ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Pages 1–11 (January 14, 2005) S 1079-6762(05)00141-1

ON GLUING FORMULAS FOR THE SPECTRAL INVARIANTS OF DIRAC TYPE OPERATORS

PAUL LOYA AND JINSUNG PARK

(Communicated by Michael Taylor)

Abstract. In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac type operators on compact manifolds and manifolds with cylindrical ends. We also explain the central ideas in their proofs.

1. The gluing problem for the spectral invariants

Since their inception, the eta invariant and the ζ-determinant of Dirac type operators have influenced mathematics and physics in innumerable ways. Especially with the development of quantum field theory, the behavior of these spectral invariants under gluing of the underlying manifold has become an increasingly important topic. However, the gluing formula for the ζ-determinant of a Dirac Laplacian has remained an open question due to the nonlocal nature of this invariant. In fact, Bleecker and Booss-Bavnbek stated that [2, p. 89] “no precise pasting formulas are obtained but only adiabatic ones.” In [23, 24], we give precise gluing formulas for ζ-determinants of Dirac type operators on compact manifolds and manifolds with cylindrical ends, respectively, and moreover we present new and unified derivations of the gluing formulas for both invariants. The purpose of this note is to announce these gluing formulas for the spectral invariants and to indicate the main ideas in their proofs. We also announce a relative invariant formula proved in [25, 24].

We begin with describing the gluing problem for compact manifolds. Let D be a Dirac type operator acting on C^{∞}(M, S) where M is a closed compact Riemannian manifold of arbitrary dimension and S is a Clifford bundle over M. Let Y be an embedded hypersurface in M and let M = M_{−} ∪ M_{+} be decomposition of M into manifolds with boundary such that ∂M_{−} = ∂M_{+} = Y. We assume that all geometric structures are of product type over a tubular neighborhood N = [−1, 1]×Y of Y where the Dirac operator takes the product form D = G(∂_{u} +D_{Y}), where G is a unitary operator on S_{0} := S|_{Y} and D_{Y} is a Dirac type operator over Y satisfying G^{2} = −Id and D_{Y} G = −GD_{Y}. Recall that the eta function of D and the zeta function of D^{2} are defined through the heat operator e^{−tD^{2}} via

(1.1)  η_{D}(s) = \frac{1}{Γ(\frac{s+1}{2})}( ∫_{0}^{1} + ∫_{1}^{∞} ) t^{\frac{s−1}{2}} Tr(De^{−tD^{2}}) dt,

\begin{matrix}c \\ ◯ \end{matrix}2005 American Mathematical Society

Reverts to public domain 28 years from publication

Received by the editors October 6, 2004.

2000 Mathematics Subject Classification. Primary 58J28, 58J52.

(1.2)  ζ_{D^{2}}(s) = \frac{1}{Γ(s)}( ∫_{0}^{1} + ∫_{1}^{∞} )t^{s−1} Tr(e^{−tD^{2}}) dt,

where the first integrals in (1.1) and (1.2) are defined a priori for Re s ≫ 0 and the second ones for Re s ≪ 0, and both extend to meromorphic functions on ℂ that are regular at s = 0; actually, the second integral in (1.1) is entire, but we present these general definitions because they work later for b-spectral invariants on manifolds with cylindrical ends (see Section 2). The eta invariant η(D) is by definition the value η_{D}(0), the reduced eta invariant is η̃(D) := (η(D) + dim ker(D))/2, and the ζ-determinant is by definition

det_{ζ}D^{2} := exp ( − \frac{d}{ds}|_{s=0}ζ_{D^{2}}(s)).

The eta invariant was introduced in the paper [1] by Atiyah, Patodi, and Singer as the boundary correction term in their index formula for manifolds with boundary. The ζ-determinant was introduced by Ray and Singer in the paper [36] on the analytic torsion. Since their introductions, these invariants have impacted geometry, topology, and physics in incredible ways; cf. Singer [40, 41].

By restriction, D induces Dirac type operators D_{+} over M_{+} and D_{−} over M_{−}. For these operators, we choose orthogonal projections P_{+}, P_{−} over L^{2}(Y, S_{0}) that provide us with well-posed boundary conditions for D_{+}, D_{−} in the sense of Seeley [39]. Then the operators

(1.3)  D_{P_{±}} : dom(D_{P_{±}}) → L^{2}(M_{±}, S),

where

dom(D_{P_{±}}) := { ϕ ∈ H^{1}(M_{±}, S) | P_{±}(ϕ|_{Y}) = 0 },

share many of the analytic properties of D; in particular, they are Fredholm and have discrete spectra, but are not necessarily self-adjoint. Amongst such projectors are the (orthogonalized) Calderón projectors C_{±} [7], which are projectors defined intrinsically as the unique orthogonal projectors onto the closures in L^{2}(Y, S_{0}) of the infinite-dimensional Cauchy data spaces of D_{±}:

{ ϕ|_{Y} | ϕ ∈ C^{∞}(M_{±}, S), D_{±}ϕ = 0 } ⊂ C^{∞}(Y, S_{0}).

In order to have well-defined eta invariants and ζ-determinants, it is necessary to restrict to a subclass of projectors. A natural class is formed by those in the smooth, self-adjoint Grassmannian Gr_{∞}^{∗}(D_{±}), which consists of orthogonal projections P_{±} such that P_{±} − C_{±} are smoothing operators and GP_{±} = (Id − P_{±})G. Examples of such projectors are the generalized APS spectral projectors P_{−} = Π_{<} + \frac{1+σ_{1}}{2} Π_{0}, P_{+} = Π_{>} + \frac{1+σ_{2}}{2} Π_{0}, where Π_{<}, Π_{>}, Π_{0} are the orthogonal projectors onto the negative, positive, and zero eigenspaces of D_{Y}, respectively, and the σ_{i}’s are involutions on ker(D_{Y}) anticommuting with G. For P_{±} ∈ Gr_{∞}^{∗}(D_{±}), the invariants η(D_{P_{±}}) and det_{ζ}D_{P_{±}}^{2} can be defined via the formulas (1.1) and (1.2); see Grubb [12, 13], Loya and Park [25], and Wojciechowski [44].

We now have all the ingredients to state the gluing problem. The gluing problem is to describe the “defects”

\frac{det_{ζ}D^{2}}{det_{ζ}D_{P_{+}}^{2} · det_{ζ}D_{P_{−}}^{2}} = ? ,  η̃(D) − η̃(D_{P_{+}}) − η̃(D_{P_{−}}) = ?

in terms of recognizable data. The gluing problem for the eta invariant has been solved by several authors. Moreover, because the variation of the eta invariant is local, a variety of formulas and proofs have appeared (many modulo ℤ); see, for instance, Brüning and Lesch [3], Bunke [5], Dai and Freed [9], Hassell, Mazzeo, and Melrose [16], Kirk and Lesch [18], Mazzeo and Melrose [28], Müller [32], Park and Wojciechowski [35], Wojciechowski [43, 44]; cf. the superb survey articles by Mazzeo and Piazza [29] or Bleecker and Booss-Bavnbek [2]. Because of the highly nonlocal nature of the ζ-determinant and its variation, according to Bleecker and Booss-Bavnbek [2, p. 90] the ζ-determinants (of a Dirac operator and its square) are “the most subtle and the most fascinating objects of our study.” Nonetheless, the gluing problem for the ζ-determinant of Laplace type operators with local boundary conditions was solved by Burghelea, Friedlander, and Kappeler [6] and has been further extended by Carron [8], Hassell [14], Hassell and Zelditch [15], Lee [19], Loya and Park [22], Vishik [42], and many others. However, the gluing formula for the ζ-determinant of a Dirac Laplacian has remained an open question, partly due to the nonlocal nature of the ζ-determinant and its variation and the technical aspects inherent in the global pseudodifferential boundary problems required for Dirac type operators. We now describe our solution to the gluing problem.

To state our main theorem, we recall that the Calderón projectors C_{±} have the matrix forms

(1.4)  C_{±} = \frac{1}{2} (\begin{matrix}Id & κ_{±}^{−1} \\ κ_{±} & Id \end{matrix})

with respect to the decomposition L^{2}(Y, S_{0}) = L^{2}(Y, S^{+}) ⊕ L^{2}(Y, S^{−}), where S^{±} ⊂ S_{0} are the (±i)-eigenspaces of G. The maps κ_{±} : L^{2}(Y, S^{+}) → L^{2}(Y, S^{−}) are isometries, so that U := −κ_{−}κ_{+}^{−1} is a unitary operator over L^{2}(Y, S^{−}), which is moreover of Fredholm determinant class. We denote by Û the restriction of U to the orthogonal complement of its (−1)-eigenspace. We also put

(1.5)  L := ∑_{k=1}^{h_{M}} γ_{0}U_{k} ⊗ γ_{0}U_{k} = ∑_{k=1}^{h_{M}} ⟨ · , γ_{0}U_{k}⟩_{L^{2}(Y,S_{0})}γ_{0}U_{k},

where h_{M} = dim ker(D), γ_{0} is the restriction map from M to Y, and {U_{k}} is an orthonormal basis of ker(D). Then L is a positive operator on the finite-dimensional vector space γ_{0}(ker(D)). The following theorem is the main result of [23].

Theorem 1.1 ([23]). The following gluing formulas hold:

\frac{det_{ζ}D^{2}}{det_{ζ}D_{C_{+}}^{2} · det_{ζ}D_{C_{−}}^{2}} = 2^{−ζ_{D^{2}_{Y}} (0)−h_{Y}} (det L)^{−2} det_{F} (\frac{2 Id + Û + Û^{−1}}{4} ),

η̃(D) − η̃(D_{C_{+}}) − η̃(D_{C_{−}}) = \frac{1}{2πi} Log det_{F} U (mod ℤ),

where h_{Y} = dim ker(D_{Y}), det_{F} denotes the Fredholm determinant, Log is the principal value of the logarithm, and the integer defect in the eta formula is given in terms of the winding numbers of the function det_{F} (K(λ)K^{c}(λ)^{−1}) appearing in Section 4.

Let us remark that the ζ-determinant formula in Theorem 1.1 is a new result, and only its adiabatic limit form [34], [35] has been proved hitherto. The formulation of the integer defect in the eta formula in Theorem 1.1 is also new. The eta formula, without integer ambiguity, in terms of Tr log U was proved by Kirk and Lesch [18, Th. 5.10] using spectral flow, the Scott-Wojciechowski comparison theorem [38], and the rotating boundary condition technique from Brüning and Lesch [3]. Our proof of the eta formula is different from and independent of their proof, and is obtained “simultaneously” as a byproduct from the proof for the ζ-determinant.

We can generalize the gluing formulas in Theorem 1.1 in terms of other boundary conditions by using Theorem 3.1 (see Section 3) proved in [25]. Let P_{1} ∈ Gr_{∞}^{∗}(D_{−}) and P_{2} ∈ Gr_{∞}^{∗}(D_{+}). Then P_{1} and P_{2} determine maps κ_{1} and κ_{2} as in (1.4), and we can define U_{1} := κ_{−}κ_{1}^{−1}, U_{2} := κ_{2}κ_{+}^{−1} and U_{12} := −κ_{1}κ_{2}^{−1} over L^{2}(Y, S^{−}). As before, we let Û_{i} denote the restriction of U_{i} to the orthogonal complement of its (−1)-eigenspace. We define the operator L_{1} over the finite-dimensional vector space Im(C_{−}) ∩ Im(Id − P_{1}) by

(1.6)  L_{1} = −P_{1} G R_{−}^{−1} G P_{1},

where R_{−} is the sum of the Dirichlet to Neumann maps on the double of M_{−}, which was introduced by Burghelea, Friedlander, and Kappeler in [6], and P_{1} is the orthogonal projection onto Im(C_{−}) ∩ Im(Id − P_{1}). Then L_{1} is a positive operator [25]. We define L_{2} in a similar way. We can now state the general gluing formulas for the spectral invariants.

Theorem 1.2. The following general gluing formulas hold:

\frac{det_{ζ}D^{2}}{det_{ζ}D_{P_{1}}^{2} · det_{ζ}D_{P_{2}}^{2}} = 2^{−ζ_{D^{2}_{Y}} (0)−h_{Y}} (det L)^{−2} det_{F} (\frac{2Id + Û + Û^{−1}}{4} )

× ∏_{i=1}^{2}(det L_{i})^{−2} · det_{F} (\frac{2Id + Û_{i} + Û_{i}^{−1}}{4} )^{−1},

η̃(D) − η̃(D_{P_{1}}) − η̃(D_{P_{2}}) = \frac{1}{2πi} Log det_{F} U_{12} (mod ℤ),

where the integer defect in the eta formula can be identified exactly in terms of winding numbers of specific naturally defined operators.

Let us remark that the adiabatic decomposition formulas presented in [34], [35], which are proved mainly using the Duhamel principle and scattering theory, can be derived from the ζ-determinant formula in Theorem 1.2 (cf. [27]).

2. The gluing problem on manifolds with cylindrical ends

We now describe the solution to the gluing problem on manifolds with cylindrical ends presented in [24]. Assume now that instead of M_{+} being compact, it is a noncompact half-infinite cylinder: M_{+} = [0, ∞) × Y, over which D is of product type. We still assume that M_{−} is compact. Then M is called a manifold with cylindrical end. For a noncompact manifold M with cylindrical end, the heat operators De^{−tD^{2}} and e^{−tD^{2}} are not of trace class. In particular, the definitions (1.1) and (1.2) cannot be used as in the compact case. There are two principal ways to make sense of these invariants. One way is to define so-called relative invariants as in Bruneau [4], Carron [8], Müller [33], and others, whereby we subtract off certain operators that make the difference of the heat operators of trace class, and the other way is to use Melrose’s b-trace [30], ^{b}Tr, which is a natural substitute for the trace. In particular, De^{−tD^{2}} and e^{−tD^{2}} are b-trace class. Moreover,^{b}Tr(De^{−tD^{2}}) and ^{b}Tr(e^{−tD^{2}}) have asymptotic expansions in half-integer powers of t as t → 0 and t → ∞. It follows that the b-eta function ^{b}η_{D}(s) and the ^{b}ζ-function ^{b}ζ_{D^{2}}(s) can be defined exactly as in formulas (1.1) and (1.2), respectively, where we replace Tr with ^{b}Tr. Furthermore, ^{b}η_{D}(s) and ^{b}ζ_{D^{2}}(s) extend to define meromorphic functions on ℂ that are regular at s = 0, so we can define the b-eta invariant of D by bη(D) := ^{b}η_{D}(0), and the b-determinant of D^{2} by det_{^{b}ζ}D^{2} := exp(−^{b}ζ′_{D^{2}}(0)).

In order to discuss the gluing formula for the b-spectral invariants into the decomposition M = M_{−} ∪M_{+}, we need to impose a boundary condition on M_{+}. Note that M_{−} is compact so D_{C_{−}} is defined as before. Hence, we need a “Calderón projector” on the noncompact manifold M_{+}, which is found by looking at the Cauchy data space of D over the whole manifold M:

{ ϕ|_{Y} | ϕ ∈ C^{∞}(M, S), Dϕ = 0 } ⊂ C^{∞}(Y, S_{0}).

Atiyah, Patodi, and Singer [1] (cf. [30, 31]) showed that an element ψ of this space is the restriction ψ = ϕ|_{Y} of an L^{2} section ϕ over M if and only if (Π_{<} + \frac{1+σ}{2} Π_{0})ψ = 0, where σ is the unitary map on ker(D_{Y}) such that σ^{2} = Id and σG = −Gσ determined by the scattering matrix. For this reason, the natural projection on M_{+} is

(2.1)  C_{+} := Π_{>} + \frac{1 − σ}{2} Π_{0}.

We can now define D_{C_{+}} exactly as before (see (1.3)) in the case when M_{+} was compact. However, since M_{+} is a manifold with cylindrical end, we need to use the b-trace to define the corresponding b-spectral invariants, ^{b}η(D_{C_{+}}) and det_{^{b}ζ}D_{C_{+}}^{2}.

We now have all the ingredients to state the gluing problem for manifolds with cylindrical end. The (b-)gluing problem is to describe the “defects”

\frac{det_{^{b}ζ}D^{2}}{det_{ζ}D_{C_{−}}^{2} · det_{^{b}ζ}D_{C_{+}}^{2}} = ? , ^{b}η̃(D) − η̃(D_{C_{−}}) −^{b}η̃(D_{C_{+}}) = ?

in terms of recognizable data. Just as before, the projectors C_{±} can be written as (1.4) for unitary operators κ_{±}, and we denote by Û the restriction of U := −κ_{−}κ_{+}^{−1} to the orthogonal complement of its (−1)-eigenspace. Also, we define L as in (1.5), using the orthonormal basis {U_{k}} of ker_{L^{2}}(D). Then L is a positive operator on the finite-dimensional vector space γ_{0}(ker_{L^{2}}(D)). The following theorem is proved in [24].

Theorem 2.1 ([24]). The following gluing formulas hold:

\frac{det_{^{b}ζ}D^{2}}{det_{ζ}D_{C_{−}}^{2} · det_{^{b}ζ}D_{C_{+}}^{2}} = 2^{−ζ_{D^{2}_{Y}} (0)−h_{Y}}(det L)^{−2} det_{F} (\frac{2 Id + Û + Û^{−1}}{4} ),

bη̃(D) − η̃(D_{C_{−}}) −^{b}η̃(D_{C_{+}}) = ^{b}η̃(D) − η̃(D_{C_{−}}) = \frac{1}{2πi} Log det_{F} U (mod ℤ),

where h_{Y} := dim ker(D_{Y}) and det_{F} denotes the Fredholm determinant.

One can show that det_{^{b}ζ}D^{2}/det_{^{b}ζ}D_{C_{+}}^{2} = det_{ζ}(D^{2}, D_{C_{+}}^{2}), the relative determinant of the pair (D^{2}, D_{C_{+}}^{2}), and ^{b}η̃(D) = η̃(D, D_{C_{+}}), the reduced relative eta invariant of the pair (D, D_{C_{+}}), so this theorem can be written with relative spectral invariants

on the left-hand side. Also, some computations show that det_{^{b}ζ}D_{C_{+}}^{2} = 2^{\frac{1}{2} ζ_{D^{2}_{Y}} (0)}, so after substituting this expression into our theorem, we can get another version of the ζ-determinant formula without the det_{^{b}ζ}D_{C_{+}}^{2} term.

3. The comparison, or relative invariant, problem

The comparison problem for the spectral invariants on M_{−} can be stated as follows: For P_{−} ∈ Gr_{∞}^{∗}(D_{−}), find formulas for the “defects”

\frac{det_{ζ}D_{P_{−}}^{2}}{det_{ζ}D_{C_{−}}^{2}} = ? ,  η̃(D_{P_{−}}) − η̃(D_{C_{−}}) = ?

in terms of recognizable data. (Of course, one can consider M_{+} too.) For generalized APS spectral projectors, the comparison problem for the eta invariant was first solved modulo ℤ by Lesch and Wojciechowski [20] and Müller [31]; later, this integer ambiguity was removed by the first author [21]. For P_{−} ∈ Gr_{∞}^{∗}(D_{−}) and assuming the invertibility of D_{P_{−}}, the comparison problems for the eta invariant and the ζ-determinant were solved by Scott and Wojciechowski [38] and Scott [37]; cf. Forman [10]. The comparison problem for the eta invariant was later solved for P_{−} ∈ Gr_{∞}^{∗}(D_{−}) without integer ambiguities by Kirk and Lesch [18] using the Scott-Wojciechowski comparison theorem [38]. In [25], we remove the invertibility assumption on D_{P_{−}}, and we present a new and unified derivation of the comparison formulas for both the eta invariant and ζ-determinant. For the eta invariant, we obtain a new formulation of the integer defect, and the ζ-determinant formula contains an additional term that is absent in the case of invertible D_{P_{−}} dealing with ker(D_{P_{−}}) and the Dirichlet to Neumann map on the double of M_{−}.

Recall that the Calderón projector C_{−} can be written as (1.4) for a unitary operator κ_{−}, and P_{−} has a similar decomposition with a unitary operator κ_{P_{−}}. We denote by Û_{P_{−}} the restriction of U_{P_{−}} := κ_{−}κ_{P_{−}}^{−1}, which is a unitary operator over L^{2}(Y, S^{−}), to the orthogonal complement of its (−1)-eigenspace. We define L_{P_{−}} := −P_{−} G R_{−}^{−1}G P_{−} just as in (1.6), where P_{−} is the orthogonal projection onto the finite-dimensional vector space Im(C_{−}) ∩ Im(Id − P_{−}). Then L_{P_{−}} is a positive operator on this space [25], so det L_{P_{−}} is a positive number. The following theorem solves the comparison problem (there is a corresponding formula for the comparison problem on M_{+}).

Theorem 3.1 ([25]). The following comparison formulas hold:

\frac{det_{ζ}D_{P_{−}}^{2}}{det_{ζ}D_{C_{−}}^{2}} = (det L_{P_{−}})^{2} · det_{F} (\frac{2Id + Û_{P_{−}} + Û_{P_{−}}^{−1}}{4} ),

η̃(D_{P_{−}}) − η̃(D_{C_{−}}) = \frac{1}{2πi} Log det_{F} U_{P_{−}} (mod ℤ),

where the integer defect in the eta formula can be identified exactly in terms of winding numbers of a specific naturally defined operator.

Assume now that M_{+} is an infinite cylinder as in Section 2. Recall that C_{+} := Π_{>} + \frac{1−σ}{2} Π_{0} (see (2.1)) is the natural Calderón projector on the cylinder M_{+}. This suggests that the natural boundary projector on M_{−} taking into consideration the infinite cylinder should be S_{−} := Π_{<} + \frac{1+σ}{2} Π_{0}, instead of the Calderón projector C_{−}. The projector S_{−} is called the augmented APS spectral projector (cf. [17]). Then S_{−} ∈ Gr_{∞}^{∗}(D_{−}), so we can define the spectral invariants of D_{S_{−}} on M_{−}. The (b-)comparison problem is to describe the “defects”

\frac{det_{^{b}ζ}D^{2}}{det_{ζ}D_{S_{−}}^{2}} = ? , ^{b}η(D) − η(D_{S_{−}}) = ?

in terms of recognizable data. Thus, the comparison problem in this context is the comparison of the b-spectral invariants on M with those on M_{−} corresponding to the “scattering Calderón projector” S_{−}. The following theorem solves the comparison problem on manifolds with cylindrical ends.

Theorem 3.2 ([24]). The following comparison formulas hold:

\frac{det_{^{b}ζ}D^{2}}{det_{ζ}D_{S_{−}}^{2}} = 2^{− \frac{1}{2} ζ_{D^{2}_{Y}} (0)−h_{Y}} (\frac{det L}{det L_{−}} )^{−2}, ^{b}η(D) = η(D_{S_{−}}) (mod 2ℤ),

where L_{−} is defined by (1.5) but with {U_{k}} an orthonormal basis for ker(D_{S_{−}}).

Thus, the b-eta invariant of D and the eta invariant of D_{S_{−}} are identical modulo 2ℤ, while the ζ-determinants differ by data over Y and global data. Because ζ-determinants are highly nonlocal compared to eta invariants, one would not expect the ratio to be unity. This is indeed the case as shown in Theorem 3.2. The ζ-determinant formula is new, and a similar formula for the eta invariants of compatible Dirac operators (without integer ambiguities and with the augmented APS projector Π_{<} + \frac{1−σ}{2} Π_{0}) was first proved by Müller [31] using a completely different method.

4. Brief sketch of the proof of Theorem 1.1

In this last section, we briefly sketch the proof of Theorem 1.1; we refer to Section 1 for the notation and to our preprint [25] for all the details. We remark that the proofs of Theorems 2.1 and 3.1 are similar in spirit, but the proof of Theorem 2.1 has surprising twists due to the presence of continuous spectrum. For λ not in the spectrum of D, the Calderón projectors of D_{±} − λ are defined by

(4.1)  P_{±}(λ) := ±γ_{0^{±}}(D − λ)^{−1}γ_{0}^{∗}G : L^{2}(Y, S_{0}) → L^{2}(Y, S_{0}),

where γ_{0^{±}} := lim_{ε→0^{±}} γ_{ε} with γ_{ε} the restriction map from M to {ε} × Y, and γ_{0}^{∗} is the adjoint map of γ_{0}, the restriction to {0} × Y. We now introduce an auxiliary model problem over N = [−1, 1]×Y. To do so, fix an involution ω over ker(D_{Y}) that anticommutes with G. We then impose boundary conditions C_{−}^{c} = Π_{<} + \frac{1−ω}{2} Π_{0} at \{1\} × Y, C_{+}^{c} = Π_{>} + \frac{1+ω}{2} Π_{0} at \{−1\} × Y. Let us denote the operator G(∂_{u} + D_{Y}) with these boundary conditions by D^{c}. Then we can define the Calderón projectors P_{±}^{c}(λ) for the restrictions D_{±}^{c} − λ of D^{c} − λ to M_{±} ∩ N exactly as in (4.1). Let V be a unitary operator on L^{2}(Y, S_{0}) that satisfies V (Id − C_{+})V^{−1} = C_{−}. Then we define operators K(λ) and K^{c}(λ) on L^{2}(Y, S_{0}) by defining their inverses as

K(λ)^{−1} = C_{+}P_{+}(λ) + (Id − C_{+})V^{−1}C_{−}P_{−}(λ),
K^{c}(λ)^{−1} = C_{+}^{c}P_{+}^{c}(λ) + C_{−}^{c}P_{−}^{c}(λ).

(The operators on the right-hand side do turn out to be invertible.) From these formulas, we see that K(λ) and K^{c}(λ) “link” the Calderón projectors of D_{±} − λ and D_{±}^{c} − λ, respectively. Moreover, these operators have quite remarkable properties as we shall see. First, K(λ)K^{c}(λ)^{−1} is of Fredholm determinant class, so det_{F} (K(λ)K^{c}(λ)^{−1}) makes sense. We henceforth fix a simply connected region of the plane consisting of the upper and lower half-planes and an interval on the real axis and fix a corresponding logarithm log det_{F} (K(λ)K^{c}(λ)^{−1}) depending holomorphically on λ in this region. Let us define D_{C} = D_{C_{+}} ⊔ D_{C_{−}} and D_{C^{c}}^{c} = D_{C_{+}^{c}}^{c} ⊔ D_{C_{−}^{c}}^{c}.

Then we prove that

(4.2) ∂_{λ} log det_{F} (K(λ)K^{c}(λ)^{−1})

= − Tr ((D − λ)^{−1} − (D_{C} − λ)^{−1} − ((D^{c} − λ)^{−1} − (D_{C^{c}}^{c} − λ)^{−1})).

The operator in parentheses turns out to be trace class! In fact, the auxiliary model problem was introduced for exactly this reason: to overcome certain trace class issues, and the fact that the gluing problem on the model problem can be solved exactly [26]. The second remarkable property of K(λ) and K^{c}(λ) is that, using (4.2) and deriving formulas for the spectral invariants in terms of resolvents, we prove that for ν ∈ ℝ the spectral invariants can be expressed as

(4.3) \frac{det_{ζ}(D^{2} + ν^{2}, D_{C}^{2} + ν^{2})}{det_{ζ}((D^{c})^{2} + ν^{2}, (D_{C}^{c})^{2} + ν^{2})}

= C · det_{F} (K(iν)K^{c}(iν)^{−1}) · det_{F} (K(−iν)K^{c}(−iν)^{−1}),

for a constant C, where det_{ζ}(D^{2} +ν^{2}, D_{C}^{2} +ν^{2}) := det_{ζ}(D^{2} +ν^{2})·(det_{ζ}(D_{C}^{2} +ν^{2}))^{−1} with a similar formula for the operators on the model cylinder, and

(4.4) η(D, D_{C})

= −\frac{1}{πi}( lim_{ν→∞}( log det_{F} (K(iν)K^{c}(iν)^{−1}) − log det_{F} (K(−iν)K^{c}(−iν)^{−1}))

− lim_{ν→0^{+}}( log det_{F} (K(iν)K^{c}(iν)^{−1})−log det_{F} (K(−iν)K^{c}(−iν)^{−1}))),

where η(D, D_{C}) := η(D) − η(D_{C}). Third, as ν → ±∞, we prove that

(4.5)  lim_{ν→±∞} \frac{det_{ζ}(D^{2} + ν^{2}, D_{C}^{2} + ν^{2})}{det_{ζ}((D^{c})^{2} + ν^{2}, (D_{C}^{c})^{2} + ν^{2})} = 1,

(4.6)  lim_{ν→±∞} det_{F} (K(iν)K^{c}(iν)^{−1}) = a_{±},

where a_{+} = 1 and a_{−} = det_{F} U, and as ν → 0^{+}, we have

(4.7) det_{F} (K(±iν)K^{c}(±iν)^{−1}) = (±ν)^{h_{M}} (det L)^{−1} det_{F} (\frac{Id + Û}{2} )(1 + o(1)).

Using the aforementioned properties, we can now prove the gluing formulas “in one shot”! For the ζ-determinant, we first find the constant C in (4.3). To do so, we take ν → ∞ on both sides of (4.3), and use the limits in (4.5) and (4.6) to obtain C = det_{F} U^{−1} = (−1)^{h_{M}} · det_{F} Û^{−1}. Substituting this value into (4.3), then using (4.7) and simplifying, we see that as ν → 0^{+},

\frac{det_{ζ}(D^{2} + ν^{2}, D_{C}^{2} + ν^{2})}{det_{ζ}((D^{c})^{2} + ν^{2}, (D_{C}^{c})^{2} + ν^{2})}

= ν^{2h_{M}} (det L)^{−2} det_{F} (\frac{2Id + Û + Û^{−1}}{4} )(1 + o(1)).

However, we prove that as ν → 0^{+}, the left-hand side has the asymptotics

\frac{det_{ζ}(D^{2} + ν^{2}, D_{C}^{2} + ν^{2})}{det_{ζ}((D^{c})^{2} + ν^{2}, (D_{C}^{c})^{2} + ν^{2})} = ν^{2h_{M}} \frac{det_{ζ}(D^{2}, D_{C}^{2})}{det_{ζ}((D^{c})^{2}, (D_{C^{c}}^{c})^{2})}(1 + o(1)).

Combining this expression with the previous equality, then setting ν = 0, we obtain

det_{ζ}(D^{2}, D_{C}^{2}) = det_{ζ}((D^{c})^{2}, (D_{C^{c}}^{c})^{2}) (det L)^{−2} det_{F} (\frac{2Id + Û + Û^{−1}}{4} ).

Finally, the main result of [26] tells us that det_{ζ}((D^{c})^{2}, (D_{C^{c}}^{c})^{2}) = 2^{−ζ_{D^{2}_{Y}} (0)−h_{Y}}, which completes the proof for the ζ-determinant.

Before proving the eta invariant formula, recall that if f(t) is a smooth nonzero complex-valued function on an interval [a, b], then the winding number W (f) ∈ ℤ of f is defined by the equality

(4.8)  log f(b) − log f(a) = Log f(b) − Log f(a) + 2πi W (f),

where log f(t) is any continuous logarithm for f(t) with t ∈ [a, b] and Log denotes the principal value of the logarithm. In view of the limits (4.6) and (4.7), by definition of the winding number (4.8), we have, modulo 2πiℤ,

lim_{ν→∞} ( log det_{F} (K(iν)K^{c}(iν)^{−1}) − log det_{F} (K(−iν)K^{c}(−iν)^{−1})) ≡ − Log det_{F} U,

where the integer defect is just the winding number of det_{F} (K(λ)K^{c}(λ)^{−1}) from −i∞ to i∞, and modulo 2πiℤ,

lim_{ν→0^{+}}( log det_{F} (K(iν)K^{c}(iν)^{−1}) − log det_{F} (K(−iν)K^{c}(−iν)^{−1}))

≡ \begin{cases}0, & h_{M} even, \\ −πi, & h_{M} odd, \end{cases}

where the integer defect is just the winding number of det_{F} (K(λ)K^{c}(λ)^{−1}) from −iν to iν for ν > 0 sufficiently small. Substituting these limits into (4.4) completes the proof for the eta invariant.

Acknowledgement. The authors thank Gerd Grubb and K. P. Wojciechowski for all their help and for taking a keen interest in our mathematical careers.

References

1. M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR0397797 (53:1655a)

2. D. Bleecker and B. Booss-Bavnbek, Spectral invariants of operators of Dirac type on partitioned manifolds, Aspects of Boundary Problems in Analysis and Geometry, Birkhäuser, Boston, 2004, pp. 1–130. MR2072498

3. J. Brüning and M. Lesch, On the η-invariant of certain nonlocal boundary value problems, Duke Math. J. 96 (1999), 425–468. MR1666570 (99m:58180)

4. V. Bruneau, Fonctions zêta et êta en présence de spectre continu, C. R. Acad. Sci. Paris Sér. I Math. 323, no. 5 (1996), 475–480. MR1408979 (97j:58154)

5. U. Bunke, On the gluing formula for the η-invariant, J. Differential Geometry 18 (1995), 397–448. MR1331973 (96c:58163)

6. D. Burghelea, L. Friedlander, and T. Kappeler, Mayer-Vietoris type formula for determinants of differential operators, J. Funct. Anal. 107 (1992), 34–65. MR1165865 (93f:58242)

7. A.-P. Calderón, Boundary value problems for elliptic equations, Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), Acad. Sci. USSR Siberian Branch, Moscow, 1963, pp. 303–304. MR0203254 (34:3107)

8. G. Carron, Déterminant relatif et la fonction Xi, Amer. J. Math. 124, no. 2 (2002), 307–352. MR1890995 (2003c:58023)

9. X. Dai and D. Freed, η-invariants and determinant lines, J. Math. Phys. 35 (1994), 5155–5195. MR1295462 (96a:58204)

10. R. Forman, Functional determinants and geometry, Invent. Math. 88 (1987), 447–493. MR0884797 (89b:58212)

11. G. Grubb, Trace expansions for pseudodifferential boundary problems for Dirac-type operators and more general systems, Ark. Math. 37 (1999), 45–86. MR1673426 (2000c:35265)

12. , Poles of zeta and eta functions for perturbations of the Atiyah-Patodi-Singer problem, Comm. Math. Phys. 215 (2001), 583–589. MR1810945 (2002a:58036)

13. , Spectral boundary conditions for generalizations of Laplace and Dirac operators, Comm. Math. Phys. 240 (2003), 243–280. MR2004987 (2004f:58030)

14. A. Hassell, Analytic surgery and analytic torsion, Comm. Anal. Geom. 6, no. 2 (1998), 255– 289. MR1651417 (2000c:58061)

15. A. Hassell and S. Zelditch, Determinants of Laplacians in exterior domains, IMRN 18 (1999), 971–1004. MR1722360 (2001a:58045)

16. A. Hassell, R. R. Mazzeo, and R. B. Melrose, Analytic surgery and the accumulation of eigenvalues, Comm. Anal. Geom. 3 (1995), 115–222. MR1362650 (97f:58132)

17. , A signature formula for manifolds with corners of codimension two, Topology 36, no. 5 (1997), 1055–1075. MR1445554 (98c:58163)

18. P. Kirk and M. Lesch, The eta invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary, Forum Math. 16 (2004), 553–629. MR2044028 (2005b:58029)

19. Y. Lee, Burghelea-Friedlander-Kappeler’s gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion, Trans. Amer. Math. Soc. 355, no. 10 (2003), 4093–4110. MR1990576 (2004e:58058)

20. M. Lesch and K. P. Wojciechowski, On the η-invariant of generalized Atiyah-Patodi-Singer boundary value problems, Illinois J. Math. 40, no. 1 (1996), 30–46. MR1386311 (97d:58194)

21. P. Loya, Dirac operators, boundary value problems, and the b-calculus, Contemp. Math. 366 (2005), 241–280.

22. P. Loya and J. Park, Decomposition of the ζ-determinant for the Laplacian on manifolds with cylindrical end, Illinois J. Math. to appear.

23. , On the gluing problem for the spectral invariants of Dirac operators, Advances in Math. to appear.

24. , On the gluing problem for Dirac operators on manifolds with cylindrical ends, Preprint, 2004.

25. , The comparison problem for the spectral invariants of Dirac type operators, Preprint, 2004.

26. , The ζ-determinant of generalized APS boundary problems over the cylinder, J. Phys. A. 37, no. 29 (2004), 7381–7392. MR2078962

27. , Asymptotics of ζ-determinant of Dirac Laplacian under adiabatic process, In preparation.

28. R. Mazzeo and R. B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal. 5, no. 1 (1995), 14–75. MR1312019 (96a:58200)

29. R. Mazzeo and P. Piazza, Dirac operators, heat kernels and microlocal analysis. II. Analytic surgery, Rend. Mat. Appl. (7) 18, no. 2 (1998), 221–288. MR1659838 (2000a:58065)

30. R. B. Melrose, The Atiyah-Patodi-Singer index theorem, A. K. Peters, Wellesley, 1993. MR1348401 (96g:58180)

31. W. Müller, Eta invariants and manifolds with boundary, J. Differential Geometry 40 (1994), 311–377. MR1293657 (96c:58165)

32. , On the L^{2}-index of Dirac operators on manifolds with corners of codimension two. I, J. Differential Geometry 44 (1996), 97–177. MR1420351 (98b:58163)

33. , Relative zeta functions, relative determinants and scattering theory, Comm. Math. Phys. 192 (1998), 309–347. MR1617554 (99k:58189)

34. J. Park and K. P. Wojciechowski, Scattering theory and adiabatic decomposition of the ζdeterminant of the Dirac Laplacian, Math. Res. Lett. 9, no. 1 (2002), 17–25. MR1892311 (2003c:58024)

35. , Adiabatic decomposition of the ζ-determinant and Scattering theory, MPI Preprint, 2002.

36. D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR0295381 (45:4447)

37. S. Scott, Zeta determinants on manifolds with boundary, J. Funct. Anal. 192, no. 1 (2002), 112–185. MR1918493 (2003g:58051)

38. S. Scott and K. P. Wojciechowski, The ζ-determinant and Quillen determinant for a Dirac operator on a manifold with boundary, Geom. Funct. Anal. 10 (1999), 1202–1236. MR1800067 (2001k:58067)

39. R. T. Seeley, Topics in pseudo-differential operators, Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968), 1969, pp. 167–305. MR0259335 (41:3973)

40. I .M. Singer The eta invariant and the index, Mathematical aspects of string theory, World Scientific, Singapore, 1988, pp. 239–258. MR0915824

41. , Families of Dirac operators with applications to physics, Astérisque, Numero Hors Serie, The mathematical heritage of Élie Cartan (Lyon, 1984), 1985, pp. 323–340. MR0837207 (88a:58192)

42. S. M. Vishik, Generalized Ray-Singer conjecture. I. A manifold with a smooth boundary, Comm. Math. Phys. 167, no. 1 (1995), 1–102. MR1316501 (96f:58184)

43. K. P. Wojciechowski, The additivity of the η-invariant. The case of a singular tangential operator, Comm. Math. Phys. 169 (1995), 315–327. MR1329198 (96k:58210)

44. , The ζ-determinant and the additivity of the η-invariant on the smooth, self-adjoint Grassmannian, Comm. Math. Phys. 201, no. 2 (1999), 423–444. MR1682214 (2000f:58071)

Department of Mathematics, Binghamton University, Binghamton, New York 13902

E-mail address: paul@math.binghamton.edu

Mathematisches Institut, Universität Bonn, Beringstraße 1, D-53115 Bonn, Germany

E-mail address: jpark@math.uni-bonn.de