denote the η-invariant of the operator G(∂u +B) and P1 , P2 ∈ Gr∞ on the manifold [−1, 1] × Y subject to the boundary condition P1 at u = −1 and the boundary condition Id − P2 at u = 1. The following Theorem is a generalization of the additivity formula for the η-invariant proved in [34] (see also [13, 20, 21, 33] for partial results and discussion of related topics). ∗ (D) one has the following formula Theorem 4.1. For any P1 , P2 ∈ Gr∞

ηD = ηD1Id−P + ηD2P + η(P1 , P2 ) mod Z.

(4.2)

2

1

Remark 4.2. (1) Theorem 4.1 extends Theorem 0.2 of [34] to boundary conditions from ∗ (D). In [34], Formula (4.2) was proved only for projections of the form 5σ , though Gr∞ the method extends without problems to all finite-dimensional Lagrangian perturbations of the Atiyah–Patodi–Singer condition. However, the Calderon projection of D is seldom ∗ (D). of this type. On the other hand it is an element of Gr∞ (2) Results analogous to Theorem 0.2 of [34] were obtained and discussed by other authors. We refer especially to the papers [9, 12, 19, 22, 23, 24]. (3) The crucial point in the proof of Theorem 4.1 is the extension of the formula on the variation of the η-invariant under a change of boundary condition from the work [21]. Let 5σ denote a projection given by Formula (1.6). The following special case of the additivity formula from [34] is the starting point of the proof of Theorem 4.1, ηD = ηD1Id−5 + ηD25 σ

σ

mod Z.

(4.3)

Now we have to study the variation of the η-invariant under a change of boundary ∗ ∗ (D) and let us choose a path {Pr }0≤r≤1 ⊂ Gr∞ (D) such that condition. Let P ∈ Gr∞ P0 = 5σ and P1 = P . It follows from Lemma 1.2 that we have a smooth family {gr } of

ζ-Determinant on the Smooth, Self-Adjoint Grassmannian

437

unitary operators of the form Id|(S|Y ) + smoothing operator which commutes with G and such that g0 = Id and g1 5σ g1−1 = P. Next, we choose a smooth non-increasing function γ(u) such that γ(u) = 1 f or u < 1/4 and γ(u) = 0 f or u > 3/4, and for each 0 ≤ r ≤ 1 use the family gr,u = grγ(u) f or 0 ≤ u ≤ 1,

(4.4)

in order to construct a corresponding unitary operator Ur on M2 (see Formula (1.7)). The operator D2Ur ,σ is unitarily equivalent to the operator D2Pr . This allows us to study the η-invariant of D2Ur ,σ instead of the η-invariant of D2Pr . It follows from Proposition 1.1 that the η-invariant of D2Ur ,σ is given by the formula (1.3), hence the variation of the η-invariant is given by the standard formula √ 2 2 d (ηD2U ,σ ) = − √ lim ε·Tr (d(D2Ur ,σ )/dr)e−εDUr ,σ mod Z. r ε→0 dr π

(4.5)

The main technical result of this section is the following theorem. ∗ (D), and any path g = {gr,u } connecting 5σ with P , Theorem 4.3. For any P ∈ Gr∞ as described above, the following formula holds:

ηD2P − ηD25

σ

∂g )|r0 = where (g −1˙ ∂u

1 =− π

Z

Z

1

1

dr 0

0

˙ ∂g du Tr G(g −1 )|r mod Z, ∂u

(4.6)

d −1 ∂g dr (g ∂u )|r=r0 .

Proof. We show that √ −εD22 1 2 Ur ,σ 0 √ lim ε·Tr (d(D2Ur ,σ )/dr)|r=r0 e = π π ε→0

Z

˙ ∂g Tr G(g −1 )|r=r0 du. ∂u (4.7)

1

0

We have ˙ ∂U

(U −1˙DU ) = G U −1

!

∂u

+ G[U

−1

BU, U

−1

˙ ∂g

U˙ ] = G g −1

∂u

Thus lim

ε→0

√

−εD22

ε·Tr (d(D2Ur ,σ )/dr)|r=r0 e

Ur ,σ 0

,

! + G[g −1 Bg, g −1 g]. ˙

438

K. P. Wojciechowski

contains two terms. Let us start with lim

ε→0

√

−εD22

ε·Tr G[g −1 Bg, g −1 g]e ˙

Ur ,σ 0

.

−εD22

Ur ,σ 0 by the operator Once again we use Duhamel’s Principle and replace e 2 2 exp(−t(−∂u + B + K2 )σ ) on the cylinder. The point here is that the kernel of this ˙ anticommutes with the operator commutes with G and the operator G[g −1 Bg, g −1 g] involution G. It follows that

˙ −εDUr ,σ = O(e−c/ε ), Tr G[g −1 Bg, g −1 g]e 2

and one is left with

˙ ∂U −εD22 √ 2 Ur ,σ 0 √ lim ε·Tr G(U −1 . )e ∂u π ε→0

−1˙ ∂g )| The term −G(U −1˙ ∂U ∂u )|r=r0 = G(g ∂u r=r0 is supported in [1/4, 3/4] × Y , and so we −εD22

Ur ,σ 0 by the kernel of the operator exp(−ε(−∂u2 + replace the kernel of the operator e B 2 )) on the infinite cylinder (−∞, +∞) × Y . More precisely, this is the kernel of the operator

exp(−εU −1 (−∂u2 + B 2 )U ) = U −1 e−ε(−∂u +B ) U = g −1 e−ε(−∂u +B ) g, 2

2

2

2

at r = r0 . Now we have ˙ ∂g √ 2 −εD22 Ur ,σ − √ lim ε·Tr G(g −1 )|r0 e ∂u π ε→0 Z 1 ˙ ∂g √ 2 2 2 = − √ lim ε du T rY G(g −1 )|r0 g −1 e−ε(−∂u +B ) g ∂u π ε→0 0 Z 1 ˙ ∂g √ 2 2 2 du T rY {g −1 G(g −1 )|r0 e−ε(−∂u +B ) g} = − √ lim ε ∂u π ε→0 0 Z 1 ˙ ∂g √ 2 2 2 du T rY G(g −1 )|r0 e−ε(−∂u +B ) = − √ lim ε ∂u π ε→0 0 Z 1 ˙ ∂g √ 2 2 1 = − √ lim ε √ du T rY G(g −1 )|r0 e−εB ∂u π ε→0 4πε 0 Z 1 Z ˙ ˙ ∂g 2 1 1 1 ∂g =− du lim T rY G(g −1 )|r0 e−εB = − du T rY G(g −1 )|r0 . ε→0 π 0 ∂u π 0 ∂u



Remark 4.4. In the paper [29] we discussed the special case of (4.6) with gr (u) given by the formula   Id 0 , gr (u) = 0 exp(irγ(u))2 where 2 : C ∞ (Y ; S − |Y ) → C ∞ (Y ; S − |Y ) is a self-adjoint operator with a smooth kernel. In this case our formula gives Z Z 1 1 1 Tr 2 dr du γ 0 (u)Tr 2 = mod Z. ηD2P − ηD25 = − σ π 0 π 0

ζ-Determinant on the Smooth, Self-Adjoint Grassmannian

439

∗ Corollary 4.5. Let P1 , P2 ∈ Gr∞ D, then

ηD2P − ηD2P = − 1

2

1 π

Z

Z

1

1

dr 0

0

˙ ∂g du Tr G(g −1 )|r mod Z, ∂u

(4.8)

where {gr,u } is any family connecting P1 with P2 in the way described above (see (4.4)). Corollary 4.6. η(P1 , P2 ) = −

1 π

Z

Z

1

dr 0

0

1

˙ ∂g du Tr G(g −1 )|r mod Z, ∂u

(4.9)

where {gr,u } is any family connecting P1 with P2 as described above (see (4.4)). The next result follows directly from the formula (4.8) which can be written as follows Z ˙ ∂g 1 1 d (ηD2P )|r=r0 = − du Tr G(g −1 )|r0 . (4.10) r dr π 0 ∂u d (ηD2P )|r=0 does not depend on the Proposition 4.7. The variation of the η-invariant dr r choice of the base projection P = P0 . It depends only on the family of unitary operators {gr }.

This result plays an important role in the proof of equality of ζ-determinant and C-determinant (see [29, 4, 30, 31]). Proof of Theorem 4.1. As a result of Corollary 4.5 and Corollary 4.6 the following sequence of equalities holds mod Z: ηD = ηD1Id−5 + ηD25 σ

σ

= (ηD1Id−5 + η(5σ , P1 )) − η(5σ , P1 ) + (ηD25 + η(P2 , 5σ )) − η(P2 , 5σ ) σ

σ

= ηD1Id−P + ηD2P + η(P1 , 5σ ) + η(5σ , P2 ) = ηD1Id−P + ηD2P + η(P1 , P2 ). 1

2

1

2

In the last line we use the elementary identities η(P1 , P2 ) = −η(P2 , P1 ) and η(P1 , P2 ) + η(P2 , P3 ) = η(P1 , P3 )

mod Z.



A. Proof of Theorem 2.1 and Proposition 3.1 We start with a discussion of Theorem 2.1. Recall the formulas for the functions gn (t; u, v) (see for instance [6], (22.33) and (22.35)) (u−v)2 (u+v)2 e−µn t gn (t; u, v) = √ ·{e− 4t − e− 4t } for n > 0, 2 πt 2

(A1)

440

K. P. Wojciechowski

and (u+v)2 e−(−µn ) t − (u−v)2 √ ·{e 4t + e− 4t } 2 πt √ u+v +(−µn )e−(−µn )(u+v) ·erf c( √ − (−µn ) t) f or n < 0, 2 t Z ∞ 2 2 2 2 e−r dr < √ e−x . erf c(x) = √ π x π 2

gn (t; u, v) =

where

We begin with the estimate of the integral most singular term is Z

∞

R∞ 0

gn (s; u0 , v)gn (t − s; v, u0 )dv. The

2 e−µn s − (u0 −v)2 e−µn (t−s) − (u4(t−s) 0 +v) 4s √ ·e · √ dv ·e 2 πs 2 π(t − s) 0 Z ∞ 2 t(u0 −v)2 1 e−µn t √ e− 4s(t−s) dv = 4π s(t − s) 0 r Z 2 2 1 1 1 e−µn t s(t − s +∞ −r2 √ 2 e dr = √ e−µn t < √ . < 4π s(t − s) t 2 πt 2 πt −∞ 2

2

We also have the inequality e−

(u+v)2 4t

≤ e−

(u−v)2 4t

,

which holds for u, v ≥ 0 and implies the estimate Z

∞ 0

2 e−µn s − (u0 ∓v)2 e−µn (t−s) − (u4(t−s) 1 0 ±v) 4s √ ·e ·e · √ dv ≤ √ . 2 πs 2 π(t − s) 2 πt 2

This gives

Z

∞ 0

2

1 gn (s; u0 , v)gn (t − s; v, u0 )dv < √ , 2 πt

for positive n. If n < 0 we also have to discuss the terms of the form Z

∞ 0

√ e−µn s − (u0 −v)2 u0 + v 4s √ ·e + µn t − s)dv. ·µn eµn (u0 +v) ·erf c( √ 2 πs 2 t−s 2

We have Z

∞

√ u+v e−µn s − (u0 −v)2 4s √ ·e + µn t − s) ·µn eµn (u+v) ·erf c( √ 2 πs 2 t−s 0 2 r −µ2n t Z ∞ −µ 2 t(u0 −v) 2 µn e n t t − s µn e − 4s(t−s) √ e dv < √ < ce−µn t . < t π s π 0 2

ζ-Determinant on the Smooth, Self-Adjoint Grassmannian

441

Finally, we have to estimate the term in which the erf c function appears twice, Z

∞

0

√ √ u0 + v u0 + v + µn t − s)dv µn eµn (u0 +v) erf c( √ + µn s) ·µn eµn (u0 +v) erf c( √ 2 s 2 t−s Z √ u0 +v √ 2 −( √ 0 +v +µ +µn t−s)2 4 ∞ 2 2µn (u0 +v) −( u2√ n s) s µn e e ·e 2 t−s dv < π 0 r Z Z 2 s(t − s) +∞ −r2 4 2 −µ2n t 4 2 −µ2n t ∞ − t(u 0 +v) 4s(t−s) e dv = µn e 2 e dr = µn e π π t 0 −∞ √√ √ 2 2 8 ≤ √ µ2n e−µn t 2 t π < c te−µn t . π

The computations given above finish the proof of (2.8). We work the same way in order to obtain the estimate (2.7). The only difference is the appearence of hn (t; u0 , v) = ∂gn ∂u (t; u0 , v). The first term we have to consider has the form Z

∞

2 e−µn s |u0 − v| − (u0 −v)2 e−µn (t−s) − (u4(t−s) 0 −v) 4s √ e · √ dv e 2 πs 2s 2 π(t − s) 0 Z ∞ 2 e−µn t |u0 − v| − (u0 −v)2 1 4s √ e dv < 4π 2 s(t − s) 0 2s Z u0 Z ∞ 2 u0 − v − (u0 −v)2 v − u0 − (u0 −v)2 1 e−µn t 4s 4s √ e e { dv + dv} = 4π s(t − s) 0 2s 2s u0 Z u0 Z ∞ 2 2 (u0 −v)2 t(u0 −v)2 1 e−µn t 1 e−µn t √ ·√ { . d(e− 4s ) − d(e− 4s } < = 4π s(t − s) 0 2π s(t − s) u0 2

2

R∞ We work on the other terms which appear in 0 h−n (s; u0 , v)gn (t − s; v, u0 )dv in the same way. The details are left to the reader. Now, we show that the function 0(s)ζDσ (s) has a simple pole at s = 0. We follow the method applied in Sect. 4 of [13] to study the η-invariant of Dσ . Let us point out that the situation is simpler in the case of the η-invariant due to the absence of the boundary contribution. Nevertheless, the result corresponding to Lemma 4.2 of [13] holds also in the present case. Namely modulo a function holomorphic on the whole complex plane, 0(s)ζDσ (s) splits into an interior contribution and a cylinder contribution. This again follows from Duhamel’s Principle. First of all, Z ∞ ts−1 Tr e−tDσ dt, 1

is a holomorphic function on the whole complex plane. For 0 < t < 1, we replace exp(−tDσ2 ) by the operator exp(−tD˜ 2 ) inside of M and by the operator exp(−t(−∂u2 + B 2 )σ ) on N . The interior contribution produces simple poles at the points sk = k−n 2 with residues given by the formula Z ak (D˜ 2 ; x)dx, M

442

K. P. Wojciechowski

˜ , the closed double of M (see where D˜ denotes the double of the Dirac operator D on M formulas (3.2) and (3.3)). In particular the contribution to the residuum at s = 0 is equal to Z an (D˜ 2 ; x)dx = 0. M

This is due to the point-wise vanishing of an (D˜ 2 ; x), which follows from the fact that n = dim M is odd (see for instance [14]). The cylinder contribution has the form Z ∞ Z ∞ X Z ∞ 2 2 ts−1 Tr γ(u)e−t(−∂u +B )5σ dt = ts−1 dt γ(u)gn (t; u, u)du, 0

0

n∈Z\{0}

0

where γ(u) denotes the cut-off function. The integral Z ∞ Z ∞ ts−1 dt γ(u)gn (t; u, u)du 0

0

consists of two terms. The first term produces the contribution Z ∞ 2 X Z ∞ u2 e−µn t s−1 √ ·(1 − sign(n)e− t )du}. t dt{ γ(u) 2 πt 0 n∈Z\{0} 0 This is convergent for Re(s) > n/2 and in fact it is equal to Z

∞ 0

=

t

s−1

R∞ 0

dt{

X Z

n∈Z\{0}

γ(u)du √ · 2 π

Z

∞

0

∞

u2 e−µn t γ(u) √ ·(1 − sign(n)e− t )du} 2 πt 2

ts−3/2 T r e−tB dt. 2

0

It follows now from (3.2) and (3.3) that the expression on the right side has a meromorphic extension to the whole complex plane with simple poles. Moreover, it is regular at s = 0. The reason is that the residuum at s = 0 is given by the formula R∞ γ(u)du 0 √ adim (Y )+1 (B 2 ), 2 π and adim (Y )+1 (B 2 ) is equal to 0 due to the fact that dim (Y ) + 1 is an odd number (see for instance [14]). We are left with Z ∞ Z ∞ X √ u ts−1 dt γ(u){ µn e2uµn erf c( √ + µn t)}du. t 0 0 n>0 We only have to show that this term produces at most a simple pole at s = 0. We can neglect the presence of the cut-off function γ(u) and then we obtain for large <(s), Z ∞ X Z ∞ √ u s−1 t dt { µn e2uµn erf c( √ + µn t)}du t 0 0 n>0 Z ∞ Z ∞ X √ d 2uµn u 1 (e ts−1 dt { )erf c( √ + µn t)}du = 2 0 du t 0 n>0

ζ-Determinant on the Smooth, Self-Adjoint Grassmannian

1 = 2

Z Z

∞

443

X

√ u {e2uµn erf c( √ + µn t)}|∞ 0 t n>0

ts−1 dt{

0

√ d u {erf c( √ + µn t)})du} du t Z0 Z ∞ X √ 2 du 1 ∞ s−1 X −µ2n t 2 √ ( t { e e−u /t √ ) − erf c(µn t)}dt = 2 0 π 0 t n>0 n>0 Z ∞ Z ∞ X √ 2 1 1 ts−1 Tr e−tB dt − ts−1 { erf c(µn t)}dt. = 2 0 2 0 n>0 −

∞

e2uµn (

The first sum on the right side is equal to 21 0(s)ζB 2 (s), hence it produces the correct asymptotic expansion with a simple pole at s = 0. We need the next identity in order to deal with the second sum. Z Z ∞ √ √ 1 ∞ d s (t )erf c(µn t)dt ts−1 erf c(µn t)dt = s 0 dt 0 √ ∞ 1 Z ∞ s −µ2 t µn 1 s t e n √ dt = { t erf c(µn t)}|0 − s s 0 2 t Z ∞ 2 0(s + 1/2) 1 ts−1/2 e−µn t dt = − (µ2n )−s . = − µn 2s 2s 0 Therefore we obtain Z √ 0(s + 1/2) 1 0(s + 1/2) 1 ∞ s−1 X ζB 2 (s) = ζB 2 (s), t { erf c(µn t)}dt = − 2 0 4s 2 8s n>0 and the expression on the right side has a meromorphic extension to the whole complex plane, with a simple pole at s = 0 with residuum √ 0(s + 1/2) π ζB 2 (s) = adim (Y ) (B 2 ). Ress=0 8s 8 R∞ 2 We have shown that the cylinder contribution to the trace integral 0 ts−1 Tr e−tD5σ dt has a meromorphic extension to the whole complex plane with an isolated simple pole at s = 0, which ends the proof of Proposition 3.1. References 1. Atiyah, M.F., Patodi, V.K., and Singer, I.M.: Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Phil. Soc. 77, 43–69 (1975) 2. Bismut, J.-M. and Freed, D.S.: The analysis of elliptic families.II. Dirac operators, eta invariants, and the holonomy theorem. Commun. Math. Phys. 107, 103–163 (1986) 3. Booß–Bavnbek, B., Morchio, G., Strocchi, F., and Wojciechowski, K.P.: Grassmannian and chiral anomaly. J. Geom. Phys. 22, 219–244 (1997) 4. Booß–Bavnbek, B., Scott, S.G., and Wojciechowski, K.P.: The ζ-determinant and C-determinant on the Grassmannian in dimension one. Lett. Math. Phys. 45, 353–362 (1998) 5. Booß–Bavnbek, B., and Wojciechowski, K.P.: Pseudo-differential projections and the topology of certain spaces of elliptic boundary value problems. Commun. Math. Phys. 121, 1–9 (1989) 6. Booß–Bavnbek, B., and Wojciechowski, K.P.: Elliptic Boundary Problems for Dirac Operators. Boston: Birkh¨auser, 1993

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