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GENERALIZED RAY-SINGER CONJECTURE. I.

arXiv:hep-th/9305184v1 31 May 1993GENERALIZED RAY-SINGER CONJECTURE. I.A MANIFOLD WITH A SMOOTH BOUNDARYS.M.

VISHIKFor my parentsAbstract. This paper is devoted to a proof of a generalized Ray-Singer con-jecture for a manifold with boundary (the Dirichlet and the Neumann boundaryconditions are independently given on each connected component of the boundaryand the transmission boundary condition is given on the interior boundary).

TheRay-Singer conjecture [RS] claims that for a closed manifold the combinatorial andthe analytic torsion norms on the determinant of the cohomology are equal. For amanifold with boundary the ratio between the analytic torsion and the combinato-rial torsion is computed.

Some new general properties of the Ray-Singer analytictorsion are found. The proof does not use any computation of eigenvalues and itsasymptotic expansions or explicit expressions for the analytic torsions of any specialclasses of manifolds.Contents1.Analytic torsion and the Ray-Singer conjecture91.1.

Analytic and combinatorial torsions norms1.2. Gluing formulas1.3.

Properties of analytic and combinatorial torsion norms1.4. Generalized Ray-Singer conjecture2.Gluing formula for analytic torsion norms.

Proof of Theorem 1.1342.1. Strategy of the proof2.2.

Continuity of the analytic torsion norms2.3. Actions of the homomorphisms of identifications on the determinant.Proof of Lemma 2.32.4.

Analytic torsion norm on the cone of a morphism of complexes. Proofof Lemma 2.42.5.

Variation formula for norms of morphisms of identifications. Proof ofLemma 2.12.6.

Variation formula for the scalar analytic torsion. Proof of Lemma 2.21

2S.M. VISHIK2.7.

Continuity of the truncated scalar analytic torsion. Proof of Proposi-tion 2.12.8.

Dependence on the phase of a cut of the spectral plane. The analytictorsions as functions of the phase of a cut.

Gluing formula for theanalytic torsions3.Zeta- and theta-functions for the Laplacians with ν-transmissioninterior boundary conditions873.1. Properties of zeta- and theta-functions for ν-transmission boundaryconditions3.2.

Zeta-functions for the Laplacians with ν-transmission interior bound-ary conditions. Proofs of Theorem 3.1 and of Proposition 3.13.3.

Theta-functions for the Laplacians with ν-transmission boundary con-ditions. Proofs of Theorem 3.2 and of Proposition 3.23.4.

Estimates for zeta-functions and for the corresponding kernels in ver-tical strips in the complex plane3.5. Appendix.

Trace class operators and their tracesReferences118Torsion invariants for manifolds which are not simply connected were introducedby K. Reidemeister in [Re1], [Re2], where he obtained with the help of such invariantsa full PL-classification of three-dimensional lens spaces. These invariants were gen-eralized by W. Franz to multi-dimensional PL-manifolds in [Fr].

As the result of thisgeneralization he obtained a PL-classification of lens spaces of any dimension. (Thesetorsions were the first invariants of manifolds which are not homotopy invariants.)J.H.C.

Whitehead in [Wh] and G. de Rham in [dR3] introduced torsion invariantsfor smooth manifolds. G. de Rham in [dR3] proved that a spherical Clifford-Kleinmanifold (i.e., the quotient of a sphere under the fixed-point free action of a finitegroup of rotations) is determined up to an isometry by its fundamental group andby its Reidemeister torsions.The Whitehead torsion for a homotopy equivalencebetween finite cell complexes was introduced in [Wh] as a generalization of the Rei-demeister torsion invariants defined in [Re1], [Fr], and [dR3].

(Its values are in theWhitehead group Wh(π1) of the fundamental group π1.) The Whitehead torsion isconnected with Whitehead’s theory of simple homotopy types ([Wh], [dRMK], [Mi],Section 7).

Some modifications of Reidemeister torsions were considered by J. Milnorin [Mi], Sections 8, 12, and by V. Turaev in [T], Section 3. The scalar Reidemeistertorsion is a global invariant of a cell decomposition of a manifold and of an acyclicrepresentation of its fundamental group.

It is an invariant of the PL-structure of amanifold. The Reidemeister torsion for an arbitrary finite-dimensional unimodularrepresentation of the fundamental group can be defined as a canonical norm on the

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY3determinant line of the cohomology of a manifold (with the coefficients in the localsystem defined by this representation).

It is some kind of multiplicative analog ofthe Euler characteristic in the case of odd-dimensional manifolds. (The Euler char-acteristic of a closed manifold is trivial in the odd-dimensional case.) Formulas forthe Reidemeister torsions of a direct product of manifolds ([KwS]) are analogous tothe multiplicative property of the Euler characteristic.The Ray-Singer analytic torsion was introduced in [RS] for a closed Riemannianmanifold (M, gM) with an acyclic orthogonal representation of the fundamental groupπ1(M).

It is equal to a product of the corresponding powers of the determinants of theLaplacians on differential forms DR•(M). These determinants are regularized withthe help of the zeta-functions of the Laplacians.

(The scalar Reidemeister torsion alsocan be written by the analogous formula, where Riemannian Laplacians are replacedby the combinatorial ones.) The Ray-Singer analytic torsion is defined with the helpof a Riemannian metric gM but it is independent of gM in the acyclic case.

(Thisassertion was proved in [RS], Theorem 2.1.) So it is an invariant of a smooth structureon M. It has the properties analogous to the properties of the Reidemeister torsion([RS], Sections 2, 7).

The Ray-Singer conjecture ([RS]) claims that for an acyclicrepresentation ρ of the fundamental group of a closed manifold M the Reidemeistertorsion of (M, ρ) (which is defined for any smooth triangulation of M) is equal to theRay-Singer analytic torsion of (M, ρ). This conjecture was independently proved byW.

M¨uller in [M¨u1] and by J. Cheeger in [Ch] for closed manifolds. The Ray-Singeranalytic torsion can also be defined for any finite-dimensional unitary representationρ of π1(M).

In this case the Ray-Singer torsion is the norm on the determinant linedet H• (M, ρ). For instance, it is defined for a trivial one-dimensional representation.So the analytic torsion norm provides us with a canonical norm on the determinantline of the de Rham complex of a manifold.

(The Ray-Singer formula for an arbitraryfinite-dimensional unitary representation ρ of π1(M) in the case, when M is a smoothclosed manifold, claims that the Ray-Singer norm on det H• (M, ρ) is equal to theReidemeister norm on det H• (M, ρ). )Let (M, gM) be a manifold with a smooth boundary ∂M and with the Dirichlet andthe Neumann boundary conditions independently given on the connected componentsof ∂M.Let Z ⊂∂M be a union of the components of ∂M where the Dirichletboundary conditions are given.

Let Fρ be a local system with a fiber Cm definedby a unitary representation ρ: π1(M) →U(m). Then the Ray-Singer torsion normT0 (M, Z; Fρ) is defined on det H• (M, Z; Fρ).

It is independent of gM (if gM is a directproduct metric near ∂M) and it depends on a flat Hermitian metric on the fibersFρ (for a general (M, Z)). A flat Hermitian structure on Fρ defines a norm on theline det (Fx, M, Z) := ⊗k (det Fxk)χ(Mk,Z∩∂Mk), where the product is over the full setof representatives Fxk of fibers of Fρ over the connected components Mk of M (withone such a fiber Fxk for each Mk, xk ∈Mk, det Fx := ∧maxFx).

The tensor product ofthis norm and of T0 (M, Z; Fρ) is a modified Ray-Singer norm on det H• (M, Z; Fρ) ⊗

4S.M. VISHIKdet (Fx, M, Z) and it does not depend on gM and on a flat Hermitian metric on Fρ([V1]).

The Ray-Singer torsion norm for the de Rham complex of (M, Z) with thecoefficients in the direct sum of any finite-dimensional local system Fρ and of the dualone F ∨ρ is defined in [V2]. In this case the Reidemeister torsion τ0M, Z; Fρ ⊕F ∨ρ(i.e., the one for (M, Z) with the coefficients in Fρ ⊕F ∨ρ ) is well-defined, becausethe fibers of the line bundle detFρ ⊕F ∨ρhave the canonical norm in accordancewith the local system structure.

In this case, the Ray-Singer torsion differs fromthe Reidemeister torsion by an explicit factor (which is computed in [V2]) but thistorsion does not depend on gM (if gM is a direct product metric near ∂M). Thisdefinition of the Ray-Singer torsion norm does not use a Hermitian structure in thefibers of Fρ.

In [M¨u2] another Ray-Singer torsion was introduced for the de Rhamcomplex of a closed (M, gM) with the coefficients in a local system Fρ, defined bya unimodular finite-dimensional representation ρ of π1(M). This torsion is definedwith the help of an arbitrary Hermitian metric hρ in the fibers of Fρ and it dependsin general on this metric.

(For a non-unitary representation ρ there are no Hermitianmetrics on Fρ, which are flat with respect to the canonical flat structure.) It wasproved in [M¨u2] that in the case of an odd-dimensional M the Ray-Singer torsion,defined with the help of a Hermitian metric hρ, is independent of (hρ, gM) and isequal to the Reidemeister torsion.

(The Reidemeister torsion is canonically definedfor any unimodular finite-dimensional representation of π1. In the case of an odd-dimensional closed M it is independent of a flat Hermitian metric on det Fρ, sincethe Euler characteristic in this case is equal to zero for each connected componentof M.) The Ray-Singer torsion, defined with the help of hρ, depends on (hρ, gM)for a general even-dimensional M. The definition of the Ray-Singer torsion for anyfinite-dimensional representation ρ of π1(M) for a closed (M, gM) equiped with aHermitian metric hρ (on the fibers of the corresponding vector bundle) is given in[BZ1], [BZ2].

In [BZ2] the Ray-Singer metric on the determinant line, correspondingto a finite flat exact sequence (F •, dF) of finite-dimensional flat vector bundles overM is computed (in terms of gM and of Hermitian metrics on F j).The Gaussian integral of exp (−(Sx, x)), where S is a positive self-adjoint operatorin a finite-dimensional Hilbert space H, dim H = n, is equal to (2π)n/2 (det S)−1/2.The Ray-Singer torsion appears naturally in the computations of asymptotic expan-sions for analogous infinite-dimensional integrals of exp (−ikI(A)), where I(A) pos-sesses an infinite-dimensional symmetry group G ([Sc], [Wi1], [Wi2]). For instance,the Chern-Simons actionI(A) := (4π)−1ZM Tr (A ∧dA + 2/3A ∧A ∧A)on a trivialized principal G-bundle PG over a closed orientable three-dimensionalmanifold (where G = SUN and Tr is the trace in the N-dimensional geometricalrepresentation of G, and where A is a connection form) is invariant under the gauge

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY5transformations A →gAg−1−dg·g−1 =: Ag for a smooth g : M →G (where Ag is thesame connection but with respect to another trivialization of PG, i.e., with respect toanother smooth section G →PG).

Stationary points of I(A) are the flat connectionsAα (i.e., such that the curvature F (Aα) is equal to zero). The asymptotic of anintegral of exp (−ikI(A)) as k →+∞, k ∈Z+, is computed by the stationary phasemethod.

The principal term of the contribution of a point Aα into this integral (inthe case when the flat connection Aα is an isolated one) has as its absolute valuethe square root of the Ray-Singer torsion of M with coefficients in the local system,defined by a flat connection Aα, with the Lie algebra g of G as its fibers (see [Wi1];[Wi2], 2.2; [BW], 2).The Reidemeister torsion was essentially used in [Wi2], 4, for the computationof the volume of a moduli space M of the fundamental group representations for aclosed two-dimensional surface. In this case the Reidemeister torsion is a section of|det| T ∗M, i.e., it is a density on M.This paper is devoted to a proof of a generalized Ray-Singer conjecture for mani-folds with a smooth boundary (and also for transmission boundary conditions givenon the interior boundaries).

We suppose that the local system is trivial. The proofof the Ray-Singer conjecture for non-unitary local systems and for manifolds withcorners will be the subject of a subsequent paper.Let (M, gM) be a Riemannian manifold with a smooth boundary ∂M and letthe Dirichlet and the Neumann boundary conditions be independently given on theconnected components of ∂M.

Let gM be a direct product metric near ∂M. Thenthe Ray-Singer torsion of (M, gM) is defined as a norm on the determinant linedet H• (M, Z).

(Here Z is the union of the connected components of ∂M where theDirichlet boundary conditions are given.) This norm is independent of gM (for directproduct metrics gM near ∂M).

The Reidemeister torsion of (M, Z) is an invariant ofthe PL-structure of (M, Z) and it is a norm on the same determinant line. The torsionnorms are defined in Section 1.

The Ray-Singer norm differs from the Reidemeisternorm on det H• (M, Z) for a general ∂M ̸= ∅. Their ratio is computed in Theorem 1.4below.Let (M, gM) be obtained by gluing two Riemannian manifoldsMj, gMjalongthe common component N of their boundaries, M := M1 ∪N M2 (where N isa closed smooth manifold of codimension one in M).Let gM be a direct prod-uct metric near N. Then, as it is proved in Theorem 1.1, the Ray-Singer torsionnorm T0(M, Z) on det H• (M, Z) is equal to the tensor product of the Ray-Singernorms T0 (M1, Z1 ∪N) ⊗T0 (M2, Z2 ∪N) ⊗T0(N) (Zk := Z ∩∂M k), where the linedet H•(M, Z) is identified with the tensor product of the lines det H• (M1, Z1 ∪N) ⊗det H• (M2, Z2 ∪N) ⊗det H•(N) by the short exact sequence of the de Rham com-plexes0→DR•(M1,Z1∪N)⊕DR•(M2,Z2∪N)→DR•(M,Z)→DR•(N)→0,(0.1)

6S.M. VISHIKwhere DR•(M, Z) is the relative de Rham complex of smooth forms with the zerogeometrical restrictions to Z, the left arrow is the natural inclusion, and the rightarrow is√2 times a geometrical restriction.

For the Reidemeister norm this assertionis also true and the identification of the determinant lines is given by the analogousexact sequence of cochain complexes. However in this case the right arrow is thegeometrical restriction of cochains (without additional factor√2).

Let (M, Z) beobtained by gluing two manifolds (M1, Z1) and (M2, Z2) along the common compo-nent N of their boundaries, M := M1 ∪N M2. Then the ratio of the square of theRay-Singer norm and the square of the Reidemeister norm for (M, Z) is equal tothe product of the same ratios for (M1, Z1 ∪N), (M2, Z2 ∪N), and for N with anadditional factor 2−χ(N).

So the assertion of Theorem 1.1 claims that it is possible tocalculate the Ray-Singer norm by cutting of a manifold into pieces which are mani-folds with smooth boundaries. The main theorems of this paper are consequences ofTheorem 1.1.

This theorem provides us with the gluing formula for the Ray-Singertorsion norms. Such a gluing formula is a new one.In the case of a manifold with a smooth boundary, the Ray-Singer torsion T0(M, Z)is a function not only of (M, Z) but also of the phase θ of a cut of the spectral planeC (because the zeta-functions ζj(s) for the Laplacians ∆j on DRj(M, Z) are definedfor Re s > (dim M)/2 as the sums P λ−s over the nonzero eigenvalues, and λ−s isdefined as λ−s(θ) := exp−s log(θ) λ, where θ−2π < Im log(θ) λ < θ, θ /∈2πZ).

In fact,T0(M, Z; θ) (as well as ζj(s)) depends only on [θ/2π]. The zeta-function regularizationof the det′ (∆j) (i.e., of the product of all the nonzero eigenvalues of ∆j, includingtheir multiplicities) is defined as exp (−∂sζj(s)|s=0).The analytic continuation ofζj(s) is regular at zero.

The zeta-function ζj(s; m) depends on m := [θ/2π], θ /∈2πZ,as follows:ζj(s; m + 1) = exp(−2πis)ζ(s; m),det′ (∆j; m + 1) = exp (2πiζj(0)) det′ (∆j; m) .The number ζj(0) is independent of m, and the number ζj(0) + dim Ker ∆j can beinterpreted as the regularized dimension of the space DRj(M).This regularizeddimension depends not only on the space DRj(M) but it also depends on a positivedefinite self-adjoint elliptic differential operator of a positive order, which acts inDRj(M). This dimension is a real number but it is not an integer in the case ofthe Laplacians on DR•(M) for a general closed even-dimensional (M, gM).

Hence,det (∆j; m) depends on m for such (M, gM).The number ζj(0) is an integer fora generalized Laplacian on a closed odd-dimensional (M, gM), according to [BGV],Theorem 2.30, or to [Gr], Theorem 1.6.1. It is equal to zero when M is closed, dim Mis odd, and dim Ker ∆j = 0.Even in such a simple case as for an interval (I, ∂I) with the Dirichlet boundaryconditions, the dependence of T0(M, Z; m) on m is nontrivial.

The ratio of the torsion

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY7T0 (M, Z; [θ/2π]) and the Reidemeister torsion norm is computed in Theorem 2.2.The paper is organized as follows.

In Section 1 we deduce a generalization of theRay-Singer conjecture from the gluing formula for Ray-Singer torsion norms. Thisformula is proved in Theorem 1.1.

The proof uses ν-transmission interior boundaryconditions on N, where ν = (α, β) ∈R2 \ (0, 0). These interior boundary prob-lems give us a smooth in ν family of spectral problems on M. Such a problem forν = (1, 1) coincides (in a spectral sense) with the spectral problem for a glued M.For ν = (0, 1) or for ν = (1, 0) it is a direct sum of spectral problems on M1 and onM2, i.e., the two pieces of M are completely disconnected.

So this family providesus with a smooth process of cutting (in a spectral sense) of M in two pieces M1and M2. Let M = M1 ∪N M2 be obtained by gluing M1 and M2 along the com-mon component N of their boundaries.

Then the Ray-Singer norm T0 (Mν, Z) onthe determinant line det H• (Mν, Z) for the de Rham complex DR• (Mν, Z) with ν-transmission conditions on N is defined. The short exact sequence for DR• (Mν, Z),similar to (0.1), has the same the first and third terms as (0.1).

The homomorphismsrν : DR• (Mν, Z) →DR•(N) are of the form rν = (αi∗1 + βi∗2) /|ν|, where i∗jωj are thegeometrical restrictions to N for the components ωj of ω = (ω1, ω2) ∈DR• (Mν, Z).Note that r(1,1) =√2 i∗. (This is the reason of the appearance of√2 i∗in the exactsequence (0.1), connected with the gluing formula.) In Lemma 1.2 we prove that thegluing property for analytic torsion norms (Theorem 1.1) is equivalent to the indepen-dence of ν of the norms on det H• (M1, Z1 ∪N) ⊗det H• (M2, Z2 ∪N) ⊗det H•(N)induced by T0 (Mν, Z).

(Here the identification of the determinant lines is definedby the short exact sequence for DR• (Mν, Z).) The latter assertion is proved in Sec-tion 2.

First we prove that the norm induced by the Ray-Singer torsion T0 (Mν, Z)is locally independent of ν in the case when αβ ̸= 0 (where ν = (α, β)). We dothis in Sections 2.3, 2.5, and 2.6 with the help of explicit variation formulas for thescalar Ray-Singer torsion T (Mν, Z) (if ν depends smoothly on a parameter).

Wedefine a family (in ν) of homomorphisms to identify finite-dimensional subcomplexesW •a (ν) of DR• (Mν, Z). (The complexes W •a (ν) are spanned by the eigenforms of theLaplacians with eigenvalues less than a fixed number a > 0.

We suppose that a is notan eigenvalue of ∆j (Mν, Z) for 0 ≤j ≤n.) Then we compute the actions of thesehomomorphisms on the determinant lines.

These identifications are not canonical;we choose some particular (quite natural) identifications for ν sufficiently close to ν0such that α0β0 ̸= 0.Then it is enough to prove the continuity in ν ∈R2 \ (0, 0) of the norm ondet H• (M1, Z1 ∪N) ⊗det H• (M2, Z2 ∪N) det H•(N), which is induced by the Ray-Singer norm T0 (Mν, Z). We prove in Section 2.7 that the truncated scalar analytictorsion T (Mν, Z; a), corresponding to the eigenvalues λ of ∆j (Mν, Z) which aregreater than a, is locally continuous in ν.

Then we prove that the norm, induced bythe analytic torsion norm T0 (W •a (ν)) of a finite-dimensional complex W •a (ν), is locally

8S.M. VISHIKcontinuous in ν.

The latter assertion is proved in Sections 2.2, 2.4, and 2.7 with theuse of the cone of the homomorphism R•ν(a): W •a (ν) →C• (Xν, Z ∩X) (where R•ν(a)is the integration of differential forms from W •a over the simplexes of a given smoothtriangulation X of M, and C• (Xν, Z ∩X) is the corresponding cochain complex).This homomorphism is a quasi-isomorphism for any ν ∈R2 \ (0, 0) (Proposition 2.3).We can conclude that the analytic torsion norm on det H• (Cone Rν(a)) = C (fora fixed ν) corresponds to an acyclic finite-dimensional complex and is defined bythe derivatives at zero of the zeta-functions for self-adjoint finite-dimensional in-vertible operators.So these norms are locally continuous in ν. (This is provedin Section 2.7.) Then the local continuity of the norm induced by T0 (W •a (ν)) ondet H• (M1, Z1 ∪N) ⊗det H• (M2, Z2 ∪N) ⊗det H•(N) follows from the continuityof the norm (on the same determinant line) induced by T0 (C• (Xν, Z ∩X)) and fromthe identity:T0 (W •a ) = T0 (C• (Xν, Z ∩X)).∥1∥2T0(Cone• Rν(a)) .This identity is proved in Lemma 2.4.The use of the cone of R•ν(a) allows us to avoid difficulties, connected with the factthat some positive eigenvalues of the Laplacians ∆• (Mν, Z) tend to 0 as ν = (α, β)tends to ν0 = (1, 0) (or to ν0 = (0, 1)).The dimensions of H• (Mν, Z) essen-tially change when ν, αβ ̸= 0, is replaced by ν0.

(Only the Euler characteristicχ (H• (Mν, Z)) does not change when ν is replaced by ν0.) It is impossible to findfor a general N the precise asymptotic expressions for the eigenvalues λ, which tendto zero as ν →ν0, and especially to find the asymptotics of the corresponding eigen-forms ωλ of ∆• (Mν, Z).

So the continuity of the norm induced by T0 (Mν, Z) (viewedas a function of ν) at the point ν0 cannot be proved for a general M (obtained bygluing two pieces M1 and M2 along N) with the help of separate computations ofthe asymptotic expressions for the scalar torsion T (Mν, Z) and for the measure ondet H• (Mν, Z) defined by harmonic forms.The proof of the classical Ray-Singerconjecture in [Ch] and the proof in [M¨u2] (in the case of unimodular representationsof π1(M)) are based on asymptotic computations of such quantities for a manifoldwith boundary Mu := M \ Su, where Su is a tubular neighborhood of an embeddedsphere Sk ֒→Mn as the radius u of the tubular neighborhood (in the normal to Skdirection) tends to zero. (It is also supposed in [Ch] that Su is a direct product onSk × Dn−k and that gM|Su is a direct product metric on Sk × Dn−k.

)To give a rigorous proof of the assertions above used in the proof of the gluingformula, it is necessary to prove a lot of analytic propositions.We do it in Sec-tions 2.2, 2.6, 2.7, and in Section 3. The theory of ζ- and θ-functions in the case ofν-transmission interior boundary conditions is elaborated in Section 3.

The precise es-timates of the corresponding ζ-functions in vertical strips are obtained in Section 3.4.These estimates allow us using the inverse Mellin transform to derive the informationabout the densities on M, N, and ∂M for the asymptotic expansions as t →+0 of

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY9θ-functions from the properties of the densities for appropriate ζ-functions.1.

Analytic torsion and the Ray-Singer conjecture1.1. Analytic and combinatorial torsions norms.

The analytic torsion normappears in the following finite-dimensional algebraic situation. Let (A•, d) be a finitecomplex of finite-dimensional Hilbert spaces.

The determinant of (A•, d) is the tensorproduct⊗j(ΛmaxAj)(−1)j+1 =: det(A•),where ΛmaxAj =: detAj is the top exterior power of the linear space Aj and whereL−1 is the dual space L∨for a one-dimensional vector space L over C. The naturalHilbert norm ∥·∥2det A is defined by the Hilbert norms on Aj.The determinant of the cohomology detH•(A) of (A•, d) is also defined and thereis a natural norm on it (since Hj(A) is the subquotient of Aj). The differential dprovides us with the identificationf(d) : det(A•) ≃detH•(A).However in the general case this identification is not an isometry of the norms ∥·∥2det Aand ∥·∥2det H•(A).

For f(d) to be an isometry it is necessary to multiply ∥·∥2det H•(A) bythe scalar analytic torsion of a complex (A•, d), which is defined asT(A•, d) = expΣ(−1)jj∂sζj(s)|s=0. (1.1)Here ζj(s) =P′ λ−s is the sum1 over all the nonzero eigenvalues λ ̸= 0 (includingtheir multiplicities) of the nonnegative (i.e., if λ ̸= 0 then λ > 0) self-adjoint operator(d∗d + dd∗)|Aj.

The derivative ∂sζj(s)|s=0 is equal to −log det′ ((d∗d + dd∗) |Aj) (i.e.,it is equal to the sum of (−log λ) ∈R over all the nonzero eigenvalues λ).It is enough to prove the assertion (1.1) in the case of a two-terms complex d: F0 →∼F1, where dim Fj = 1, ej ∈Fj, de0 = µe1, µ ̸= 0, and where ∥e0∥2 = 1 = ∥e1∥2.In this case the element e1 ⊗e−10∈det (F •) is of the unit norm and the square ofthe norm of the corresponding element µ−1 ∈C from C = det 0 = det H•(F) isequal to |µ−2|2. If the norm |µ−1|2 is multiplied by the scalar analytic torsion forF •, namely by exp (log det (d∗d)) = exp (log det (dd∗)) = |µ|2 then the isomorphismbetween det(F •) and C = det 0 (defined by d) becomes an isometry.This finite-dimensional definition make sense also for the infinite-dimensional deRham complex of a closed smooth manifold.In this case the analytic torsion isthe norm on the determinant of the cohomology of this manifold.

Let (DR•(M), d)be the de Rham complex of smooth differential forms (with the values in C) on aclosed manifold M. The scalar analytic torsion for a closed Riemannian manifold(M, gM) is defined by the same formula (1.1), where d∗= δ (relative to gM) and1The function λ−s is defined as exp (−s log λ) where log λ ∈R for λ ∈R+.

10S.M. VISHIK(d∗d + dd∗)|DRj(M) is the Laplace-Beltrami operator ∆j.

In this case the series,which defines ζj(s), converges for Re s > (dim M)/2. The analytic function ζj(s)can be analytically (meromorphically) continued to the whole complex plane.

It isknown that ζj(s) has simple poles and that it is regular at zero ([Se2]).The cohomology H• (DR(M)) are canonically identified (by the integration of theforms over the simplexes) with the cohomology H•(M) of M. This follows from thede Rham theorem. The Hodge theorem claims that each element of Hj (DR(M)) hasone and only one representative in the space of harmonic forms Ker ∆j.

The naturalnorm on Ker ∆j (defined by the Riemannian metric gM) provides us with the norm∥·∥2det H•(M) on det H•(M). For an odd-dimensional M this norm depends on gM.Definition.

The analytic torsion norm T0(M) on detH•(M) is the normT0(M) := ∥·∥2det H•(M) · expΣ(−1)jj∂sζj(s)|s=0. (1.2)The main property of this norm is its independence of a Riemannian metric gM.So it is an invariant of a smooth structure on M. Let us suppose that gM = gM(γ)depends smoothly on a parameter γ ∈R1.

Then the variation formulas in [RS],Theorems 2.1, 7.3 (or in [Ch], Theorem 3.10, (3.22)), claim that∂γXj(−1)jj∂sζj,γ(s)s=0=Xj(−1)j−Tr(exp (−t∆j,γ) α)0+Tr(Hj,γα). (1.3)Here Hj,γ is the kernel of the orthogonal projection operator from DRj(M) ontoKer ∆j(M, gM(γ)), α := ∗−1γ ∂γ (∗γ) (∗γ corresponds to gM(γ)) and Tr(exp(−t∆j,γ)α)0is the constant coefficient in the asymptotic expansion as t →+0 (n := dim M):Tr (exp (−t∆j,γ) α) =lXk=0mj,kt−n/2+k + otl.

(1.4)The existence of the asymptotic expansion (1.4) follows from [Gr], Theorem 1.6.1,or from [BGV], Theorem 2.30. For a family of norms ∥·∥2 (γ) on det H•(M) definedby the harmonic forms Ker (∆j (M, gM(γ))) the following equality holds for any fixedµ ∈det H•(M), µ ̸= 0 ([RS], Section 7):∂γ log ∥µ∥2det H•(M) (γ) = −X(−1)j Tr (Hj,γα) .Hence, (1.3) involves the equality∂γ log T0 (M, gM) =X(−1)j+1 mj,n/2.

(1.5)Since k in (1.4) are integers, we see that the right side of (1.5) is zero for odd n.For even n, n = 2l, the right side of (1.5) is also equal to zero, since mj,l = −m2l−j,l.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY11This fact follows from the equalities∂γ∗−1γ ∗γ= 0,α = −∗α∗−1,Tr (exp (−t∆j) α) = Tr∗exp (−t∆j) ∗−1(−α)= −Tr (exp (−t∆n−j) α) ,(since they involve the equalities mj,k = −mn−j,k, where n is even and k ∈Z+ ∪0).The analytic torsion norm can be interpreted (in an intuitional sense) as the norm,corresponding to an element v ∈det DR•(M) (v is defined up to a multiplicative con-stant c ∈C, |c| = 1, and its “torsion norm” is equal to one).

The space det DR•(M)and L2-norm on it are not defined but the space det H•(M) and the analytic torsionnorm T0(M) on it are rigorously defined. For a finite-dimensional complex the an-alytic torsion norm on the determinant of its cohomology corresponds to the normon the determinant of the complex defined by the Hilbert structures on the termsof this complex.

The analytic torsion norm is (in some sense) a multiplicative Eulercharacteristic useful for odd-dimensional manifolds.The same definition of T0(M) make sense also in the case when M is a compactRiemannian manifold with a smooth boundary ∂M = ∪Ni and with the Dirichlet orthe Neumann boundary conditions given independently on each connected componentNi of ∂M. Let the metric gM be a direct product metric near ∂M.

Then T0(M) isindependent of gM as in the case of a closed manifold (this is proved below).Let X be a smooth triangulation of M and let (C•(X), dc) be a cochain complex ofX ( with complex coefficients). Then each Cj(X) has the Hilbert structure definedby the orthonormal basis of basic cochains {δe}, where δe(e1) is 1 for e1 = e and 0for e1 ̸= e. Hence the scalar torsion T (C•(X), dc) is also defined.The combinatorial torsion τ0(X) is defined as the following norm on the determi-nant of the cohomology H• (C(X), dc) = H•(M) :τ0(X) := ∥·∥2det H•(C(X)) · T (C•(X), dc)(1.6)(where Hj (C(X)) is the subquotient of Cj(X) and so it has the natural Hilbertstructure induced from Cj(X)).

The norm (1.6) is invariant under any regular subdi-visions of X. So this norm is an invariant of the combinatorial structure of M (whichis completely defined by a smooth structure on M).

This norm corresponds to theHilbert norm on det C•(X), defined by the basic cochains.Let M be a manifold with a smooth boundary ∂M = ∪Ni, where Ni are theconnected components of ∂M. Let Z be the union of Ni where the Dirichlet boundaryconditions are given.

Set V := X∩Z. Then (1.6) (where H•(C(X)) and T (C•(X), dc)are replaced by H•(C(X, V )) and by T (C•(X, V ), dc)) provides us with the definitionof the norm τ0(X, V ).

This norm is an invariant of the combinatorial structure on(M, Z) ([Mi], Sections 7, 8, 9).

12S.M. VISHIK1.2.

Gluing formulas. The Ray-Singer conjecture claims that for a closed smoothmanifold M the norms τ0(M) and T0(M) on the same one-dimensional space detH•(M)are equal2τ0(M) = T0(M).

(1.7)How to prove such a formula in a natural way? It is necessary to find a generalproperty of the analytic torsion which involves the equality (1.7).

Such a propertycan be formulated as follows. Let (M, ∂M) be a Riemannian manifold with a smoothboundary and with the Dirichlet or the Newmann boundary conditions given in-dependently on the connected components of ∂M.Let a closed codimension onesubmanifold N of M, N ∩∂M = ∅, divides M in two pieces M1 and M2 (glued alongN), M = M1 ∪N M2, and let a metric gM be a direct product metric near N and near∂M.

Let T0(Mk, N) be the analytic torsion norm for Mk (with the Dirichlet bound-ary conditions on N), and let the boundary conditions on the connected componentsof ∂M belonging to ∂Mk be the same as for T0(M). The following assertion centralin this paper.Theorem 1.1 (Gluing property).

The analytic torsion norm T0(M, Z) is the ten-sor product of the analytic torsion norms for (M1, Z1 ∪N), (M2, Z2N), and for NϕanT0(M, Z) = T0(M1, Z1 ∪N) ⊗T0(M2, Z2 ∪N) ⊗T0(N),(1.8)where Zk := Z ∩∂M k.The identification ϕan (in (1.8)) of det H•(M, Z) with the tensor product of thethree one-dimensional spaces:ϕan : det H•(M, Z) →det H•(M1, Z1 ∪N) ⊗det H•(M2, Z2 ∪N) ⊗det H•(N) ==: Det(M, N, Z)(1.9)is defined by the long cohomology exact sequence corresponding to the following shortexact sequence of the de Rham complexes:0→DR•(M1,Z1 ∪N)⊕DR•(M2,Z2 ∪N)→DR•(M1,1,Z) r→DR•(N)→0. (1.10)The relative de Rham complex (DR•(Mk, Zk ∪N), d) (where d is the exteriorderivative of differential forms) consists of the smooth forms ω on Mk, having thezero geometrical restriction to N : i∗kω = 0 (where ik : N ⊂∂Mk ֒→Mk) and alsohaving the zero restrictions to the components of ∂M ∩Mk, where the Dirichletboundary conditions are given (i.e., to Zk).

The complex (DR (M1,1) , d) consists ofthe pairs (ω1, ω2) of smooth differential forms ωk ∈DR• (Mk, Zk) (i.e., ωk have the2The cohomology H• (DR(M)) and H• (C(X)) are identified (according to the de Rham theorem)by the homomorphism of the integration of forms from DR•(M) over the simplexes of a smoothtriangulation X of M.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY13zero geometrical restrictions to the corresponding components of ∂M ∩M k), whichhave the same geometrical restrictions to N:i∗1 ω1 = i∗2 ω2.The differential d(ω1, ω2) in DR• (M1,1) is defined as (dω1, dω2).

The left arrow in(1.10) is the natural inclusion of ⊕kDR• (Mk, Zk ∪N) into DR• (M1,1, Z). The rightarrow r in (1.10) is not a usual geometrical restriction but is the one multiplied by√2 :r(ω1, ω2) =√2 i∗kωk ∈DR•(N).

(1.11)To define ϕan it is necessary to introduce a natural identification of H• (DR(M, Z))with H• (DR (M1,1, Z)). (The short exact sequence (1.10) provides us with the iden-tificationϕan : det H• (DR (M1,1, Z)) →∼Det(M, N, Z),but not with the identification of det H• (DR(M, Z)) with Det(M, N, Z).) We showin Proposition 1.1 (for any given metric gM) that not only all the eigenvalues withtheir multiplicities but also all the eigenforms of the natural Laplacian ∆1,1 onDR• (M1,1, Z) are the same as for the Laplacian on DR•(M, Z).

Thus, the oper-ator ∆1,1(gM) in a very strict spectral sense is the same as ∆(gM).The homotopy operator between the identity operator on DR• (M1,1, Z) and theprojection operator from DR• (M1,1, Z) onto Ker• ∆1,1 = Ker• ∆is obtained withthe help of the Green function G1,1 for the operator ∆1,1 (Lemma 1.1). This ho-motopy operator provides us with the canonical identification of H• (DR (M1,1, Z))with Ker ∆•1,1.

So it defines the identification of H• (DR (M1,1, Z)) with Ker ∆• =H• (DR(M, Z)) (since Ker ∆• is canonically identified with Ker ∆•1,1).To prove Theorem 1.1 we introduce a family of interior boundary conditions on Nand show that the induced norm ϕνanT0 (Mν, Z) on Det(M, N, Z) is independent ofν (where ν = (α, β) ∈R2 \ (0, 0) are the parameters of interior boundary conditionson N). Namelyϕanν T0 (Mν, Z) = c0T0 (M1, Z1 ∪N) ⊗T0 (M2, Z2 ∪N) ⊗T0(N)(1.12)with some positive c0 which may depend on (M, gM, ∂M) and on the boundary con-ditions on ∂M but does not depend on the parameters (α, β) = ν.

Suppose thatthe formula (1.12) holds for any gluing two pieces M1 and M2 along a closed N,M = M1 ∪N M2, where the factor c0 is independent of ν. Then it is easy to con-clude that c0 = 1 (Lemma 1.2).

In (1.12) T0(Mν, Z) is the analytic torsion normfor the de Rham complex (DR•(Mν, Z), d). This complex consists of the pairs ofsmooth forms (ω1, ω2) such that ωk ∈DR• Mk, Zkhas the zero geometrical restric-tions to Zk := Z ∩∂Mk3 and that the following transmission condition holds for the3Z is the union of the components of ∂M where the Dirichlet boundary conditions are given.

14S.M. VISHIKgeometrical restrictions i∗kωk of ωk to Nαi∗1 ω1 = βi∗2 ω2.

(1.13)The analytic torsion norm T0 (Mν, Z) is defined for an arbitrary ν = (α, β) ∈R2 \ (0, 0). There is a canonical identification of Hj (DR• (Mν, Z)) with the spaceof the corresponding harmonic forms Ker (∆ν|DRj (Mν, Z)) (Lemma 1.1).Thisidentification (similarly to the case of DR• (M1,1, Z)) is obtained by the homotopyoperator, which is defined using the Green function for the Laplacian ∆ν.

(ThisLaplacian is an elliptic self-adjoint operator by Theorem 3.1.) The boundary con-ditions for ∆ν on N and ∂M are elliptic (and differential).The Green functionGν for ∆ν exists (and depends smoothly on ν ̸= (0, 0)) according to Theorem 3.1and to Proposition 3.1.

This identification provides us with the natural norms onHj (DR• (Mν, Z)) =: Hj (Mν, Z) and on det H• (Mν, Z). The scalar analytic torsionT (Mν, Z) is defined by ζν,j(s) :=P′λ−sifor Re s > (dim M)/2 (where the sum isover all the nonzero eigenvalues λi of the Laplacian ∆ν,j := ∆ν|DRj (Mν, Z) withtheir multiplicities).

These functions ζν,j can be continued to meromorfic functionson the whole complex plane with simple poles and regular at zero. (This statementis proved in Theorem 3.1 and in Proposition 3.1 below.

)The analytic torsion norm on det H• (Mν, Z) is the normT0(Mν, Z) = ∥·∥2det H•(Mν,Z) expX(−1)jj∂sζν,j(s)s=0.The identification ϕanνin (1.12 ) is defined by the short exact sequence of the deRham complexes (where Zk := Z ∩∂M k):0 →DR• (M1, Z1 ∪N) ⊕DR• (M2, Z2 ∪N) →DR• (Mα,β,Z)rα,β−−→DR•(N) →0. (1.14)The left arrow in (1.14) is the natural inclusion and the right arrow rα,β isrα,β(ω1, ω2) := (α2 + β2)−1/2(βi∗1ω1 + αi∗2ω2).

(1.15)For (α, β) = (1, 1) we have r1,1 =√2 i∗kωk. This corresponds to (1.11).

Hence, ϕan isequal to ϕanα,β for (α, β) = (1, 1).The complex DR• (Mν, Z) for the values (0, 1) and (1, 0) of ν is the direct sum of thede Rham complexes of all the smooth forms (with the zero geometrical restriction toZk) on one of the manifolds Mk and of all the smooth forms with the zero geometricalrestriction to Zj ∪N on another piece Mj of the manifold M. Thus, the two pieces ofM are completely disconnected with respect to DR• (Mν, Z) for these special valuesof ν. The family of spectral problems on DR• (Mν, Z) for ν ∈R2 \ (0, 0) provides uswith a smooth deformation between a spectral problem on M (without any interiorboundary conditions) and the direct sum of spectral problems on (M1, Z1) and on

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY15(M2, Z2 ∪N).

So this family of interior boundary problems is (in a spectral sense) akind of a smooth cutting of M in two disconnected pieces.Let (M1, N) be a compact smooth Riemannian manifold (M1, gM1) with a smoothboundary ∂M1 and let N be a union of some connected components of ∂M1. Let ametric gM1 be a direct product metric near the boundary.

Then (as it follows fromthe equality (1.8)) the analytic torsion norm T0 (M1, N) on det H• (DR (M1, N))does not depend on gM1. To prove this it is enough to take as (M, gM) a closedmanifold M = M1 ∪N M1 with a mirror symmetric (with respect to N) Riemannianmetric gM which coincides with gM1 on each piece M1 of M (gM1 is a direct productmetric near N and so gM is smooth on M).

Since the torsions T0(M) and T0(N) areindependent of gM1 and of gN = gM1|TN we see that T0 (M1, N) does not depend ongM1.It follows from the equality (1.12) with c0 = 1 that T0 (Mν, Z) does not depend ongM. Indeed, T0 (Mj, Zj ∪N) and T0(N) are independent of gM, and the identificationϕanνis also independent of gM.

(Here M is a manifold with a smooth boundary ∂M,N ∩∂M = ∅, the Dirichlet boundary conditions are given on a union Z of somecomponents of ∂M, the Neumann boundary conditions are given on ∂M \ Z, and gMis a direct product metric near ∂M and near N, Zk := Z ∩∂M k.)Since DR• (M0,1, Z) is the direct sum DR• (M1, Z1) ⊕DR• (M2, Z2 ∪N) of the deRham complexes (Zk := Z ∩∂M k), we see that the analytic torsion norm T0 (M0,1)is canonically equal to the tensor product of norms:T0 (M0,1, Z) = T0 (M1, Z1) ⊗T0 (M2, Z2 ∪N) . (1.16)The determinant line in (1.16) is the tensor productdet H• (M0,1, Z) = det H• (M1, Z1) ⊗det H• (M2, Z2 ∪N)(where H• (M1, Z1) and H• (M2, Z2 ∪N) are the relative cohomology).The formula (1.8) claims for ν = (0, 1) thatϕan0,1T0 (M0,1, Z) = T0 (M1, Z1 ∪N) ⊗T0 (M2, Z2 ∪N) ⊗T0(N).

(1.17)It follows from the definition of the exact sequence (1.14) that ϕan0,1 is the identity onthe component det H• (M2, Z2 ∪N) of det H• (M0,1, Z). The following theorem is animmediate consequence of (1.16) and (1.17).

Let N be a union of some connectedcomponents of ∂M1, let M1 be a compact Riemannian manifold with a smooth bound-ary ∂M1 and let Z1 be a union of some connected components of ∂M1 not belongingto N. Suppose that the metric gM1 is a direct product metric near ∂M1.Theorem 1.2 (Gluing of boundary components). The equality holdsϕanT0 (M1, Z1) = T0 (M1, Z1 ∪N) ⊗T0(N).

(1.18)

16S.M. VISHIKThe identification of the determinant lines in (1.18)ϕan : det H• (M1, Z1) →∼det H• (M1, Z1 ∪N) ⊗det H•(N)(1.19)is defined by the short exact sequence of the de Rham complexes:0 →DR• (M1, Z1 ∪N) →DR• (M1, Z1) →DR•(N) →0,(1.20)where the left arrow is the natural inclusion, and the right arrow is the geometricalrestriction.Example 1.1.

Formula (1.18) contains the Lerch formula ([WW], 13.21, 12.32) forthe derivative at zero of the zeta-function of Riemann ζ(s) (defined for Re s > 1 asPn≥1 n−s):∂sζ(s)|s=0 = −2−1 log 2π.Indeed, let M be an interval (0, b] ⊂R with the Dirichlet boundary conditions at0 and the Neumann conditions at b. Set N be a point b.

Then the formula (1.18)claims in this case thatT0 ((0, b]) = T0 ((0, b)) ⊗T0(b). (1.21)The cohomology H• ((0, b]) = H• ([0, b], 0) are trivial.

The scalar analytic torsionT((0, b]) is equal to exp (−∂sζ1(s; M)|s=0), where ζ1(s; M) is the zeta-function for theLaplacian on DR1 ((0, b]). This zeta-function for Re s > 1/2 is defined by the seriesζ1(s; M) =Xn≥0((π/2b)(2n + 1))2−s .So ζ1(s; M) = (π/2b)−2s (1 −2−2s) ζ(2s) for Re s > 1/2, where ζ(s) is the zeta-function of Riemann.

Hence, the latter equality between the analytic continuationsof ζ1(s; M) and of ζ(2s) holds for all s ∈C, and ∂sζ1(s)|s=0 = 2ζ(0) log 2.The determinant line det H•(M) on the left in (1.21) is canonically isomorphic toC, and the T0(M)-norm of the element 1 ∈C is equal to∥1∥2T0(M) = exp (−ζ′1(0; M)) = exp (−2 ζ(0) log 2) = 2. (Note, that the function 2 ζ(2s) is the zeta-function for the Laplacian ∆= (−∂2/∂x2)on functions on the circle of the length 2π.

As the circle is odd-dimensional, thenthe value of 2 ζ(2s) at zero is equal to −dim Ker ∆= −1. Hence, 2 ζ(0) = −1.

)The scalar analytic torsion T ((0, b)) is equal to exp(−∂sζ1(s;M,N)), where ζ1(s;M,N)for Re s > 1/2 is defined by the seriesζ1(s; M, N) =Xn≥1((π/b)n)2−s = (π/b)−2sζ(2s).

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY17Hence, this equality holds for all s ∈C, and the scalar analytic torsion is equal toT ((0, b)) = exp (−2 ζ′(0) + 2 ζ(0) log(π/b)) = exp (−log(π/b) −2ζ′(0)) .The identification of the determinant lines on the right and on the left in (1.21) isdefined by the cohomology exact sequence0 →H0(b) →H1 ((0, b)) →0.

(1.22)The element 1 ∈H0(b) (of the norm 1) is mapped by (1.22) to the element (dx/b)of the norm ∥dx/b∥2 = b−1. The element h = 1−1 ⊗(dx/b), corresponding to theelement 1 ∈C = det H• ((0, b]), has the norm b−1.

So the equality (1.21) claims thatlog 2 = −log b −log(π/b) −2 ζ′(0).Thus the equality ζ′(0) = −2−1 log(2π) is a particular case4 of Theorem 1.2.The natural L2-norm on ⊕jDR• M jis defined by(v1, v1) :=ZM (v1 ∧∗¯v1) ,(1.23)where (v1 ∧∗¯v1) is a real density on M, corresponding to v1 ∧∗¯v1.Lemma 1.1. The Green functions Gν for the Laplacians ∆•ν provide us with thehomotopy operator in the complex DR• (Mν, Z)Kν := δGν(1.24)between the orthogonal projection operator pH : DR• (Mν, Z) →Ker (∆•ν) and theidentity operator on DR• (Mν, Z).

The following equality holds in DR• (Mν, Z) :dKν + Kνd = id −pH.Proof. The Green function for ∆•ν maps the L2-completion (DR•(M))2 of DR•(M)5into the Dom (∆•ν) (Theorem 3.1).

The Dom (∆•ν) is defined as the domain of defi-nition D (∆•ν) for ∆•ν in DR• (Mν, Z) completed with respect to the graph topologynorm ∥ω∥2graph := ∥ω∥22 + ∥∆•νω∥22 for ω ∈D (∆•ν) (where ∥ω∥22 := (ω, ω) is the L2-norm (1.23)).The Green function Gν maps DR• (Mν, Z) into D (∆•ν) (since, by4In this paper the proofs of the equality (1.18), of Theorem 1.1, and of the equality (1.12) withc0 = 1 do not use the Lerch formula. So we have obtained (by the way) a new proof of the Lerchformula.5(DR•(M))2 coincides with the L2-completion of DR• (Mν, Z) and with the L2-completion of⊕jDR• M j.

18S.M. VISHIKTheorem 3.1, ∆•ν is a nonnegative elliptic differential operator with elliptic boundaryconditions).

The definition of the Green function claims that∆•νGν = id −pH,(1.25)on (DR•(M))2 (where ∆•νω for ω ∈Dom (∆•ν) is defined as limi ∆•νωi for ωi ∈D (∆•ν),∥ω −ωi∥2graph →0). In particular, this equality holds on DR• (Mν, Z) ⊂(DR•(M))2.The D (∆•ν) ⊂(DR•(M))2 is defined as follows.

The adjoint to dν operator δν in⊕jDR• M jis defined on elements v2 = (ω1, ω2), where ωk are smooth differentialforms on M k and the linear functionallv2(v1) =< dv1, v2 >=ZM(dv1 ∧∗¯v2)is continuous in DR• (Mν, Z) with respect to the L2-norm (1.23) of v1 ∈DR• (Mν, Z).For such an element v2 = (ω1, ω2) the form ∗v2 = (∗ω1, ∗ω2) has the zero geometricalrestriction to ∂M \ Z, and the following transmission condition has to hold on N forv2:β i∗N,1(∗ω1) = α i∗N,2(∗ω2),(1.26)where iN,k : N ⊂∂Mk ֒→Mk. (These boundary conditions for v2 are consequencesof Stokes’ formula.

)The domain D (∆•ν) ⊂DR• (Mν, Z) is defined as the set of ω ∈DR• (Mν, Z) suchthatω ∈D(δν),dω ∈D(δν),δω ∈DR• (Mν, Z) . (1.27)Note that dGνω = Gνdω for ω ∈DR• (Mν, Z) (this equality follows from Stokes’formula).

Hence the identity (1.25) can be represented on DR• (Mν, Z) asKd + dK = id −pH.Thus the lemma is proved.□Corollary 1.1. The homotopy operator ( 1.24) defines a canonical identification be-tween the cohomology H• (DR (Mν, Z)) and the space of harmonic forms Ker• (∆ν).Let for simplicity gM be a direct product metric near N. Let the Dirichlet boundaryconditions be given on a union Z of some connected components of ∂M and theNeumann conditions be given on ∂M \ Z.

Then the following holds.Proposition 1.1. The eigenforms of ∆(M, Z; gM) are the same as the eigenformsof ∆1,1.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY19Proof.

Let ν be equal to (1, 1). The conditions (1.27) for ω =(ω1, ω2)∈DR• (M1,1,Z)are equivalent on N to the following ones:i∗N,1ω1 = i∗N,2ω2,i∗N,1(∗ω1) = i∗N,2(∗ω2),(1.28)i∗N,1(∗dω1) = i∗N,2(∗dω2),i∗N,1(∗d ∗ω1) = i∗N,2(∗d ∗ω2),(1.29)where ∗is the star-operator for the Riemannian metric gM.The equalities (1.28) claim that the restrictions to N of the forms ω1 and ω2 arethe same (i.e., they are the same smooth sections of ∧•T ∗M|N).

The equalities (1.28)and (1.29) are equivalent to the assertion that the following pairs of forms have thesame restrictions to N (as the smooth sections of ∧•T ∗M|N):{dω1, dω2} ,{δω1, δω2} ,{ω1, ω2} . (1.30)Any eigenform for ∆• (M, Z; gM) belongs to D∆•1,1.

So it is an eigenform for ∆•1,1.Let ω = (ω1, ω2) ∈D∆•1,1be an eigenform for ∆1,1:∆•1,1ω = (∆•ω1, ∆•ω2) = λ(ω1, ω2). (1.31)Then6 ωk are C∞-forms on Mk and (as it follows from (1.30), (1.31)) the restrictionsof the following pairs of forms are the same as the sections of ∧•T ∗M|N (for k =0, 1, 2 .

. .

):n∆kω1, ∆kω2o,nd∆kω1, d∆kω2o,nδ∆kω1, δ∆kω2o. (1.32)So ω = (ω1, ω2) is a C∞-form on M = M1 ∪N M2.

In fact, it follows from (1.28)and from the identity of the restrictions to N of ∆ω1 and ∆ω2 that (∆I ⊗id) ωkhave (for k = 1, 2) the same restrictions from Mk to N. (The Laplacian ∆is equalto idI ⊗∆N + ∆I ⊗idN with respect to the direct product structure I × N in theneighborhood of N = 0 × N ֒→I × N ֒→M, 0 ∈I \ ∂I.) Hence, according to (1.28)and (1.29), the 2-jets of ω1 and of ω2 are the same on N. The identity between the(2k + 1)-jets of ωk on N follows (by induction) from (1.32).

Thus, ω is an eigenformfor ∆M: ∆Mω = λω. The proposition is proved.□1.3.

Properties of analytic and combinatorial torsion norms. One of themain properties of the analytic torsion norm is as follows.

Let M be a manifoldM1 × M2 with a direct product metric.One of these Riemannian manifolds, forinstance M1, can have a nonempty boundary ∂M1. In this case let gM1 be a directproduct metric near ∂M1, and let the Dirichlet boundary conditions be given on thecomponents Z = Z1×M2 of (∂M1)×M2 = ∂(M1 × M2).

Let the Neumann boundaryconditions be given on ∂(M1 × M2) \ (Z1 × M2) = (∂M1 \ Z1) × M2.6All the eigenforms of ∆•ν (for ν ∈R2 \ (0, 0)) are C∞-smooth on Mk, as it follows from Theo-rem 3.1.

20S.M. VISHIKThe K¨unneth formula for the cohomology claims thatHj(M, Z) = ⊕i+k=jHi (M1, Z1) ⊗Hk (M2) .

(1.33)So the determinant of the cohomology of DR•(M, Z) is the tensor productdet H•(M, Z) = (det H• (M1, Z1))χ(M2) ⊗det H• (M2)χ(M1,Z1) . (1.34)Proposition 1.2.

The identification ( 1.34) induces the isomorphism of the analytictorsion norm T0(M, Z) with the tensor productT0(M, Z) = T0 (M1, Z1)⊗χ(M2) ⊗T0 (M2)⊗χ(M1,Z1) ,(1.35)where χ(M1, Z1), χ (M2) are the Euler characteristics.Remark 1.1. It is shown above that the analytic torsion norms T0(M, Z), T0 (M1, Z1),and T0 (M2) are independent of Riemannian metrics gM, gMk which are supposed tobe direct product metrics near ∂M, ∂M1.

So, if the equality (1.35) holds for a directproduct metric on (M1, ∂M1) ×M2, (where gM1 is a direct product metric near ∂M1)then this equality holds for any metric gM (which is supposed to be a direct productmetric near ∂M).Proof. The scalar analytic torsion T(M) for a direct product metric on M = M1×M2is equal toT(M, Z) = T (M1, Z1)χ(M2) T (M2)χ(M1,Z1) .

(1.36)This statement is proved in [RS], Theorem 2.5, in the case of an acyclic local systemover M1. In the general case, (1.36) follows from the proof of Theorem 2.5 in [RS]and from the following equality (where λ ̸= 0, m(i, λ, M2) is the dimension of theλ-eigenspace for ∆M2,i, m (j, 0, M1) := dim Ker ∆M1,Z1;j):X(−1)i+j (i + j)m (i, λ, M2) m (j, 0, M1) =X(−1)i im (i, λ, M2)χ (M1, Z1) ,which holds, since the alternating sum over i of m(i, λ, M2) is equal to zero (for anynonzero λ).For such a metric on M the following canonical identifications are the isometriesbetween the natural Hilbert structure on the space of harmonic forms Ker ∆j(M, Z)and the tensor products (and the direct sums) of the Hilbert structures on harmonicforms for ∆• (M1, Z1) and ∆• (M2):Ker ∆j(M, Z) = ⊕i+k=j Ker ∆i (M1, Z1) ⊗Ker ∆k (M2) .

(1.37)These Hilbert structures induce the norms ondet H•(M, Z) = det Ker ∆• (M, Z; gM) , det H• (M1, Z1) = det Ker ∆• (M1, Z1; gM1)

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY21and on det H• (M2) = det Ker ∆• (M2, gM2) such that the identification (1.34) of thedeterminant lines is an isometry:∥·∥2det Ker ∆M =∥·∥2det Ker ∆M1,Z1χ(M2) ∥·∥2det Ker ∆M2χ(M1,Z1) .

(1.38)The equality (1.35) follows from (1.36),(1.34), and (1.38).□The following lemma makes it possible to use the variations on ν in the proofof Theorem 1.1. Let (M, gM) be a compact Riemannian manifold with a smoothboundary ∂M and let N ֒→M \∂M be a smooth closed codimension one submanifoldof M with a trivial normal bundle (TM|N) /TN such that M = M1∪N M2 is obtainedby gluing two its pieces M1 and M2 along N. Let gM be a direct product metric near∂M and near N. Let Z be a union of some connected components of ∂M where theDirichlet boundary conditions are given and let the Neumann boundary conditionsbe given on ∂M \ N.Lemma 1.2.

Let us suppose that the norm ϕanν T0 (Mν, Z) is independent of ν ∈R2 \ (0, 0) for any such (M, gM, N, Z)7 (where the identification ϕanνis defined by theexact sequence ( 1.14) of the de Rham complexes and by Lemma 1.1). Then the factorc0 in the gluing formula ( 1.12) for ϕanν T0 (Mν, Z) is equal to one.Remark 1.2.

Theorem 1.1 is a direct consequence of Lemma 1.2 and of the assertionthat ϕanν T0 (Mν, Z) is independent of ν ∈R2 \ (0, 0). Indeed, T0 (M1,1, Z) coincideswith T0(M, Z) (according to Proposition 1.1) and the identifications ϕanν and ϕan arethe same.

Hence the formula (1.12), where c0 is equal to one and ν = (1, 1), is thegluing formula of Theorem 1.1.Remark 1.3. The assertion that the norm ϕanν T0 (Mν, Z) does not depend on ν isequivalent to the independence of ν of the factor c0 in (1.12).Proof.

The factor c0 in (1.12) lies in R+. If c0 is independent of ν for (M, gM, N, Z)thenϕan1,0 T0 (M1,0, Z) = ϕan0,1 T0 (M0,1, Z) .

(1.39)It follows from (1.39) and from (1.16), (1.19), and (1.14) that there are the equalitieswith the same positive constant c0 as in (1.12) for (M, gM, N, Z) (where Zk := Z ∩∂Mk):ϕanT0 (M1, Z1) = c0T0 (M1, Z1 ∪N) ⊗T0(N),(1.40)ϕanT0 (M2, Z2) = c0T0 (M2, Z2 ∪N) ⊗T0(N). (1.41)7The equivalent formulation is as follows.

Let M be obtained by gluing along N, i.e., M =M1∪N M2, and let it be equiped with a Riemannian metric gM, which is a direct product metric near∂M and near N. Then it is supposed that the norm ϕanν T0 (Mν, Z) is independent of ν ∈R2 \(0, 0).

22S.M. VISHIKWe can conclude from (1.40) and (1.41) that the factor c0 = c0(N, gN) is defined byN, gN and that it does not depend on M1, M2, M, and gM (it is independent also ofν).Let M1 in (1.40) be a manifold M1 = N × I with a direct product metric.

Then∂M1 = N ∪N and (1.40) claims in this case thatϕanT0(N × I) = c0(N, gN)2 T0(N × I, N × ∂I) ⊗T0(N)⊗2,(1.42)where the identification ϕan is defined by the exact sequence (1.20). It follows from(1.42) and from the multiplicative property (1.35) thatT0(N)χ(I) ⊗T0(I)χ(N) = c20 T0(N)χ(I,∂I) ⊗T0(I, ∂I)χ(N) ⊗T0(N)2,(1.43)where c0 := c0(N, gN) depends on N and on gN only.

Then the following equality is aconsequence of (1.43) and of the identification (1.19) (defined by the exact sequence(1.20)):T0(I)χ(N) = c0(N, gN)2 T0(I, ∂I)χ(N) ⊗T0(∂I)χ(N). (1.44)Note that T0(∂I) is the standart norm on det H•(∂I) which is canonically identifiedwith C (up to a possible factor (−1) in the identification).

Namely ∥1∥2T0 = 1 for1 ∈C. An immediate consequence of the equality (1.41) for M1 = I, N = ∂I and of(1.44) is the following:c0 (N, gN)2 = c0(∂I)χ(N).

(1.45)Hence, it is enough to prove that c0(∂I) = 1, and it will be done now.Let I be an interval [0, a]. The scalar analytic torsions for I and for (I, ∂I) areequal: T(I) = T(I, ∂I), sinceζ1(s; I) = ζ0(s; I, ∂I),(1.46)ζ1(s; I, ∂I) = ζ0(s; I, ∂I),(1.47)(where ζj(s; M, Z) is the ζ-function of the Laplacian on (DRj(M, Z), gM)).Theequality (1.47) follows from the identification of the eigenforms, defined by the exte-rior derivative d, and the equality (1.46) follows from the identification of the eigen-forms defined by the Riemannian ∗on I.The cohomology exact sequence for the pair (I, ∂I) is0 →H0(I) →H0(∂I) →H1(I, ∂I) →0.

(1.48)The complex (1.48) is acyclic and so the determinant D of its cohomology is canoni-cally isomorphic to C. The components of (1.48) are equiped with the natural Hilbertstructures (because they are the spaces of harmonic forms on the interval I ⊂R withthe standart metric). Hence, there is the induced norm ∥·∥D on D = C. We have toprove that ∥1∥2D = 1 for 1 ∈C = D. This equality is equavalent to the assertion thatc0(∂I) is equal to one.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY23The norm of the element a−1/2 · 1 ∈DR0(I) is equal to 1.

(It is a harmonic formand it represents an element from H0(I)). Its image in H0(∂I) is as follows:−a−1/2 · [0] + a−1/2 · [a] ∈H0(∂I) = H0(0 ∪a).The norm of the element a−1/2 · dx ∈DR1(I, ∂I) is equal to 1 in H1(I, ∂I) and anelement −a1/2 · [0] is mapped by the differential of the exact sequence (1.48) to theharmonic form a−1/2dx ∈H1(I, ∂I).

(The arrows in (1.48) are of the topologicalnature. So the latter statement is obtained usinga1/2 =Z[0,a] a−1/2 dx =a−1/2dx, (I, ∂I),where (I, ∂I) is the fundamental class of H1(I, ∂I).

)The corresponding volume element−a−1/2[0] + a−1/2[a]∧−a1/2[0]= [0] ∧[a]in det H0(∂I) is an element with the norm one. Hence, c0(∂I) = 1.

The equalityc0 (N, gN) = 1 (for a union N of some connected components of ∂M1) follows fromthe equality c0(∂I) = 1 and from (1.45). The lemma is proved.□Let M = M1 ∪N M2 be obtained by gluing M1 and M2 along N (as in Theorem1.1), and let X be a smooth triangulation of M such that Mk and N are invariantunder X. Namely X = X1 ∪W X2, where Xk is a smooth triangulation of a manifoldMk with a smooth boundary ∂Mk = N ∪∂M ∩M k. (Here N ⊂M is a smoothclosed manifold of codimension one in M such that N divides M in two pieces M1and M2 as in Theorem 1.1, N ∩∂M = ∅, and W := X ∩N = Xk ∩N.

)Let Z be a union of the connected components of ∂M, where the Dirichlet boundaryconditions are given. Set V := X ∩Z, Zk := ∂Mk ∩Z, Vk := Xk ∩Zk.

The exactsequence of cochain complexes0 →⊕k=1,2C•(Xk, W ∪Vk) →C•(X, V )i∗N−→C•(W) →0(1.49)(where the left arrow is the natural inclusion and the right arrow is the geometricalrestriction of cochains) provides us with the identificationϕc : det H•(X, V ) →∼det H•(X1, W ∪V1) ⊗det H•(X2, W ∪V2) ⊗det H•(W).By the definition of the combinatorial torsion norm on the determinant line (de-termined by the prefered basises of the basic cochains) the following statement holds.Proposition 1.3. Under the conditions above, the combinatorial torsion norms areequal:ϕcτ0(X, V ) = τ0(X1, W ∪V1) ⊗τ0(X2, W ∪V2) ⊗τ0(W).

(1.50)This combinatorial equality is analogous to the gluing formula of Theorem 1.1. Butit is necessary to note as follows.

24S.M. VISHIKRemark 1.4.

The formulas (1.50) and (1.8) correspond to the different identificationsϕc and ϕan = ϕan1,1 between one pair of the canonically identified8 one-dimensionalspacesdet H•(X, V ) = det H•(M, Z),and the triple tensor products of three other pairs of the canonically identified spacesdet H•(Xk, W ∪Vk) = det H•(Mk, N ∪Zk),det H•(W) = det H•(N). (Note that ϕc is defined by the exact sequence (1.49), where the right arrow i∗N is therestriction of the cochains.

However, in the exact sequence (1.10), which defines ϕan,the right arrow is equal to√2 i∗N for the common geometrical restriction i∗N to N ofpairs ω = (ω1, ω2) of smooth differential forms ωk on M k such that i∗N,1ω1 = i∗N,2ω2. )Let X be a smooth triangulation of a compact manifold with boundary (M, ∂M).Let Z and Y be disjoint unions of some connected components of ∂M such thatZ ∩Y = ∅.

Let V = X ∩Z, F = X ∩Y . Then the exact sequence0 →C•(X, V ∪F) →C•(X, V ) →C•(F) →0(where the left arrow is the natural inclusion of cochains and the right arrow is therestriction of cochains) defines the identificationϕFc : det H•(X, V ) →∼det H•(X, V ∪F) ⊗det H•(F).The following assertion is an immediate consequence of the definition of the com-binatorial torsion norm.Proposition 1.4.

The combinatorial torsion norm of (X, V ) is equal to the tensorproduct of the following combinatorial torsion norms:ϕcτ0(X, V ) = τ0(X, V ∪F) ⊗τ0(F).This combinatorial equality is similar to the gluing formula of Theorem 1.2.Let e(M, Z) be the logarithm of the ratio between the analytic and the combina-torial torsion norms:e(M, Z) := log2 (T0(M, Z)/τ0(X, V ))(where T0(M, Z)/τ0(X, V ) := ∥l∥2T0(M,Z) /∥l∥2τ0(X,V ) for an arbitrary nonzero elementl of the determinant line det H•(M, Z) = det H•(X, V )).Remark 1.5. It is proved above that e(M, ∂M) does not depend on a metric gM, ifgM is a direct product metric near ∂M.8The cohomology are identified according to the de Rham theorem by the integration over thesimplexes of X of the corresponding differential forms.

The spaces of harmonic forms Ker ƥ(M, Z)and Ker ƥ (M1,1, Z) are canonically identified by Proposition 1.1.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY25Lemma 1.3.

1. Let (S, gS) be a closed Riemannian manifold.

Then the followingidentity holds, if gM×S is a direct product metric near ∂(M × S) = ∂M × S:e(M × S, Z × S) = χ(M, Z)e(S) + e(M, Z)χ(S)(1.51)(χ(M, Z) is the relative Euler characteristic of M modulo Z ⊂∂M).2. Let Y be a union of some connected components of ∂M \ Z. Thene(M, Z) = e(M, Y ∪Z) + e(Y ).(1.52)Proof.

The equality (1.52) follows from Theorem 1.2 and from Proposition 1.4. (Inthis case, ϕc = ϕan.

)The equality (1.51) follows from Proposition 1.2 and fromthe multiplicative property of the combinatorial torsion norms. Namely let K bea smooth triangulation of S and let V = X ∩Z.

Then the identification of thedeterminants of the cohomology defined by (1.33) and (1.34) is an isometry of thecombinatorial torsion norms:τ0(X × K, V × K) = τ0(X, V )χ(K) ⊗τ0(K)χ(X,V ).The same identification of the cohomology is the isometry (1.35) of the analytictorsion norms, if the metric gM×S is a direct product metric near ∂(M × S). Hence,the identity (1.51) holds for such metrics gM×S.□Remark 1.6.

It follows from (1.52) and from Remark 1.5 that e(M, Z) does not de-pend on gM for any union Z of the connected components of ∂M (in particular forZ = ∅).1.4. Generalized Ray-Singer conjecture.1.4.1.

Properties of the ratio of the analytic and the combinatorial torsion norms.Lemma 1.2 claims that Theorem 1.1 follows from (1.12) with c0 independent of ν. Soit is enough to prove that the norm ϕanν T0 (Mν, Z) is independent of ν ∈R2 \ (0, 0)(under the same conditions on M, gM, N, and Z as in (1.12) and in Lemma 1.2). Thelatter assertion is proved in Section 2.

In the remaining part of Section 1 we prove ageneralization of the Ray-Singer conjecture for manifolds with boundary (and withthe transmission condition (1.13) on the interior boundary) using the gluing formulaof Theorem 1.1. This formula has the following consequence.Let M = M1 ∪N M2 be obtained by gluing M1 and M2 along N.Lemma 1.4.

Under the conditions of Lemma 1.2, on (gM, N, Z) the following holds:e(M, Z) = e(M1, Z1 ∪N) + e(M2, Z2 ∪N) + e(N) −χ(N).

26S.M. VISHIKProof.

This identity is an immediate consequence of Theorem 1.1 and of the followingcommutative diagram:det H•(M, Z)ϕan=ϕan1,1−−−−−→^Det(M, Z, N)∥R∥yRdet H•(X, V )ϕc−−→gDet(X, V, W)AHW−−→gDet(X, V, W)x≀dcx≀dcx≀dcdet C•(X, V )(1.49)−−−→gDetC•(X, V, W)AW−−→gDetC•(X, V, W)(1.53)HereDet(M, Z, N) := (⊗k=1,2 det H• (Mk, N ∪Zk)) ⊗det H•(N),Det(X, V, W) := (⊗k=1,2 det H• (Xk, W ∪Vk)) ⊗det H•(W),DetC•(X, V, W) := (⊗k=1,2 det C• (Xk, W ∪Vk)) ⊗det C•(W),(1.54)AW := idX ⊕√2 idW ∈Aut (⊕k=1,2C• (Xk, W ∪Vk)) ⊕Aut C•(W),AHW is the induced by AW operator on the determinant of the cohomology,R is the identification induced by the integration of differential forms over the sim-plexes of X (by the de Rham theorem),ϕc and ϕan are the identifications induced by (1.49) and by (1.10) in a view of Propo-sition 1.1.The commutativity of (1.53) follows from the commutativity of the diagram0 →⊕k=1,2 DR• (Mk, N ∪Zk) →DR• (M1,1, Z)√2 i∗N−−−→DR•(N) →0yRyRyR0 →⊕k=1,2C• (Xk, W ∪Vk)→C•(X, V )i∗N−→C•(W)√2 id−−−→C•(W)→0The induced action of√2 id on lW ∈det C•(W) is lW →2−χ(W )/2lW (whereχ(W) = χ(N) is the Euler characteristic). So the induced action of AW and of AHWon l ∈Det(M, Z, N) = (⊗k=1,2 det C• (Xk, W ∪Vk)) ⊗det C•(W) isl →2−χ(N)/2l.

(1.55)(The identification of the determinant lines is defined by R and by dc in the rightcolumn of (1.53). )For an arbitrary nonzero m ∈Det(M, Z, N) the following equality is deduced from(1.55) and from the commutativity of (1.53):(ϕanT0(M, Z)) (m) = 2−χ(N) (ϕcT0(M, Z)) (m).

(1.56)

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY27Theorem 1.1 and Proposition 1.3 claim thatϕanT0(M, Z) = T0 (M1, N ∪Z1) ⊗T0 (M2, N ∪Z2) ⊗T0(N),ϕcτ0(X, V ) = τ0 (X1, W ∪V1) ⊗τ0 (X2, W ∪V2) ⊗τ0(W).

(1.57)The isometries (1.56) and (1.57) involve the equalitye(M, Z) = log2 (T0(M, Z)/τ0(X, V )) == −χ(N)+Xk=1,2log2 (T0 (Mk, N ∪Zk) /τ0 (Xk, W ∪Vk))+log2 (T0(N)/τ0(W)) .Thus the lemma is proved.□Let ν = (α, β) ∈R2 \ (0, 0) and let (C• (Xν, V ) , dc) be the complex of pairsof cochains (c1, c2), ck ∈C• (Xk, Vk), with the ν-transmission boundary condition(similar to (1.13)) on W ⊂∂Xk between their geometrical restrictionsαi∗W,1c1 = βi∗W,2c2. (1.58)The integration over the simplexes provides us with a quasi-isomorphism of thecomplexes:Rν : (DR• (Mν, Z) , d) →(C• (Xν, V ) , dc)(i.e., Rν induces an isomorphism between the corresponding cohomology).The morphism of complexes rν,c : (C• (Xν, V ) , dc) →C• (W, dc) is defined by anal-ogy with the definition of rν.

Its value on each element (c1, c2) ∈C• (Xν, V ) isrν,c (c1, c2) = (α2 + β2)−1/2 βi∗W,1c1 + αi∗W,2c2.The vertical arrows in the following diagram of complexes are quasi-isomorphisms90 →⊕k=1,2 DR• (Mk, N ∪Zk) →DR• (Mν, Z)rν−→DR•(N) →0yRyRνyR0 →⊕k=1,2C• (Xk, W ∪Vk)j→C• (Xν, V )rν,c−→C•(W)→0(1.59)This diagram is commutative. The left horisontal arrows in it are the natural inclu-sions.

Let ϕcν be the identificationϕcν : det H• (C(Xν, V )) Rν= det H• (Mν, Z) →Det(M, Z, N),(1.60)defined by the bottom row of this diagram.9Rν is a quasi-isomorphism according to Proposition 2.3.

28S.M. VISHIKRemark 1.7.

The equalityϕcν = ϕanν(1.61)follows from the commutativity of (1.59). But ϕc1,1 ̸= ϕc (in contrast with the identityϕan1,1 = ϕan).

According to (1.56) it holds thatϕc1,1 = ϕan = 2−χ(N)ϕc.The space Cj(Xν, V ) is a subspace of ⊕k=1,2Cj (Xk, Vk). The Hilbert structure onCj (Xk, Vk) is defined by the orthonormal basises of cochains {δe} (parametrized byj-dimensional simplexes e of Xk\Vk).

So the Hilbert structures on C• (Xν, V ) and ondet C• (Xν, V ) are defined. The scalar combinatorial torsion is defined as in (1.1):T (C• (Xν, V ) , dc) := expX(−1)j j∂sζcj,ν(s)s=0,where ζcj,ν(s) := Tr′∆cj,ν−sis the sum P′λ−s over all the nonzero eigenvalues λof the finite-dimensional operator ∆cj,ν = (d∗cdc + dcd∗c|Cj (Xν, V )) (with their multi-plicities), d∗c is adjoint to dc in Cj (Xν, V ) with respect to the Hilbert structure inC• (Xν, V ).The combinatorial torsion is the following norm on det H• (C (Xν, V ))10:τ0 (Xν, Z) := ∥·∥2det H•(C(Xν,V )) · T (C• (Xν, V ) , dc) ,(1.62)where the norm on det H• (C (Xν, V )) is defined by the Hilbert structures on thesubquotions Hj (C (Xν, V )) of the Hilbert spaces Cj (Xν, V ).Remark 1.8.

For each ν = (α, β) ∈R2 \ (0, 0) the combinatorial torsion τ0 (Xν, V ) isan invariant of the combinatorial structure defined by a smooth triangulation of thetriplet [(M, ∂M); Z; N], where M is a manifold with a smooth boundary ∂M, Z isa union of some connected components of ∂M, and N is a smooth codimension oneclosed submanifold of M with a trivial normal bundle (TM|N) /TN.Proposition 1.5. The combinatorial torsion norm τ0 (Xν, Z) is isometric under theidentification ( 1.60) to the tensor product of the combinatorial torsion norms:ϕcντ0 (Xν, V ) = τ0 (X1, W ∪V1) ⊗τ0 (X2, W ∪V2) ⊗τ0(W).10It is isomorphic to det H• (Mν, Z) under the quasi-isomorphism Rν.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY29Proof.

Under the identification (1.60), the Hilbert space C• (Xν, V ) is isometric tothe tensor product of the Hilbert spaces on det C• (Xk, W ∪Vk) (for k = 1, 2) andon det C•(W). (The Hilbert structures on on C• (Xν, V ), C• (Xk, W ∪Vk), and onC•(W) are defined above.) Indeed, let ρν,c : Cj(W) →Cj (Xν, V ) be linear mapsdefined for w ∈Cj(W) byρν,c(w) =α2 + β2−1/2 (βw, αw) ∈Cj (Xν, V )(1.63)Then rν,cρν,c = id on C•(W), ρν,c is an isometry between Cj(W) and Im ρν,c, andIm ρν,c is the orthogonal complement in C• (Xν, V ) to the image of the natural in-clusion j : ⊕k=1,2 C• (Xk, W ∪Vk) ֒→C• (Xν, V ) (where j is an isometry onto Im j).So the identification ϕcν is the isometry of the combinatorial torsion norms.□The number e (Mν, Z) ∈R is defined as the logarithm of the ratio between theanalytic and the combinatorial torsion norms:e (Mν, Z) := log2 (T0 (Mν, Z) /τ0 (Xν, V )) .Corollary 1.2.

Under the conditions of Lemma 1.2, the equality holds:e (Mν, Z) = e (M1, Z1 ∪N) + e (M2, Z2 ∪N) + e(N),(1.64)where Z is a union of some connected components of ∂M and Zk = Z ∩∂Mk.Corollary 1.3. e (Mν, Z) is independent of ν ∈R2 \ (0, 0).Corollary 1.4. For an arbitrary ν ∈R2 \ (0, 0) the equality holds:e (Mν, Z) −e(M, Z) = χ(N).

(1.65)This equality follows from Lemma 1.4 and from (1.64).Remark 1.9. Even for ν = (1, 1) the number e (Mν, Z) differs from e(M, Z) in thecase χ(N) ̸= 0.1.4.2.

Ratio of the analytic torsion norm and the combinatorial torsion norm forspheres and disks. Spherical Morse surgeries.

The values of e(M) and e(M, ∂M),where M is a sphere Sn or a disk Dn (with a direct product metric near ∂Dn = Sn−1)are deduced now from Lemma 1.4.Lemma 1.5. 1.

For all the spheres, e (Sn) is zero.2. For even-dimensional disks, e (D2n) and e (D2n, ∂D2n) are zero.3.

For all odd-dimensional disks, e (D2n+1) and e (D2n+1, ∂D2n+1) are equal to one.

30S.M. VISHIKProof.

A closed interval D1 is obtained by gluing two intervals D1 = D1 ∪pt D1 intheir common boundary point. Lemma 1.4 claims in this case thateD1= 2eD1, pt+ e(pt) −χ(pt).

(1.66)Since e(pt) = 0, we see that (1.52) involves the equalitieseD1= eD1, pt= eD1, ∂D1. (1.67)Hence (1.66) involves e (D1) = χ(pt) = 1.A circle S1 is obtained by gluing two intervals, namely S1 = D1 ∪∂D1 D1.

So,according to Lemma 1.4 and to (1.67), we haveeS1= 2eD1+ e∂D1−χ∂D1= 0. (1.68)Suppose (by the induction hypothesis) that e(Sm) = 0 for m ≤n −1.

The sphereSn (for n ≥2) is the union (Dn−1 × S1) SSn−2×S1 (D2 × Sn−2) = Sn Indeed, Sn =n(x1, . .

. , xn+1) ∈Rn+1:P x2j = 1o, the disk D2 in D2 × Sn−2 in the decompositionabove corresponds to {(x1, x2): x21 + x22 ≤ε} and Sn−2 = {(xj) ∈Sn, x1 = x2 = 0}.Lemma 1.4 claims in this case that (since χ (Sn−2 × S1) = 0)e (Sn) = eDn−1 × S1, Sn−2 × S1+ eD2 × Sn−2, S1 × Sn−2+ eSn−2 × S1.The equalities below are deduced from the induction hypothesis, from Lemma 1.3((1.52), (1.51)), and from (1.68):eDn−1 × S1, Sn−2 × S1= eDn−1 × S1−eSn−2 × S1,eD2 × Sn−2, S1 × Sn−2= eD2 × Sn−2−eS1 × Sn−2,eS1 × Sn−2= 0,eDn−1 × S1= 0,eD2 × Sn−2= χSn−2e(D2).

(1.69)Hence the combinatorial torsion norm is equal to the analytic torsion norm for allodd-dimensional spheres S2m+1:eS2m+1= 0,T0S2m+1= τ0S2m+1. (1.70)It follows from Lemma 1.3 and from (1.68) that e (D2) = e (D2, ∂D2).

It is deducedfrom Lemma 1.4 and from (1.68) that e (S2) = 2e (D2). According to (1.69) theequality e (S2m) = 0 for all even-dimensional spheres is a consequence of the equalitye (S2) = 0.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY31Let (M, gM) be any closed Riemannian manifold of even dimension 2n.

Then thescalar analytic torsion T (M, gM) is equal to 1. (This equality was proved in [RS],Theorem 2.1, with the help of the equalityX(−1)jj m(λ, j) = 0,where λ is an arbitrary nonzero eigenvalue of ∆j on DRj(M) and m(λ, j) is itsmultiplicity.The latter assertion follows from the symmetry relation m(λ, j) =m(λ, 2n −j), which is obtained applying the operator ∗for a Riemannian metricgM to the λ-eigenforms for ∆j.) So (in particular) the torsion norm T0 (S2) is equalto ∥·∥2det H•(S2), where the norm on H• (S2) is the norm defined by gM on the harmonicforms Ker ∆•.

(The unduced norm ∥·∥2det H•(S2) does not depend on the metric gS2,as it follows from the invariance of T0 (M, gM) with respect to gM, proved above. )Let v be a volume of S2 relative to a Riemannian metric gS2.

Then the elementh ∈det H• (S2) defined below is of the norm 1:h =v−1/2 · 1S2−1 ⊗v−1/2(∗1S2)−1 ,∥h∥2det H•(S2) = 1(here 1S2 is the constant 1 ∈DR0 (S2) and ∗1S2 is the gS2-volume form).The sphere S2 has a cell decomposition11 XS2 : X := D2∪∂D2 pt. Hence the elementhc ∈det C• (XS2) defined below is of the norm 1:hc = (δpt)−1 ⊗(δD2)−1 , ∥hc∥2det C•(XS2) = 1.

(For this cell-decomposition dc = 0, and so det C• (XS2) is the same as det H• (S2)without the dc-identification. The cochains δpt, δD2 are the basic elements in H0 (S2),H2 (S2).

)The integration homomorphism R: DR• (S2) →C• (XS2) maps 1S2 to δpt and ∗1S2to v · δD2. So R(h) = hc and we haveeS2= 0,eS2m= 0.

(1.71)The equalities below follow from Lemmas 1.3, 1.4, and from (1.70), (1.71):0 = e (Sn) = 2e (Dn, ∂Dn) + eSn−1−χSn−1,e (Dn, ∂Dn) = e (Dn) −eSn−1= e (Dn) ,e (Dn, ∂Dn) = 2−1χSn−1= e (Dn) .Lemma 1.5 is proved.□11This CW-complex (cell stratification) has a subdivision which is a C1-triangulation of S2. So asthe combinatorial torsion is defined also for CW-complexes and as it is invariant under subdivisions,τS2can be computed from this cell stratification ([Mi], Sections 7, 8, 12.3).

32S.M. VISHIKThe equality e (Dm+1 × Sn) = e (Dm+1 × Sn, ∂(Dm+1 × Sn)) holds by Lemmas 1.3and 1.5.Corollary 1.5.

For arbitrary n, m ≥0 the equality holds:eDm+1 × Sn= eSm × Dn+1. (1.72)(According to Lemma 1.5, each side of (1.72) is equal to 2 in the case of a pair ofeven numbers (m, n) and it is equal to zero for other pairs (m, n).

)Let M be a compact manifold with a smooth boundary ∂M and let Z be a unionof some connected components of ∂M. Let fM be obtained by some spherical Morsesurgery (with a trivial normal bundle) of M (i.e., there exists a manifold (M1, ∂M1) ⊂M\∂M, M1 ≃Dm+1×Sn, m+n+1 = dim M, with ∂M1 = Sm×Sn, M = M1∪∂MM2,such that fM = fM1 ∪∂eM1 M2 is obtained by gluing fM1 = Sm × Dn+1 and M2 by adiffeomorphism f : ∂fM1 →∼∂M2).Let the metrics gM and g eM be direct product metrics near ∂M and ∂fM.

(It isproved above that the numbers e(M, Z) and e fM, Zdo not depend on the metricsgM, g eM supposed to be direct product metrics near ∂M and near ∂fM. )Lemma 1.6.

The number e(M, Z) is invariant under the spherical Morse surgerieswith a trivial normal bundle, i.e., the equality holdse(M, Z) = e fM, Z.(1.73)Proof. The metrics gM and g eM can be replaced by Riemannian metrics on M and fMwhich are direct product metrics on ∂M1 × I and ∂fM1 × I near ∂M1 ⊂M and near∂fM1 ⊂fM (and which are direct product metrics near ∂M and ∂fM).

Lemma 1.4claims in this case thate(M, Z) = e (M1, ∂M1) + e (M2, ∂M1 ∪Z) + e (∂M1) −χ (∂M1) ,e( fM, Z) = e fM1, ∂fM1+ eM2, ∂fM1 ∪Z+ e∂fM1−χ∂fM1. (1.74)The smooth closed manifolds ∂fM1 and ∂M1 are diffeomorphic.

Hencee (∂M1) = e∂fM1,χ (∂M1) = χ∂fM1,e (M2, ∂M1 ∪Z) = eM2, ∂fM1 ∪Z.Corollary 1.5 and Lemmas 1.3 and 1.5 claim that e (M1, ∂M1) = e fM1, ∂fM1. Sothe equality (1.73) follows from (1.74).□

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY331.4.3.

Proof of the generalized Ray-Singer conjecture.Theorem 1.3 (Classical Ray-Singer conjecture). For any closed Riemannianmanifold (M, gM) its analytic torsion norm is equal to the combinatorial torsion normT0(M) = τ0(M).Proof.There is a smooth Morse function f on a direct product M × I (i.e., afunction with the nondegenerate isolated critical points with different critical values)such that the following holds.

Its minimum value is equal to zero, f −1(0) = M × ∂I,and the zero is not a critical value of f. Its maximum value maxM×I f equals 1 andthe maximum value level is the only one point. Namely f −1(1) is an interior point ofM × (I, ∂I).As f −1(1 −ε) (where ε > 0 is very small) is a sphere Sn (n = dim M), thereexists a sequence of spherical Morse surgeries (given by transformations of levelsf −1(x), x ∈(0, 1 −ε) for x divided by critical values) such that their composition isa transformation of a manifold12 M ∪M = M × ∂I = f −1(0) into Sn = f −1(1 −ε).Aa a consequence of Lemma 1.6 in this case we get2e(M) = e(M ∪M) = e (Sn) .Lemma 1.5 claims that 0 = e (Sn) = e(M).

Thus, the Ray-Singer conjecture isproved.□Let (M, gM) be a compact Riemannian manifold with a smooth boundary ∂M. LetZ be a union of some connected components of ∂M and let gM be a direct productmetric near ∂M.

The following two theorems are generalizations of the Ray-Singerconjecture.Theorem 1.4. Under the conditions above, the following equality holds for a mani-fold with a smooth boundary:T0(M, Z) = 2χ(∂M)/2τ0(M, Z).(1.75)Proof.

Lemma 1.3 claims that e(M, Z) = e(M, ∂M) + e(∂M \ Z). According toTheorem 1.3, e(∂M \ Z) is equal to zero.

Hence e(M, Z) = e(M, ∂M). In the caseof ∂M ̸= ∅there is a mirror-symmetric closed Riemannian manifold P = M ∪∂M Mobtained by gluing two copies of (M, gM) along ∂M.

According to Lemma 1.4, wehavee(P) = 2e(M, ∂M) + e(∂M) −χ(∂M).12The manifold M is not supposed to be orientable.

34S.M. VISHIKTheorem 1.3 claims that e(∂M) = 0 = e(P).

Thus, we gete(M, Z) = e(M, ∂M) = 2−1χ(∂M),which is equivalent to (1.75).□Let (M, Z, gM) be as in Theorem 1.4. Let N be a codimension one in M two-sided in M closed submanifold N ⊂M \ ∂M.

Let M be obtained by gluing M1 andM2 along N. Let gM be a direct product metric near N and let the ν-transmissionboundary conditions (1.13) be given on N (where ν = (α, β) ∈R2 \ (0, 0)).Theorem 1.5. The analytic torsion norm is expressed by the combinatorial torsionnorm (in the case of the ν-transmission interior boundary condition on N) as followsT0(Mν, Z) = 2χ(∂M)/2+χ(N)τ0(Mν, Z).Proof.

The equality (1.65) claims that e(Mν, Z) = e(M, Z)+χ(N). So the assertionof the theorem follows from Theorem 1.4 and from the equality (1.65).□Remark 1.10.

This proof of the generalizations of the Ray-Singer conjecture does notuse any explicit expressions for the scalar analytic torsions of any special classes ofmanifolds. The proofs in [M¨u1], [Ch] of the classical Ray-Singer conjecture essentiallyused the explicit expressions for the scalar analytic torsions for spheres and lensspaces.

(The latter expressions were obtained by D.B. Ray in [Ra].

He computedthere the scalar analytic torsion for lens spaces and spheres with homogeneous metricsby explicit calculations of the ζ-functions for the corresponding Laplacians usingGegenbauer’s polynomials.) The proof in [M¨u1] used the precise estimates of [DP]for the eigenvalues of the corresponding combinatorial Laplacians.

The proof of [Ch]used the Lerch formula for the derivative at zero of the zeta-function of Riemann([WW], 13.21, 12.32). We don’t use this formula.

(Its new proof is obtained here. )Our proof of the generalized Ray-Singer formula is based on a gluing property forthe analytic torsion norms.

This property is proved here for a general gluing twoRiemannian manifolds by a diffeomorphism of some connected components of theirboundaries. It is proved without any computations of asymptotics of eigenvalues andeigenforms for the corresponding Laplacians.2.

Gluing formula for analytic torsion norms. Proof of Theorem 1.12.1.

Strategy of the proof. In Section 1 the generalized Ray-Singer conjecturefor a manifold with a smooth boundary is deduced from Theorem 1.1.

Namely it isdeduced from the gluing formulaϕanT0(M, Z) = T0 (M1, Z1 ∪N) ⊗T0 (M2, Z2 ∪N) ⊗T0(N),(2.1)which holds under the conditions of Theorem 1.1 (where Zk := Z ∩∂Mk).Theidentification ϕan in (2.1) is defined in (1.9) with the help of the exact sequence

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY35(1.10) of the de Rham complexes.

It is proved in Lemma 1.2 that the equality (2.1)follows from the assertion that under the conditions of Theorem 1.1, the inducedanalytic torsion norm13 ϕanν T0 (Mν) does not depend on a parameter ν of the interiorboundary conditions. The latter statement means that the equalityϕanν T0 (Mν, Z) = c0T0 (M1, Z1 ∪N) ⊗T0 (M2, Z2 ∪N) ⊗T0(N),(2.2)holds with a positive constant c0 which is independent of ν ∈R2 \ (0, 0).

(However,it is not supposed in Lemma 1.2 that c0 is independent of M, N, gM, and Z. )The strategy of the proof of the equality (2.2) is as follows.

First we prove that c0is constant on each of four connected componentsUj ⊂U := {(α, β) ∈R2: αβ ̸= 0}. (2.3)Then it is enough to prove that c0(ν) is continuous as a function of ν for ν ∈R2 \ (0, 0).

These two assertions provide us with a proof of the equality (2.2).Let ν0 ∈U and let a > 0 be a number not belonging to the spectrum S(ν0) :=Si Spec ∆i(Mν0, gM) ⊂R+ of the Laplacians on DR• (Mν, Z).This spectrum isdiscrete according to Theorem 3.1. In particular, each its eigenvalue is of a finitemultiplicity.

Let W ia(ν) be a subspace of DRi (Mν, Z), spanned by all the eigenformsωλ for ∆ν,i := ∆i(Mν, gM) with their eigenvalues λ ≤a. Then dW ia(ν) ⊂W i+1a(ν).So (W •a (ν), d) is a finite-dimensional subcomplex of (DR• (Mν, Z) , d) equiped withthe natural Hilbert structures on W •a (ν) ֒→DR• (Mν, Z) (defined by gM).Let ∥·∥2det W •a (ν) be the induced norm on det W •a (ν).For ν very close to ν0 itholds also that a /∈S(ν) (Proposition 3.1).By the definition of W •a (ν), its co-homology Hj (W •a (ν)) are canonically identified with the space of harmonic formsKer ∆j (Mν, gM).

The differential d in W •a (ν) induces the identificationdW : det W •a (ν) →∼det Ker ∆• (Mν, gM) . (2.4)According to Lemma 1.1 there is a canonical identification between the harmonicforms and the cohomology of the de Rham complex (the latter one is independent ofgM):Ker ∆i (Mν, gM) = Hi (DR (Mν, Z)) .

(2.5)So there is the induced canonical identification of the determinant lines:det Ker∆• (Mν, gM) = det H• (DR (Mν, Z) , d) . (2.6)Let ∥·∥2det H•(Mν) be a norm on det H• (Mν, Z) := det H• (DR (Mν, Z) , d) inducedby the identifications (2.5) and (2.6) from the Hilbert structure on the harmonicforms Ker ∆• (Mν, gM).

(This structure is defined by the Riemannian metric gM. )13The identification ϕanνis defined by the short exact sequence (1.14).

36S.M. VISHIKThe identification (2.4) is not an isometry of the norms ∥·∥2det H•(Mν) and ∥·∥2det W •a (ν)in general.

The norm ∥·∥2det H•(Mν) has to be multiplied by an additional factor for theidentification (2.4) to become an isometry. This factor is the scalar analytic torsion ofa complex (W •a (ν), d), defined by the general formula (1.1).

We can conclude that theanalytic torsion norm T0 (Mν, Z) on the determinant of H• (DR (Mν, Z)) is isometric(under the identifications (2.4) and (2.6)) to the normT0 (Mν, Z; a) := ∥·∥2det W •a (ν) expX(−1)jj∂sζν,j(s; a)|s=0. (2.7)The zeta-function ζν,j(s; a) is defined for Re s > (dim M)/2 by the series Pλ>0 λ−s,where the sum is over all the eigenvalues λ of ∆j ((Mν, gM)) (including their multi-plicities), such that λ > a.

This ζ-function can be continued meromorphically to thewhole complex plane C and it is regular at zero. The latter assertion follows fromTheorem 3.1 and from the equality (which is obvious for Re s > (dim M)/2):ζν,j(s; a) = ζν,j(s) −X0<λ≤aλ−s.

(2.8)(The series for ζν,j(s), Re s > (dim M)/2, is the sum over all the nonzero eigenvaluesof ∆j (Mν, gM) with their multiplicities, where λ−s := exp(−s log λ) and log λ ∈Rfor λ > 0).The identifications dW (2.4) and ϕνan (the latter one is defined with the help of(1.14)) provide us (under the conditions of Lemma 1.2) with the identification:ϕanν (a): det W •a (ν) →∼Det(M, N, Z)(2.9)(Det(M, N, Z)14 is defined in (1.9)). The assertion that c0(ν) is independent of νon each connected component Uj of U (2.3) is equivalent to the following one.

Theanalytic torsion norm T0 (Mν, Z; a) is transformed (under the identification (2.9))into the norm on Det(M, N, Z):ϕanν (a)◦T0(Mν, Z; a)=c0(ν)T0(M1, Z1 ∪N)⊗T0(M2, Z2 ∪N)⊗T0(N),(2.10)where c0(ν) is constant on each connected component Uj.The action of ϕνan(a) is as follows (by its definition):ϕνan(a)T0 (Mν, Z; a) = T (Mν, Z; a) ϕνan(a) ◦∥·∥2det W •a (ν) ,where the scalar analytic torsion T (Mν, Z; a) is defined as the scalar factor in (2.7):T (Mν, Z; a) := expX(−1)jj∂sζν,j(s; a)|s=0. (2.11)Let ν(γ), γ ∈(ε, ε) ⊂R, be a smooth curve in U (2.3) and let ν(0) = ν0.

LetΠj(ν0; a) be an ortogonal projection operator from (DRj(M))2 onto W ja(ν0) (relative14To remind, Z is the union of the connected components of ∂M, where the Dirichlet boundaryconditions are given. The Neumann boundary conditions are given on ∂M \ Z.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY37to the natural Hilbert structure (1.23) in (DRj(M))2).

Let p1 be a linear operatorin (DR•(M))2, mapping (ω1, ω2) ∈(DR•(M))2 to (ω1, 0). (Respectively p2 maps(ω1, ω2) to (0, ω2).

)Let ν and ν0 be arbitrary points from U. Then the following isomorphism of thede Rham complexes is defined (where kν := α/β for ν = (α, β) ∈U):vν =vνν0 : DR•(Mν0, Z) →∼DR•(Mν, Z) , vν(ω1, ω2):=(ω1,(kν/kν0)·ω2) .

(2.12)Thus the induced isomorphism is defined:vν∗: H• (DR (Mν0, Z)) →H• (DR (Mν, Z)) .Let a be a positive number from R+\S(ν0). Then for ν very close to ν0 the numbera is also from R+ \ S(ν) (Proposition 3.1).

The complexes W •a (ν) and W •a (ν0) areisomorphic as abstract finite-dimensional complexes (and (2.12) provides us with anatural but not canonical isomorphism of these complexes). We have to computethe action of ϕνan on the norms ∥·∥2det W •a (ν) for ν very close to ν0.

However ∥·∥2W •a (ν)are the norms on different complexes. So it is necessary to define some isomorphismbetween W •a (ν0) and W •a (ν) and then to compute its action on ∥·∥2det W •a (ν0) and onthe space Det(M, N, Z).

The choice (2.13) of such an identification is done below.For ν very close to ν0 the subspaces W •a (ν) and W •a (ν0) are very close in the L2-completion (DR•(M))2 of DR• (Mν, Z) =: DR•(ν), according to Proposition 3.1. Sothe following isomorphism of these finite-dimensional complexes is well-defined:gν = Π•(ν; a) · vνν0 · jν0 :W •a (ν0) ֒→(DR•(ν0), d)vνν0−−→g(DR•(ν), d) −−−−→Π•(ν;a)(W •a (ν), d) ,(2.13)where jν0 is the natural inclusion of W •a (ν0) and Π•(ν; a) is the orthogonal projectionoperator onto W •a (ν).

Its action on the norm ∥·∥2det W •a (ν0) is computed by the followinglemma.Lemma 2.1. Let l be an arbitrary nonzero element of det W •a (ν0).

Then the equalityholds for any smooth variation ν(γ) of ν0 = ν(0):∂γ log ∥gνl∥2det W •a (ν)γ=0 = −2 ∂γ log(kν)γ=0X(−1)j Trp2Πj(ν0; a). (2.14)In (2.14) the rank (i.e., the dimension of the image) of the operator p2Πj(ν0; a) isless or equal to dim W •a (ν0).

This operator acts in (DR•(M))2.Then the following lemma provides us with the variation formula for T (Mν, Z; a).

38S.M. VISHIKLemma 2.2.

For γ = 0 the equality holds:∂γ log T (Mν, Z; a) = 2 ∂γ log(kν)γ=0X(−1)jb1,j (Mν0, Z; a) ,where kν := α/β for ν = (α, β) ∈U. Here b1,j(Mν0, Z; a) is a constant coefficient(i.e., t0-coefficient q0) in the asymptotic expansion as t →+0 of the trace of theoperator below (acting in (DRj(M))2):Trp1nexp (−t∆ν0,j)1 −Πj(ν0; a)o∼q−nt−n/2 + q−n+1t−(n−1)/2 + .

. .

+ q0t0 + . .

. (2.15)Remark 2.1.

The operators exp (−t∆ν0,j) and Πj(ν0; a) acting in the L2-completion(DRj(M))2 of DRj (Mν0, Z) (which coincides with the L2-completion of DRj(M))have their images in the domain of definition of the Laplacian D(∆ν0,j)⊂DRj(Mν0, Z).The existence of the asymptotic expansion (2.15) follows from Theorem 3.2. The co-efficients qm with m ≤−1 in (2.15) are independent of a.

The coefficients ˜qm of theasymptotic expansion for Trp1 exp(−t∆ν0,j)are equal to the sums of the integralsover M1 and over ∂M1 ⊃N of the locally defined densities on M1 and on ∂M1 (byTheorem 3.2). However, in the general case we cannot represent Tr (p1Πj(ν0; a)) asan integral of a locally defined density (because there are no universal local formu-las for the eigenforms ωλ of ∆ν0,j).

Hence there is no universal local formula for acoefficient q0 in (2.15) but there are such formulas for qm = ˜qm with m < 0.Corollary 2.1. For an arbitrary nonzero l ∈det W •a (ν0) the equality holds∂γlog ∥gνl∥2T0(Mν,Z)=2∂γ(log kν)γ=0X(−1)jb1,j(Mν0, Z)−dim W ja, (2.16)where b1,j (Mν0, Z) is a constant coefficient (i.e., the t0-coefficient) of the asymptoticexpansion of Tr (p1 exp (−t∆ν0,j)) relative to t →+0 and p1 exp (−t∆ν0,j) is the op-erator acting in (DRj(M))2.Remark 2.2.

Note that in the right side of (2.16) there are the Euler characteristicχ (Mν, Z) :=P(−1)j dim W ja and the alternating sum of the integrals b1,j (Mν0, Z)(over M1 and over ∂M1) of the locally defined densities (Remark 2.1). (Here Z is theunion of the connected components of ∂M, where the Dirichlet boundary conditionsare given).

The number χ (Mν, Z) is also equal to the sum of the integrals over M,N, and over ∂M of the locally defined densities.Let ν(γ) be a smooth variation of a point ν0 ∈U (2.3). Let l(γ) ∈det W •a (ν(γ))be a variation of an arbitrary nonzero element l ∈det W •a (ν0) such that ϕanν(γ)(a)◦l(γ)is a fixed (nonzero) element of Det(M, N, Z).

Then the equality (2.10) (where thefactor c0(ν) is constant on each connected component of U (2.3)) is equivalent to the

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY39assertion that for any such a variation l(γ) its analytic torsion norm is independentof γ:∂γ ∥l(γ)∥2T0(Mν(γ),Z;a)γ=0 = 0.

(2.17)Corollary 2.1 provides us with the formula (2.16) for a variation of the analytictorsion norm ∥gν(γ)l∥2T0(Mν(γ),Z;a) (where l ∈det W •a (ν0)).The assertion (2.17) isequivalent to the following identity:∂γ loggν(γ)l2T0(Mν(γ),Z;a)γ=0 = ∂γ log ∥gν∗f∥2 γ=0,(2.18)where f is an arbitrary nonzero element of Det(M, N, Z) (for instance, f = ϕanν0 (a)◦l)and gν∗= ϕanν (a) ◦gν ◦ϕanν0 (a)−1 is defined by the following commutative diagram,where ν ∈U is very close to ν0:Det(M, N, Z)−−−→gν∗Det(M, N, Z)ϕanν0 (a)x≀≀≀≀xϕanν (a)det W •a (ν0)−−−−→gν=gνν0det W •a (ν)The norm on the right in the equality (2.18) is an arbitrary Hilbert norm in one-dimensional space Det(M, N, Z). The value of the expression on the right in (2.18)is independent of such a norm.The action of the isomorphism gν = gνν0 : W •a (ν0) →W •a (ν) on Det(M, N, Z) isdescribed by the following lemma.Lemma 2.3.

For an arbitrary element f ∈Det(M, N, Z) the equality holds:∂γ log ∥gν∗f∥2 γ=0 = −2∂γ (log kν)γ=0X(−1)jb2,j (Mν0, Z) ,(2.19)where b2,j (Mν0, Z) is the constant coefficient (i.e., the t0-coefficient) in the asymptoticexpansion (relative to t →+0) for the trace of the operator p2 exp (−t∆ν0,j) actingin (DRj(M))2.Here p2 is the operator p2: (ω1, ω2) →(0, ω2) for ωk ∈DRM k2.Remark 2.3. Note that Tr exp (−t∆ν0,j) = Pj Tr (pj exp (−t∆ν0,j)).

So we have−X(−1)jb2,j (Mν0, Z) =X(−1)jb1,j (Mν0, Z) −χ (Mν0, Z) .Hence the equality (2.18) follows from (2.16) and (2.19).Thus Lemmas 2.1–2.3 provide us with a proof of the assertion that the factor c0(ν)is independent of ν on each connected component Uj of U (2.3).

40S.M. VISHIK2.2.

Continuity of the analytic torsion norms. To prove that c0(ν) is inde-pendent of ν ∈R2 \ (0, 0), it is enough15 to show that the norm ϕanν ◦T0(Mν) onDet(M, N, Z) is continuous in ν ∈R2 \(0, 0).

The following norms on Det(M, N, Z)are the same for an arbitrary a ≥0:ϕanν (a) ◦T0 (Mν, Z; a) = ϕanν ◦T0 (Mν, Z) . (2.20)Let us prove the continuity of ϕanν T0 (Mν, Z) as a function of ν at a point ν0 ∈R2\(0, 0).

(The series of lemmas above provides us with the proof of this assertion inthe case when ν0 ∈U (2.3). But now this will be proved at an arbitrary ν0 ∈R2\(0, 0),for instance, at ν0 ∈R2\(U ∪(0, 0)).) By (2.20), it is enough to obtain the continuityin ν at ν = ν0 of the norm ϕanν (a) ◦T0(Mν; a) on Det(M, N, Z) for a fixed a > 0such that a /∈S(ν0) := ∪j Spec (∆ν0,j).

Since a /∈S(ν0), we see that a /∈S(ν) for νvery close to ν0. (The latter assertion follows from Proposition 3.1.

It claims thatthe resolvents G•λ(ν) := (∆•ν −λ)−1 for λ /∈Spec(∆ν,•) form a smooth in (λ, ν) familyof bounded operators in (DR•(M))2 and that Spec (∆ν,•) is discrete. As G•a(ν0) isbounded in (DR•(M))2, the operator G•a(ν) is also bounded for ν, close to ν0, andso a /∈Spec(∆ν,•) for such ν.) The assertion below claims that the truncated scalaranalytic torsion (2.11) is a locally continuous function.16Proposition 2.1.

The scalar analytic torsion T (Mν, Z; a) is continuous in ν at ν0.Thus, the continuity of ϕanν T0 (Mν, Z) (as a function of ν) at ν0 is equivalent to thecondition that the norm on Det(M, N, Z)ϕanν (a) ◦∥·∥2det W •a (ν)(2.21)is continuous in ν at ν0. The continuity of the norm (2.21) is deduced from thefollowing finite-dimensional algebraic lemma.

Letf : (A•, dA) →(V •, dV )(2.22)be a quasi-isomorphism of finite complexes of finite-dimensional Hilbert spaces. Letf∗: det H•(A) →∼det H•(W) be the induced identification of the determinant lines.Let T0(A•) and T0(V •) be the analytic torsion norms (1.2) on the determinant linesidentified by f∗: det H•(A) = det H•(V ).

Let (Cone• f, d), Conejf = Aj−1 ⊕V j, bea simple complex, associated with the bicomplex (2.22):dCone: Conej →Conej+1,dCone(x, y)=(−dAx, fx + dV y)(2.23)15The factor c0(ν) is constant on each connected component Uj of U (2.3), and U is dense inR2 \ (0, 0).16This truncated scalar analytic torsion is a continuous function on the set of ν ∈R2 \ (0, 0) suchthat a /∈∪i Spec (∆ν,i).

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY41(for (x, y) ∈Aj+1⊕V j).

Then Cone•f is an acyclic finite complex of finite-dimensionalHilbert spaces (Conejf is the direct sum of Hilbert spaces Aj+1 and V j), H•(Conef)=0. Hence det H•(Cone f) is canonically identified with C and the analytic torsionnorm for Cone• f is a norm on C .

The ratio T0(V )/T0(A) ∈R+ is defined as theratio between the two norms on the one-dimensional spaces det H•(V ) and det H•(A)identified by f∗.Lemma 2.4. Under the conditions above, the equality holds:∥1∥2T0(Cone• f) = T0(V •)/T0(A•),(2.24)where the left side is the analytic torsion norm of 1 ∈C = det H•(Cone f).Let a > 0 be a number from R+ \ S(ν0).

Then there exists an open neighborhoodUν0(a) of ν0 ∈Uν0(a) ⊂R2\(0, 0) such that a /∈S(ν) for ν ∈Uν0(a) (Proposition 3.1).The family of complexes (W •a (ν), d) of Hilbert spaces is continuous on Uν0(a) in thefollowing sense.The operator Πja(ν) := Πj(ν; a) is a finite rank projection operator in (DRj(M))2with its image W ja(ν):Πja(ν):DRj(M)2 →W ja(ν) ⊂DRj (Mν, Z) ⊂DRj(M)2 .Proposition 2.2. The family of operators Π•a(ν) is continuous in ν for ν ∈Uν0(a)with respect to the operator norm in (DR•(M))2.

The same is true for the familiesdΠ•a(ν): (DR•(M))2 →DR•+1(Mν) ⊂DR•+1(M)2 ,δΠ•a(ν): (DR•(M))2 →DR•−1(Mν) ⊂DR•−1(M)2 .These are the families of finite rank operators.Proof. It follows from Proposition 3.1 that if a /∈S(ν0) then there exists an ε > 0such that (a −ε, a + ε) ∩S(ν) = ∅for ν sufficiently close to ν0.

Hence {λ: a −ε <|λ| < a + ε} ∩S(ν) = ∅for such ν (since S(ν) ⊂R+ ∪0 by Theorem 3.1). Thus,according to Proposition 3.1, the operatorsΠ•a(ν) = i2πZΓa G•λ(ν)dλform a smooth in ν (for such ν) family of finite rank operators in (DR•(M))2 (wherethe circle Γ = {λ: |λ| = a} is oriented opposite to the clockwise).

The operatorsdΠ•a(ν): (DR•(M))2 →(DR•+1(M))2 form (for such ν) a smooth in ν family offinite rank operators (according to Proposition 3.1.□

42S.M. VISHIKCorollary 2.2.

For ν sufficiently close to ν0 the family of operators Π•a(ν) identifiesthe graded linear spaces W •a (ν0) := Im Π•a(ν0) and W •a (ν). Such an identificationnearly commutes with d in the following sense:dΠ•a(ν)w −Π•+1a(ν)dw2 ≤c(ν, ν0)∥w∥2(2.25)(for any w ∈W •a (ν0)), where c(ν, ν0) →+0 as ν →ν0.

This identification also nearlycommutes with δ:δΠ•a(ν)w −Π•−1a(ν)δw2 ≤c(ν, ν0) ∥w∥2for w ∈W •a (ν0) (∥·∥2 is the L2-norm in (DR•(M))2).The estimate (2.25) follows from the continuity (in ν) of the families dΠ•a(ν) andΠ•+1a(ν) since the following operator norms tend to zero as ν →ν0:∥dΠ•a(ν) −dΠ•a(ν0)∥2 →+0,Π•+1a(ν) −Π•+1a(ν0)2 →+0.Indeed, for an arbitrary w ∈W •a (ν0) we have dΠ•aw = dw. Hence the estimatesΠ•+1a(ν)dw −dw2 ≤Π•+1a(ν) −Π•+1a(ν0)2 · ∥dw∥2 ≤≤C ·Π•+1a(ν) −Π•+1a(ν0)2 · ∥w∥2are true because the differential d: W •a (ν0) →W •+1a(ν0) of a finite complex of finite-dimensional spaces is bounded (with respect to the Hilbert norm induced from(DR•(M))2).For each ν ∈R2 \ (0, 0) the combinatorial cochain complex (C• (Xν, V ) , d) (withV := X ∩Z) is defined by the ν-transmission condition (1.58).

A homomorphism ofthe integration of forms from W •a (ν) over the simplexes of XRν(a): (W •a (ν), d) →(C•(Xν, V ), d)(2.26)is also defined for all ν ∈R2 \ (0, 0). For every such ν the following variant of the deRham theorem holds.Proposition 2.3.

Rν(a) is a quasi-isomorphism.Proof. 1.

Let Rν : (DR•(Mν, Z), d) →(C•(Xν, V ), d) be the integration homomor-phism of pairs of forms (ω1, ω2) ∈DR•(Mν, Z) over the simplexes of Xj \ Vj. ThenRν is a quasi-isomorphism.1717This assertion claims that the analogy of the classical de Rham theorem is true in the case ofthe ν-transmission interior boundary conditions.

The classical de Rham theorem for smooth closedmanifolds was proved in [dR1] (see also [dR4], Ch. IV, [W], Ch.

IV, § 29). The explicit isomorphismbetween the ˇCech cohomology for a good cover of a smooth closed M and the de Rham cohomologyof M is defined with the help of the de Rham-ˇCech complex ([BT], Ch.

II, § 9).

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY43Indeed, in the commutative diagram (1.59) the left and the right vertical arrowsare quasi-isomorphisms according to the de Rham theorem for a closed manifold Nand for manifolds M1 and M2 with smooth boundaries.

(The proof of the latter oneis given in [RS], Proposition 4.2.) The cohomology exact sequences provide us withthe commutative diagram∂D−→H∗(⊕k=1,2DR• (Mk, N ∪Zk)) →H∗(DR• (Mν, Z)) −−→(rν)∗H∗(DR•(N)) →↓R∗↓(Rν)∗↓R∗∂c−→H∗(⊕k=1,2C• (Xk, W ∪Vk))−→j∗H∗(C• (Xν, V ))(rν,c)∗−−−→H∗(C•(W))→(2.27)with the exact rows, where the vertical arrows R∗on the left and on the right areisomorphisms (according to the de Rham theorem) and where ∂D = ∂c under theidentifications R∗.

Hence (Rν)∗is also an isomorphism.The exactness of the top row in (2.27) can be interpreted and proved as follows.The sheaf F •ν := DR•ν (ν = (α, β) ∈R2\(0, 0)) of germs (ω1, ω2) of pairs of C∞-formsωj on Mj such that18 αi∗1ω1 = βi∗2ω2 (here i∗j are the geometrical restrictions fromMj to N ֒→∂Mj) is a c-soft sheaf. (The latter notion means that the restrictionΓ (M, F jν ) →ΓK, i−1K F jνis surjective for any compact iK : K ֒→M, [KS], Defini-tion 2.5.5.) The sheaf Fν is c-soft since appropriate smooth partitions of unity existon M. The sequence of complexes of global sections0 →Γc (M \ N, F •ν ) →Γ (M, F •ν ) →ΓM, iN,∗i−1N F •ν→0(here iN : N ֒→M) has the terms which possess the following properties:1) Γ (M, F •ν ) = DR• (Mν, Z),2) Γc (M \ N, F •ν ) is a subcomplex of ⊕k=1,2DR• (Mk, N ∪Zk) and its natural inclu-sion is a quasi-isomorphism.

Indeed, if ω ∈DR• (Mk, N ∪Zk) is a closed form thenω = dv in a neighborhood of N in Mk (where v is a smooth form with the zero geo-metrical restriction to N). So ω −d(ϕv) = 0 in some neighborhood of N in Mk (ϕ isan appropriate cutting function).

We have Γc (M \ N, F •ν ) = Γ (M, j!j−1F •ν ), wherej : M \ N ֒→M and j! is the direct image with proper supports, j−1F •ν ≃DR•|M\N.The sheaf j!j−1F •ν is c-soft according to [KS], Proposition 2.5.7.3) ΓM, iN,∗i−1N F •νhas a natural homomorphism qν := rν ◦(i∗1, i∗2) onto DR•(N)(where rν is defined in (1.15)) and qν is a quasi-isomorphism.

In fact, if the formu = dt ∧ωN(t) on I × N is closed then it is exact, because then dNωN(t) = 0 andso u = dR t0 ωN(τ)dτ. (Here t is the coordinate on I and t = 0 is the equation of18It is supposed that ωj has the zero geometrical restriction to Zk (at the points x ∈Zk ⊂∂Mk).

44S.M. VISHIKN = 0 × N ֒→I × N, 0 ∈I \ ∂I.) Hence qν is a quasi-isomorphism.

(This asser-tion follows also from the Poincar´e lemma.) The sheaf iN,∗i−1N F •ν is c-soft by [KS],Proposition 2.5.7.For a compact manifold M the category of c-soft sheaves on M is injective withrespect to the functor of global sections Γ(M; ) ([KS], Proposition 2.5.10).Thecomplex F •ν is a c-coft resolvent of a constructible sheaf ([KS], Chapter VIII) Cν onM, which is isomorphic to CM\N on M \ N and to CN on N (where CX is a constantsheaf on X), and the gluing map for Cν is |ν|−1/2 (α, β): CN →i−1N j∗CM\N = CN⊕CN(i.e., c →|ν|−1 (βc, αc)).

The complexes j!j−1F •ν and iN,∗i−1N F •ν are c-soft resolventsof constructible sheaves j!j−1Cν = j!CM\N and of iN,∗i−1N Cν. (The latter one isisomorphic to iN,∗CN under rν.) So the exactness of the cohomology sequence in thetop row of (2.27) follows from [KS], (2.6.33), Remark 2.6.10.2.The projection operator pH : DR• (Mν, Z) ֒→(DR•(M))2 →Ker (∆•ν) pro-vides us with the isomorphism pH∗: H• (DR (Mν, Z)) →Ker (∆•ν) (by Lemma 1.1).So the inclusion ia : (W •a (ν), d) ֒→(DR• (Mν, Z) , d) is a quasi-isomorphism and(ia)∗: Ker (∆•ν) →∼H• (DR (Mν, Z)) is equal to (pH∗)−1 (since pHia = id on Ker (∆•ν)).From an obvious equality Rν(a) = Rνia it follows that Rν(a) is a quasi-isomorphism.□Thus the assertion of Lemma 2.4 can be applied to the bicomplex (2.26).

Theresult is as follows.Corollary 2.3. The equality holds:T0(C•(Xν, V ))/ ∥1∥2T0(Cone• Rν(a)) = T0 (W •a ) .

(2.28)The identifications ϕanν (a) (for an arbitrary a > 0) and ϕanνare defined such thatthe following norms on Det(M, N, Z) are equal:ϕanν (a) ◦∥·∥2det W •a (ν) = ϕanν ◦T0(Wa). (2.29)Hence, as it follows from (2.28), (2.29), we haveϕanν (a) ◦∥· ∥2det W •a (ν) = (ϕanν ◦T0 (C• (Xν, V ))) ◦∥1∥2−1T0(Cone•(Rν(a))).

(2.30)Proposition 2.4. The factor∥1∥2T0(Cone•(Rν(a)))−1 in ( 2.30) is a continuous func-tion of ν ∈Uν0(a).Proof.

The complex Cone• (Rν(a)) is acyclic according to Proposition 2.3. Its scalaranalytic torsion∥1∥2T0(Cone•(Rν(a))) := expXj≥−1(−1)jjζ′j(0)(2.31)

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY45is defined as in (1.1) by the ζ-functions of the “Laplacians” Lν := d∗νdν + dνd∗ν ofthe complex (Cone• (Rν(a)) , dν).19 Since the complex Cone• (Rν(a)) is acyclic we seethat these Laplacians are positive definite.

(So they have the zero kernels.) Theirdeterminants det(∆•ν) are continuous positive functions of ν on Uν0(a) (and so theexpression on the right in (2.31) is a continuous function of ν ∈Uν0(a)).

The latterstatement is derived as follows.Proposition 2.5. Let m ∈Z+ and m ≥m0 := 1 + min{k ∈Z+ : 4k ≥dim M}.Then there exists a positive constant C = C(M, N, Z, gM) independent of ν ∈R2 \(0, 0) (and of m also) such that the following estimate holds uniformly with respectto x ∈M 1 ∪M 2:|ω(x)|2 < CmXi=0∆iνω22(2.32)for all ω such that20ω ∈DR•(Mν, Z),ω ∈D(∆•ν),∆νω ∈D(∆•ν), .

. .

, ∆mν ω ∈D(∆•ν). (2.33)(Here |ω(x)|2 is the norm at ∧•TxM defined by gM and ∥·∥22 is the L2-norm in(DR•(M))2.

)Corollary 2.4. If w ∈W •a (ν) then w ∈D(∆mν ) for an arbitrary m ∈Z+.

So thefollowing estimate holds uniformly with respect to x ∈M 1 ∪M 2 and to ν ∈R2 \(0, 0)|w(x)|2 < C1 ∥w∥22 ,(2.34)where C1 = C1 (M, N, Z, gM) > 0.The graded Hilbert space C•(Xν, V ) is isomorphic to the direct sumC•(Xν, V ) = C• (X, NX ∪V ) ⊕C•(NX),(2.35)where V := X ∩Z, NX := X ∩N, C• (X, NX ∩V ) is a graded linear subspace ofC• (Xν, V ) (with respect to the natural inclusion), and the inclusion jν : C•(NX) ֒→C• (Xν, V ) ⊂⊕iC•(Xi) is defined as jν := (α2 + β2)−1 (β id, α id). The space onthe right in (2.35) is independent of ν.

Hence (2.35) provides us with the isometricidentification of the graded Hilbert spacespν : C•(Xν0, V ) →∼C•(Xν, V ). (2.36)19The spaces W •a (ν) are equiped with the Hilbert structure from ((DR•(M))2 , gM).

The spacesC• (Xν, V ) ⊂C•(X1)⊕C•(X2) are equiped with the Hilbert structure defined by the basic cochainsin ⊕C•(Xk) and Cone•(Rν) = W •−1a(ν)⊕C• (Xν, V ) is the orthogonal direct sum of Hilbert spaces.20The domain of definition of D(∆•ν) for ∆•ν is defined by (1.27) and (1.26).

46S.M. VISHIKCorollary 2.5.

Let ν ∈Uν0(a) be sufficiently close to ν0.Let W •a (ν) be identi-fied with W •a (ν0) byΠ•a(ν): W •a (ν0) →W •a (ν). Let C•(Xν0, V ) be identified withC•(Xν, V ) by pν ( 2.36).

Then the estimate ( 2.34) involves that for such ν the familyof homomorphisms of the integration over the simplexes of XR•ν(a): (W •a (ν), d) →(C•(Xν, V ), dc)is a continuous in ν family of quasi-isomorphisms between finite complexes of finite-dimensional Hilbert spaces.Let fν : (F •(ν), dF(ν)) →(K•(ν), dK(ν)) be a family of homomorphisms betweenfinite complexes of finite-dimensional Hilbert spaces. Let the trivialization of thesetwo families of complexes be defined by the identifications of the graded linear spacesΠν : F •(ν0) →F •(ν),pν : K•(ν0) →K•(ν).Let these idetifications be chosen such that fν becomes a continuous family of thehomomorphismsfν : (F •, dF(ν)) →(K•, dK(ν))between the continuous families of complexes with the fixed underlying graded linearspaces F • := F •(ν0) and K• := K•(ν0).

Let the Hilbert structures on F j and Kjare continuous functions of ν for all j. In this case, fν is called a continuous family.Then the following assertion is true.Proposition 2.6.

Let fν be a continuous family. Then the determinants det(L•ν) ofthe Laplacians L•ν = d∗νdν + dνd∗ν on (Cone• fν, dν) are continuous functions of ν.Proof.

The operator d∗ν adjoint to the differential dν of Cone• fν (relative to theHilbert structure on Cone• fν = F •−1⊕K•)21 is defined on the whole finite-dimensionalspace Cone• fν. Since dν tends to dν0 (for instance, in the operator norm22) as ν →ν0we see that d∗ν also tends to d∗ν0.

Thus L•ν →L•ν0 as ν →ν0 and det L•ν →det L•ν0(since the space Cone• fν is finite-dimensional).□Corollary 2.6. The functions det (L•ν) of ν for fν = R•ν(a) are continuous and pos-itive.The positivity of det (L•ν) is equivalent to the acyclicity of (Cone• Rν(a), dν) (wheredν := dCone(Rν(a))).Proposition 2.4 is proved.□21Cone• f is the direct sum of Hilbert spaces F •+1 ⊕K• (with the Hilbert structures on F •+1and Kj depending continuously on ν).22As Cone• fν is a finite-dimensional space, the weak convergence of the operators acting in it isequivalent to the convergence with respect to the operator norm.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY47Remark 2.4.

Propositions 2.2, 2.5, and Corollary 2.5 claim that under the identifica-tions (2.36) and Π•a(ν), the Hilbert structures on det Cone• (Rν(a)) and the differen-tials dν in Cone• (Rν(a)) are continuous in ν at ν0. Hence the analytic torsion norms∥·∥2T0(Cone•(Rν(a))) on C = det H• (Cone (Rν(a))) = det 0 are also continuous in ν atν0.According to (1.61) we have ϕanν= ϕcν, where ϕcν is defined by the bottom row ofthe commutative diagram (1.59).

So the continuity of the norm ϕanν (a) ◦∥·∥2det W •a (ν)on Det(M, N, Z) can be deduced from (2.30) and from the following lemma.Lemma 2.5. The norm ϕcνT0 (C (Xν, V )) on Det(M, N, Z) does not depend on ν ∈Uν0(a).The continuity in ν of the norm ϕanν T0 (Mν, Z) on Det(M, N, Z) follows from (2.20),(2.21), and from the continuity of the norm ϕanν (a)◦∥·∥2det W •a (ν).

(The latter assertionis proved above.) The equality (1.12) holds with c0(ν) which is constant and posi-tive on each connected component Uj of U (2.3).

Because the norm ϕanν T0(Mν) onDet(M, N, Z) is continuous in ν ∈R2 \ (0, 0), the equality (1.12) holds for all such νwith c0 independent of ν. Theorem 1.1 follows from (1.12) and from the assertion ofLemma 1.2.□Remark 2.5. It is not important for the proofs of Theorem 1.1 and of (1.12) that thefamily of finite-dimensional complexes (C• (Xν, V ) , dc) in (2.30) is of a combinatorialnature.

It is enough for the proof to have a family of finite-dimensional complexes(F •ν , dF) which are defined locally in ν (i.e., for ν in a neighborhood of an arbitraryν0 ∈R2 \ (0, 0)) together with the data as follows. Continuous families of quasi-isomorphisms fν(a): (W •a (ν), d) →(F •ν , dF) and of Hilbert structures hν on F •ν aredefined.

A family (F •ν , hν) may depend on a and on ν0 but it has to possess theproperty as follows.The norm ϕanν◦(fν(a)∗)−1 ◦T0(F •ν , hν) on Det(M, N, Z) iscontinuous in ν at ν0. (Here fν(a)∗: det H•(Mν, Z) →∼det H•(Fν, dF) is the inducedidentification.

)Proof of Lemma 2.5. Let ψν be the identification of the determinant lines definedby the bottom row of the commutative diagram (1.59):ψν : det C•(Xν, V ) →∼(⊗k=1,2 det C•(Xk, W ∪Vk)) ⊗det C•(W) =: DetC•(X, V, W)(where V is the induced smooth triangulation Z ∩X of Z ⊂∂M, Vk := V ∩∂Mk

48S.M. VISHIKand W := X ∩N = NX).

The following diagram is commutative:det C•(Xν, V )ψν−−−→gDetC•(X, V, W)dcy ≀dcy ≀det H•(Xν, V )ϕcν−−−→gDet(X, V, W)∥R∥Rdet H•(Mν, Z)ϕanν−−−→gDet(M, Z, N)(2.37)(The determinant lines on the right in (2.37) are defined by (1.54). The identificationdc on the right in (2.37) is a triple tensor product of the identifications induced by dcon C•(Xk, W ∪Vk) and on C•(W).

The identification R is defined by the integrationover the simplexes of X.) The commutativity of the diagram (2.37) is equivalentto the definition (1.60) of ϕcν.

Since the identification dc on the right in (2.37) isindependent of ν we see that the statement of Lemma 2.5 is a consequence of thefollowing proposition.Proposition 2.7. The identification ψν in ( 2.37) is an isometry between the combi-natorial norm ∥·∥2det C•(Xν,V ) and the triple tensor product of the combinatorial normson det C• (Xk, W ∪Vk) (k = 1, 2) and on det C•(W).

(The Hilbert structures on ⊕k=1,2C• (Xk, Vk) and on C•(W) are defined by the or-thonormal basis of the basic cochains.)Proof. Let ρν,c : C•(W) →Cj(Xν, V ) be defined by (1.63).

Then rν,cρν,c = id andρν,c is an isometry onto Im(ρν,c) (relative to the Hilbert structures, defined above).The subspace Im(ρν,c) is the orthogonal complement to Im j (⊕k=1,2C• (Xk, W ∪Vk))in Cj (Xν, V ) and j is an isometry onto Im j. (Here, rν,c and j are the same as in thebottom row of (1.59)).23.Thus Lemma 2.5 is proved.□2.2.1.

Uniform Sobolev inequalities for ν-transmission interior boundary conditions.Proof of Proposition 2.5. Let I × N ⊂M (where I = [−1, 1]) be a neighborhood ofN = 0 × N ⊂M and let gM be a direct product metric on I × N. Proposition 2.5 isa consequence of the assertions as follows.Proposition 2.8.

The inequality ( 2.32) holds uniformly with respect to ν ∈R2\(0, 0)for all ω ∈DR•(Mν) of the class ( 2.33) and such that supp ω ⊂[−4/5, 4/5]×N ⊂M.23This proposition is essentially equivalent to Proposition 1.5, Section 1.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY49Proposition 2.9.

The inequality ( 2.32) holds for all ω ∈DR•(M, Z) such thatsupp ω ⊂M \ ([−1/3, 1/3] × N) and such that24ω ∈D (∆M,Z) , ∆ω ∈D (∆M,Z) , . .

. , ∆mω ∈D (∆M,Z) .

(2.38)The last assertion is well known ([Ch], Section 5).Let f be a smooth function on M, 0 ≤f ≤1, f ≡1 on [−1/2, 1/2] × N and f ≡0on M \ ([−3/4, 3/4] × N). The 2m-Sobolev norm on the right in (2.32), defined as∥ω∥2(2m) :=mXk=0∆kνω22 ,is equivalent uniformly in ν ∈R2 \ (0, 0) (i.e., with constants c3, c4 > 0 independentof ν and of ω) to the norm25: ∥ω∥2(2m),f := Pmk=0∆kν(fω)22 +∆k ((1 −f)ω)22:c3 ∥ω∥2(2m) < ∥ω∥2(2m),f ≤c4 ∥ω∥2(2m) .

(2.39)It is enough to verify the upper estimate (with c4 independent of ν) for fω. It istrue for m = 1, since the estimate holds:∥∆ν(fω)∥22 ≤C1 ·∥∆νω∥22 + ∥ω∥22 + ∥dω∥22 + ∥δω∥22≤3/2C1∥∆νω∥22 + ∥ω∥22,where C1 depends on f but it is independent of ν.

Hence the following estimate holdsfor ω ∈D∆kν(with C2 independent of ν and of ω):∆kν(fω)22 ≤C2 ·∆kνω22 +∆k−1νω22 + . .

. + ∥ω∥22.The upper estimate is done.Thus Proposition 2.5 follows from (2.39) and fromPropositions 2.8 and 2.9.Proof of Proposition 2.8.The form ω on I × N is the sum ω0 + ω1, whereωi is an i-form in the direction of I (where I = [−1, 1]).It is enough to provethe inequality (2.32) separately for ω0 and for ω1.Let us prove it for ω0.Forν = (α, β) ∈R2 \ (0, 0) the Green function G(ν) for the Laplacian ∆ν,I on functions24For ω with supp ω ⊂M \ N the conditions (2.33) and (2.38) are equivalent.

The domainD (∆M,Z) of ∆M,Z consists of smooth forms on M with the Dirichlet boundary conditions on Zand the Neumann ones on ∂M \ Z.25The lower estimate with c3 in (2.39) is obvious.Note that supp ((1 −f)ω)⊂M \([−1/3, 1/3] × N).Then the upper estimate with a constant c′4 for ∥(1 −f)ω∥(2m) by ∥ω∥(2m)is well known ([H¨o], Appendix B and Proposition 20.1.11).

50S.M. VISHIKon I with the ν-transmission boundary condition at 0 ∈I and the Dirichlet boundaryconditions on ∂I = {−1, 1} is given by the kernel(GI(ν))x1,x2 = gx1,x2 + β2 −α2α2 + β2 g−x1,x2 for x1, x2 ∈Q1 = [−1, 0],(GI(ν))x1,x2 = gx1,x2 + α2 −β2α2 + β2 g−x1,x2 for x1, x2 ∈Q2 = [0, 1],(2.40)(GI(ν))x1,x2 =2αβα2 + β2 gx1,x2for x1, x2 from different Qk.Here, gx1,x2 is the Green function for the Laplacian on functions on I with the Dirichletboundary conditions on ∂I:gx1,x2 =c · (x2 + 1)(1 −x1),−1 ≤x2 ≤x1 ≤1,c · (x1 + 1)(1 −x2),−1 ≤x1 ≤x2 ≤1,(2.41)where c ̸= 0 is a constant.It follows from (2.40) and (2.41) that GI(ν) has a continuous kernel on Qr1 × Qr2and that it is estimated uniformly with respect to ν ∈R2 \ (0, 0) and to x1, x2:supx1,x2,ν(GI(ν))x1,x2 < c2.

(2.42)Since supp ω0 ⊂(I \ ∂I) × N and since the Laplacian ∆ν,I has the zero kernel onfunctions with the Dirichlet boundary conditions on ∂I, we haveω0 = (idI ⊗Π•0(N)) ω0 + GI(ν) ⊗Gm2N ((∆ν,I ⊗∆m2N ) ω0) ,(2.43)where GN is the Green function for ∆•N and where Π•0(N) is the orthogonal projectionoperator in (DR•(N))2 onto Ker ∆•N. The operator Gm2Non a closed Riemannianmanifold (N, gN) has a square-integrable kernel (relative to the second argument) form2 > (n −1)/4 (where n −1 = dim N) and it has a continuous on N × N kernel form2 > (n −1)/2.The following estimate holds uniformly with respect to ν ∈R2 \ (0, 0) for anym2 ∈Z+, m2 > (n −1)/4.

From (2.43), (2.42), and from the Cauchy inequality wehave|ω0(x)|2 ≤c5∥ω0∥22 + ∥(∆ν,I ⊗id) ω0∥22 + ∥(∆ν,I ⊗∆m2N ) ω0∥22. (2.44)Indeed, the following two Banach norms on the finite-dimensional space Ker ∆•N∥h∥2B := maxx∈N |h(x)|2and∥h∥22,N

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY51are equivalent.

So we get (where x = (x1, xN) ∈I × N and I = [−1, 1]):|((idI ⊗Π•0(N)) ω0) (x)|2 ≤∥(Π•0(N)ω0(x1, ∗))∥2B ≤c6 ∥ω0(x1, ∗)∥22,N ,(2.45)∥ω0(x1, ∗)∥22,N ≤2 supx2(GI(ν))x1,x22·∥(∆ν,I ⊗id) ω0∥22,M ≤2c22∥(∆ν,I ⊗id)ω0∥22 . (2.46)The following estimate is obtained by the similar method:|((GI(ν) ⊗Gm2N ) (∆ν,I ⊗∆m2N ) ω0) (x1, xN)|2 ≤≤2c22 supy1(Gm2N )y1,∗22,N · ∥(∆ν,I ⊗∆m2N ) ω0∥22,M .

(2.47)Hence the estimate (2.44) holds for ω0 (even without the first term on the right in(2.44)), as follows from (2.43), (2.45), (2.46), and (2.47).Since ∆ν = idI ⊗∆N + ∆ν(I) ⊗idN and since ∆N and ∆ν,I are nonnegative self-adjoint operators, we have for m2 ∈Z+:∥(∆ν,I ⊗∆m2N ) ω0∥2 ≤∆m2+1νω02 . (2.48)The inequality (2.32) for ω0 follows from (2.44) and (2.48).

For ω1 the analogousto (2.43) equality holds:ω1 =Π10(Iν) ⊗idN −Π•−10(N)ω1 +idI −Π10(Iν)⊗Π•−10(N)ω1 ++Π10(Iν) ⊗Π•−10(N)ω1 + (GI(ν) ⊗Gm2N ) (∆ν,I ⊗∆m2N ω1) ,(2.49)where Π10(Iν) is the projection operator of (DR1(I))2 onto the one-dimensional spacec · dx1 and GI(ν)1 is the Green function for the Laplacian ∆ν,I on DR1(Iν) (withthe Dirichlet boundary conditions on ∂I = {−1, 1} and with the ν-transmissionboundary conditions at 0). The kernel GI(ν)1 is continuous on Qr1 × Qr2 because itcan be written in a form similar to (2.40).

It is written through the Green functiong1 of ∆I on DR1(I) with the Dirichlet boundary conditions on ∂I where the kernel(g1)x1,x2 of g1 is continuous on I × I. Hence the second term on the right in (2.49)is estimated similarly to (2.45) and to (2.46).

The kernel Π10(Iν)x1,x2 is expressed ina form analogous to (2.40) through the kernel 2−1dx1 ⊗dx2 on I × I (correspondingto Π10(I1,1)). So the kernel of Π10 (Iν) is continuous on Qr1 × Qr2, and it satisfies theestimate (2.42) (with the upper bound c).

The first and the third terms in (2.49) areestimated as follows:Π1(Iν) ⊗idN −Π•−10(N)ω1(x1, xN)2 ≤2c2 supy1(Gm2N )y1,∗22,N∥(id ⊗∆m2N ) ω1∥22,M ,Π1(Iν) ⊗Π•−10(N)ω1(x)2 ≤2c2c6 ∥ω1∥22,M .Hence the estimate (2.44) holds uniformly with respect to ν ∈R2 \ (0, 0) for anym ∈Z+, m ≥m0 := 1 + min{k ∈Z+, 4k ≥n}. Thus Proposition 2.8 is proved.□

52S.M. VISHIK2.3.

Actions of the homomorphisms of identifications on the determinant.Proof of Lemma 2.3. The most simple method to compute the action of gν∗onDet(M, N, Z)26 is to obtain the expression for the action of vcν∗on the determi-nant line DetC•(X, V, W) (1.54), induced by the identifications of the correspond-ing cochain complexes vcν = vcνν0 : C•(Xν0, V ) →C•(Xν, V ) (where vcν(c1, c2) :=(c1, (kν/kν0)c2) for ν, ν0 ∈U (2.3)), and then to use Proposition 2.10 below.

Theaction of vcν∗is defined by identifications ψν and ψν0, where ψν : det C•(Xν, V ) →∼DetC•(X, V, W) are defined by the exact sequence in the bottom row of the diagram(1.59). The following diagram of the identifications is commutative:det C•(Xν0, V )vcν−−−→gdet C•(Xν, V )ψν0y ≀ψνy ≀DetC•(X, V, W)vcν∗−−−→gDetC•(X, V, W) .Proposition 2.10.

Under the conditions of Lemma 2.3, the equality holds:gν∗= vcν∗. (2.50)The proof of the equality (2.50) is done just after the end of the proof of Lemma 2.3.The expression for the action of vcν∗on the determinant line can be obtained as follows.Let ν ∈U and let j be the natural inclusion j : ⊕k=1,2 C•(Xk, W ⊕Vk) →C•(Xν, V ).Then vcν acts on C•(X1, W ∪V1) as the identity operator and it acts on C•(X2, W ∪V2)as the operator (kν/kν0) id.

Proposition 2.7 claims that the identification ψν is anisometry between the combinatorial norm on det C•(Xν, V ) and the triple tensorproduct of the combinatorial norms on the components of DetC•(X, W, V ). It isenough to compute the action of vcν∗on the component det C•(W) of the tensor prod-uct DetC•(X, W, V ).

The inclusion ρν,c : C•(W) ֒→C•(Xν, V ) (defined by (1.63)) isan isometry onto orthogonal complement to Im j and rν,cρν,c = id on C•(W). So theaction of vcν on this orthogonal complement (Im j)⊥(identified with C•(W) by rν0,cand by rν,c) can be expressed as the compositionm ∈C•(W) −−→ρν0,c(β0, α0).qα20 + β20m −→vcν(β0, α0(kν/kν0)).qα20 + β20m ==(1, kν)/q1 + k2ν0m −−→rν,cq1 + k2ν.q1 + k2ν0m ∈C•(W).

(2.51)26This action is multiplying by a nonzero factor.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY53(Here, the signs are written for positive β and β0.27) The expression for vcν∗followsfrom (2.51) and from the assertion that vcν acts on C• (X2, W ∪V2) as (kν/kν0) id.Namelyvcν∗l = (kν/kν0)−χ(M2,N∪Z2) 1 + k2ν/1 + k2ν0−χ(N)/2 l.(2.52)for l ∈DetC•(X, V, W).

It follows from (2.52) that the equality holds (for l ̸= 0):∂γ log ∥vcν∗l∥2 =−2χ(M2, N ∪Z2)∂γlog(kν)−2χ(N)1 + k−2ν−1∂γlog(kν). (2.53)Proposition 2.10 claims that the same identity holds also for the action of gν∗onDet(M, N, Z).The right side in the formula (2.19) (i.e., in the assertion of Lemma 2.3) is de-fined in analytic terms while the right side in (2.53) is defined in topological terms.Each b2,j (Mν0, Z) on the right in (2.19) is the sum of integrals over M2 and overN of the locally defined densities, according to Theorem 3.2.So it is enough tocompute (in topological terms) the expression on the right in (2.19) in the case of amirror-symmetric fM = M2 ∪N M2 with a mirror-symmetric metric g eM (which is adirect product metric near ∂fM and near N) and with mirror-symmetric boundaryconditions on the connected components of ∂fM.

In this case, the expression in (2.19)is the same as for a general M (if the piece M2 of M, gM|TM2, and the boundaryconditions on ∂M ∩M2 are the same as in the mirror-symmetric case on each pieceM2 of fM). It is supposed from now on in the proof of Lemma 2.3 that M and allthe data on M are mirror-symmetric relative to N. In this case the kernel E•t,x,y(ν)(ν ∈R2 \ (0, 0)) of the operator exp (−t∆•ν) with the Dirichlet boundary conditionson Z = Z2 ∪Z2 ⊂∂M and with the Neumann conditions on ∂M \ N is expressedthrough the fundamental solution E•t,x,y for ∂t + ∆• on DR•(M, Z) (with the sameboundary conditions on ∂M)28 as follows29:E•t,x,y(ν)=E•t,x,y+α2 −β2/α2 + β2(σ∗1E•)t,x,y for x ∈M2, y ∈M2, (2.54)E•t,x,y(ν) =2αβ/α2 + β2E•t,x,y for x ∈M1, y ∈M2.

(2.55)Note that the kernel (Et + σ∗1Et)x,y =: ENeut,x,y for x, y ∈M2 is the fundamentalsolution for ∂t+∆•M2, where ∆•M2 is the Laplacian on DR•(M2, Z2) with the Neumannboundary conditions on N and the kernel (Et −σ∗1Et)x,y =: EDirt,x,y is the kernel of27The signs are not important for the transformations of the norm on the determinant line underthe actions of vcν∗.28It is proved in Proposition 1.1 that ∆• on DR•(M, Z) for M obtained by gluing two pieces,M = M1 ∪N M2, has the same eigenvalues (including their multiplicities) and eigenforms as ∆•1,1in DR•(M1,1, Z). The analogous assertion is true for the operators exp(−t∆•1,1) and exp(−t∆•) in(DR•(M))2 and for their kernels.29These formulas are analogous to (2.40).

54S.M. VISHIKexp(−t∆•M2,N), where ∆•M2,N is the Laplacian on DR•(M2, N ∪Z2) i.e., with theDirichlet boundary conditions on N. It follows from (2.54) that the alternating sumof zero-order terms (in the asymptotic expansions of the traces of the heat equationoperator relative to t →+0) on the right in (2.19) can be represented in the followingform (where mν0 := 2−1 1 −k−2ν0 .

1 + k−2ν0):X(−1)jb2,j (Mν0, Z) =X(−1)jZM2trEjt,x,x(ν0)0 == 2−1 X(−1)jZM trEjt,x,x0 + mν0X(−1)jZM2trEj,Neut,x,x0 −trEj,Dirt,x,x0==2−1χ(M, Z)+mν0(χ(M2, Z2)−χ(M2, Z2 ∪N))=χ(M2, Z2∪N)+1+k−2ν0−1χ(N). (2.56)Hence the expression on the right in (2.53) is equal to the right side of (2.19), andthe assertion of Lemma 2.3 follows from Proposition 2.10.

The zero superscripts in(2.56) denote the densities on Mj, N, ∂M, corresponding to the constant terms (i.e.,the t0-coefficients) in the asymptotic expansions as t →+0 for Tr (pj exp (−t∆•)),where ∆• is the Laplacian with appropriate boundary conditions. In (2.56)RMj tr()0denotes the sum of the integrals over Mj, N, and over ∂Mj \ N of the correspondingdensities.

We use the following equalities to produce (2.56):X(−1)jZM trEjt,x,x0 = χ(M, Z),(2.57)X(−1)jZM2trEj,Neut,x,x0 = χ(M2, Z2),X(−1)jZM2trEj,Dirt,x,x0 = χ(M2, N ∪Z2).These equalities are consequences of the analogous equalities without the zero super-scripts and of the existence of asymptotic expansions in powers of t for the corre-sponding traces as t →+0 ([Se2], Theorem 3, or Theorem 3.2 below).□Proof of Proposition 2.10. The identificationsDetC•(X, V, W)dc−−→gDet(X, V, W)R←−−gDet(M, N, Z)(2.58)do not depend on ν (the determinant lines in (2.58) are defined in (1.54)).

So theactions of vcν∗on Det(X, V, W) and on DetC•(X, V, W) are the same (i.e., they multi-ply by the same number). To prove (2.50) it is enough to show that the correspondingoperators on Det(M, Z, N) are the same (i.e., that gν∗= vcν∗on det(M, N, Z)).The proof of Proposition 2.10 uses the following assertion.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY55Proposition 2.11.

Let ϕ: (F •0 , dF0) →(F •1 , dF1) be an isomorphism of finite com-plexes of finite-dimensional linear spaces. Then the diagram is commutative:det F •0ϕ−−−→gdet F •1d ≀d ≀det H•(F0)ϕ∗−−−→gdet H•(F1)(2.59)Proof.

The identifications det F •jd≂det H•(Fj) are defined with the help of differ-entials d = dFj. Hence the commutativity of (2.59) holds.□The commutativity of the following diagram of the identifications (for ν sufficientlyclose to ν0 such that vν is an isomorphism) follows from (2.59):det W •a (ν0)vν−−→gdet vν (W •a (ν0))Πa−−→gdet W •a (ν0)d ≀dy ≀ ≀ddet H•(Mν0, Z)−→vν∗det H•(Mν, Z)=det H•(Mν, Z)(2.60)where the identification j∗: H• (vν (W •a (ν0))) →H• (DR(Mν, Z)) = H•(Mν, Z) isdefined by the natural inclusion j : vν (W •a (ν0)) ֒→DR•(Mν, Z) of a quasi-isomorphicsubcomplex.The commutativity of the left square in (2.60) follows from (2.59).The commutativity of the right square in (2.60) also follows from (2.59) because theoperator induced by the projection operator Πa on H• (DR(Mν, Z)) is the identityoperator.The commutativity of the following diagram is a consequence of the commutativityof the diagram (2.60):det W •a (ν0)=det W •a (ν0)vν−−→gdet vν (W •a (ν0))d ≀d ≀ϕν0(a) ≀det H•(Mν0, Z)−→vν∗det H•(Mν, Z)ϕanν0 ≀ϕanν ≀Det(M, N, Z)=Det(M, N, Z)gν∗−−→gDet(M, N, Z)(2.61)The action of vν∗: H• (Mν0, Z) →H• (Mν, Z) coincides with the combinatorialaction vcν∗: H• (Xν0, V ) →H• (Xν, V ) under the identification of the cohomologyR: H• (Mν, Z) →H• (Xν, V ) induced by the integration R of closed differential

56S.M. VISHIKforms over the simplexes of X.

Hence the commutativity of (2.60) involves also thecommutativity of the diagram:det W •a (ν0)vν−−→gdet vν (W •a (ν0))d ≀d ≀det H•(Mν0, Z)=det H•(Mν0, Z)−→vν∗det H•(Mν, Z)dc ≀dc ≀ϕcν0=ϕanν0y ≀det C•(Xν0, V )−→vcνdet C• (Xν, V )ψν0 ≀ψν ≀DetC•(X, V, W)−→vcν∗DetC•(X, V, W)(2.58) ≀(2.58) ≀Det(M, N, Z)=Det(M, N, Z)−→vcν∗Det(M, N, Z)(2.62)The equality (2.50) follows immediately from the commutativity of the right bot-tom square in (2.61) and from the commutativity of (2.62).Proposition 2.10 isproved.□2.4. Analytic torsion norm on the cone of a morphism of complexes.

Proofof Lemma 2.4. Lemma 2.4 is a particular case of the following assertion.

Let f bea morphism (2.22) of finite complexes of finite-dimensional Hilbert spaces.30 ThenCone• f is defined by (2.23). The exact sequence of complexes:310 →V • →Cone• f −→p A•[1] →0,(2.63)(where the left arrow maps y ∈V • into (0, y) ∈Cone• f and p(x, y) = x for (x, y) ∈Aj+1 ⊕V j) defines the identification of the determinants of its cohomology:ϕHCone• f : det H•(Cone f) →∼det H•(V ) ⊗(det H•(A))−1 .

(2.64)Let the Hilbert spaces Conej f be the direct sums Aj ⊕V j+1 of the Hilbert spaces.Lemma 2.6. The analytic torsion norm on the determinant of the cohomology ofCone• f is isometric under the identification ( 2.64) to the tensor product of the an-alytic torsions norms32:T0(Cone• f) = T0(V •) ⊗T0(A•)−1.

(2.65)30The morphism f does not supposed to be a quasi-isomorphism.31A•[1] is a complex with components A[1]j = Aj+1 and with dA[1] = −dA.There are thecanonical identifications: det A•[1] = (det A•)−1 and det H•(A[1]) = (det H•(A))−1.32The analytic torsion norm on det H•(A[1]) = det H•(A)−1 is the dual norm T0(A•)−1.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY57Remark 2.6.

Let h ∈det H•(Cone f) be identified by (2.64) with h1 ⊗h−12for h1 ∈det H•(V ) and h2 ∈det H•(A). Namely ϕCone• fh = h1⊗h−12 , where h−12is an elementof the dual one-dimensional space det H•(A)−1 such that h−12 (h2) = 1.

In this case,the equality (2.65) claims that∥h∥2T0(Cone• f) = ∥h1∥2T0(V •).∥h2∥2T0(A•) . (2.66)Remark 2.7.

The identity (2.65) (Lemma 2.6) and the equality (2.66) also hold underweaker assumptions. Let the Hilbert structures on A•, V •, and on Cone• f be suchthat the identificationϕCone• f : det Cone• f →∼det V • ⊗(det A•)−1(2.67)(induced by the exact sequence (2.63)) is an isometry.

Then the equality (2.66) holds.Corollary 2.7. Let f : A• →V • be a quasi-isomorphism.

Then H•(Cone f) = 0.Hence det H•(Cone f) is canonically C and 1 ∈C = det H•(Cone f) is identified by( 2.64) with h1 ⊗h−12 . Here, f∗h2 = h1 under the identification induced by f:f∗: det H•(A) →∼det H•(V ).

(2.68)In this case, the equality ( 2.66) claims that∥1∥2T0(Cone f) = ∥h1∥2T0(V •).∥h2∥2T0(A•) = T0(V •)/T0(A•),(2.69)where T0(V •)/T0(A•) is the ratio of the two norms on the same determinant line(since det H•(V ) and det H•(A) are identified by ( 2.68)).The equality (2.69) is the assertion of Lemma 2.4.Proof of Lemma 2.6. The identification (2.67) is an isometry of norms on thedeterminant lines.

Let u be a nonzero element of det Cone• f and letϕCone• fu = u1 ⊗u−12 ,(2.70)where u1 ∈det V • and u2 ∈det A•. Let h, h1, h2 be the images of u, u1, u2 under theidentifications (defined by the differentials of the corresponding complexes):det Cone• f dCone• f^ det H•(Cone f),det A•dA^det H•(A),det V •dV^det H•(V ).Then by the definition of ϕHCone• f we have ϕHCone• fh = h1 ⊗h−12 .The analytic torsion norm on the determinant of the cohomology of a finite-dimensional complex is the norm, corresponding to the L2-norm on the determinant

58S.M. VISHIKof this complex defined by the Hilbert structures on its components.Hence theequalities hold:∥h∥2T0(Cone• f) = ∥u∥2det(Cone• f) ,∥h1∥2T0(V •) = ∥u1∥2det(V •) ,∥h2∥2T0(A•) = ∥u2∥2det(A•) .

(2.71)Since the identification (2.67) is an isometry, we see that the equality∥h∥2T0(Cone• f) = ∥h1∥2T0(V •).∥h2∥2T0(A•)(2.72)follows from (2.71) and from (2.70).□The equality (2.24) in Lemma 2.4 is a particular case of (2.72) (by Corollary 2.7)corresponding to the case of a quasi-isomorphism f. Lemma 2.4 is proved.□2.5. Variation formula for norms of morphisms of identifications.

Proof ofLemma 2.1. The assertion (2.14) of Lemma 2.1 can be deduced from the definitionof gν and from the following proposition.Proposition 2.12.

There is a neighborhood Uν0, ν0 ∈Uν0 ⊂R2 \ (0, 0), such that afamily of finite rank projection operators Πja(ν) := Πj(ν; a) in (DRj(M))2 is smoothon Uν0 ∋ν.Its proof follows just after the proof of Lemma 2.1. Let l ∈det W •a (ν0), l ̸= 0, andlet l = ⊗jl(−1)j+1j, where lj ∈det W ja(ν0), lj ̸= 0.

Then we havelog ∥gνl∥2det W •a (ν) =X(−1)j+1 log ∥gνlj∥2det W ja(ν). (2.73)Proposition 2.13.

For every j the following equality holds (under the conditions ofLemma 2.1):∂γ log ∥gνlj∥2det W ja(ν)γ=0 = 2 ∂γ log(kν)γ=0 Trp2Πja(ν0). (2.74)Thus the assertion (2.14) of Lemma 2.1 is a consequence of (2.74) and of (2.73).Proof of Proposition 2.13.

It is enough to prove the equality (2.74) in the casewhen ∥lj∥2det W •a (ν0) = 1, i.e., when l = e1 ∧. .

. ∧em, where {ei} is an orthonormalbasis in W •a (ν0) (ν0 = ν(0)).

In this case, we have∂γ log ∥gνl∥2det W ja(ν)γ=0 = tr (∂γ (aij(ν)))γ=0,(2.75)where a(ν) = (aij(ν)) is a matrix of scalar products inDRj M 1⊕DRj M 2, gMof the images of the basis elementsaij(ν) :=< gνei, gνej > .

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY59The formula (2.75) is deduced as follows.

Since ∥gνl∥2det W ja(ν) = det a(ν), we have∂γ log det a(ν)γ=0 = tr∂γa · a−1(ν) γ=0 = tr(∂γa)γ=0 .The family of the operators Πja(ν) is smooth in ν for ν ∈Uν0 (Proposition 2.12).Hence the operator ∂γΠja(ν) exists. Since Πja(ν) are projection operators, we haveΠ2a = Πa,∂γΠa · Πa = (id −Πa)∂γΠa.So ei are orthogonal to ∂γΠaγ=0ej and we get∂γaij(ν)γ=0 = ∂γ < Πagνei, Πagνej >γ=0 =< ∂γ(Πagν)ei, ej >γ=0 ++ < ei, ∂γ(Πagν)ej >γ=0 = ∂γ(log kν)γ=0(< p2ei, ej > + < ei, p2ej >).

(2.76)Since {ei} is an orthonormal basis in W •a (ν0), we havetr (< p2ei, ej >) = trp2Πja(ν0)(2.77)Hence (2.74) follows from (2.75), (2.76), and (2.77).Thus Proposition 2.13 andLemma 2.1 are proved.□Proof of Proposition 2.12. Let Uν0 ⊂R2 \ (0, 0) be the set of ν ∈R2 \ (0, 0)such that the Laplacians ∆ν,j on DRj(Mν, Z) (for all j) have no eigenvalues λ from(a −ε, a + 2ε) ⊂R+ (where ε > 0 is small enough).

Then for ν ∈Uν0 the pro-jection operator Πja(ν) from (DRj(M))2 onto a linear space W ja(ν) (spanned by theeigenforms of ∆ν,j with the eigenvalues λ from [0, a]) is equal to the contour integralΠja(ν) = i/2πZΓa+εGjλ(ν)dλ,where Gjλ(ν) = (∆ν,j −λ)−1 is the resolvent for the Laplacian ∆ν,j and Γa+ε is acircle Γa+ε = {λ : |λ| = a + ε} oriented opposite to the clockwise. For λ ∈Γa+ε andν ∈Uν0 the operators Gjλ(ν) form a smooth in (λ, ν) family of bounded operatorsin (DRj(M))2.

(It is an immediate consequence of Proposition 3.1 below. Indeed,Spec(∆ν,j) is discrete and it is a subset of R+ ∪0, according to Theorem 3.1.

Thusif (a −ε, a + 2ε) ∩Spec(∆ν,j) = ∅then Γa+ε ∩Spec(∆ν,j) = ∅and Gjλ(ν) form asmooth in ν ∈Uν0 and in λ ∈Γa+ε family of bounded in (DRj(M))2 operators byProposition 3.1.) Proposition 2.13 is proved.□2.6.

Variation formula for the scalar analytic torsion. Proof of Lemma 2.2.First the lemma is proved in the case when a > 0 is less than 4−1λ1(ν0) (where λ1(ν)is the minimal positive eigenvalue of the Laplacian ∆ν on (⊕jDRj(Mν, Z), gM)).

LetU1(a) be a neighborhood of ν0, ν0 ∈U1(a) ⊂U (2.3), such that for ν ∈U1(a) wehave a < 2−1λ1(ν). (Such a neighborhood exists according to Theorem 3.1 and toProposition 3.1.)

60S.M. VISHIKLet ν(γ) be a smooth local map (R1, 0) →(U(a), ν0).Proposition 2.14.

For t > 0 the following variation formula holds33:∂γX(−1)jj Tr exp (−t∆ν,j)γ=0 == 2∂γ log(kν)γ=0X(−1)j (−t ∂/∂t) Tr (p1 exp (−t∆ν0,j)) ,(2.78)where kν := α/β for ν = (α, β) ∈U (i.e., for αβ ̸= 0).Proposition 2.15. Let Re s > (dim M)/2 and let 0 < a < λ1(ν0).

Then the follow-ing equalities hold:∂γΓ(s)−1Z ∞0ts−1 X(−1)jj (Tr exp (−t∆ν,j) −dim Ker ∆ν,j) dt γ=0 == Γ(s)−1Z ∞0ts X(−1)j(−∂t) Trp1exp (−t∆ν0,j) −Πja(ν0)dt == 2 ∂γ log(kν)γ=0s.Γ(s) X(−1)jZ ∞0ts−1 Trp1exp (−t∆ν0,j) −Πja(ν0)dt. (2.79)Proof of Proposition 2.15.

To produce the second row of (2.79) from (2.78), it isenough to prove ([Bo], II.26, Proposition 7) that for Re s > (dim M)/2 the integralZ ∞0ts(−∂t) Tr (p1 exp(−t∆ν,j)) dt =Z ∞0ts Tr (p1∆ν,j exp(−t∆ν,j)) dt(2.80)converges uniformly in ν for ν from a small neighborhood Uν0 of ν0 and to prove theconvergence of the integralZ ∞0ts−1 Tr (exp(−t∆ν0,j) −dim Ker ∆ν0,j) dt(2.81)together with the uniform convergence in ν for an arbitrary ν1 ∈Uν0 (as ν →ν1) ofthe functionts+1X(−1)jTr(p1∆ν,jexp(−t∆ν,t))→ts+1X(−1)jTr(p1∆ν1,jexp(−t∆ν1,t))(2.82)for t from any closed finite interval t ∈I ⊂(0, +∞). (To apply the theorem from[Bo], quoted above, it is useful first to do the transformation R+ ∋t →h = log t ∈R,dt →tdh.

)33The operator exp (−t∆ν,j) acts fromDRj(M)2 into the domain D (∆ν,j) of ∆ν,j definedby (1.27).The operators exp (−t∆ν,j) and p1 exp (−t∆ν,j) are of trace class.Their traces areequal to the integrals over the diagonal of the traces of their kernels restricted to the diagonals (byProposition 3.8).

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY61The following estimates are satisfied|Tr (p1∆ν,j exp(−t∆ν,j))| ≤Tr (∆ν,j exp(−t∆ν,j)) ≤≤∥∆ν,j exp(−t∆ν,j/2)∥2 (Tr (exp(−t∆ν,j/2) −dim Ker ∆ν,j)) ,(2.83)∥∆ν,j exp(−t∆ν,j/2)∥2 ≤maxλ≥0 (λ exp(−tλ/2)) = 2/(te),(2.84)(where ∥·∥2 are the operator norms in (DRj(M))2).The first estimate in (2.83)follows from the Mercer theorem.

(The applying of this theorem here is similar to itsapplying in the proofs of Propositions 3.8 and 3.9 below. )Let t0 be any positive number.

Then for t ≥2t0 and for ν sufficiently close to ν0we have the following uniform with respect to ν estimateTr exp (−t∆ν,j/2) −dim W ja(ν) < C exp(−c1t),(2.85)where C and c1 are positive constants. Indeed, according to Theorem 3.2, for anyt0 > 0 there is a constant L > 0 (depending on t0) such that the inequalityTr exp (−t0∆ν,j/2) ≤L.holds uniformly with respect to ν ∈R2 \ (0, 0).

So for all ν ∈U sufficiently close toν0 and such that λ1(ν) > 2a, the estimate (2.85) is true for t ≥2t0 with C = L andc1 = a/4:Tr exp (−t∆ν,j/2) −dim W ja(ν) ≤L exp(−ta/4). (2.86)The uniform convergence (with respect to ν) of the integral (2.80) for Re s >(dim M)/2 follows from the asymptotic expansion (2.15) in powers of t as t →+0 andfrom the estimates (2.83), (2.84), and (2.85).

The convergence of the integral (2.81)for Re s > (dim M)/2 follows from (2.85) and from the existence of the asymptoticexpansion of the trace as t →+0 (by Theorem 3.2):Tr exp(−t∆ν0,j) = f−dim Mt−(dim M)/2 + f1−dim Mt(1−dim M)/2 + · · · ++ fktk/2 + Ot(k+1)/2,(2.87)where k ∈Z+ and fi := fi(ν; j) are smooth in ν ∈R2 \ (0, 0). This asymptoticexpansion (2.87) is differentiable with respect to ν, according to Proposition 3.2.The uniform convergence of (2.82) for t ∈I ⊂(0, ∞) (if s is fixed and Re s >(dim M)/2) follows from Proposition 3.8 and from the uniform convergence (withrespect to ν) of the functions of tts+1∂tZM1Tr Ejt,x,x(ν) →ts+1∂tZM1tr Ejt,x,x(ν1)for t ∈I.

(The latter assertion follows from Proposition 3.2 and from Theorem 3.2.)

62S.M. VISHIKThe last equality in (2.79) is true for Re s > (dim M)/2 according to the asymptoticexpansion (2.87), to the estimate (2.86), to the absolute convergence of the integral(2.80), and to the following estimates (where 0 < a < λ1(ν0)):0 < Trp1exp (−t∆ν0,j) −Πja(ν0)≤Trexp (−t∆ν0,j) −Πja(ν0).

(2.88)(These estimates are deduced from the Mercer theorem the same way as in the proofsof Propositions 3.8 and 3.9.) Thus Proposition 2.15 is proved.□Proposition 2.16.

For 0 < a < λ1(ν0) the assertion of Lemma 2.1 is true, i.e., itholds:∂γlogT(Mν, Z)γ=0 =2∂γlog(kν)γ=0X(−1)jTrp1exp(−t∆ν0,j)−Πja(ν0)0 . (2.89)(The zero superscript denotes the constant coefficient in the asymptotic expansionas t →+0 for the operator trace.)Proof.

The equality (2.79) claims that for Re s > (dim M)/2 and for 0 < a <λ1(ν0)/4 we have∂γX(−1)jjζν,j(s) γ=0 ==2∂γlog(kν)γ=0(s/Γ(s))X(−1)jZ ∞0 ts−1Trp1exp(−t∆ν0,j)−Πja(ν0)dt. (2.90)The final expressions in (2.90) and (2.79) are analytic functions of s for Re s >(dim M)/2 according to (2.88), (2.86), and to (2.15).The meromorphic continu-ation to the whole complex plane C ∋s of this function of s can be producedwith the help of the asymptotic expansion (2.15) (or of the expansion (2.91) belowfor Tr (p1 (exp (−t∆ν0,j) −Πja(ν0)))) using the estimates (2.88) and (2.86).

Let theasymptotic expansion as t →+0 for the trace of the operator below be as follows(n := dim M):Trp1exp (−t∆ν0,j)−Πja(ν0)=q−nt−n/2+. .

.+q0t0+ˆq1t1/2+. .

.+ˆqktk/2+rk,j(t),(2.91)where rk,j(t) is Ot(k+1)/2as t →+0 and rk,j(t) is a C[k/2]-smooth function oft ∈[0, 1].34 Then the analytic continuation to Re s > −(k + 1)/2 of the integral on34This asymptotic expansion exists and is differentiable with respect to t by Proposition 3.2.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY63the right in (2.90) is given by the expressionZ ∞0ts−1 Trp1exp (−t∆ν0,j) −Πja(ν0)dt == q−n.

(−n/2 + s) + q−n+1. (−n/2 + 1/2 −s) + · · · + q0.s + ˆq1.

(1/2 + s) + . .

.+ ˆqk. (k/2 + s) +Z 10 ts−1rk,j(t)dt +Z ∞1 ts−1 Trp1exp (−t∆ν0,j)−Πja(ν0)dt.

(2.92)The latter integral in (2.92) is an analytic function on the whole complex place C ∈s(according to (2.88) and (2.86)). The integral of rk,j in (2.92) is an analytic functionof s for Re s > −(k + 1)/2.

The asymptotic expansions (2.87) for Tr exp (−t∆ν,j)can be differentiated with respect to ν according to Proposition 3.2. They provideus with the analytic continuation of P(−1)jjζν,j(s) to Re s > −(k + 1)/2 as follows:X(−1)jjζν,j(s) = Γ(s)−1F−n.

(−n/2 + s) + · · · + Fk. (k/2 + s) ++Z 10 ts−1mk,ν(t)dt +Z ∞1 ts−1 X(−1)jj Tr(exp (−t∆ν,j)−dim Ker ∆ν,j) dt,(2.93)where Fk :=Pj(−1)jj (fk(ν, j) −δ0,k dim Ker ∆ν,j) and the functions mk,ν(t) areC[k/2]-smooth in t ∈[0, 1] and in ν (for ν ∈R2 \ (0, 0)) and are such that mk,ν =Ot(k+1)/2uniformly with respect to ν as t →+0.The latter integral in (2.93) is an analytic function of s ∈C.We obtain itsderivative with respect to γ taking into account (2.78), (2.86), and (2.88):∂γZ ∞1ts−1 Xj(−1)jj Tr (exp (−t∆ν,j) −dim Ker ∆ν,j) dtγ=0 == 2 ∂γ log(kν)γ=0sZ ∞1ts−1 Xj(−1)j Trp1exp (−t∆ν0,j) −Πja(ν0)dt ++Xj(−1)j Trp1exp (−∆ν0,j) −Πja(ν0),where 0 < a < λ1(ν0).35 Using (2.87), (2.93), and (2.92), we get the equalities (whereqk(j, ν0) := qk for k ≤0, qk(j, ν0) := ˆqk for k > 0 and qk, ˆqk are from (2.91)):∂γFlγ=0 = 2 ∂γ log(kν)γ=0(−l/2)X(−1)jql(j, ν0),∂γmk,ν(t)γ=0 = 2 ∂γ log(kν)γ=0(−t)∂tXj(−1)jrk,j(t).

(2.94)35For such a the operator Πja(ν0) is the projection operator fromDRj(M)2 onto the space ofharmonic forms Ker (∆ν0,j).

64S.M. VISHIKHence the equality (2.90) holds on the whole complex plane C ∋s.

In particular, weobtain∂γ log T (Mν, Z)γ=0 = 2 ∂γ log(kν)γ=0X(−1)jq0(j, ν0).Thus Proposition 2.16 is proved.□Remark 2.8. A consequence of (2.94) is as follows.

For any smooth local map ν(γ) :(R1, 0) →(U, 0) (where U is defined by (2.3)) the identity ∂γF0 ≡0 holds accordingto (2.94). Hence the function on UF0(ν) =X(−1)jj (f0(ν, j) −dim Ker ∆ν,j)is independent of ν.

The dimension of Ker ∆ν,j is independent of ν for ν ∈U asit follows from the cohomology exact sequence in the top row of (2.27). Indeed, forν ∈U the dimension of Im ∂D (where ∂D : Hi(N, C) →⊕k=1,2Hi+1(Mk, Zk ∪N; C) isa differential in this exact sequence) is independent of ν because for different ν = ν0and ν1 from U the maps ∂D,k(ν): Hi(N, C) →Hi+1(Mk, N ∪Zk; C) for fixed k = 1, 2(and for a fixed i) differ by the nonzero scalar constant factor (depending on ν0 andν1).

HenceP(−1)jjf0(ν, j) is a constant function on U.Proof of Proposition 2.14. Let Ejt,x,y(ν) (where t > 0) be the heat kernel of theoperator exp (−t∆ν,j).

By the Duhamel principle, a variation in ν of E•t,x,y(ν) can bewritten as follows. LetE•t1,x1,∗(ν1), E•t2,∗,y(ν)M =:RM E•t1,x1,z(ν1) ∧∗zE•t2,z,x2(ν) be ascalar product (1.23) (with respect to the variable z).

We haveE•t,x,y(ν) −E•t,x,y(ν0) = limε→+0Z t−εεdτ∂/dτE•τ,x,∗(ν), E•t−τ,∗,y(ν0)== limε→+0Z t−εεdτh−∆∗E•τ,x,∗(ν), E•t−τ,∗,y(ν0)+E•τ,x,∗(ν), ∆∗E•t−τ,∗,y(ν0)i. (2.95)Stokes’ formula claims that for any two smooth forms ω1, ω2 ∈DRj(M 1) on a mani-fold M1 with boundary we have(dω1, ω2)M1 −(ω1, δω2)M1 = (rω1, Aω2)∂M1 :=Z∂M1rω1 ∧∗∂M1Aω2,(2.96)where the density rω1 ∧∗∂MAω2 on ∂M1 =: N and the operators r and A are definedas follows.

Let any local orientations be chosen on TM|N and on TN. Then thefollowing forms on N are locally defined:rω1 := [N : M1] i∗N,M1ω1,Aω2 := ∗−1N i∗N,M1 ∗M1 ω2,(2.97)where i∗N,M1 : DR•(M 1) →DR•(N) is the geometrical restriction to N (and [N : M1] =1 if N is locally oriented as ∂M1 and [N : M1] = −1 in the opposite case).

The lo-cally defined form rω1 ∧∗∂M1Aω2 is a globally defined density on N = ∂M36. So the36It does not depend on a local orientation on T M and if a local orientation on T N changes tothe opposite then this local form changes its sign.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY65equality (2.95) can be written in the formE•t,x,y(ν) −E•t,x,y(ν0) = limε→+0Z t−εεdτXk=1,2−(rδ)∗E•τ,x,∗(ν), A∗E•t−τ,∗,y(ν0)∂Mk ++A∗E•τ,x,∗(ν), (rδ)∗E•t−τ,∗,y(ν0)∂Mk+(Ad)∗E•τ,x,∗(ν), r∗E•t−τ,∗,y(ν0)∂Mk −−r∗E•τ,x,∗(ν), (Ad)∗E•t−τ,∗,y(ν0)∂Mk,(2.98)where r = rk, A = Ak are the corresponding operators for pairs (Mk, ∂Mk).Let any local orientations be chosen on TM|N and on TN.

Then the conditions(1.27) for the domain D (∆•ν) claim that for the kernels E•(ν) and E•(ν1) (whereν = (α, β) and ν1 = (α1, β1) are from R2 \ (0, 0)) the following equalities hold on N:α rz(M1, N) ◦E•t,x,z(ν) = −β rz(M2, N) ◦E•t,x,z(ν),β Az(M1, N) ◦E•t,x,z(ν) = α Az(M2, N) ◦E•t,x,z(ν),α rz(M1, N) ◦δzE•t,x,z(ν) = −β rz(M2, N) ◦δzE•t,x,z(ν),β Az(M1, N) ◦dzE•t,x,z(ν) = α Az(M2, N) ◦dzE•t,x,z(ν),(2.99)The analogous equalities hold for E•t (ν1) (where (α, β) are replaced by (α1, β1)).Hence the equality (2.98) for ν ∈U(a) ⊂U can be written in the formE•t,x,y(ν) −E•t,x,y(ν0) == limε→+0Z t−τεdτ−(1 −(kν/kν0))(rδ)∗E•τ,x,∗(ν), A∗E•t−τ,∗,y(ν0)∂M1 ++ (1 −(kν0/kν))A∗E•τ,x,∗(ν), (rδ)∗E•t−τ,∗,y(ν0)∂M1 ++ (1 −(kν0/kν))A∗d∗E•τ,x,∗(ν), r∗E•t−τ,∗,y(ν0)∂M1 −−(1 −(kν/kν0))r∗E•τ,x,∗(ν), A∗d∗E•t−τ,∗,y(ν0)∂M1. (2.100)The kernel E•t,x,y(ν) is smooth in ν ∈R2 \ (0, 0) as a smooth double form onM r1 × M r2 (where the limit values on N ⊂∂M r are taken from the side of Mr)according to Proposition 3.2.

So we can conclude from (2.100) that∂γE•t,x,y(ν)γ=0 = ∂γ log(kν)γ=0 limε→+0Z t−εεdτ(rδ)∗E•τ,x,∗(ν0), A∗E•t−τ,∗,y(ν0)∂M1 ++A∗E•τ,x,∗(ν0), (rδ)∗E•t−τ,∗,y(ν0)∂M1 +A∗d∗E•τ,x,∗(ν0), r∗E•t−τ,∗,y(ν0)∂M1 ++r∗E•τ,x,∗(ν0), A∗d∗E•t−τ,∗,y(ν0)∂M1,(2.101)where the limits of the kernels are taken from the side of M1.

66S.M. VISHIKBy Proposition 3.8, we have the equalityTr exp (−t∆ν0,j) =Xr=1,2ZMri∗MrEjt,x1,x2(ν0),(2.102)where the exterior product of double forms on the diagonals iMr : Mr ֒→Mr × Mrare implied.We deduce from the semigroup property for the kernels of the operators exp (−t∆ν0,j)thatQ1,z1Ejt1,x,z1(ν0), Q2,z2Ejt2,z2,x(ν0)M :==ZM Q1,z1Ejt1,x,z1(ν0) ∧x ∗xQ2,z2Ejt2,z2,x(ν0) = Q1,z1Q2,z2Ejt1+t2,z2,z1(ν0),(2.103)where Qi,zi are differential operators acting on differential forms, ti > 0 and theintegration is with respect to the variable x.We deduce the following variationformula from (2.101) using (2.102) and (2.103):∂γ Tr exp (−t∆ν,j)γ=0 = 2 ∂γ log(kν)γ=0tAz1(M1, N)dz1rz2(M1, N)Ejt,z1,z2(ν0)N++Az1(M1, N)rz2(M1, N)δz2Ejt,z1,z2(ν0)N,(2.104)where the summands are the integrals of the densities on N (as it is defined in (2.96)and (2.97)).

For instance,37Az1dz1rz2Ejt,z1,z2(ν0)N :=Z Az1dz1 ∧∗z2,Nrz2Ejt,z1,z2(ν0). (2.105)The local forms F1 =i∗M1 ∗z2 dz1dz2E•t,z1,z2(ν0)and F2 =i∗M1 ∗z2 δz1dz1E•+1t,z1,z2(ν0)are smooth on the diagonal iM1 : M 1 ֒→M 1 × M 1.

(The limit values on N ⊂∂M1 ofthese double forms are taken from the side of M1 and the exterior product of thesedouble forms is implied.) The local form i∗N ∗N,z2 Az1dz1rz2E•t,z1,z2(ν0) is a smoothdensity on the diagonal N ⊂∂M1, iN : N ֒→N × N. Hence we can apply Stokes’formula (2.96) to the expression on the right in (2.104).

The result transformed withthe help of the equality dxE•t,x,y(ν) = δyE•+1t,x,y(ν) is as follows:∂γ Tr exp (−t∆ν,j)γ=0 == 2∂γ log(kν)γ=0t−δz1dz1Ejt,z1,z2(ν0)M1 +dz1δz1Ej+1t,z1,z2(ν0)M1 ++dz2δz2Ejt,z1,z2(ν0)M1 −δz2dz2Ej−1t,z1,z2(ν0)M1,(2.106)37We imply (but do not write) the restriction to the diagonal N ֒→N × N in (2.105).

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY67where the summands in (2.106) are the integrals of the densities of the type:δz1dz1E•t,z1,z2(ν0)M1 :=ZM1i∗M1 ∗z2 δz1dz1E•t,z1,z2(ν0)(2.107)(and the exterior product of double forms is implied in (2.107) and (2.106)).The formula (2.78) is an immediate consequence of (2.106) if we take into accountthe equalities∂tE•t,x,y = −∆xE•t,x,y = −∆yE•t,x,y,E•t,x,y = E•t,y,xand use the formula (3.67) in Proposition 3.8.

Thus Proposition 2.14 is proved. HenceLemma 2.2 is proved in the case of a < 4−1λ1(ν0).□Let now a > 0 be an arbitrary number such that a /∈∪j Spec (∆ν0,j).

Then for anynonzero element l ∈det W •a (ν0) we have by Lemma 2.1 that∂γ log ∥gνl∥2det W •a (ν0)γ=0 = −2 ∂γ log(kν)γ=0X(−1)j Trp2Πja(ν0),where gν = Π•avν is defined by (2.13). We know that under the identifications (2.4)and (2.6), the analytic torsion norm T0 (Mν, Z; a) on det W •a (ν) transforms into theanalytic torsion norm T0 (Mν, Z) on det H•(Mν, Z), ∥gνl∥2T0(Mν,Z;a) = ∥(gνl)H∥2T0(Mν,Z).

(Here, (gνl)H ∈det H•(Mν, Z) corresponds to gνl under these identifications.) Let(gνl)H be fixed.

Then the analytic torsion norm of gνl ∈det W •a (ν)∥gνl∥2T0(Mν,Z;a) := ∥gνl∥2det W •a (ν) · T(Mν, Z; a)(2.108)is independent of a > 0. Let ν0 = (α0, β0) ∈U (i.e., α0β0 ̸= 0).

Then using (2.108),(2.89) we obtain (µ := ∂γ log(kν)γ=0):∂γ log T(Mν, Z; a)γ=0 = limε→+0 ∂γ log T(Mν, Z; ε)γ=0 ++ 2µ−limε→+0X(−1)j Trp2Πjε(ν0)+X(−1)j Trp2Πja(ν0)== 2µX(−1)j (Tr p1 exp(−t∆ν0,j))0 −X(−1)j Tr Πjε(ν0) +X(−1)j Trp2Πja(ν0). (2.109)Note that for arbitrary c > 0, a > 0 we haveX(−1)j Tr Πjc(ν0) = χ(Mν0; Z) =X(−1)j Tr(p1 + p2)Πja(ν0).So the final expression in (2.109) is equal to2 ∂γ log(kν)γ=0X(−1)j Trp1exp(−t∆ν0,j) −Πja(ν0)Thus Lemma 2.2 is proved for an arbitrary a > 0 such that a /∈∪j Spec (∆ν0,j).□

68S.M. VISHIK2.7.

Continuity of the truncated scalar analytic torsion. Proof of Propo-sition 2.1.

Taking into account the definition (2.11) of the truncated scalar analytictorsion T(Mν, Z; a), we see that Proposition 2.1 is a consequence of the assertion asfollows. For Re s > (dim M)/2 the truncated ζ-function (2.8) for ∆ν,j is defined bythe integral38ζν,j(s; a) = Γ(s)−1Z ∞0ts−1 Trexp (−t∆ν,j)1 −Πja(ν)dt(2.110)Proposition 2.17.

For a > 0 the truncated determinant for the ν-transmission in-terior boundary conditionsdet (∆•ν; a) := exp−∂/∂sζν,•(s; a)s=0is a continuous function of ν for ν ∈R2 \ (0, 0) such that a /∈Spec (∆•ν).Proof. Let E•t,x,y(ν), t > 0, be the kernel of exp (−t∆•ν).

According to Proposition 3.8we haveTr exp (−t∆•ν) =Xk=1,2ZMktr∗x2i∗kE•t,x1,x2,(2.111)where i: M k ֒→M k × M k are the diagonal immersions. (The exterior product of therestriction to the diagonal of double forms is implied in (2.111).) Set I = [−1, 1].

LetI ×N ⊂M be the inclusion of the neighborhood of N = 0×N into M and let gM bea direct product metric on I ×N. Let ∆•ν;0 be the Laplacian on DR•(I ×N) with theν-transmission boundary conditions on N = 0 × N and with the Dirichlet boundaryconditions on ∂I × N. Let E•t,x,y(ν; 0), t > 0, be the kernel of exp−t∆•ν;0.

Theequality analogous to (2.111) holds also for Tr exp−t∆•ν;0(where M k is replacedby Qk × N, Q1 := [−1, 0], Q2 := [0, 1]).Let ν0 ̸= (0, 0) and let a /∈Spec∆•ν0be a fixed positive number. Then fromTheorem 3.2, Proposition 3.1, and from the estimate (2.86) it follows that for anarbitrary ε > 0 there are a neighborhood U0(ε) of ν0 and T > 0 such that forν ∈U0(ε) and for t ≥T the estimate holds|Tr (exp (−t∆•ν) (1 −Π•a(ν)))| ≤ε exp(−at/2).

(2.112)To prove the continuity of det (∆•ν) in ν at ν = ν0, it is enough to obtain thefollowing estimate.39 For a given b, 1 > b > 0, and for an arbitrary ε > 0 there exists38The analytic continuation of this integral from Re s > (dim M)/2 to the whole complex planecoincides with the meromorphic continuation of ζν,j(s; a).39The integrals (2.110) for the values ν0 and ν of the transmission parameter have the analyticcontinuations from Re s > (dim M)/2 to the whole complex plane. It follows from the estimates(2.113) and (2.112) that the difference of these integrals multiplied by Γ(s) is an absolutely conver-gent integral for Re s > −1.

Hence this difference is an analytic function of s for such s and it isequal to the difference of the analytic continuations of the integrals (2.110) for ν and ν0.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY69a neighborhood Uε ∋ν0 such that the estimate holds for ν ∈Uε, −(1 −b) < s < 1:Z T0Tr(exp(−t∆•ν)(1−Π•a(ν)))−Trexp−t∆•ν0(1−Π•a(ν0))ts−1dt<ε.

(2.113)The spectrum Spec (∆•ν) is discrete and it depends continuously on ν (by Propo-sition 3.1). Since a /∈Spec (∆•ν) we see that Tr Π•a(ν) = Tr Π•a(ν0) (= rk Π•a) in aneighborhood of ν0 and that the following estimate is satisfied uniformly with re-spect to s, −1/2 < s < 1 and to ν for ν sufficiently close to ν0:Z T0Tr (exp (−t∆•ν) Π•a(ν)) −Trexp−t∆•ν0Π•a(ν0) ts−1dt < ε/2 (2.114)The inequalityZ T0Tr exp (−t∆•ν) −Tr exp−t∆•ν0 ts−1dt < ε/2(2.115)for ν sufficiently close to ν0 and for −1/2 < s < 1 is obtained as follows.

Accordingto Proposition 3.1, Tr exp (−t∆•ν) is equal to the integral over ∪M j of the densitydefined by the restriction to the diagonal of the corresponding kernel. So it is enoughto estimate in (2.115) the integrals of the difference between the densities definedby exp (−∆•ν) and by exp−t∆•ν0separately over a fixed neighborhood U of N =0×N ֒→M and over M\U.

The estimate of the integral over U ⊃N is obtained withthe help of the kernel E•t,x,y(ν; 0) of exp−t∆•ν;0. Set e•t,x,y(ν) := E•t,x,y(ν)−E•t,x,y(ν; 0)for x, y ∈I × N.Proposition 2.18.

For an arbitrary m ∈Z+ there is a neighborhood of ν0 such thatfor all x, y ∈M[−1/2,1/2] := [−1/2, 0] × N ∪[0, 1/2] × N ֒→M1 ∪M2 and for t ∈(0, 1]the estimate is satisfied uniformly with respect to ν ∈R2 \ (0, 0)E•t,x,y(ν) ≤cmtm,(2.116)(where cm is independent of t and of ν).Proposition 2.19. The following estimate holds uniformly with respect to s for−(1 −b) < s < 1 and to ν for ν sufficiently close to ν0Z T0 ts−1dtZM[−1/2,1/2]tr∗i∗ke•t,x1,x2(ν)−tr∗i∗ke•t,x1,x2(ν0) < ε/4.

(2.117)

70S.M. VISHIKRemark 2.9.

For x, y ∈[−1, 1] × N the equalities hold (analogous to (2.54), (2.55),(2.40)):E•t,x,y(ν; 0) =E•t,x,y +β2 −α2.β2 + α2(σ∗1E•)t,x,yfor x, y ∈Q1 × N,E•t,x,y +α2 −β2.β2 + α2(σ∗1E•)t,x,yfor x, y ∈Q2 × N,2αβ.α2 + β2E•t,x,yfor x, y from different Qk×N. (2.118)Here, E•t,x,y is the kernel of exp (−t∆•) on I × N with the Dirichlet boundary condi-tions on ∂I × N and σ1 is the mirror symmetry with respect to N = 0 × N actingon the variable x of the kernel.

So we getZM[−1/2,1/2]tr∗x2i∗ke•t,x1,x2(ν)=ZM[−1/2,1/2]tr∗x2i∗kE•t,x1,x2(ν)−E•t,x1,x2. (2.119)From this equality and from the estimate (2.117) it follows that the integral overM[−1/2,1/2] of the difference between the kernels for ν and for ν0 gives the term in(2.115) which is less than ε/4.Proposition 2.20.

The following estimate holds uniformly with respect to s, −(1 −b) < s < 1, and to ν for ν sufficiently close to ν0:Z T0ZM\M[−1/2,1/2]tr∗x2i∗E•t,x1,x2(ν)−tr∗x2i∗E•t,x1,x2(ν0) dt < ε/4. (2.120)The estimate (2.115) is a consequence of (2.117), (2.119), and (2.120).

The estimate(2.113) follows from (2.114), (2.115). The estimate (2.115) together with (2.112) givesus the continuity of ∆• (ν; a) in ν at ν0.

Thus we get the proofs of Propositions 2.17and 2.1.□Proof of Proposition 2.18. The following equality is obtained similarly to (2.98)by using of (2.99):et,x,y(ν) = −limε→+0Z t−εεdτ∂/∂τZ∂I×N(rδ)∗E•τ,x,∗(ν), A∗E•t−τ,∗,y(ν; 0)∂I×N ++r∗E•τ,x,∗(ν), (Ad)∗E•t−τ,∗,y(ν; 0)∂I×N,(2.121)where the operators r and A correspond to the pair (I × N, ∂I × N).

So the estimate(2.116) follows from the analogous estimates for the kernelsrzE•t,x,z(ν),(rδ)zE•t,x,z(ν),AzE•t,z,y(ν; 0),(Ad)zE•t,z,y(ν; 0),(2.122)where x, y ∈M[−1/2,1/2] and z ∈∂I × N = {−1, 1} × N ֒→M. Such estimates arederived with the help of Proposition 2.5 for ∆•ν and ∆•ν;0 as follows.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY71Let m ∈Z+ be taken large enough.

Then there is an approximate fundamentalsolution P •(m)(ν) for∂t + ∆•ν,xwhich is the sum of an interior term P •(m)intand ofterms, defined near the boundaries ∂M and N. The kernel P •(m)(ν) is a good approx-imation for E•t,x,y(ν) for small t > 0. Its interior term is defined as follows.

For anyclosed Riemannian (M, gM) there is a locally defined parametrix p•(m)t,x,y (i.e., an approx-imate fundamental solution for (∂t + ∆•M)) such that the differencep•(m) −E•t,x,y(where E•t,x,y is the fundamental solution for (∂t + ∆•M)) is a C∞-double form for t > 0and such that the following estimates hold uniformly with respect to (x, y) ∈M ×Mand to t ∈(0, T]:p•(m)t,x,y −E•t,x,y ≤cmt−n/2+m+1,(∆•x)k p•(m)t,x,y −E•t,x,y ≤cm,kt−n/2+m+1−k(∂t + ∆•M) p•(m)t,x,y < c(m)t−n/2+m,(∆•x)k (∂t + ∆•M) p•(m)t,x,y < c(m,k)t−n/2+m−k(2.123)([RS], Proposition 5.3, [BGV], Theorems 2.20, 2.23, 2.26, 2.30). Such a parametrixcan be represented in the following form (n := dim M):p•(m)t,x,y = (4πt)−n/2 exp−d(x, y)2/4tf (d(x, y))mXi=0tiΦi(x, y),(2.124)where d(x, y) is the geodesic distance between x and y, f ∈C∞0 (R+), f(τ) ≡1 for0 ≤τ ≤ε and f ≡0 for τ > 2ε.

The injectivity radius i (M, gM) is supposed tobe greater than 2ε, i.e., the exponential map expx B2ε is a diffeomorphism on itsimage for any x ∈M (where B2ε := {ξ ∈TxM, |ξ| ≤2ε}). The coefficients Φi(x, y)in (2.124) are smooth double forms on M × M whose germs on the diagonal M ֒→M ×M are unique.

The principal term Φ0(x, y) is the kernel of the parallel transportin ∧•TM along the geodesic line expy ξ = x from y into x (and it is defined ford(x, y) < i (M, gM)). Each Φi(x, y) is determined through Φi−1(x, y) in differentialgeometry terms and it is well-defined for d(x, y) < i (M, gM) ([RS], Sect.

5, [BGV],Theorem 2.26, Lemma 2.49).Let 0 × N ֒→I × N ֒→M be a neighborhood of N = 0 × N ֒→M, where themetric gM is a direct product. The fundamental solution for (∂t + ∆•ν) on I ×N (withthe Dirichlet boundary conditions on ∂I × N) isP•(ν) =Xi=0,1EiI,t(ν) ⊗E•−iN,t ,(2.125)where E•I,t(ν) is defined by the formulas completely analogous to (2.40) and (2.118).

(Here, E•N,t is the fundamental solution for (∂t + ∆•N). The operator corresponding

72S.M. VISHIKto the kernel EiI,t(ν) ⊗E•−iN,t acts on DRi(I, ∂I) ⊗DR•−i(N).

The kernel EiI,t(ν)corresponds to the Laplacian on I with the Dirichlet boundary conditions on ∂I andwith the ν-transmission boundary conditions at 0 ∈I. The term in (2.125) withi = 1 is equal to zero for • = 0.

)The parametrix P •(m)t,x,y (ν) for (∂t + ∆•ν) on M is defined byP •(m)t,x,y (ν) = ψP•t,x,y(ν)ϕ + ψ1p•(m)t,x,y (1 −ϕ). (2.126)Here, ϕ = ϕ(y1), ψ = ψ(x1) (in the coordinates (x1, x′) = x and (y1, y′) = y of pointsin I × N), ϕ, ψ ∈C∞0 (I, ∂I); ϕ, ψ ≥0, ϕ(y1) ≡1 for |y1| ≤2−1 + ε, ϕ(y1) ≡0for |y1| ≥5/8, ψ ≡1 in a neighborhood of supp ϕ, and ψ ≡0 for |x1| ≥3/4,ψ1 ∈C∞0 (M \ N), ψ1 ≡1 in a neighborhood of supp(1 −ϕ) ⊂M \ M[−1/2,1/2].

Hencethe parametrix P •(m)t,x,y is equal to zero for y ∈M[−1/2,1/2], x ∈M \ ([−3/4, 3/4] × N).The term ψ1p•(m)(1 −ϕ) in (2.126) is defined from now on as P •(m)int . (In the case of∂M ̸= ∅the terms, completely analogous to P•(ν) for ν ∈{(0, 1), (1, 0)}, have to beadded to P •(m).

Their supports are in ([0, 1] × ∂M)2 ֒→M × M and gM is a directproduct metric on [0, 1] × ∂M. )Proposition 2.21.

1. The boundary condition for P •(m)t,x,y (ν) on N and on ∂M andthe boundary condition for (∆•x)k P •(m)t,x,y (ν) (k ∈Z+) are the same as for E•t,x,y(ν) andfor∆•ν,xk E•t,x,y(ν).

Namely P •(m)t,x,y (ν) is a smooth in t > 0 and in (x, y) ∈M j1 ×M j2kernel, ∆kν,xP (m)t,x,y(ν) ⊂D(∆•ν,x) for fixed y, t > 0, and for any k ∈Z+ ∪0. HereD∆•ν,x⊂DR• (Mν, Z) is the domain of definition of ∆•ν,x on pairs (ω1, ω2) ofsmooth forms on Mj.

It is defined by ( 1.27)).2. The following estimates (analogous to ( 2.123)) hold for t ∈(0, T] uniformlywith respect to ν ∈R2 \ (0, 0) and to (x, y) ∈M j1 × M j2 (with Cm, Cm,k independentof t):∂t + ∆•ν,xP •(m)t,x,y (ν) < Cmt−n/2+m,(2.127)∆•ν,xk ∂t + ∆•ν,xP •(m)t,x,y (ν) < Cm,kt−n/2+m−k.

(2.128)The kernel r•(m)t,x,y (ν) :=∂t + ∆•ν,xP •(m)t,x,y (ν) is smooth in (x, y) ∈M × M j and itsC2l-norm on M × M j is estimated through cm,lt−n/2+m−l. For any linear differentialoperator F of order d = d(F), acting on double forms on M × Mj, and for anyT > 0 the kernel F ◦r•(m)t,x,y (ν) satisfies the estimate as follows when t ∈(0, T].

It holduniformly with respect to (x, y) ∈M × M j and to ν ∈R2 \ (0, 0)F ◦r•(m)t,x,y (ν) < c(F)t−(n+d)/2+m. (2.129)

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY733.

For 0 ≤k ≤[−n/2] + m −1 the following condition is satisfied uniformly withrespect to ν ∈R2 \ (0, 0) and to (x, y) ∈M j1 × M j2:limt→+0∆•ν,xk E• −P •(m)t,x,y (ν) = 0. (2.130)Corollary 2.8.

1. For k ∈Z+, 0 ≤k ≤[−n/2] + m −1 the following estimates ofL2-norms (with respect to the variable x) of the kernelP •(m)(ν) −E•(ν)t,x,y holdfor t ∈(0, T] uniformly in y ∈∪j=1,2M j and in ν ∈R2 \ (0, 0), where Cm and Cm,kare the same as in ( 2.127) and ( 2.128):E•(ν) −P •(m)(ν)t,∗,y2≤Cm(−n/2 + m + 1)−1t−n/2+m+1,(2.131)∆kν,∗E•(ν)−P •(m)(ν)t,∗,y2≤Cm,k (−n/2+m+1−k)−1t−n/2+m+1−k.(2.132)2.

The following estimate for E•t,x,y(ν) holds for t ∈(0, T] and for an arbitraryq ∈Z+ uniformly with respect to ν ∈R2 \ (0, 0), to x ∈M \ ([−3/4, 3/4] × N), andto y ∈M[−1/2,1/2]:E•t,x,y(ν) ≤Cqt−n/2+q. (2.133)It holds according to ( 2.131), ( 2.132), and ( 2.32).

(Indeed, for such x, y and foran arbitrary m ∈Z+ we have E•t,x,y(ν) =E•(ν) −P •(m)(ν)t,x,y. Hence m can bechosen large enough to get ( 2.133).

)The estimate (2.131) is a consequence of (2.127) and of the following equalityE• −P •(m)t,x,y (ν) = limε→+0Z t−εεdτ∂τE•t−τ,x,∗(ν),E• −P •(m)τ,∗,y (ν)M == limε→+0Z t−εεdτE•t−τ,x,∗(ν),∂τ + ∆•ν,∗ E• −P •(m)τ,∗,y (ν)M . (2.134)This equality follows from the assertions thatE• −P •(m)t,x,y (ν) →0 as t →+0and thatE• −P •(m)τ,x,y (ν) ∈D∆•ν,xwith respect to the variable x.

74S.M. VISHIKThe estimate (2.131) is a consequence of (2.134), (2.127), and (2.135), since theoperator exp (−t∆•ν) for t > 0 is bounded in (DR•(M))2 and its operator norm isless or equal to one:∥exp (−t∆•ν)∥2 ≤1.

(2.135)(This inequality follows from Theorems 3.1 and 3.2. They claim that ∆•ν is a nonneg-ative self-adjoint unbounded operator in (DR•(M))2 and that exp (−t∆•ν) is a traceclass operator.

)The inequality (2.132) for 1 ≤k ≤[−n/2] + m −1 is a consequence of (2.127),(2.135), and of the following equality (which is a generalization of (2.134)):∆kν,xE• −P •(m)t,x,y (ν) == limε→+0Z t−εεE•t−τ,x,∗(ν),∆•ν,∗k (∂τ + ∆ν,∗)E• −P •(m)τ,∗,y (ν)M .This equality holds since ∆kν,xE• −P •(m)t,x,y (ν) →0 as t →+0 (for 0 ≤k ≤[−n/2] + m −1) and since∆•ν,xk E• −P •(m)t,x,y (ν) ∈D∆•ν,xfor fixed y andt > 0.The proof of Proposition 2.21 is preceded by the proof of Proposition 2.18.The estimates analogous to (2.133) (with t ∈(0, T] and q ∈Z+) hold also forthe kernel E•t,x,y(ν; 0) = P•t,x,y of exp−t∆•ν;0, where x ∈(I \ [−3/4, 3/4]) × N, y ∈M[−1/2,1/2]. Indeed, such estimates are true (n is replaced by 1) forEiI,t(ν)x1,y1 withx1 ∈I \ [−3/4, 3/4] and y1 ∈[−1/2, 1/2], and the kernelE•−iNx′,y′ is Ot−(n−1)/2for t ∈(0, T].The desired estimates for (rδ)E•(ν) and AdE•(ν; 0) are obtained from the estimates(2.131), (2.132), and from the generalization of the inequality (2.32) as follows.

LetK be an arbitrary first order differential operator acting on DR• Mj. Let ω ∈DR• (Mν, Z) obeys the conditions (2.33) with m1 = 1 + min{l : 4l ≥dim M + 1}.Then the inequality is satisfied uniformly with respect to ω and to x ∈M1 ∪M2:|Kω(x)|2 < CKm1Xi=0∆iνω22 ,(2.136)where CK > 0 is independent of ν ∈R2 \ (0, 0).

The proof of (2.136) is exactly thesame as the proof of (2.32) given above except the kernel (GI(ν) ⊗Gm2N )x,y has to bereplaced by Kx (GI(ν) ⊗Gm2N )x,y. Thus Proposition 2.18 is proved.□Proof of Proposition 2.21.

1. For x from a neighborhood of N = 0 × N ֒→M,where ψ1 ≡0, the parametrix P •(m)t,x,∗(ν) is equal to P•(m)t,x,∗(ν)ϕ(∗).

So P •(m)t,x,∗(ν) ∈

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY75D∆•ν,xkwith respect to the variable x, since P•(m)t,x,∗(ν) ∈D∆•ν,xkfor k ∈Z+.2.

The estimates (2.127) and (2.128) hold for the term ψP•(ν)ϕ of P •(m)(ν) withan arbitrary m ∈Z+, since∂t + ∆•ν,xP•(ν) = 0 (for x ∈(I \ ∂I) × N) and sinceminx1∈A miny1∈supp ϕ (|x1 −y1|, |x1 + y1|) > δ > 0 (where A := supp (∂x1ψ) and thenumber δ > 0 is fixed). For x1 ∈A and y1 ∈supp ϕ the estimate (2.133) (n isreplaced by 1) holds with an arbitrary q ∈Z+ for the fundamental solutionE•I,tx,yof (∂t + ∆•I) with the Dirichlet boundary conditions on ∂I.

The same estimate forsuch x1, y1 holds for the kernelsσ∗1E•I,tx1,y1 andE•I,t(ν)x1,y1.So this estimateholds also for the kernel (P•(ν))(x1,x′),(y1,y′) (defined by (2.126)), since the kernelE•N,tx′,y′ is Ot−(n−1)/2uniformly with respect to (x′, y′) ([RS], Proposition 5.3,[BGV], Theorem 2.23). By the analogous reasons, for such x1, y1 the estimate (2.133)with an arbitrary q ∈Z+ holds for the kernel (F ◦P•(ν))(x1,x′),(y1,y′), where F is alinear differential operator of finite order d(F) on M ×M, acting on double forms onM j1 × Mj2, and n in (2.133) is replaced by n + d(F).So the estimate (2.133) with an arbitrary q ∈Z+ is satisfied by∂t + ∆•ν,x(ψP•ϕ)and by∆•ν,xk ∂t + ∆•ν,x(ψP•ϕ) with k ∈Z+.The estimates (2.127) and (2.128) hold for ψ1p•(m)(1 −ϕ) =: P •(m)int , since theyhold for p•(m)t,x,y and since the distance on M between the closure B of B (whereB := {x: dxψ1 ̸= 0}) and the support supp(1 −ϕ) is greater than a positive numberδ.

Hence the uniform with respect to (x, y) ∈B × supp(1 −ϕ) estimate (2.133)(with an arbitrary q ∈Z+) is satisfied by p•(m)t,x,y . This estimate holds also for thekernel Fp•(m)t,x,y , where F is a linear differential operator of finite order d(F), actingon the smooth kernels, defined on M × M. (For instance, the function f (d(x, y)) inthe definition (2.124) of p•(m)t,x,y can be chosen such that f(τ) ≡0 for τ ≥δ.

Theseestimates follow also from [RS], Proposition 5.3, estimates (5.5), and from [BGV],Theorem 2.23(2).)403. The differenceE• −P •(m)t,x,y (ν) can be written as the Volterra series (anal-ogous to [BGV], 2.4):E• −P •(m)t,x,y (ν) ==Xl≥1(−t)lZ∆lZ(y1,...,yl)∈(M1∪M2)l P •(m)σ0t,x1,y1(ν)r(m)σ1t,y1,y2(ν) .

. .

r(m)σlt,yl,y(ν),(2.137)40For the sake of brievity the proof of Proposition 2.21 is written in the case of ∂M = ∅.

76S.M. VISHIKwhere ∆l = {(σ0, .

. .

, σl): 0≤σi ≤1, P σi =1}41and r(m)t,x,y(ν) := (∆ν,x+∂t) P •(m)t,x,y (ν) (ascalar product tr(ω1∧∗ω2) with its values in densities on M is implied in (2.137)). Theassertion (2.130) follows from the convergence of the series (2.137) in the topologyof uniform convergence of smooth kernels on Mj1 × M j2 together with their partialderivatives of orders ≤2k on Mj1 × M j2 (i.e., in the C2k-topology on M j1 × M j2).Indeed, the definition of P •(m)t,x,y (2.126) involves that the kernel r(m)t,x,y(ν) is equal tozero for x from a fixed (independent of t, ν, and m) neighborhood of N = 0×N ֒→Min M.So r(m)t,x,y(ν) is a smooth kernel on M × Mj, and the inequalities (2.129)claim that the C2k-norm of r(m)t,x,y(ν) is Ot−n/2+m−kfor t ∈(0, T] uniformly withrespect to ν ∈R2 \ (0, 0).

It is O(t) for 0 ≤k ≤[−n/2] + m −1, and the series(2.137) is convergent in the C2k-topology for such k, since the volume of ∆l is (l! )−1and since the following assertion is true.

For any T > 0 the parametrix P •(m)t,x,y (ν)defines a family of bounded operators from the space of smooth forms DR•(M)(equiped with a C2k-norm) into the space ⊕j=1,2DR• M j(equiped with a C2k-normon DR• Mj). These operators are bounded uniformly with respect to t ∈(0, T] andto ν ∈R2 \ (0, 0).

This assertion for the operators, corresponding to P •(m)int,t , is provedin [BGV], Lemma 2.49. It is also true for the operators corresponding to ψP•(m)t(ν)ϕ.Indeed, it holds for the operators exp (−t∆•N) in DR•(N) (equiped with a C2k-norm)and for the operators ψ exp (−t∆•I) ϕ in DR•(I) equiped with a C2k-norm.

(Here ∆•Iis defined on forms with the Dirichlet boundary conditions on ∂I.) It holds also forthe operators with the kernels ψ(x1)σ∗1E•I,tx1,ya ϕ(y1), acting from smooth forms onI into smooth forms on [0, ±1] (where the Dirichlet boundary conditions are impliedon ∂I and σ1 is the mirror symmetry with respect to 0 ∈I).The C2k-norm of the kernelP •(m)N,t−E•N,ton N × N is Otm−(n−1)/2−kfort ∈(0, T], where P •(m)N,tand E•N,t are the parametrix of the type (2.124) and the fun-damental solution for (∂t + ∆•N) ([BGV], Theorem 2.30).

The operators in DR•(N)corresponding to P •(m)N,tare uniformly bounded for t ∈(0, T] with respect to a C2k-norm ([BGV], Lemma 2.49). So the operators exp (−t∆•N) in DR•(N) are uniformlybounded for t ∈(0, T] with respect to a C2k-norm.

The convergence of the serieson the right in (2.137) with respect to C2k-norms (k ≤[−n/2] + m −1) for thekernels on Mj1 × M j2 involves also a proof of the equality (2.137) (in the case ofm ≥−[−n/2] + 2). Indeed, we have42∂t + ∆•ν,x P •(m)lt,x,y =r•(m)l+ r•(m)l+1t,x,y ,(2.138)41Here σi = ti+1 −ti for 1 ≤i ≤k −1, σ0 = t1, σk = 1 −tk.

The volume of ∆k with respect tothe density dt1 . .

. dtk is equal to 1/k!.42The proof of (2.138) is analogous to the one given in [BGV], Lemma 2.22.

It follows from theformula for ∂tR t0 f(x, t)dx.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY77where (−1)mP •(m)lis the term with the number l in the right side of (2.137), (−1)mr•(m)lis the same term in which P •(m)σ0tis replaced by r•(m)σ0t , P •(m)0:= P •(m).

For any fixedy and t > 0 and for any k ∈Z+ ∪0 we have∆kν,x P •(m)lt,x,y ⊂D∆•ν,x. Theseries P •t = Pl≥0(−1)mP •(m)lfor m ≥[−n/2] + 2 is the fundumental solution for(∂t + ∆•ν) since∂t + ∆•ν,xP •t = 0 for t > 0 and since the operator correspondingto P •(m)ttends in a weak sense to the identity operator in (DR•(M))2 as t →+0(i.e., P •(m)tω →ω as t →+0 for ω ∈(DR•(M))2).

The latter assertion holds forP •(m)t,int (1 −ϕ)ω and for P•(m)ϕω. Proposition 2.21 is proved.□Proof of Proposition 2.19.

Proposition 2.18 involves the following conclusion. Forany ε > 0 there exist a δ > 0 and a neighborhood U := U(ν0, ε) ⊂R2 \ (0, 0) of ν0such that the estimate holds uniformly with respect to ν ∈U and to s, −1 < s < 1:Z δ0 ts−1dtZM[−1/2,1/2]Xk=1,2tr (∗i∗ke•t(ν)) < ε/20.So it is enough to prove the existence of a neighborhood U1 of ν0 such that for anyν ∈U1 the following estimate holds uniformly with respect to s for −(1 −b) < s < 1(b, 0 < b < 1, is fixed):Z Tδ ts−1dtZM[−1/2,1/2]Xk=1,2(tr (∗i∗ke•t(ν)) −tr (∗i∗ke•t(ν0))) < ε/10.

(2.139)For e(ν) and e(ν0) the equalities (2.121) hold. So the estimate (2.139) takes placefor ν sufficiently close to ν0 since the convergenceE•t,x,y(ν) →E•t,x,y(ν0)(2.140)is uniform with respect to t∈[δ1, T] (where δ1 >0 is fixed), to x∈M \((−3/4,3/4)×N),and to y ∈M[−1/2,1/2].

The convergence of the kernelsdxE•t,x,y(ν) →dxE•t,x,y(ν0),δxE•t,x,y(ν) →δxE•t,x,y(ν0). (2.141)is a uniform one for such (t, x, y).

All the double forms in (2.140) and (2.141) areuniformly bounded on the set of such (t, x, y) and their norms at (t, x, y) satisfy theupper estimate for t ∈(0, δ1] (obtained in Proposition 2.18 above) through cmtm withan arbitrary m ∈Z+ and with cm independent of ν. The uniform convergence of thekernels in (2.140) and (2.141) on the compact set of (t, x, y) defined above followsfrom the continuity in t, x, y, and ν for (x, y) ∈M j1 × Mj2 of the correspondingdouble forms.

(See Proposition 3.2, where it is proved that these double forms areC∞-smooth in t, x, y, and ν for Re t > 0 and ν ̸= (0, 0). )□

78S.M. VISHIKProof of Proposition 2.20.

If (α0, β0) = ν0 ∈U (i.e., if α0 · β0 ̸= 0) then wecan suppose that ν ∈U in (2.120). In this case, the identity (2.100) holds for thedifference(E•(ν) −E•(ν0))t,x1,x2 .

(2.142)Let ν0 ∈R2 \ (U ∪(0, 0)).For example, let ν0 := (α0, 0), α0 ̸= 0.Then theidentities (2.98) and (2.99) claim in the cases of ν0 and of ν := (α, β) that (2.142) isequal to βα!limε→+0Z t−εεdτ−r∗,∂M2δE•τ,x1,∗(ν), A∗,∂M1E•t−τ,∗,x2(ν0)N ++A∗,∂M1E•τ,x1,∗(ν), r∗,∂M2δ∗E•t−τ,∗,x2(ν0)N+A∗,∂M1d∗E•τ,x1,∗(ν), r∗,∂M2E•t−τ,∗,x2(ν0)N−−r∗,∂M2E•τ,x1,∗(ν), A∗,∂M1d∗E•t−τ,∗,x2(ν0)N.(2.143)The factor k−1ν= β/α in (2.143) tends to zero as ν tends to ν0.The factors(1 −kν/kν0) and (1 −kν0/kν) in (2.100)) also tend to zero as ν →ν0 in the caseν0, ν ∈U. The estimate (2.120) follows from (2.143) and (2.100).

Indeed, there arethe uniform with respect to ν upper estimates (analogous to (2.116)) for the kernels(2.122), where x, y ∈M \ M[−1/2,1/2] and z ∈N = 0 × N ⊂∂Mj. These estimatesfollow from (2.32) and their proofs are completely analogous to the proof of (2.116).The main step in these proofs is using of the parametrix P (m)43 and of the estimates(2.127), (2.128), and (2.132) for t ∈(0, T].□2.8.

Dependence on the phase of a cut of the spectral plane. The analytictorsions as functions of the phase of a cut.

Gluing formula for the analytictorsions. The scalar analytic torsion (2.11) depends not only on (M, gM, Z, ν) butalso on the phase θ of a cut on the spectral plane C ∋λ.

A zeta-function ζν,•(s; θ)is defined for Re s > n/2 (n := dim M) as the sum of absolutely convergent seriesP m(λj)λ−sj,θ, where the sum is over nonzero λj ∈Spec (∆•ν) and m(λj) are the multi-plicities of λj. The function λ−sj,θ := exp−s log(θ) λj, θ > Im(log(θ) λj) > θ −2π, isdefined for θ /∈arg λj +2πZ.

(For positive self-adjoint operators this condition meansthat θ /∈2πZ.) All the results for the analytic torsion norm are obtained above inthe case of 0 < θ < 2π (for instance, for θ = π).The zeta-function ζν,•(s; θ) does not depend on θ /∈2πZ, if [θ/2π] does not change.However we haveζν,•(s; θ + 2π) = exp(−2πis)ζν,•(s; θ)for Re s > n/2.

(2.144)43The parametrix P (m),•(ν) for E•(ν) can be chosen such that P (m),•t,x,z (ν) = 0 for x /∈[−1/3, 1/3]×N ֒→M and z ∈[−1/6, 1/6] × N ֒→M.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY79Since ζν,•(s; π) can be meromorphically continued to the complex plane C ∋s (The-orem 3.1 below), we see that ζν,•(s; θ) for θ /∈2πZ also can be meromorphicallycontinued to C with at most simple poles at sj := (n −j)/2.

The continuation ofζν,•(s; θ) is regular at sj for (−sj) ∈Z+ ∪0. So the equality (2.144) holds for allθ /∈2πZ, s ∈C.

Hence for such θ we haveζν,•(s; θ) = ζν,•(s) exp(−2πism),(2.145)where ζν,•(s) := ζν,•(s; π) and m := [θ/2π], θ /∈2πZ. From now on this ζ-functionwill be denoted as ζν,•(s; m) with m = [θ/2π].

The value ζν,•(0, m) is independent ofm (according to (2.145)).The dependence of the scalar analytic torsion (2.11) on m is given byT (Mν, Z; m) = T(Mν, Z) exp (−2πimF (Mν, Z) , )(2.146)where44 F (Mν, Z) := P (−1)j jζν,•(0)(mod Z) and T (Mν, Z) := T (Mν, Z; 0), i.e.,T (Mν, Z) corresponds to θ = π and it is the scalar analytic torsion defined by (2.11).Here Z is the union of the connected components of ∂M where the Dirichlet boundaryconditions are given. The Laplacian ∆•ν is defined on ω ∈DR•(Mν, Z) with the ν-transmission boundary conditions (1.27) on N, with the Dirichlet and the Neumannboundary conditions on Z and on ∂M \ Z.

The equality (2.146) is obtained by usingof∂sζν,•(s; m)s=0 = −2πimζν,•(0) + ∂sζν,•(s)s=0.The number F(M, Z) is defined also in the case of a manifold M without an interiorboundary N. In this case, ζν,•(0) in the definition of F(M, Z) is replaced by ζ•(0)for the Laplacians on DR•(M, Z). The dependence of T(M, Z; m) on m is given by(2.146) with F (Mν, Z) replaced by F(M, Z).

In particular, F(M) is defined for aclosed M and also in the case ∂M ̸= ∅, Z = ∅. Let M be obtained by gluing two piecesM1 and M2 along the common component N of their boundaries, M = M1 ∪N M2,where N ⊂M is closed and of codimension one.Then F(M, Z) = F (M1,1, Z),according to Proposition 1.1.The class of F (Mν, Z) in C/Z is the same as the class (mod Z) of the numberF1 (Mν, Z) ∈C, whereF1 (Mν, Z) :=X(−1)j j (ζν,j(0) + dim Ker (∆ν,j)) .The Laplacian ∆ν,j (Mν, Z) with its domain Dom (∆ν,j) ⊂(DRj(M))2 is self-adjoint according to Theorem 3.1.

For Re s > 2−1 dim M the zeta-function ζν,j(s)is defined by the absolutely convergent series P m (λk) exp (−s log λk) ([θ/2π] = 0,θ ̸= 0), where the sum is over λk ∈Spec (∆ν,j), λk ̸= 0, and with the branch oflogarithm −π < Im log λ < π. Because log λk ∈R for λk > 0, the function ζν,j(s) is44By the definition, F (Mν, Z) ∈C/Z but it can be also defined as P (−1)j jζν,j(0) ∈C.

80S.M. VISHIKreal for real s. Hence F1 (Mν, Z) ∈R.

It is supposed from now on that a metric gMon M = M1 ∪N M2 is a direct product metric near N and near ∂M.Proposition 2.22. 1.

For a closed manifold M the number F1(M) is an integer.2. Let M = M1 ∪N M2 be obtained by gluing two its pieces M1 and M2 alongthe common component N of their boundaries.Let the ν-transmission boundaryconditions ( 1.27) be given on N, the Dirichlet boundary conditions be given on aunion Z of some connected components of ∂M and the Neumann boundary conditionsbe given on ∂M \ Z.

Let L be a closed Riemannian manifold. Then the followingholds:45F1 (Mν × L, Z × L) = χ(L)F1 (Mν, Z) + F1(L)χ (Mν, Z) .

(2.147)3. Let K ⊂∂M \ Z be a union of some connected components of ∂M.

Then thefollowing holds under the conditions on M above:F1 (Mν, Z) = F1 (Mν, Z ∪K) + F1(K) + 2−1χ(K).(2.148)4. Under the conditions on M above, the number F1 (Mν, Z) obeys a gluing propertyanalogous to the gluing property ( 2.1) for the analytic torsion norms.

Namely thefollowing holds:F1 (Mν, Z) = F1(M1, Z1 ∪N) + F1(M2, Z2 ∪N) + F1(N) + 2−1χ(N), (2.149)where Zk := Z ∩∂M k.Corollary 2.9. 1.

For a closed M the scalar analytic torsion T (M, [θ/2π]) is inde-pendent of θ /∈2πZ.2. Under the conditions of ( 2.147), ( 2.148), and ( 2.149), the following holds inR/Z:F (Mν × L, Z × L) = χ(L)F (Mν, Z) .F (Mν, Z) = F (Mν, Z ∪K) + 2−1χ(K).F (Mν, Z) = F (M1, Z1 ∪N) + F(M2, Z2 ∪N) + 2−1χ(N).Example 2.1.

The number F1(S1) is equal to −f0;1 = −f0;0.46 The latter one isequal to zero because the asymptotic expansion for Tr exp (−t∆0 (S1)) as to t →+0(where ∆0 is the Laplacian on functions) is f−1;0t−1/2 + f1;0t1/2 + f3;0t3/2 + . .

. .45The Euler characteristic χ (Mν, Z) := P(−1)i dim Hi (Mν, Z) is equal to the Euler character-istic of the complex (C•(Xν, X ∩Z), dc) (as it follows from Proposition 2.3).

Hence it is equal toχ(M, Z) and is independent of ν ∈R2 \ (0, 0).46The coefficients fk,j := fk,j(M, Z) are the coefficients in the asymptotic expansion P fk,jtk/2(k ≥−n) for Tr exp (−t∆j(M, Z)) as t →+0 for the Laplacian on DRj(M, Z) (n := dim M).

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY81Example 2.2.

The number F1(I, ∂I) is equal to −f0;1(I, ∂I) = −f0;0(I). Since S1has a mirror symmetry relative to its diameter, we have, taking into account (2.54)and (2.118),f0,0S1= f0;0(I) + f0;0(I, ∂I) = 0.

(2.150)Since f0;0(I, ∂I) −f0;1(I, ∂I) = χ(I, ∂I) = 1 and (2.150) holds, we see thatF1(I, ∂I) = −f0;1(I, ∂I) = −f0;0(I) = f0;0(I, ∂I) = −2−1. (2.151)Remark 2.10.

The equality (2.151) means that the analytic torsion T (I, ∂I; [θ/2π])is multiplied by the factor exp (−2πi · (−2−1)) = −1, if θ is replaced by θ + 2π(θ /∈2πZ).It is necessary to note the following.The scalar analytic torsion T(I, ∂I) :=T(I, ∂I; π) is the factor in the analytic torsion norm. But the latter one is the squareof the norm on the determinant line det H1(I, ∂I).

So the factor, corresponding tothe norm itself, is multiplied in the case of (M, Z) = (I, ∂I) by the factorexp (πiζ1(I, ∂I)|s=0) = exp(−πi/2) = −i,if θ is replaced by θ + 2π, θ /∈2πZ. Indeed, we have−2−1 = F1(I, ∂I) = −dim H1(I, ∂I) + ζ1(I, ∂I)|s=0,and so ζ1(I, ∂I)|s=0 = −2−1 = −F(I, ∂I).

For M = S1 it holds that −F(S1) =ζ1 (S1) |s=0 = −1, and so exp (−πiF (S1)) = −1.It follows from Proposition 2.23 below that F(Mν, Z) and F(M, Z) have a form(1/2) + Z, if the numbers n := dim M and χ(M, Z) are odd. So in this case thefactors exp (−πiF (Mν, Z)) and exp (−πiF(M, Z)) are equal to {±i}.Proof of Proposition 2.22.

1. Theorem 3.2 1. claims that the number ζν,j(0) +dim Ker (∆ν,j) is equal to the constant term f0;j (Mν, Z) in the asymptotic expansion(2.87) for Tr exp (−t∆ν,j) as t →+0.

So according to Theorem 3.2 1. the numberF1(Mν, Z) is equal to the sum of the integrals over M1, M2, N, and ∂M of the locallydefined densities. Then we haveF1 (Mν, Z) =X(−1)j jf0;j(Mν, Z).

(2.152)If (M, gM) is a closed Riemannian manifold then f0;j(M) = f0;n−j(M).Hencetaking into account (2.152) and (2.57), we get (for even n := dim M)F1(M) =X(−1)j jf0;j(M) = (n/2)X(−1)jf0;j(M) = (n/2)χ(M). (2.153)Let n be odd.

Then f0;j(M) is equal to zero since the asymptotic expansion fortr exp (−t∆j (M, gM)) (as t →+0) is t−n/2 P tlf2l−n;j (M, gM), where the sum is over

82S.M. VISHIKl ∈Z+ ∪0 ([Gr], Theorem 1.6.1; [BGV], Theorem 2.30).Hence F1(M) = 0 =(n/2)χ(M) for an odd n also.This number (n/2)χ(M) is an integer for any closed M. (The assertion that F1(M)is an integer follows also from the equality which holds for any closed even-dimensionalRiemannian (M, gM):Xj (−1)j ζj(M, s) =Xj(−1)jζn−j(M, s) = (n/2)X(−1)j ζj(M, s) = 0,because ζj(M, s) = ζn−j(M, s).)2.

Let λ ∈Spec (∆ν,j(M, Z)), µ ∈Spec (∆i(L)) and let mλ(j; Mν, Z), mµ(i; L) betheir multiplicities. If λ ̸= 0 and µ ̸= 0 then we haveX(−1)j+i (j + i)mλ(j; Mν, Z)mµ(i; L) = 0,(2.154)since the subcomplexes (V •λ (Mν, Z) , d) ֒→(DR• (Mν, Z) , d) andV •µ (L), d֒→(DR•(L), d), corresponding to the λ-eigenforms for ∆ν,•(M, Z) and to the µ-eigenformsfor ∆•(L), are acyclic.

If λ ̸= 0 but µ = 0 then the right side of (2.154) is equal toX(−1)j jmλ (j; Mν, Z) X(−1)i dim Ker ∆i(L)== χ(L)X(−1)j jmλ (j; Mν, Z) .So under the conditions of 2, by using (1.37), we haveX(−1)j jζν,j (Mν × L, Z × L)s=0 == χ(L)X(−1)j jζν,j (Mν, Z)s=0 + χ (Mν, Z)X(−1)j jζj(L)s=0,(2.155)X(−1)j j dim Ker (∆ν,j (Mν × L, Z × L)) == χ(L)X(−1)j j dim Ker (∆ν,j (Mν, Z)) + χ (Mν, Z)X(−1)j j dim Ker (∆j(L)) . (2.156)The equality (2.147) follows now from (2.155) and (2.156).3.

The numbers F1 (Mν, Z) and F1(Mν, Z ∪K) are the sums of the integrals overM1, M2, N, and ∂M of the locally defined densities (as it follows from (2.152) andfrom Theorem 3.2). The densities, corresponding to the pairs (Mν, Z) and (Mν, Z ∪K), differ only on K. So the difference F1(Mν, Z) −F1(Mν, Z ∪K) depends only onK and on gM near K. Thus taking into account that gM is a direct product metricnear K, we get2 (F1 (Mν, Z) −F1 (Mν, Z ∪K)) = F1(K × I) −F1 (K × (I, ∂I))(2.157)for any fixed metric on K in all the terms of this equality.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY83According to (2.147) we haveF1(K × I) = F1(K)χ(I) + F1(I)χ(K),F1 (K × (I, ∂I)) = F1(K)χ(I, ∂I) + F1(I, ∂I)χ(K),F1(K × I) −F1 (K × (I, ∂I)) = 2F1(K) + χ(K) (F1(I) −F1(I, ∂I)) .

(2.158)Since ζ1(s; I) = ζ1(s; I, ∂I) (on the same I) we haveF1(I) −F1(I, ∂I) = −ζ1(I)s=0 + ζ1(I, ∂I)s=0 + dim H1(I, ∂I) = 1.By (2.157) and (2.158) we getF1(Mν, Z) = F1(Mν, Z ∪K) + F1(K) + 2−1χ(K).4. The number F1(Mν, Z) is the sum of the integrals over M1, M2, N, and over∂M of the locally defined densities.

(It is a consequence of Theorem 3.2). So thedensities on Mj, N, and on ∂M ∩M j are the same as for the number F1M(2)j,ν , Z(2)j,where M(2)j:= Mj ∪N Mj, Z(2)j:= Zj ∪Zj, and gM(2)jare mirror symmetric withrespect to N (the ν-transmission boundary conditions are given on N ֒→M(2)j ) andgM(2)j |Mj = gM|Mj.

Thus we have2F1(Mν, Z) =Xj=1,2F1M(2)j,ν , Z(2)j.Since pairsM(2)j , Z(2)jare mirror symmetric with respect to N, it follows from (2.54)and (2.55) thatF1M(2)j,ν , Z(2)j= F1M(2)j , Z(2)j= F1(Mj, Zj) + F1 (Mj, Zj ∪N) ,(2.159)F1(Mν, Z) = 2−1 Xj=1,2(F1 (Mj, Zj) + F1 (Mj, Zj ∪N)) . (2.160)The equality (2.149) follows from (2.148) with M = Mj, Z = Zj, K = N, andν = (1, 1) and from (2.160), because F1(Mj, Zj) = F1(Mj, Zj∪N)+F1(N)+2−1χ(N).Thus Proposition 2.22 is proved.□The analytic torsion T0 (Mν, Z; m) (where Z ⊂∂M is a union of some connectedcomponents of ∂M, m := [θ/2π], θ /∈2πZ) is defined as the product of the norm∥·∥2det H•(Mν,Z) (given by the natural norm on harmonic forms for ∆•ν(Mν, Z)) and ofthe scalar analytic torsion T(Mν, Z; m):T0 (Mν, Z; m) := ∥·∥2det H•(Mν,Z) T (Mν, Z; m) .

(2.161)The analytic torsion T0 (M, Z; m) is the norm ∥·∥2det H•(M,Z) T (M, Z; m), wherethe norm on the determinant line is given by the harmonic forms for ∆•(M, Z).

84S.M. VISHIK(If N is the interior boundary and if gM is a direct product metric near N thenT0 (M, Z; m) = T0 (M1,1, Z; m) according to Proposition 1.1.

)Theorem 2.1. 1.

Let M be obtained by gluing two pieces along N, M = M1 ∪N M2,where N is a closed of codimension one submanifold in M with a trivial normalbundle TM|N/TN and gM is a direct product metric near N and near ∂M. Thenfor ν ∈R2 \ (0, 0) the following gluing formula holds:ϕanν T0 (Mν, Z; m) :== (−1)mχ(N)T0 (M1, Z1 ∪N; m) ⊗T0 (M2, Z2 ∪N; m) ⊗T0(N; m),(2.162)where the identification ϕanνof the determinants lines is defined by the short exactsequence ( 1.14) of the de Rham complexes and by Lemma 1.1, Zk := Z ∩∂M k. Thefactor T0(N; m) := T0(N) is independent of m (according to Proposition 2.22 1).2.

Let K ⊂∂M \ Z be a union of some connected components of ∂M. Then theformula holds for gluing K and (Mν, Z ∪K):ϕanT0 (Mν, Z; m) = (−1)mχ(K) T0 (Mν, Z ∪K; m) ⊗T0(K; m).

(2.163)Here the identification ϕan is defined by the short exact sequence (analogous to ( 1.20)):0 →DR• (Mν, Z ∪K) →DR• (Mν, Z) →DR•(K) →0(2.164)(the left arrow in ( 2.164) is the natural inclusion and the right arrow is the geomet-rical restriction) and by Lemma 1.1. The factor T0(K; m) := T0(K) is independentof m. The analogous formula holds for gluing K and (M, Z ∪K):ϕanT0 (M, Z; m) = (−1)mχ(K) T0 (M, Z ∪K; m) ⊗T0(K),(2.165)where ϕan is defined by the short exact sequence ( 2.164) with Mν replaced by M.Proof.

1. For T0 (Mν, Z) := T0 (Mν, Z; 0) the following gluing formula holds (ac-cording to (1.12) and to Lemma 1.2):ϕanν T0 (Mν, Z) = T0 (M1, Z1 ∪N) ⊗T0 (M2, Z2 ∪N) ⊗T0(N).

(2.166)By the definition of F (Mν, Z) we haveT0 (Mν, Z; m + 1) = exp (−2πiF (Mν, Z)) T0 (Mν, Z; m) . (Analogous equalities are true for T0 (Mj, Zj ∪N; m).

The differences F1 (Mν, Z) −F(Mν, Z), F1(Mj, Zj∪N)−F(Mj, Zj∪N), and F1(N)−F(N) are integers and F1(N)is an integer (according to Proposition 2.22 1). Hence (2.162) is a consequence of(2.149) and (2.166).2.

The gluing formula holds for T0(M, Z) := T0(M, Z; 0) according to (1.18) (The-orem 1.2): ϕanT0(M, Z) = T0(M, Z ∪K) ⊗T0(K). So the gluing formula (2.165)follows from (2.148) since T0(M, Z; m + 1) = exp (−2πiF(M, Z)) T0(M, Z; m) andsince the difference F1(M, Z) −F(M, Z) is an integer.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY85Let N ⊂M be a disjoint union N1 ∪N2 of two closed codimension one submani-folds of M with trivial normal bundles and let the νj-transmission interior boundaryconditions be given on Nj.

Let M = M1 ∪N1 M2 and let N2 ⊂M1. Under theseconditions, the equality (1.12) and the assertion of Lemma 1.2 are also true.

Theirproofs are similar to the given above. The resulting formula isϕanν1 T0 (Mν1,ν2, Z) = T0 (M1,ν2, Z1 ∪N1) ⊗T0 (M2, Z2 ∪N1) ⊗T0(N1).

(2.167)(Here Z ⊂∂M is a union of some connected components of ∂M, Zk = Z ∩∂M kand gM is a direct product metric near Nj and ∂M.) As a consequence of (2.167)(obtained by the same method as Theorem 1.2 is obtained from (1.16) and (1.17))we get the following equalityϕanT0 (M1,ν2, Z1) = T0 (M1,ν2, Z1 ∪N1) T0(N1).

(2.168)The equality (2.163) follows from (2.168) (where M1,ν2, Z1, N1 are replaced by Mν,Z, K) and from (2.148) since T0 (Mν, Z; m + 1) = exp (−2πiF (Mν, Z)) T0 (Mν, Z; m).Theorem 2.1 is proved.□Proposition 2.23. Let M = M1 ∪N M2 be obtained by gluing along N and let theν-transmission boundary conditions (ν ∈R2 \(0, 0)) be given on N. Set n := dim M.Then the number F1 (Mν, Z) for the scalar analytic torsion T (Mν, Z; m) is expressedbyF1 (Mν, Z) = 2−1nχ (Mν, Z) = 2−1nχ (M, Z) .

(2.169)The number F (Mν, Z) := P (−1)j jζν,j(0) is as follows:F (Mν, Z) =X(−1)j −j + 2−1ndim Hj (Mν, Z) .Proof.1.Proposition 2.3 claims that χ (Mν, Z) (i.e., the Euler characteristicP (−1)j dim Hj (DR (Mν, Z))) is equal to the Euler characteristic for the finite-dimensional complex (C• (Xν, Z ∩X) , dc).Note that dim Cj(X, Z ∩X) is equalto dim Cj (Xν, Z ∩X). Hence χ (Mν, Z) =P (−1)j dim Cj (X, Z ∩X).

This sum isequal to χ(M, Z) by the de Rham theorem ([RS], Proposition 4.2). Thus χ (Mν, Z) =χ(M, Z).2.

According to (2.149), the gluing formula holds:F1 (Mν, Z) = F1 (M1, Z1 ∪N) + F1 (M2, Z2 ∪N) + F1(N) + 2−1χ(N).For the Euler characteristics the analogous formula holds:χ (Mν, Z) = χ(M, Z) = χ (M1, Z1 ∪N) + χ (M2, Z2 ∪N) + χ(N). (2.170)

86S.M. VISHIKThe number F1(N) for a closed manifold N is equal to χ(N)(dim N)/2 by (2.153).So F1(N) + 2−1χ(N) = nχ(N)/2.

Let (M, gM) be a closed Riemannian manifold,mirror symmetric with respect to N = ∂M1, M = M1 ∪N M1. Let gM be a directproduct metric near N. Then the equality F1 (M1, N) = 2−1nχ (M1, N) follows from(2.153), which claims that F1(M) = 2−1nχ(M), and from (2.170).

So (2.169) holdsfor pairs (M, Z), where Z = ∂M. For any union Z of some connected components of∂M the equality (2.169) follows from (2.148) for K = ∂M \Z, as F1(K)+2−1χ(K) =nχ(K)/2, according to (2.153).

The equality (2.169) for F1 (Mν, Z) follows from itsparticular cases for F1 (Mj, Zj ∪N) by using (2.149) and (2.170).□Corollary 2.10. 1.

The analytic torsion T0 (Mν, Z; m) (defined by ( 2.161)) is thefollowing function of m = [θ/2π] (θ /∈2πZ):T0 (Mν, Z; m) = (−1)mnχ(M,Z) T0(Mν, Z). (2.171)Here T0 (Mν, Z) := T0 (Mν, Z; 0), n := dim M.2.

The analytic torsion T0(M, Z; m) is equal to T (Mν0, Z; m) for ν0 = (1, 1) ac-cording to Proposition 1.1. The formula ( 2.171) holds also for T0(M, Z; m), whereT0 (Mν, Z) is replaced by T0(M, Z) := T0(M, Z; 0).Let (M, gM) be obtained by gluing two Riemannian manifolds M1 and M2 alonga common component N of their boundaries, M = M1 ∪N M2.

Let gM be a directproduct metric near N and near ∂M and let Z ⊂∂M be a union of some connectedcomponents of ∂M. The following main theorem is an immediate consequence ofTheorems 1.4, 1.5 and of Corollary 2.10.Theorem 2.2 (Generalized Ray-Singer conjecture).

1. The analytic torsionT0 (Mν, Z; m) is expressed through the combinatorial torsion norm ( 1.62) as follows:T0 (Mν, Z; m) = 2χ(∂M)+χ(N) (−1)mnχ(M,Z)/2 τ0 (Mν, Z)(where m = [θ/2π], θ /∈2πZ is the phase of a cut of the spectral plane C ∋λ andn = dim M).2.

The analytic torsion T0(M, Z; m) is expressed through the combinatorial torsionnorm:T0(M, Z; m) = 2χ(∂M) (−1)mnχ(M,Z)/2 τ0(M, Z).Remark 2.11. The combinatorial torsion norms τ0(M, Z) and τ0 (Mν0, Z) (where ν0 =(1, 1)) on the determinant line det H• (Mν0, Z) = det H•(M, Z) are different, ifχ(N) ̸= 0 (by Remarks 1.7 and 1.9).

The canonical identifications H• (Mν0, Z) =Ker∆•ν0= Ker (∆•) = H•(M, Z) are given by Proposition 1.1 and by the de Rhamtheorem.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY873.

Zeta- and theta-functions for the Laplacians with ν-transmissioninterior boundary conditions3.1. Properties of zeta- and theta-functions for ν-transmission boundaryconditions.

Let M be a compact manifold with boundary obtained by gluing mani-folds M1 and M2 along a common component N of their boundaries, M = M1 ∪N M2(N ⊂M is a closed codimension one submanifold of M with a trivial normal bundleTM|N/TN). Let gM be a direct product metric near N = 0×N ֒→I ×N ֒→M.

Letthe Dirichlet boundary conditions be given on a union Z of some connected compo-nents of ∂M, the Neumann ones be given on ∂M \ Z and the ν-transmission interiorboundary conditions (1.27) be given on N.The operator ∆•ν is originally defined on the set D (∆•ν) of all the pairs of smoothforms ω = (ω1, ω2) ∈DR• M 1⊕DR• M2such that the Dirichlet boundaryconditions hold for ω on Z, the Neumann boundary conditions hold on ∂M \ Zand the interior boundary conditions (1.27) hold for ω on N.Let Dom (∆•ν) bethe closure of the D (∆•ν) in (DR•(M))2 in the topology given by the graph norm47∥ω∥22 + ∥∆•νω∥22 =: ∥ω∥2graph. The closure of the operator ∆•ν (with respect to thegraph norm) is an operator with the domain of definition Dom (∆•ν).

If ωj →ω inthe graph norm topology, ωj ∈D (∆•ν), then ∆•ν(ω) is defined as limj ∆νωj in theL2-topology in (DR•(M))2.Theorem 3.1. 1.The operator ∆•ν with the domain Dom (∆•ν) is self-adjoint in(DR•(M))2.

Its spectrum Spec (∆•ν) ⊂R+ ∪0 is discrete.482. Its zeta-function is defined for Re s > (dim M)/2 by the absolutely convergent se-ries (including the multiplicities) ζν,•(s) := Pλj∈Spec(∆•ν)\0 λ−sj .

This series convergesuniformly for Re s ≥(dim M)/2 + ε (for an arbitrary ε > 0). The zeta-functionζν,• can be contunued to a meromorphic function on the whole complex plane with atmost simple poles at the points sj := (j −dim M)/2, j = 0, 1, 2, .

. .

. It is regular ats = 0, 1, 2, .

. ..3.

The residues ress=sj ζν,•(s) and the values ζν,•(m) + δm,0 dim Ker (∆•ν) are equal tothe sums of the integrals over M, ∂M, and N of the densities locally defined on thesemanifolds.Proposition 3.1. 1.

Let λ /∈Spec (∆•ν). Then the resolvent G•λ(ν) := (∆•ν −λ)−1,G•λ(ν) : (DR•(M))2→∼Dom(∆•ν) ֒→(DR•(M))2, is the isomorphism (in algebraicand topological senses) onto the closure Dom(∆•ν) of D(∆•ν) with respect to the graphnorm.49 The operators G•λ(ν) for pairs (λ, ν) such that λ /∈Spec (∆•ν) form a smooth47The L2-completion (DR•(M))2 of DR•(M) coincides with the L2-completion of DR• M1⊕DR• M2.48A spectrum is discrete if it consists entirely of isolated eigenvalues with finite multiplicities.49The topology on (DR•(M))2 is given by ∥ω∥22, and on Dom(∆•ν) it is given by ∥ω∥2graph.

88S.M. VISHIKin (λ, ν) family of bounded operators in (DR•(M))2.2.

The families d ◦G•λ(ν) and δ ◦G•λ(ν) for λ /∈Spec (∆•ν) form a smooth in (λ, ν)family of bounded operators (DR•(M))2 →(DR•±1(M))2.Theorem 3.2. 1.

The operator exp (−t∆•ν) in (DR•(M))2 for an arbitrary t > 0is of trace class. For its trace the asymptotic expansion ( 2.87) (relative to t →+0)holds.

The coefficients f−dim M+j of this expansion are the sums of the integrals overM, ∂M, and N of the locally defined densities. If j ̸= dim M + 2m, m ∈Z+ ∪0, thedensities on M, ∂M, and on N for f−dim M+j are the same as forΓ((dim M −j)/2) ress=sj ζν,•(s).If j = dim M + 2m, m ∈Z+ ∪0, these densities are the same as for(m!

)−1(−1)m (ζν,•(m) + δm,0 dim Ker(∆•ν)) .2.Let p1: (DR•(M))2 →(DR•(M1))2 ֒→(DR•(M))2 be the composition ofthe restriction to M1 and of the extension by zero of L2-forms. Then the operatorp1 exp (−t∆•ν) in (DR•(M))2 for t > 0 is of trace class.

For its trace the asymptoticexpansion relative to t →+0 holdsTr (p1 exp(−t∆•ν)) = q−nt−n/2 + · · · + q0t0 + q1t1/2 + · · · + qmtm/2 + rm(t), (3.1)where rm(t) is Ot(m+1)/2uniformly with respect to ν and it is smooth in t for t > 0(n := dim M). The coefficients qj are equal to the sums of the integrals over M1 andover ∂M1 = N ∪(∂M ∩M 1) of the locally defined densities.

The coefficients qj in( 3.1) depend only on (j, M1, gM|TM1, Z ∩∂M 1, N, ν) and do not depend on M2 andZ ∩∂M 2, gM|TM2.3. For any t > 0 the traces of exp (−t∆•ν) are bounded uniformly with respect toν ∈R2 \ (0, 0):|Tr exp (−t∆•ν)| < C(t).

(3.2)The traces Tr (pj exp (−t∆•ν)) are also bounded uniformly with respect to ν for anyt > 0.Proposition 3.2. 1.

The kernel E•t,x1,x2(ν) for exp (−t∆•ν) (where t > 0) is smoothin xj ∈M rj, t, and in ν ∈R2 \ (0, 0).2. The asymptotic expansions ( 3.1), ( 2.87) are differentiable with respect to ν ∈R2 \ (0, 0).

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY893.2.

Zeta-functions for the Laplacians with ν-transmission interior bound-ary conditions. Proofs of Theorem 3.1 and of Proposition 3.1.

Let A bean elliptic differential operator on a manifold with boundary (M, ∂M). Let the dif-ferential elliptic boundary conditions be given for A on ∂M such that A with theseboundary conditions satisfies Agmon’s condition (formulated below) for λ from a sec-tor θ1 < arg λ < θ2 in the spectral plane C ∋λ.

Properties of zeta-functions for Awith these boundary conditions can be investigated with the help of the parametrixfor (A −λ)−1. The analogous statement is true also for elliptic interior boundaryconditions.50The parametrix P mλfor (∆•ν −λ)−1 is defined locally in coordinatecharts.

NamelyP mλ =XψjP mλ,Ujϕj,(3.3)where ϕj is a partition of unity subordinate to a finite cover {Uj} of M by coordinatecharts, ψjϕj = ϕj, ψj ∈C∞0 (Uj). If Uj ∩(∂M ∪N) = ∅then the operator P mλ,Uj isa pseudodifferential operator (PDO) with parameter λ ([Sh], Chapter II, § 9) andits symbol is equal to θ(ξ, λ)s(m)(∆• −λ)−1(x, ξ, λ).

This symbol is defined asfollows. Let s(∆• −λ) = ((b2 −λ) id +b1) (x, ξ) be the symbol of ∆• −λ (where ∆•is the Laplacian on DR• (Uj)) and lets(∆• −λ)−1:=Xj∈Z+∪0a−2−j(x, ξ, λ)be the symbol of (∆• −λ)−1 as of a PDO with parameter (a−k is positive homoge-neous of degree −k inξ, λ1/2).

Set s(m)(∆• −λ)−1:= Pmj=0 a−2−j(x, ξ, λ). Thecondition s(∆• −λ) ◦s ((∆• −λ)−1) = 1, where ◦is the composition of symbols withparameter ([Sh], § 11.1), is equivalent to the system of equalitiesa−2(x, ξ, λ) = (b2 −λ)−1,a−3 = −(b2 −λ)−1[b1a−2 +XiDξib2∂xia−2],(3.4)a−2−j = −(b2 −λ)−1X|γ|+i+l=j1γ!Dγξ b2−i∂γxa−2−l.The sum in the last equation of (3.4) is over (γ, i, l) such that γ = (γ1, .

. .

, γn) ∈(Z+ ∪0)n, |γ| := γ1 + · · · + γn, 0 ≤|γ| ≤j for bj, |γ| + i ≥1 (D := i−1∂). Thefunction θ(ξ, λ) (in the symbol of P mλ ) is smooth, θ(ξ, λ) ≡1 for |ξ|2 + |λ| ≥1, andθ is equal to zero for |ξ|2 + |λ| ≤ε.50Theorem 3.1 is analogous to the results of [Se1], [Se2] with modifications connected with the ν-transmission interior boundary conditions.

In [Sh], Ch.II, the theory [Se1], [Se2] of the zeta-functionsis written in detail in the case of a closed manifold.

90S.M. VISHIKLet Uj ∩N ̸= ∅.

Then the term P mλ,U of the parametrix is the sum of the interiorterm (which is a PDO with parameter and its symbol is defined with the help of(3.4)) and of the correction terms. (Here U := Uj.) The latter terms correspond tothe ν-transmission interior boundary conditions on N and to the Dirichlet and theNeumann boundary conditions, given on the connected components of ∂M.

First ofall we’ll verify that these ν-transmission boundary conditions are Agmon’s conditionson any ray arg λ = ϕ in the spectral plane not coinciding with R+.Let (t, y) ∈I ×UN be the coordinates on U := Uj near N = 0×N ֒→I ×N ֒→M,I = [−2, 2], and let t > 0 on M1.From now on it is supposed that ϕj(t, y) =ϕj,I(t)ϕj,N(y) and that ϕj,I(t) ≡1 for |t| ≤1. It is supposed also that ψj(t, y) =ψj,I(t)ψj,N(y) and that ϕj,I, ψj,I are even functions: ϕj,I(−t) = ϕj,I(t), ψj,I(−t) =ψj,I(t).

The forms dyc and dt dyf (where c = (c1, . .

. , cn−1), f = (f1, .

. .

, fn−1), ci, fi ∈{0, 1}) provide us with a trivialization of ∧•TM|I×UN. Namely ωj =P|c|=• ωj,cdyc +P|f|+1=• ωj,(1,f)dt dyf.Let ω = (ω1, ω2) ∈D(∆•ν) ⊂DR•(Mν).Then on UN =0 × UN ֒→I × UN the conditions ω ∈D(∆•ν) can be written as follows.

Set |ν| =(α2 + β2)1/2. Let L be the transformation (t ≥0)v1,c(t, y) := |ν|−1 (αω1,c(t, y) −βω2,c(−t, y)) ,v2,c(t, y) := |ν|−1 (βω1,c(t, y) + αω2,c(−t, y)) ,w2,(1,f)(t, y) := |ν|−1 αω1,(1,f)(t, y) + βω2,(1,f)(−t, y),w1,(1,f)(t, y) := |ν|−1 −βω1,(1,f)(t, y) + αω2,(1,f)(−t, y).

(3.5)Then the conditions ω ∈D (∆•ν) are equivalent on UN tov1,c(0, y) = 0,∂tv2,ct=0 = 0,w1,(1,f)(0, y) = 0,∂tw2,(1,f)t=0 = 0. (3.6)The inverse to (3.5) transformation L−1 is ω1,c(t, y)ω2,c(−t, y)!=L−1 v1,cv2,c!

(t, y), ω2,(1,f)(−t, y)ω1,(1,f)(t, y)!=L−1 w1,(1,f)w2,(1,f)! (t, y), (3.7)L := |ν|−1 α−ββα!.Agmon’s conditions on a ray l := {arg λ = ϕ} in the case of ν-transmission bound-ary conditions claim that for (ξ′, λ) ̸= (0, 0) and λ ∈l the equation on R+ ∋t−∂2t + b2(y, ξ′) −λv(t) = 0,v(t) →0 for t →+∞(3.8)has a unique solution for each of the initial conditions vt=0 = v0 or ∂tv|t=0 = v1.

(Here ξ′ are dual to y and b2(∆N) = b2(y, ξ′) id is the scalar principal symbol of∆•N on UN.) Agmon’s conditions for the ν-transmission boundary value problem are

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY91satisfied on each ray arg λ = ϕ not coinciding with R+ because the equation (3.8)with each of the initial conditions given above has a unique solution for any λ /∈R+,(ξ′, λ) ̸= (0, 0).It is convenient to compute the contribution to P mλ,U from ν-transmission bound-ary conditions in the coordinates vj,c(t, y), wj,(1,f)(t, y) defined by (3.5) with t ≥0.

(Then the ν-transmission boundary conditions are transformed into the con-ditions (3.6). )These contributions are defined with the help of the symbol d =Pj∈Z+∪0 d−2−j(t, y, τ, ξ′, λ), which is the solution of the equation51−∂2t + (b2(y, ξ′) −λ) id +b1(y, ξ′)◦d(t, y, τ, ξ′, λ) = 0(3.9)(with the composition ◦of symbols of (y, ξ′) in it).

The equation (3.9) holds fort ̸= 0. The boundary conditions for (3.9) are: d−k →0 as |t| →∞and(Ld−k)1,ct=0 =(La−k)1,ct=0,∂t (Ld−k)2,ct=0 =iτ (La−k)2,ct=0,(Ld−k)1,(1,f)t=0 =(La−k)1,(1,f)t=0,∂t (Ld−k)2,(1,f)t=0 =iτ (La−k)2,(1,f)t=0(3.10)Here the transformation L acts on the columns of the matrix valued functions d,a (depending on t and on τ).52The equation (3.9) is the recurrent system−∂2t d−k + (b2 −λ)d−k +X 1γ!Dγξ′bi∂γy d−m = 0,(3.11)where the sum is over m < k and γ such that m + |γ| + 2 −i = k, 0 ≤|γ| ≤i for bi.For t = 0 the symbol d−k over Mj ∩UN is positive homogeneous of degree (−k)in (τ, ξ′, λ1/2).

The boundary contribution to P mλ,U is an operator Dm correspondingto53 θ1(ξ, λ) Pmj=0 d−2−j(t, y, τ, ξ′, λ). This operator acts on f ∈DR•c(Rt ×Rn−1y) suchthat supp f ∩(0 × Rn−1y) = ∅as follows:(Dmf)(y, t)=(2π)−nZZexp(i(y, ξ′))mX0θ1d−2−j(t,y,τ,ξ′,λ)(Ff)(τ,ξ′)dξ′dτ (3.12)(where (Ff)(τ, ξ′) =RR exp (−i (tτ + (x, ξ′))) f(t, x)dxdt is Fourier transform of f).The term of the parametrix, corresponding to U (if U ∩N ̸= ∅) is defined byP mλ,U = P mλ,int −Dm,(3.13)51Hereb2(y, ξ′) id +b1(y, ξ) is the symbol s(∆•N) on UN of the Laplacian on DR•(N) for thecomponents ωtan,N(t, y) and on DR•−1(N) for ωnorm,N(t, y).

The variable τ is dual to t.52Note that the function a−2−j(t, y, τ, ξ′, λ) is continuous in N and nonsingular for λ /∈R+ and(τ, ξ′, λ) ̸= (0, 0, 0). (It is also independent of t for |t| small enough.) So the right sides of (3.10) canbe simplified for a−k.

In (3.9)–(3.11) it is used that gM is a direct product metric on I × N ֒→M.53θ1(ξ′, λ) ∈C∞(Rn−1 × C), θ1 ≡0 for |ξ′|2 + |λ| < ε and θ1 ≡1 for |ξ′|2 + |λ| ≥1; n := dim M.

92S.M. VISHIKwhere Pλ,int is the PDO with the symbol s(m)(t, y, τ, ξ′, λ) defined by (3.4) (x isreplaced by (t, y) and ξ = (τ, ξ′)).For Uj ∩∂M ̸= ∅the boundary term in P mλ,Uj for the Dirichlet or the Neumannboundary conditions on the connected components of ∂M is defined similarly.The following assertions are true:1.

For m ≥n the operator (∆• −λ) P mλ −id (where (∆• −λ) acts on the restric-tions of forms to M 1 and to M 2) has a continuous on M j1 × M j2 kernel which isO1 + |λ|1/2n−mfor λ ∈Λε := {λ ̸= 0, ε < arg λ < 2π −ε}, where π > ε > 0 isfixed ([Se1], Lemma 5, p. 901). This estimate is satisfied uniformly with respect toν since the families d−2−j are smooth in ν ∈R2 \ (0, 0) and since the estimates forLd−2−j by [Se1], (29), p. 900, are uniform with respect to ν ̸= (0, 0).2.

Let m ≥n and ν = (α, β) ∈R2 \ (0, 0). Let Aj := A (Mj, N) be the same asin (2.97) and (2.99) and Rj be the geometrical restrictions to N ⊂∂Mj of forms onMj.

Then the operators|ν|−1(αR1 −βR2)P mλ ,|ν|−1(αR1δ −βR2δ)P mλ ,|ν|−1(βA1 −αA2)P mλ ,|ν|−1(βA1d −αA2d)P mλhave smooth kernels on N × M j which are O1 + |λ|1/2n−mfor λ ∈Λε, whereπ > ε > 0 and ε is fixed ([Se1], Lemma 6). These estimates are uniform with respectto ν ∈R2 \ (0, 0).3.

Set B1,ν := |ν|−1(αR1−βR2): ⊕jDR•(M j) →DR•(N). Let p1: [0, 1]×N →N,p2: [−1, 0] × N →N be the natural projections.

The operatorq1,ν : DR•(N) →⊕DR•(Mj),q1,ν(ωN) := |ν|−1ϕ(t)(αp∗1ωN, −βp∗2ωN) (3.14)(where ϕ(t) ∈C∞0 (I), ϕ(t) ≡1 for t ∈[−1/2, 1/2]) is the right inverse to B1,ν sinceB1,νq1,ν = id. The analogous right inverse operators qk,ν are naturally defined forBk,νB2,ν := |ν|−1(βA1 −αA2),B3,ν := |ν|−1(αR1δ −βR2δ),B4,ν := |ν|−1(βA1d −αA2d),(3.15)Bi,νqj,ν = δij · id .

(3.16)

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY93For instance, q3,ν ◦ωN := ε|ν|−1ϕ(t)t (αdt ∧p∗1ωN, −βdt ∧p∗2ωN), ε = ±1.

Set Bν :=(Bj,ν), qν := (qj,ν), and qνBν := (qj,νBj,ν). For m ≥n the operator is defined54Rmλ := P mλ −qνBνP mλ .

(3.17)It maps (according to [Se1], Lemma 12, p. 912) C∞-forms ω ∈DR•(M), supp ω∩N =∅, to C∞-forms on Mj. Moreover Rmλ : DR•c(M\N) →D (∆•ν).4.

For m ≥n and λ /∈R+ ∪0 the operator Rλ can be continued to a boundedoperator in (DR•(M))2, Rλ : (DR•(M))2 →Dom (∆•ν).Indeed, for any fixed differential operator F of order d = d(F) ≤2 the operatorFRλ is bounded in (DR•(M))2 with its norm O1 + |λ|1/2d−2in a sector λ ∈Λε(π > ε > 0 and ε is fixed) according to [Se1], Lemmas 7, 13, 14. This estimate isuniform with respect to ν ∈R2 \ (0, 0).

The continuation of Rλ to (DR•(M))2 isas follows. If ωj ∈DR•c(M \ N) and ωj →ω in (DR•(M))2 then Rλωj convergesin (DR•(M))2 and Rλω is defined as its limit.

We see that Rλωj ∈D (∆•ν) and(∆•ν −λ) Rλωj converges in (DR•(M))2. Hence Rλω ∈Dom (∆•ν).5.The operator G•λ(ν) := (∆•ν −λ)−1: (DR•(M))2 →Dom(∆•ν) exists55 forλ ∈Λε := {λ ̸= 0: ε < arg λ < 2π −ε} and |λ| sufficiently large.

Its operator normis O(|λ|−1) for such λ uniformly with respect to ν ∈R2 \ (0, 0) ([Se1], Lemma 15).6.The Laplacian ∆•ν is a closed unbounded operator in (DR•(M))2 with itsdomain of definition Dom (∆•ν).Actually, if {ui} ⊂Dom (∆•ν) and if the limitslimi ui =: u and limi ((∆•ν −λ) ui) =: v exist in (DR•(M))2 then for sufficientlylarge λ ∈Λε we have u = limi G•λ(ν) ((∆•ν −λ) ui) = Gλ(ν)v ∈Dom (∆•ν). Hence(∆•ν −λ) u = (∆•ν −λ) (G•λ(ν)v) = v, i.e., the operators ∆•ν −λ id and ∆•ν are closedin (DR•(M))2.

The operator ∆•ν is defined on Dom (∆•ν). It is a self-adjoint un-bounded operator in (DR•(M))2.

Indeed, the domain of definition Dom ((∆•ν −λ)∗)of the adjoint operator (∆•ν −λ)∗in (DR•(M))2 is the set of v ∈(DR•(M))2 suchthat the linear functional ((∆•ν −λ) ω, v) is continuous on Dom (∆•ν) ∋ω in the L2-topology of (DR•(M))2.If v ∈Dom (∆•ν) then for any ω ∈Dom (∆•ν) we have((∆•ν −λ0) ω, v) = (ω, (∆•ν −λ0) v) for λ0 ∈R−.Indeed, for each ω and v fromDom (∆•ν) there exist sequences {ωj} and {vj} of elements D (∆•ν) whose limits inthe graph norm topology are ω and v. Hence we havelimj (∆νωj, v)2 = limj limi (∆νωj, vi)2 = limj limi (ωj, ∆νvi)2 = (ω, ∆νv)2 .54For simplicity we’ll suppose from now on here that ∂M = ∅. For the Dirichlet or the Neumannboundary conditions on the components of ∂M the appropriate terms have to be added to Rmλ .55This means that ∆•ν −λ maps Dom•(∆ν) one-to-one to (DR•(M))2.

It is equivalent to theexistence of (∆•ν −λ)−1 : (DR•(M))2 →Dom(∆•ν), (∆•ν −λ) ◦(∆•ν −λ)−1 = id on (DR•(M))2.

94S.M. VISHIKSo ((∆•ν −λ0) u, v) is a continuous linear functional on Dom (∆•ν) ∋u with respectto the L2-topology of (DR•(M))2 for any v ∈Dom (∆•ν).Hence Dom (∆•ν) ⊂Dom ((∆•ν −λ0)∗) and (∆ν −λ0)∗v = (∆ν −λ0) v for v ∈Dom (∆•ν).Let λ0 ∈R−and |λ0| be sufficiently large.

Then for any w ∈Dom ((∆•ν −λ0)∗)there exists an element v ∈Dom (∆•ν) such that (∆•ν −λ0)∗w = (∆•ν −λ0) v =(∆•ν −λ0)∗v (since Im(∆•ν −λ0) = (DR•(M))2). So w −v ∈Ker ((∆•ν −λ0)∗) and forany u ∈Dom (∆•ν) we have 0 = (u, (∆•ν −λ0)∗(w −v)) = ((∆•ν −λ0) u, w −v).

Thenw −v = 0, as Im (∆•ν −λ0) = (DR•(M))2. Hence Dom ((∆•ν −λ0)∗) = Dom (∆•ν) =Dom ((∆•ν)∗), and ∆•ν is a self-adjoint unbounded operator in (DR•(M))2.The operator ∆•ν is nonnegative, (∆•νω, ω)2 ≥0 for any ω ∈Dom (∆•ν), since thereexists a sequence {ωj}, ωj ∈D (∆•ν), such that its limit in the graph norm topologyis ω.

So we have limj (∆•νωj, ωj)2 = limj(dνωj, dνωj)2 + limj(δνωj, δνωj)2 ≥0.7. The spectrum Spec (∆•ν) of the operator ∆•ν is discrete because the operator(∆•ν −λ) (∆•ν −λ0)−1 = id + (∆•ν −λ0)−1 · (λ0 −λ)differs from the identity operator in (DR•(M))2 by a compact operator.

Here, λ0 ∈Λεand |λ0| is large enough. The assertion 5 above claims that (∆•ν −λ0)−1 exists forsuch λ0.The operator Gλ0(ν) := (∆•ν −λ0)−1 is compact since it is bounded in(DR•(M))2 and since the operators I −(∆•ν −λ0) Rmλ0 (for m ≥n) and Rmλ0 arecompact in (DR•(M))2 ([Se1], Lemmas 4, 5, 9 (iv)).

So the operator(∆•ν −λ0)−1 = Rmλ0 + (∆•ν −λ0)−1 I −(∆•ν −λ0) Rmλ0is compact in (DR•(M))2. Since Gλ0(ν) is a compact operator for λ0 ∈Λε, |λ0| >> 1,and since ∆•ν is a closed operator in (DR•(M))2, it follows that ∆•ν is an operatorin (DR•(M))2 with compact resolvent.

So (according to [Ka], Ch. 3, Theorem 6.29)its spectrum Spec (∆•ν) consists of isolated eigenvalues with finite multiplicities (i.e.,Spec (∆•ν) is discrete) and the operator G•λ(ν) is compact in (DR•(M))2 for λ ∈C \ Spec(∆•ν).

The operator ∆•ν is nonnegative. Hence Spec (∆•ν) ⊂R+ ∪0.If λ0 /∈Spec (∆•ν) then Ker (∆•ν −λ0) = 0 and Im (∆•ν −λ0) = (DR•(M))2.

Henceind (∆•ν −λ0) is equal to 0 (as the index of the operator fromDom (∆•ν) , ∥·∥2graphinto(DR•(M))2 , ∥·∥22). The operator (λ0 −λ) id from Dom (∆•ν) into (DR•(M))2is compact (since G•λ0(ν) is a compact operator in (DR•(M))2 and since it is a topo-logical isomorphism Gλ0(ν): (DR•(M))2→∼Dom (∆•ν)).

So ind (∆•ν −λ) = 0 for anarbitrary λ ∈C (according to [Ka], Ch. 4, Theorem 5.26, Remarks 1.12, 1.4).8.

The operator (∆•ν)−s for Re s > 0 is defined by the integrali2πZΓ λ−s (∆•ν −λ)−1 dλ =: T−s(ν),(3.18)

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY95where the contour Γ is{λ = reiπ, ∞> r > ε} ∪{λ = εeiϕ, π > ϕ > −π} ∪{λ = re−iπ, ε < r < ∞}.Here the number ε > 0 is such that Spec (∆•ν) ∩(0, ε] = ∅.

The integral (3.18) isabsolutely convergent (with respect to the operator norm ∥·∥2 in (DR•(M))2) becausethe estimate(∆•ν −λ)−12 ≤C|λ|−1 is satisfied as λ →−∞for λ ∈R−. So T−s(ν)is a bounded operator in (DR•(M))2 for Re s < 0.For −k ≥Re s > −(k + 1), k ∈Z+, the operator T−s is defined as ∆k+1νT−(s+k+1).Its domain is Dom(T−s) =nω ∈(DR•(M))2 , T−(s+k+1)ω ∈Dom(∆•ν)k+1o, whereDom(∆•ν)k+1:=nω ∈Dom (∆•ν) , ∆•νω ∈Dom (∆•ν) , .

. .

, (∆•ν)k ω ∈Dom (∆•ν)o.The restriction T 0−s of Ts to the orthogonal complement L0 of Ker (∆•ν) in (DR•(M))2is defined on DomT 0−s:= D (T−s) ∩L0, T 0−s := T−s|L0. Then T−s is the directsum56 of T 0s and of the zero operator on Ker (∆•ν).

Theorem 1 in [Se2] claims that thefamily T 0−s of operators in the Hilbert space L0 for Re s2 > 0 satisfies the equationT 0−s1T 0−s2 = T 0−(s1+s2) and that the same is true for −s1 ∈Z+ and for each s2. Thistheorem claims also thatT 00 = id on L0,T 0−l =(∆•ν)−1 |L0l , for l ∈Z+, and T 01 = ∆•ν|L0(the domain of T 01 is Dom(∆•ν) ∩L0) and that T 0−s for Re s > 0 is a holomorphicfunction57 with its values in a Banach space B(L0) of bounded operators in L0 wherethe Banach norm is the operator norm as the norm.9.

For Re s > n/2 the kernel of T−s(ν) is continuous on M j1 ×M j2 and analytic ins ([Se2], Theorem 2(i)). For Re s > n/2 the zeta-function ζν,•(s) is equal to the sumof integrals over the diagonals M j ֒→Mj × M j (j = 1, 2) of the densities defined bythe restrictions to these diagonals of the kernel T−s(ν), according to Proposition 3.9below.

So ζν,•(s) is holomorphic for Re s > n/2.The operator G•λ(ν) −P mλfor m ≥n (where P mλis the parametrix (3.3)) is abounded in (DR•(M))2 operator with a continuous on M j1 × M j2 kernel (rmλ )x1,x2which is(rmλ )x1,x2 = O1 + |λ|1/2−(2+m)+n(3.19)56If v ∈L0 and Re s > 0 then we have T−sv ∈L0 since for h ∈Ker (∆•ν) and λ ∈Γ it holds0 = (v, h) = ((∆•ν −λ) G•λ(ν)v, h) = −λ (G•λ(ν)v, h)and since the integral (3.18) is absolutely convergent. For h ∈Ker (∆•ν) and for Re s > 0 we haveT−sh = 0 because for such s the integralRΓ λ−s−1dλ is absolutely convergent and is equal to zero.Since T−sh = (∆•ν)k+1 T−(s+k+1)h = 0 for −k ≥Re s > −(k + 1), we get T−sh ≡0 for all s.57A function with the values in a Banach space is holomorphic in a strong sense if it is weaklyholomorphic ([Ka], Ch.

3, § 1, Theorem 1.37, p. 139)

96S.M. VISHIKas |λ| →+∞, λ ∈Λε ([Se1], Theorem 1, or also the assertions 5, 1, 2, 7 above).

Sothe operatori2πZΓ λ−s (G•λ(ν) −P mλ ) dλ(3.20)for Re s > (n −m)/2 is of trace class and its kernel is continuous on M j1 × M j2 andanalytic in s.The trace of the operator (3.20) is holomorphic in s for Re s > (n −m)/2. Letus denote by Kintx,y(s) the kernel of the operator (i/2π)RΓ λ−sP mλ,intdλ (where P mλ,int :=P ψjP mλ,Uj,intϕj is a term of (3.3) and P mλ,Uj,int is a PDO with symbol θs(m), definedby (3.4)).

This kernel is continuous on M j1 × M j2 for Re s > n/2. Offthe diagonalsM j ֒→M j × M j it extends to a kernel which is an entire function of s ∈C equal tozero for (−s) ∈Z+ ∪0.

The density on ∪jM j defined by the restriction of this kernelto the diagonals also can be continued to a meromorphic in s ∈C density. Thisdensity has at most simple poles at sj = (n −j)/2 for (−sj) /∈Z+ ∪0, 0 ≤j ≤m,and it is regular at sj for (−sj) ∈Z+ ∪0.The residue at s = sj is completely defined by the component a−2−j(x, ξ, λ) of thesymbol s(∆• −λ)−1([Se2], Lemma 1 or [Sh], Theorem 12.1).

These componentsare given by (3.4). The value of this density at s = sj for (−sj) ∈Z+∪0 is completelydefined by a−2−j (by the formulas (11), (12) in [Se2] with changing of the sign in (11)to the opposite one).

Here, j = n + 2m, m ∈Z+ ∪0.The kernel K∂x,y(s) of the operator58 (i/2π)RΓ λ−sDm,λdλ for Re s > n/2 is con-tinuous on M j1 × M j2 and analytic in s ([Se2], Lemma 4). Let (x, y) be offthediagonals or let either x or y be not from ∪j∂M j ⊃N.

Then K∂x,y(s) is an entirefunction of s ∈C and it is equal to zero at s for (−s) ∈Z+ ∪0 ([Se2], Lemma 4). ForRe s > n/2 the densities defined by the restriction K∂x,x(s) of K∂x,y(s) to the diagonalsM j are integrable over the fibers of the natural projections p1: [0, 1] × N →N andp2: [−1, 0]×N →N.

These integrals are densities on N. They can be continued59 tomeromorphic on s ∈C densities (on N) with at most simple poles at sj = (n −j)/2,1 ≤j ≤m, such that (−sj) /∈Z+ ∪0. Its residues at sj for 1 ≤j ≤m + 1 arecompletely defined by a term d−2−j+1 in d ([Se2], Theorem 2(iv), formula (II)).

Thevalues of these densities at sj for (−sj) ∈Z+ ∪0 (where n ≤j ≤m + 1) are alsocompletely defined by d−2−j+1 ([Se2], Lemmas 2, 3, 4, Theorem 2(iv), formula (II′)58Here Dm,λ = P ψjDm,U(λ, ν)ψj. The operator Dm,U from (3.13) is defined for U ∩N ̸= ∅by(3.12), (3.9), and (3.10).59Let LDm,U(λ, ν) be an operator acting on Lω as L (Dm,U(λ, ν)ω) (for any ω ∈DR•c(Rn) suchthat supp ω ∩Rn−1 = ∅, where Rn−1 are local coordinates on N).

All the assertions about thekernels, analogous to K∂x,y(s) in the case of LDm,U(λ, ν), and about the corresponding densitiesin this case, are proved in [Se2]. Thus the transformation (3.7) provides us with all the assertionsabout the kernel K∂x,y(s) (and about the corresponding densities) connected with DU,λ(ν).

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY97with changing of the sign to the opposite one).10.

The kernel of the operator (3.20) is the difference of the kernels(T−s)x,y −Kintx,y(s) −K∂x,y(s). (3.21)For Re s > (n−m)/2 it is holomorphic in s and continuous on Mj1 ×M j2.

The termKint −K∂x,y (s) is equal to zero for x ̸= y and (−s) ∈Z+ ∪0. The term (T−s)x,yis equal to zero for x ̸= y and (−s) ∈Z+ (according to the assertions of 8 above,since ∆•ν is a differential operator).

We have (T0)x,y = −H•x,y, where H• is the kernelof the orthogonal projection operator in (DR•(M))2 onto Ker (∆•ν) (the assertion 8).The properties of ζν,•(s)60 formulated in Theorem 3.1, follow from the assertions of9 and 10. The theorem is proved.□Remark 3.1.

The kernel (3.21) of the operator (3.20) is holomorphic in s and con-tinuous in (x, y) ∈M j1 × Mj2 for Re s > (n −m)/2. It is equal to zero for x ̸= yat s = −k, k ∈Z+, and to −H•x,y(ν) at s = 0.

So the analytic continuations of thedensities on Mj and on N defined by the kernels (T−s)x,y andKint(s) −K∂(s)x,yhave the same residues at s = sj, 0 ≤j ≤m, (−sj) /∈Z+ ∪0, and the same valuesat s = sj, (−sj) ∈Z+, n + 2 ≤j ≤m. They differ at s = 0 (i.e., for j = n)by the densities on M j, defined by −H•x,x(ν).

Hence the densities on Mj and onN, corresponding to the residues and to the values at s = sj, 0 ≤j ≤m −1, ofKint(s) −K∂(s)x,x are the same for all the parametrixes P mλ defined by (3.3) (withdifferent covers {Uj}, partitions of unity {ϕj} subordinate to {Uj}, and {ψj}).And back, the values and the residues of the analytic continuation for the integralRMj tri∗j(T−s)+ δs,0RMj tri∗jH•(ν)at sj, 0 ≤j ≤m, are defined by an arbitraryparametrix P mλ .Proof of Proposition 3.1.Let m ≥n := dim M, m ∈Z+, and λ ∈Λε.Then the parametrix Rmλ for G•λ(ν)61 (defined by (3.17)) is a bounded operator6260The values of ζν,•(s) at (−s) ∈Z+ and the residues of ζν,• at s = sj can be also expressedin terms of noncommutative residues ([Wo] or [Kas]). The density on M whose integral over Mis equal to a volume term in Ress=sj ζν,•(s) can be written as 2−1 resx, ∆−sjθ=π.

Here res is anoncommutative residue for the symbol of PDO ∆−sjθ=π. This symbol is defined with the help ofthe symbol P a−2−j(x, ξ, λ) of (∆• −λ)−1 ([Sh], 11.2).

The boundary term in Ress=sj ζν,•(s) for(−sj) /∈Z+ ∪0 is expressed similarly.61The statement that G•λ(ν): (DR•(M))2 →Dom (∆•ν) is an isomorphism for λ /∈Spec (∆•ν) isproved in Theorem 3.1.62The terms P (m)λ,int, ψjDmϕj, and qνBνP mλ in Rmλ are bounded operators with the same estimateof their norms in (DR•(M))2 for λ ∈Λε (the proof of Theorem 3.1). For the sake of brevity theproof of Proposition 3.1 is given in the case of ∂M = ∅.

98S.M. VISHIKin (DR•(M))2 with its norm estimated by O1 + |λ|1/2−2for λ ∈Λε.

It holdsthat Rλ : (DR•(M))2 →Dom (∆•ν). For a linear differential operator F of orderd = d(F) ≤2 the operator FRλ is defined on smooth forms ω ∈DR•c(M \ N)and its closure in (DR•(M))2 is a bounded operator in (DR•(M))2 with its normestimated by O1 + |λ|1/2d−2for λ ∈Λε.

(All these estimates are uniform withrespect to ν ∈R2 \ (0, 0).) The only terms of Rmλ depending on ν are the termsD −qνBνP mλ (ν), where63 D = D(ν) := −P ψjDm,Ujϕj has the kernel with supportin the neighborhood I × N of the interior boundary N ֒→M.64 (The parametrixP mλ (ν) := P mλ,int −D(ν) is defined by (3.13)).

We need the following assertion now.Proposition 3.3. The operators D(ν) and qνBνP mλ (ν) depend smoothly on ν ∈R2 \ (0, 0) as bounded operators in (DR•(M))2.

For a C∞-map ν = ϕ(γ) : [−a, a] →R2\(0, 0) the operator ∂γD(ν) is a bounded operator in (DR•(M))2 whose norm is uni-formly with respect to γ estimated by O1 + |λ|1/2−2for λ ∈Λε. Let F be a lineardifferential operator of order d = d(F) ≤2 from DR•(M j) into DR•+k(Mj), k ∈Z.Then FD(ν), F∂γD(ν) are bounded operators from (DR•(M))2 intoDR•+k(M)2whose norms are estimated by O1 + |λ|1/2d(F )−2for λ ∈Λε.The operator∂γ (qνBνP mλ (ν)) is uniformly with respect to γ estimated by O1 + |λ|1/2−1forλ ∈Λε.Proof.

The kernel of the operator LψjDm,UjϕjL−1 (where L = L(ν) and L−1 aredefined by (3.5) and (3.7)) has a support in ((Uj ∩N) × [0, 1])2. The operator Dm,Ujis defined in (Uj ∩N) × R+ by (3.11) and (3.10).

The right sides of the boundaryconditions (3.10) depend on ν only by their dependence on L(ν) (where L is thematrix defined by (3.7)).Since gM is a direct product metric near N, a mirrorsymmetry (relative to N) acts as the identity operator on the symbol P a−2−j(x, ξ)of the Laplacian ∆• on M for x = (t, x′) from the neighborhood I × N of N. ThesymbolP a−2−j(t, x′, τ, ξ′, λ) is independent of t for t ∈I.So the symbol L P a−2−j (for t ∈I) is expressed as LaL−1, where L and L−1 acton the components of a matrix valued functions a−2−j in the coordinates ωj,c andωj,(1,f) as follows (according to (3.5)):(La)1,c;∗= |ν|−1(α −β)(a)c;∗,(La)2,c;∗= |ν|−1(β + α)(a)c;∗,(La)1,(1,f);∗= |ν|−1(−β + α)(a)(1,f);∗,(La)2,(1,f);∗= |ν|−1(α + β)(a)(1,f);∗. (3.22)63Dm = Dm,Uj is defined by (3.11), (3.10), and (3.12).64In the case ∂M ̸= ∅the terms connected with the Dirichlet and the Neumann boundaryconditions are added to D(ν).

Then the corresponding kernel has its support in a neighborhood of∂M in M.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY99The boundary conditions (3.10) (according to (3.22)) depend on ν only by thematrix transformation whose coefficients are independent of (t, x′) and smooth in ν.This transformation acts separately on each homogeneous component a−2−j.

Theright sides in (3.22) are nonsingular in (x′, τ, ξ′, λ) for b2(x′, τ, ξ′) −λ ̸= 0 (where b2is the principal symbol of the Laplacian on M for (t, x′) ∈I × N). Hence Lemma 2in [Se1] holds also for the symbol ∂γ (L(ν) P d−2−j).

Thus the desired estimates forthe norm of ∂γD(ν) in (DR•(M))2 and for the norms of FD(ν) and of F∂γD(ν) areconsequences of [Se1], Lemma 7.The operators qj,ν (1 ≤j ≤4) from (3.14) and (3.16) (for ϕ(t) even on t) can bedefined such that65LqνBνP mλ (ν)f(t)L−1 =XqjBjLP mλ (ν)f(t)L−1,where f ∈C∞0 (I), f(t) ≡1 for t ∈[0, 1/2] and f(t) ≡0 for t > 3/4. The operatorsBj and qj are independent of ν ∈R2 \ (0, 0) and correspond to Bj,ν and qj,ν from(3.15), (3.16).

(Here B1, B2, B3, B4 are the operators (3.6) acting respectively onv1,c, w2,(1,f), w1,(1,f), w2,c.) These operators are such that Biqj = δij id.The operatorP qjBjL (∂γP mλ (ν)) fL−1 is equal toP qjBjL (−∂γD(ν)f) L (sinceP mλ,int is independent of ν).

The operator BjL (∂γD(ν)) L−1 is defined on smoothforms ω ∈DR•c ((0, 1) × N) and its L2-norm is estimated by O1 + |λ|1/2−1forλ ∈Λε ([Se1], Lemma 7). The operators ∂γ(qνBν)P mλ,int are defined on smooth formsω ∈DR•c(M \ N) and their operators L2-norms are estimated by O1 + |λ|1/2−1for λ ∈Λε uniformly with respect to ν (according to [Se1], Lemma 7).

The propositionis proved.□Let λ ∈Λε and |λ| be large enough.Then the Green function G•λ(ν) can berepresented by the seriesG•λ(ν) = Rmλ∞Xi=0(Lmλ )i ,(3.23)where (Lmλ ) := id −(∆•ν −λ) Rmλ is a bounded operator in (DR•(M))2 for λ ∈Λε. The norm of Lmλ in (DR•(M))2 is O1 + |λ|1/2n−m+2(where n := dim M)because the norm of (id −(∆• −λ)P mλ ) is O1 + |λ|1/2n−mand the norm of(∆• −λ)qνBνP mλ is O1 + |λ|1/2n−m+2(according to the proof of Theorem 3.1).Hence if m > n + 2 and if λ ∈Λε with |λ| large enough then the series (3.23) is65The operator BνP mλ (ν) has a continuous on M j1 × M j2 kernel which is estimated uniformlywith respect to (x1, x2) ∈N ∩M j1×Mj2 and to ν ∈R2\(0, 0) by O1 + |λ|1/2n−mfor λ ∈Λε([Se1], Lemma 6).

Such an estimate holds also for the kernel of qνBνP mλ (ν).

100S.M. VISHIKconvergent with respect to the operator norm in (DR•(M))2.

The operator Lmλ de-pends smoothly on ν ∈R2 \ (0, 0). The norm of ∂γLmλ is estimated by O1 + |λ|1/2(according to Proposition 3.3).

Let ν : [−a, a] →R2 \ (0, 0) be a smooth map. Thenthe series for ∂γG•λ(ν) is convergent (in the operator norm) if λ ∈Λε and if |λ| is largeenough.

Hence for such λ the resolvent G•λ(ν) := (∆•ν −λ)−1 depends smoothly onγ. So the family G•λ(ν) of bounded operators in (DR•(M))2 is smooth in (ν, λ) forsuch λ.

Their operator norms are estimated by O (|λ|−1) uniformly with respect toν ∈R2 \ (0, 0). Let F be a linear differential operator of degree d(F) ≤2.

Then theoperators FG•λ(ν) for such λ are bounded in (DR•(M))2 with their operator normsestimated by O|λ|(d−2)/2uniformly with respect to ν.These operators dependsmoothly on ν for such λ and we have∂γFG•λ(ν) = F∂γG•λ(ν). (3.24)Hence for a given ν0 ∈R2 \ (0, 0) there exists λ1 ∈Λε such that G•λ1(ν) dependssmoothly on ν for ν sufficiently close to ν0.

For λ ∈C \ Spec∆•ν0the resolventG•λ(ν0) can be represented as follows:G•λ(ν0) = −(λ −λ1)−1 −(λ −λ1)−2R(λ −λ1)−1, G•λ1(ν0),(3.25)where Rη, G•λ1(ν0):=G•λ1(ν0) −η−1 is the resolvent of a bounded operatorG•λ1(ν0) in (DR•(M))2 ([Ka], Ch. IV, (3.6), Ch.

III, (6.18)). The bounded opera-tor R(η, B) is an analytic function of a bounded operator B and of η for η /∈Spec B(i.e., near (η0, B0), η0 /∈Spec B0, it is locally defined by a convergent double powerseries in (η −η0) and (B −B0)).

The operator G•λ1(ν) depends smoothly on ν for νsufficiently close to ν0. Then it follows from (3.25) that G•λ(ν) depends smoothly onν for λ ∈C \ Spec (∆•ν) and for ν sufficiently close to ν0.Let F : ⊕j DR•(M j) →⊕jDR•+k(M j), k ∈Z, be a linear differential oper-ator of degree d(F) ≤2.Then for λ ∈Λε and |λ| large enough the operatorsFG•λ(ν): (DR•(M))2 →DR•+k(M)2 are defined, bounded, and smooth in ν ∈R2 \ (0, 0).

(It is proved above.) For example, dG•λ(ν) and δG•λ(ν) are smooth in ν.According to (3.24) we have ∂γ (dG•λ(ν)) = d∂γG•λ(ν), ∂γ (δG•λ(ν)) = δ∂γG•λ(ν).The operators dG•λ(ν) are defined for λ /∈Spec(∆•ν).

From (3.24) and (3.25) weget∂γ (dG•λ(ν))γ=0 = d∂γG•λ(ν)γ=0== −(λ −λ1)−2d ∂∂B R(λ −λ1)−1, B B=G•λ1(ν0)∂γG•λ1(ν)γ=0!

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY 101for a C∞-local mapR1γ, 0→(R2 \ (0, 0), ν0), where λ1 ∈Λε with |λ1| large enoughand λ /∈Spec∆•ν0.

Proposition 3.1 is proved.□3.3. Theta-functions for the Laplacians with ν-transmission boundary con-ditions.

Proofs of Theorem 3.2 and of Proposition 3.2. Let ε be fixed, 0 < ε <π/2.

The operator exp (−t∆•ν) is defined for Re t > 0, π/2 −ε > arg t > −(π/2 −ε),by the integralexp (−t∆•ν) = i2πZΓL,εexp(−λt)G•λ(ν)dλ,(3.26)where ΓL,ε = Γ1L,ε ∪Γ2L,ε, Γ1L,ε = {λ = −L + x exp(iε), +∞> x ≥0}, Γ2L,ε = {λ =−L + x exp(−iε), 0≤x <+∞}, L > 0. The integral (3.26) is absolutely convergentbecause the operator norm in (DR•(M))2 of the operator G•λ(ν) (which is boundedin (DR•(M))2) is estimated by O (|λ|−1) for λ ∈ΓL,ε, according to Theorem 3.1.66This integral is independent of L > 0 and of ε, π/2 > ε > 0, for t such that| arg t| < π/2 −ε, since the spectrum of ∆•ν is discrete and since Spec (∆•ν) ⊂R+ ∪0.With the help of the inverse Mellin transform f →M−1fM−1f(t) := (2πi)−1ZRe s=c Γ(s)t−sf(s)dsit is possible to obtain the results about the asymptotic expansion for Tr exp (−t∆•ν)as Re t →+0 (when π/2 −ε > | arg t|) from the results about ζν,•(−m), m ∈Z+ ∪0, and about ress=sj ζν,•(s) obtained in Theorem 3.1.

The integral (3.26) can betransformed as followsexp(−t∆•ν) = H•(ν) + i2πZΓ−δ,εexp(−tλ)G•λ(ν)dλ,where H•(ν) is the kernel of the orthogonal projection operator of (DR•(M))2 ontoKer ∆•ν and where δ > 0 and ρ, ρ ≥δ, is such that Spec (∆•ν) ∩(0, ρ] = ∅. Theoperator exp (−t∆•ν) for | arg t| < π/2 −ε can be represented as follows (where Γ is66The constant factor in this estimate depends on ε.

102S.M. VISHIKthe same as in (3.18) and c > 0):exp(−t∆•ν) = H•(ν) + i2πZΓ−δ,εexp(−tλ)G•λ(ν)dλ == H•(ν) + i2πZΓ−δ,ε(2πi)−1G•λ(ν)ZRe s=c(λt)−sΓ(s)dsdλ == H•(ν) + (2πi)−1ZRe s=c t−sΓ(s) i2πZΓ−δ,ελ−sG•λ(ν)dλ!ds == H•(ν) + (2πi)−1ZRe s=c t−sΓ(s) i2πZΓ λ−sG•λ(ν)dλds == H•(ν) + (2πi)−1ZRe s=c Γ(s)t−sT−s(ν)ds.

(3.27)Here, the integration is over Re s = c from c −i∞to c + i∞(where c > 0). Theoperator T−s(ν) for Re s > 0 is defined by the integral (3.18).

The transformationswe apply in (3.27) are correct by the Fubini theorem since the estimate∥G•λ(ν)∥2 < C · |λ|−1 ,λ ∈Γ−δ,ε,is satisfied by the operator norm of G•λ(ν) in (DR•(M))2 and since for Re s > 0 thegamma-function can be estimated as follows. We haveΓ(s) =Z ∞0ts−1 exp(−t)dt =Z ∞0ts−1 exp(iϕs) exp (−t exp(iϕ)) dtfor Re s > 0 and for an arbitrary ϕ ∈R such that π/2 > |ϕ|.

So the estimate holdsfor any ε1, 0 < ε1 ≤π/2 and for Re s > 0:|Γ(s)| ≤(sin ε1)−Re sΓ(Re s) exp−π2 −ε1|Im s|. (3.28)The kernel of (T−s(ν))x1,x2 is continuous in (x1, x2) ∈Mj1 × M j2 for Re s > n/2(according to Theorem 3.1).

The equality (3.27) holds also for c = Re s > n/2.For such s the integralRRe s=c Γ(s)t−s (T−s(ν))x1,x2 ds is absolutely convergent (byProposition 3.5 below and by (3.28)). Hence it defines a continuous on M j1 × M j2kernel.

So the kernel E•t,x1,x2(ν) of exp(−t∆•ν) is continuous on M j1 × M j2 becausewe haveE•t,x1,x2(ν) = H•(ν)x1,x2 + (2πi)−1ZRe s=c Γ(s)t−s (T−s(ν))x1,x2 dt,(3.29)where c > n/2. (The integral in (3.29) converges uniformly with respect to x1, x2 forany fixed c > n/2 by Proposition 3.5.

)From the functional equation Γ(s) = s−1(s + 1)−1 . .

. (s + l −1)−1Γ(s + l) it followsthat |Γ(s)| for Re s > −l is also estimated by exp (−(π/2 −ε1) | Im s|) as | Im s| →∞(with any fixed ε1, 0 < ε1 ≤π/2).

The operator exp (−t∆•ν) for Re t > 0 is a

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY 103trace class operator.

Namely its kernel is continuous on M j1 × Mj2 (as it followsfrom (3.29)). Hence it is a trace class operator and its trace is equal to the sumof the integrals over the diagonals M j of the corresponding densities (according toProposition 3.8 below).The theta-function θν,•(t) for ∆•ν is defined as the trace of exp (−t∆•ν) for Re t > 0.The analogous theta-function θν,•(t; pj) is defined as the trace Tr (pj exp (−t∆•ν)) forRe t > 0 (where pj : (DR•(M))2 →(DR•(Mj))2 ֒→(DR•(M))2 is the compositionof the natural restriction and of the prolongation by zero).

Proposition 3.8 claimsthat θν,•(t; pj) is equal to the integral over M j of the density tr∗x2i∗MjE•t,x1,x2(ν).The zeta-function ζν,•(s) is defined by (2.8) for Re s > n/2 (n := dim M). It isequal to Tr T−s(ν) for Re s > n/2 (according to Theorem 3.1 and to Proposition 3.9).The zeta-function ζ(s; pj) := Tr (pjT−s(ν)) is equal for such s to the integral over thediagonal iM j֒→M j × M j of the density, corresponding to the restriction of thekernel T−s(ν) to iM j.

The integral of this density over M j can be representedas the sum of the integrals of densities on M j and on ∂Mj (they are defined bythe parametrix (3.3) and can be continued to meromorphic functions on the wholecomplex plane C ∋s) and of a density on M j, which is holomorphic for Re s >(n −m)/2. (This assertion follows from the proof of Theorem 3.1.) The contourof the integration in (3.27) can be moved to Re s = a for an arbitrary a such that(−2a) /∈Z+ ∪0 (according to the estimates of |Γ(s)| as | Im s| →+∞and toProposition 3.4 below).

Then it follows from (3.27) thatθν,•(t; pj) =Xt−(n−k)/2 ress=sk (Γ(s)ζν,•(s; pj)) ++ (2πi)−1ZRe s=a t−sΓ(s)ζν,•(s; pj)ds + Tr (pjH•(ν)) ,(3.30)where the sum is over k such that sk := (n −k)/2 > a. The estimate of the integralover Re s = a in (3.30) is obtained with the help of (3.28) and (3.40) as follows.

ForRe t > 0, | arg t| < π/2 −ε (ε, 0 < ε < π/2, is fixed) and for Re s = a the estimateis satisfied:t−sΓ(s)ζν,•(s; pj)

The latter estimate is a consequenceof (3.28) and (3.40)), where ε1 and ε are replaced by ε/4. We see thatZRe s=a t−sΓ(s)ζν,•(s; pj)ds < C1(ε, a)|t|−a,(3.32)

104S.M. VISHIKwhere Re t > 0, | arg t| < π/2 −ε, π/2 > ε > 0, a < 0, and (−2a) /∈Z+.

Theassertions of Theorem 3.2 about the asymptotic expansion (3.1) for θν,•(t; pj) (relativeto t →+0 when | arg t| < π/2 −ε) follow from the equality (3.30) and from theestimate (3.32). The estimates analogous to (3.40) below and to (3.32) are satisfiedalso by the analytic continuation to C ∋s of the densities (on M j and on ∂Mj)defined by the parametrix P mλ (ν) (as in Proposition 3.5 below).

Thus we see that theequalities between the densities in the integral representation for the coefficients ofthe expansion (3.1) and the corresponding densities for the residues and the valuesof ζν,•(s; pj) are satisfied.The uniform with respect to ν ∈R2 \ (0, 0) estimate (3.2) for the traces ofexp (−t∆•ν) (for a fixed t, Re t > 0) follows from (3.30) and (3.31) because fora = −m −1/4, m ∈Z+, m >> 1, the integral over Re s = a on the right in(3.30) is absolutely convergent. The estimate67 (3.31) and the equality (3.30) pro-vide us with the uniform in ν upper estimate for Tr (pj exp (−t∆•ν)), ν ∈R2 \ (0, 0).Indeed, the estimate68 dim Ker ∆•ν < C is satisfied uniformly with respect to ν. Theformulas q−n+k = ress=sk (Γ(s)ζν,•(s; pj)) + δn,k Tr (pjH•(ν)) for the coefficients q−n+kof the asymptotic expansion (3.1) are consequences of (3.30) for a = −m −1/4,where m ∈Z+.

For a = −m−1/4 the absolute value of the integral over Re s = a in(3.30) is estimated (with the help of (3.31)) by C|t|m+1/4 uniformly with respect toν ∈R2 \ (0, 0) (where Re t > 0, | arg t| < π/2 −ε, π/2 > ε > 0 and ε is an arbitrarybut fixed). So it holdsθν,•(t; pj) =n+2m−1Xk=0q−n+kt−(n−k)/2 +nq−n+2mtm + O|t|m+1/4o.

(3.33)The latter two terms in (3.33) are O (|t|m) relative to t →+∞uniformly withrespect to ν ̸= (0, 0) (for | arg t| < π/2 −ε). The statements about the structureof the values and the residues of ζν,•(s; pj) (Theorem 3.1) provide us with the de-sired information about coefficients q−n+k in (3.1).

These values and residues (up toδn,k Tr (pjH•(ν))) are the sums of the integrals over Mj, ∂M, and N of the densitieswhich are defined by the absolutely convergent integrals of the components a−2−k andd−2−k+1 ([Se2], Theorem 2, and the proof of Theorem 3.1 above). The latter symbolsare defined by (3.4), (3.11), and (3.10).

These integrals are smooth in ν ∈R2 \ (0, 0).Hence the coefficients q−n+k in (3.1) are smooth in ν ̸= (0, 0) (and are invariant underν →cν, c ̸= 0). Theorem 3.2 is proved.□67For a < 0 the function Γ2(1 −a), cε/4ρ1/2tends to Γ (2(1 −a)) as ρ →+0 and so it isbounded for 0 < ρ < 1.68It follows from the exact sequence (1.14) (where Zj := Z ∩∂Mj) and from Lemma 1.1 thatdim Ker ∆•ν = dim H•(Mν, Z) ≤Xjdim H•(Mj, N ∪Zj) + dim H•(N).

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY 105Remark 3.2.

The coefficients q−n+k of (3.1) for 0 ≤k ≤m are completely defined(according to (3.33) and to Remark 3.1) by an arbitrary parametrix P mλ (ν) (3.3) for(∆•ν −λ)−1.Proof of Proposition 3.2. The parametrix69 P •(m)t,x,y (ν) for E•t,x,y(ν) (defined by(2.126)) is such that it is smooth in (x, y) ∈Mj1 × M j2 and in ν ∈R2 \ (0, 0).

Theν-transmission boundary conditions (1.27) are satisfied for∆•ν,xk P •(m)t,x,y (ν) (i.e., theimage of (DR•(M))2 under the action of the operator with the kernel P •(m)t,x,y (ν) belongsto D(∆•ν)kfor an arbitrary k ∈Z+). The uniform with respect to ν ∈R2 \ (0, 0)estimates (2.127), (2.128) are satisfied and for x from an appropriate neighborhoodU of N ⊂M we have∂t + ∆•ν,xP •(m)t,x,y (ν) ≡0 (where U is independent of ν).Set r(m)t,x,y(ν) :=∂t + ∆•ν,xP •(m)t,x,y (ν).

Then the estimates are satisfied for any k ∈Z+ ∪0∆kν,xr(m)t,x,y(ν) < Cm,kt−n/2+m−k,(3.34)where Cm,k is independent of ν ∈R2 \ (0, 0) and of t ∈(0, T] (n := dim M).The kernel E•t,x,y(ν) can be represented as the Volterra series70E•t,x1,x2(ν) ==Xk≥0(−t)kZ∆kZ(y1,...,yk)∈(M1∪M2)k P •(m)σ0t,x1,y1(ν)r(m)σ1t,y1,y2(ν) . .

. r(m)σkt,yk,x2(ν),(3.35)where ∆k = {(σ0, .

. .

, σk): 0 ≤σi ≤1, P σi = 1} (and the scalar product tr(ω1 ∧∗ω2)with the values in densities on M is implied in (3.35)). The proof of (3.35) (or of(2.137)) is given in the proof of Proposition 2.21.Let ϕ: I →R2 \ (0, 0), ν = ϕ(γ), be a C∞-map (where γ ∈[−a, a] =: I).

Thenthe only term in P •(m)t,(x1,x′),(y1,y′)(ν) depending on γ isEN,t ⊗ψ(x1)EI,t(ν)ϕ(y1) =: E•N,I(ν)(3.36)but it does not depend on m.(Here EI,t corresponds to ∆I with the Dirichletboundary conditions on ∂I, ϕ, ψ ∈C∞0 (I \ ∂I), ψ ≡1 in a neighborhood of supp ϕ ⊂I \ ∂I, ϕ(x1) ≡1 for x1 ∈[−1/2, 1/2] and ϕ, ψ are even: ϕ(−x1) = ϕ(x1), ψ(−x1) =ψ(x1).) So, as it follows from the explicit formulas (2.40) for (GI(ν))x1,y1 (and from69The properties of such a parametrix are summarized in Proposition 2.21.

For the sake of brevitythe proof of Proposition 3.2 is given in the case of ∂M = ∅. The terms of P (m)t(ν), connected withthe Dirichlet and the Neumann boundary conditions on the components of ∂M, are independent ofν and the proof in the case of ∂M ̸= ∅does not contain any additional difficulties.70This series was used in the case of a closed manifold M in [BGV], 2.4, 2.7.

See also the formula(2.137) above.

106S.M. VISHIKthe analogous formulas (2.54) and (2.55) for (EI,t(ν))x1,y1), the uniform with respectto γ estimates are satisfied for any k, q ∈Z+∂kγr(m)t,x,y(ν) < c(q,k)t−n/2+q.

(3.37)because they are true for (∂t + ∆x′ + ∆x1)n(EN,t)x′,y′ ⊗ψ(x1) (EI,t(ν))x1,y1 ϕ(y1)oand for (∂t + ∆x′ + ∆x1)n(EN,t)x′,y′ ⊗ψ(x1) (σ∗1EI,t(ν))x1,y1 ϕ(y1)o(where σ1 is thereflection of I = [−1, 1] with respect to 0 ∈I which acts on the variable x1). Thekernels ∂kγE•N,I(ν) are the linear combinations of these two kernels with the coefficientsindependent of x and y.

(These coefficients are smooth in γ). The estimate (3.37)is satisfied for t ∈(0, T) and for an arbitrary q ∈Z+ uniformly with respect to γ,because if ∂x1ψ(x1) ̸= 0 then ρ(x1, supp ϕ) > δ > 0.Let DR•M(l) be the space of forms on M of a class Cl (i.e., of forms with l contin-uous derivatives on M) equiped with a Cl-norm.71 Let DR•M,N(l) := DR•(l)M 1⊕DR•(l)M 2be the space of pairs (ω1, ω2) of forms ωj of a class Cl on M j with a Cl-norm.

The operators with the kernels P •(m)t,x,y (ν) for ν ∈R2 \ (0, 0) and the operatorscorresponding to ∂kγP •(m)t,x,y (ν(γ)) (for a fixed k ∈Z+ ∪0) are families of uniformly(with respect to ν and to t ∈(0, T]) bounded operators acting from DR•M(l) intoDR•M,N(l). For the operators, corresponding to the interior terms in P •(m)t,x,y (ν) thisassertion is proved in [BGV], Theorem 2.29, Lemma 2.49.

This proof uses that thisstatement is local in x ∈M (for a closed M) and it uses also the explicit definition ofP (m)int over a geodesic ball expx B ⊂M (where B is a ball ∥v∥≤c in TxM and expxis the exponential map for (M, gM) from TxM).The kernel (3.36) (i.e., the term of P •(m)t,x,y (ν) corresponding to the interior boundaryN) is equal (up to the factor ψ(x1)ϕ(y1)) to a linear combination given by (2.54) and(2.55) of the kernels E•t for N × I and σ∗1E•t (σ1 is the mirror symmetry with respectto N × 0).72 Its coefficients depend on (j1, j2, ν), where (x, y) ∈Mj1 × Mj2. Thesecoefficients and their derivatives of a fixed order on γ are uniformly bounded.For a closed N the operators defined byP (m)Nt :=P (m)int,Nt are uniformly boundedfor 0 < t ≤T with respect to a Cl-norm in the space DR•N(l) of Cl-smooth formson N. The equality (3.35) is satisfied by E•N,t, P •(m)N,tand r(m)N,t .

Since the estimate(3.38) below (as well as the analogous estimate (3.34)) is satisfied by r(m)N,t (wheren is replaced by n −1) we see that the series of operators on the right in (3.35) is71This norm corresponds to a smooth partition of unity {ϕi} subordinate to a finite cover {Ui}of M 1 and of M2 by coordinate charts (i.e., ϕi ∈C∞0 (Ui)). For v ∈DR•M,N(l) its Cl-norm ∥v∥lis equal to P supx∈Ui sup|α|≤l |Dαx(ϕiv)| i.e., to the sum of the suprema of partial derivatives oforders ≤l.

The Cl-norm for an arbitrary smooth finite cover {U ′i} and for a partition of unity {ϕ′i}subordinate to {U ′i} is equivalent to the one defined by {Ui} and by {ϕi}.72The operators ∂kγP (m)(ν) = ∂kγEN,I(ν) for k ∈Z+ are expressed similarly.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY 107convergent for m ≥(n + l −1)/2 relative to a Cl-norm in DR•N(l).

Hence the sum ofthis series defines a family of the operators E•N,t in DR•N(l) bounded uniformly withrespect to t ∈(0, T]. The analogous assertion is also true for a family of operatorsdefined by the kernels ψ(x1)E•I,tx1,y1 ϕ(y1) acting on smooth forms with compactsupports on (I, ∂I) (with respect to a Cl-norm).

So the kernels E•N,I(ν)t,(x1,x′),(y1,y′)and ∂qγE•N,I(ν(γ))t,(x1,x′),(y1,y′) (for a fixed q ∈Z+) define collections of uniformly (int ∈(0, T] and ν or in t and γ) bounded with respect to a Cl-norm operators fromDR•M(l) into DR•M,N(l).The kernel r(m)t,x,y(ν) is smooth on M × M j according to the definition (2.126) ofP •(m)t,x,y (ν). The Cl-norm of r(m)t,x,y(ν) on each M × M j satisfies the estimate (analogousto (3.34))r(m)t,x,y(ν)l ≤Clt−n/2+m−l/2.

(3.38)uniformly with respect to ν ∈R2 \ (0, 0) and to t ∈(0, T]. The estimates analogousto (3.37) and to (2.129) are satisfied uniformly with respect to ν and to t ∈(0, T]also by the Cl-norms of ∂kγr(m)t,x,y(ν) on each M × M j for any k, q ∈Z+:∂kγr(m)t,x,y(ν)l ≤c(q,k,l)t−n/2+q.

(3.39)Leibnitz’s rule claims that P (m)t,x,y(ν(γ)) is a bounded operator from the space ofCp-maps ω : [−a, a] →DR•M(l) (equiped with the normPpi=0 supγ∈[−a,a]∂iγωl) intoCp [−a, a], DR•M,N(l). Indeed, the kernel ∂kγP (m)t,x,y(ν(γ)) depends smoothly on γ ∈[−a, a] on M j1 × M j2 for k ∈Z+ ∪0, since P (m)t,x,y(ν) :=P (m)intt,x,y + E•N,I(ν) andsince E•N,I(ν) is smooth in ν ∈R2 \ (0, 0).

For 0 < t ≤T the operatorsP (m)t(ν(γ))from Cp ([−a, a], DR•M(l)) into Cp [−a, a], DR•M,N(l)are uniformly bounded, be-cause ∂kγP (m)t(for a fixed k ∈Z+ ∪0) are the operators from DR•M(l) into DR•M,N(l)bounded uniformly in γ and t, 0 < t ≤T, with respect to a Cl-norm . Hence, ac-cording to (3.38), (3.39), and to the fact that the volume of ∆k is equal to (k!

)−1,the series (3.35) for the derivative ∂pγEt,x1,x2 (ν(γ)) is convergent in the Cl-norm onS M j1 × M j2for m > (n + l)/2. (The number m in the definition P (m) is greaterthan (n + l)/2.

)This proves that ∂kγE•t,x1,x2(ν(γ)) is C∞-smooth on M j1 × M j2. (For instance, fork = 0 this proves that E•t,x1,x2(ν) is C∞-smooth on M j1 × M j2.

)So the restrictions i∗jE•t,x,x(ν(γ)) to the diagonals ij : M j ֒→Mj × M j are C∞-smooth double forms on M j which are C∞-smooth in γ . Since rm(t, ν) in (3.1)are Ot(m+1)/2uniformly with respect to ν ∈R2 \ (0, 0) and since qi are C∞-

108S.M. VISHIKsmooth in ν we see that the asymptotic series (3.1) can be differentiated on γ.Actually, the equality (3.35) holds for E•t (ν), P •(m)t(ν), and r(m)t(ν).The kernelr(m)t(ν) satisfies the estimates (3.34), (3.37), and (3.39) and the kernel P •(m)t(ν(γ))defines a family of uniformly with respect to t ∈(0, T] and to γ bounded opera-tors from Cp ([−a, a], DR•M(l)) into Cp [−a, a], DR•M,N(l).

Hence the power termst(−n+j)/2, 0 ≤j ≤2m, in the asymptotic expansion ofRMj tr i∗j∂γE•t (ν) as t →+0are equal to the appropriate terms in the asymptotic expansion ofRMj i∗j∂γP •(m)t(ν). (The kernel ∂γE•t (ν) −P •(m)t(ν)x1,x2 is Ot−n/2+m+1, according to (3.35).) Butthe coefficients qi, 0 ≤i ≤2m, in (3.1) are completely defined by i∗jP •(m)t(ν), be-cause the kernelE•t (ν) −P •(m)t(ν)x1,x2 is Ot−n/2+m+1uniformly with respect to(x1, x2) ∈M j1 × M j2 and to t ∈(0, T], according to (3.35).

Thus Proposition 3.2 isproved.□3.4. Estimates for zeta-functions and for the corresponding kernels in ver-tical strips in the complex plane.Proposition 3.4.

The meromorphic continuation of the zeta-function ζν,•(s; pj) :=Tr (pjT−s(ν)) for Re s > n/2 is estimated by C(ε) exp (ε| Im s|) as | Im s| →+∞forany fixed ε > 0. Namely for any ε > 0 and for an arbitrary a ∈R the followingestimate is satisfied if Re s ≥a:|ζν,•(s; pj)| ≤C(a, ε) exp (ε| Im s|) ××c2(Re s−1)εΓ2(1 −Re s), cερ1/2+ maxρ−Re s, 1 1 +X|s −sj|−1,(3.40)where ρ > 0 is such that Spec (∆•ν) ∩(0, ρ] = ∅and the sum is over sj := (n −j)/2,−sj /∈Z+ ∪0, sj ≥a.

The constants C(a, ε) and cε are positive and independent ofν ∈R2 \ (0, 0), and Γ(u, x) :=R ∞x tu−1 exp(−t)dt for x > 0.Proposition 3.5. For Re s > n/2 (n := dim M) and for any ε > 0 the followingestimate is satisfied (where ρ > 0 is such that Spec (∆•ν) ∩(0, ρ] = ∅):(T−s(ν))x1,x2 ≤Cερ−Re s exp (ε| Im s|) Re s −n2−1+ 1!.

(3.41)Proof of Proposition 3.4. It is proved in Theorem 3.1 that the operator norm∥G•λ(ν)∥2 in (DR•(M))2 of the Green function G•λ(ν) for the Laplacian ∆•ν is esti-mated by Cε|λ|−1 for λ ∈Λε := {λ ∈C, ε ≤arg λ ≤2π −ε}, where ε, 0 < ε ≤π isfixed.

The spectrum Spec (∆•ν) is a discrete subset of R+ ∪0 by Theorem 3.1. So the

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY 109operator T−s(ν) defined by the integral (3.18) is equal to the same integral with thecontour Γ replaced by Γ(ε) := Γ1,ε ∪Γερ ∪Γ2,ε,Γ1,ε = {λ = x exp(iε), ∞> x ≥ρ},Γερ = {λ = ρ exp(iϕ), ε > ϕ > −ε},Γ2,ε = {λ = x exp(−iε), ρ ≤x < ∞}.

(3.42)There is a constant c > 0 such that the principal symbol (b2(x, ξ) −λ) id of ∆•−λ idon M is invertible for|ξ|2 > c|λ|(3.43)in the coordinate charts Ul (of the same finite cover {Ul} of M as in (3.3)). Theintegral (3.18) over the contour Γ(ε) (the latter one is defined by (3.42)) does notdepend on ρ for all ρ > 0 such that (0, ρ] ∩Spec(∆•ν) = ∅.

We suppose from now onthat0 < ρ < (2c + 1)−1(3.44)and that (0, ρ] ∩Spec(∆•ν) = ∅.The kernel (rmλ )x1,x2 of the operator rmλ := G•λ(ν) −P mλ for m ≥n is continuous onM j1 × M j2for λ ∈Λε \0. (The parametrix P mλ is defined by (3.3).) It is estimatedfor |λ| ≥ρ, λ ∈Λε (according to (3.19)) uniformly with respect to ν ∈R2 \ (0, 0) by(rmλ )x1,x2 < Cε,ε11 + |λ|1/2−(2+m)+n+ε1(3.45)for any ε1 > 0.Since (3.45) is satisfied for all λ ∈Γ(ε), we have for Re s >2−1 (−m + n + ε1) thati2πZΓ(ε)λ−s (rmλ )x1,x2 dλ =i2πXjZΓj,ε+ZΓερλ−s (rmλ )x,x dλ ≤≤2Cε,ε1 exp (ε| Im s|) ××Z ∞ρ|λ|−Re s 1 + |λ|1/2−(2+m)+n+ε1 d|λ| + πρ maxρ−Re s, 1.

(3.46)The estimate (3.46) claims that for the proof of (3.40) in the domain Re s ≥ait is enough to prove the analogous estimate for the analytical continuation of thedensities on M j, N, and on ∂M, corresponding (for Re s > n/2) to the kernel ofi2πZΓ(ε)λ−sP mλ dλ,(3.47)where m = m(a) ∈Z+ is sufficiently large. (These densities were introduced in theproof of Theorem 3.1, and the sum of their integrals is equal to the trace of (3.47).

)Let Re s > n/2 and ps,εint(x) be the density on Mj, corresponding to the restric-tion to the diagonal ij : Mj ֒→Mj × Mj of the kernel P s,εint(x1, x2) of the operator

110S.M. VISHIKRΓ(ε) λ−s pjPi ψiP mλ,int,Uiϕidλ (where P mλ,int is defined by (3.13) and by (3.4)).

Thenps,εint(x) can be continued to a whole complex plane C ∋s as a meromorphic density([Se2], Lemma 1, or [Sh], Theorem 12.1).Proposition 3.6. The density ps,εint(x) satisfies the following estimate for any ε > 0:|ps,εint(x)| < Cε maxρ−Re s, 1exp (ε| Im s|)X|s −sk|−1 ,(3.48)where the sum is over 0 ≤k ≤m such that (−sk) /∈Z+ ∪0.Proof.

The density ps,εint(x) for Re s > n/2 corresponds to the sum of the integrals(2π)−n Xjϕj(x)Zdξ i2πZΓ(ε)λ−sθ(ξ, λ)mXj=0a−2−j(x, ξ, λ)dλ,(3.49)where a−k is a positive homogeneous of degree (−k) inξ, λ1/2component of thesymbol s(∆• −λ)−1in the coordinate chart Ui defined by (3.4). The integral (3.49)is the sum of the integrals Js0,ε(x) + Js1,ε(x) + Js2,ε(x) over the three correspondingdomains:K0 :=n(ξ, λ): |ξ|2 ≤1 −ρ, λ ∈Γ(ε), |λ| ≤1 −|ξ|2o,I1 :=n(ξ, λ): |ξ|2 ≤1 −ρ, λ ∈Γ(ε), |λ| >1 −|ξ|2o,(3.50)I2 :=n(ξ, λ): |ξ|2 > 1 −ρ, λ ∈Γ(ε)o,Since K0 is compact and since λ−sθ Pm0 a−2−j(x, ξ, λ) is continuous on Mj × K0,the density Js0,ε(x) is holomorphic in s ∈C and it is estimated byJs0,ε(x) ≤≤Cε exp (ε| Im s|) maxρ−Re s, 1maxMj×K0Xiϕi(x)θmX0a−2−j(x, ξ, λ) .

(3.51)The latter factor on the right in (3.51) does not depend on s and ρ. Hence it isestimated by a constant.Set Jsk(x) := Jsk,ε(x) from now on.

For Re s > n/2 the density Js2(x) does notchange if the interior integral in (3.49) is replaced by the integral overΓ(ε),|ξ| := |ξ|2(1 −ρ)−1Γ(ε),(3.52)because θPm0 a−2−j is holomorphic in λ in the domain between the contours Γ(ε) andΓ(ε),|ξ|. Indeed, this symbol is holomorphic in λ for (ξ, λ) such that |ξ|2 + |λ| > 1 and

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY 111|ξ|2 > c|λ| (where c > 0 is the same as in (3.43) and (3.44)).

Since 0 < ρ < (2c+1)−1,we have for λ between Γ(ε) and Γ(ε),|ξ|ρ ≤|λ| ≤|ξ|2 ρ (1 −ρ)−1 ,c|λ| ≤c|ξ|2ρ(1 −ρ)−1 < 2−1|ξ|2 < |ξ|2,and|ξ|2 + |λ| > 1for(ξ, λ) ∈I2.The density Js2(x) is represented as the sum Js2,ρ(x) + Pj=1,2 Js2,j(x), where Js2,jand Js2,ρ correspond to (3.49), with the interior integral replaced by the integral overΓεj,|ξ| := |ξ|2 (1 −ρ)−1 Γj,ε and over Γερ,|ξ| := |ξ|2(1 −ρ)−1Γερ. The density Js1(x) isequal to the sum Pj=1,2 Js1,j(x), where the interior integral in (3.49) for the term Js1,jis over the contour Γj,ε \ D1−|ξ|2 (Dr := {λ, |λ| < r}).Set λ := expiε (−1)j+1t2 on Γεj,|ξ| (for |ξ|2 > 1 −ρ) and on Γj,ε \ D1−|ξ|2 (for|ξ|2 ≤1 −ρ), where t > 0 is a new variable.

Then we haveJs1,j + Js2,j(x) ==2(2π)−(n+1)iexp(−1)jiε(s−1)Xlϕl(x)ZFt−2s+1tr mX0a−2−k(x, ξ, λε(t))!dt dξ,(3.53)where λε(t) := exp ((−1)j+1iε) t2, Re s > n/2, and F is the domainn(ξ, t): |ξ|2 + t2 ≥1, |ξ|2 ≤1 −ρ, t ≥ρ12o∪n(ξ, t): |ξ|2 ≥1 −ρ, t ≥|ξ|ρ12. (1 −ρ)12o.Since a−k(x, ξ, λ) are nonsingular for λ ∈Γj,ε and since r · (ξ, t) ∈F for (ξ, t) ∈Fand r ≥1, we see that (3.53) can be written as follows:Js1,j + Js2,j(x) = 2 (2π)−(n+1) i exp(−1)jiε(s −1)××XlmXk=0(2s + k −n)−1 ϕl(x)ZF1t−2s+1 tr a−2−k (x, ξ, λε(t)) dt ωn+1,(3.54)where F1 = F ∩{(ξ, t): ξ2+t2 = 1} and ωn+1 is the volume form on the unit sphere inRn+1ξ,t .

The integral over the compact F1 in (3.54) is an entire function of s ∈C. So theright side of (3.54) realizes the analytic continuation of the densityJs1,j + Js2,j(x)to a meromorphic in s ∈C density with no more than simple poles at the pointssk = (n −k)/2, 0 ≤k ≤m.Since λ = ρ (1 −ρ)−1 |ξ|2 exp(iϕ) on Γερ,|ξ| (where ε ≥ϕ ≥−ε), we haveJsρ(x) = −i (2π)−(n+1) XlmXk=0ϕl(x)ZIε,ρl,k (s, ξ)dξ,(3.55)

112S.M. VISHIKwhere the integral is over {ξ : |ξ|2 ≥1 −ρ} andIε,ρl,k (s, ξ) :=Z ε−εexp(−iϕ(s −1))a−2−k(x, ξ, λ(ϕ, |ξ|))dϕ |ξ|−2(s−1)ρ−(s−1)+, (3.56)λ(ϕ, |ξ|) := |ξ|2ρ+ exp(iϕ),ρ+ := ρ(1 −ρ)−1.The symbol (3.56) is positive homogeneous of degree (−2s −k) in ξ.

It is analyticin s ∈C and nonsingular. So (3.55) realizes a meromorphic continuation of Jsρ(x) tothe whole complex plane C ∋s.

NamelyJsρ(x)=2−1(2π)−(n+1)XlmXk=0ϕl(x) Z|ξ|=1Iε,ρl,k (s, ξ)xωn! (s−sk)−1(1−ρ)−(s−sk) ,(3.57)where ωn is the volume form on the unit sphere in Rn.

The formulas (3.54) and (3.57)provide us with a meromorphic continuation of the density P s,εint(x). Together withthe estimate (3.51) they provide us with the estimate (3.48).

(However, with the sumin it over all sk, 0 ≤k ≤m.) The analytic continuation of the density defined bythe sum of the integrals (3.49) with the interior integral over the contour Γ(π) (i.e.,with ε = π) is regular in s = sk for (−sk) ∈Z+ ∪0.

(This assertion is obtained inthe proof of Theorem 3.1.) For |ξ|2 > 1 −ρ the interior integral over Γ(ε) in (3.49)is equal to the integral over Γ(π).

So the estimate (3.48) is satisfied, where the sumover sk, 0 ≤k ≤m, such that (−sk) /∈Z+ ∪0.□Let Re s > n/2 and let ps,εj,∂(x) be the density on Mj corresponding to the term in(3.47) determined by the ν-transmission interior boundary conditions.73 It is definedby the restrictions to the diagonal M j ֒→Mj × Mj of the kernel for the operatori2πZΓ(ε)λ−sdλ pjXlψlDm,Ulϕl,(3.58)where Dm = Dm(λ, ν) is defined by (3.9), (3.10), and (3.12).The operator Dm is defined by the symbols P d−2−k, 0 ≤k ≤m, in a coordinatechart (x′, t), Ui ⊂Rn−1 × R1 (where N ∩Ui = (Rn−1 × 0) ∩Ui and the structureRn−1 × R1 corresponds to the direct product structure of the metric gM near N). Itsaction on f, f ∈DR•c (Rn−1 × (R+ \ 0)), can be represented for t > 0, t1 > 0, as73From now on we’ll suppose that ∂M = ∅.

Estimates of the contributions into ζν,•(s; pj) fromthe Dirichlet and the Neumann boundary conditions on the components of ∂Mj \ N are analogousto the estimates for the contributions from the ν-transmission interior boundary conditions.

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY 113follows ([Se1], (26)–(28)):mXk=0Op(θ1d−2−k)f(x′, t) :=mX0(2π)−nZZexp (i(ξ′, x′)) θ1d−−2−kf −(ξ′, t1)dt1dξ′,d−−2−k(x′, t, ξ′, t1, λ) := −ZΓ−exp(−iτt1)d−2−k(x′, t, ξ′, τ, λ),f −(ξ′, t1) =Zexp (−i(ξ′, y)) f(y, t1)dy,where Γ−= Γ−(ξ′, λ) is a simple contour in the half-plane Im τ < 0 which once goesround (in the direction opposite to the clockwise) the only zero of the principal symbol(b2(x′, ξ′, τ) −λ) id of the Laplacian (∆• −λ id).74Lemmas 2 and 3 of [Se2] andLemma 2 of [Se1] claim that the integral over R+ (where Ui∩(Rn−1 × R+) = Ui∩Mj)R d−−2−k(x′, t, ξ′, t, λ)dt is a symbol of (x′, ξ′, λ) positive homogeneous of degree (−2−k)inξ′, λ1/2.It claims also that the kernel on Rn−1 defined by the integral over[T, ∞) ⊂R+ (for an arbitrary T > 0)Zexp (i(ξ′, x′ −y′)) dξ′ZΓ dλZ ∞Tdt θ1(ξ′, λ)λ−sd−2−k(x′, t, ξ′, t, λ)is an entire function of s ∈C, smooth in x′, y′, s and vanishing at s for (−s) ∈Z+∪0.The latter assertion is an immediate consequence of the estimate ([Se1], (29)) ford−−2−k:Dγx′Dqξ′trtl1Dmt Dnt1Dpλd−−2−k(x′, t, ξ′, t1, λ) ≤C1 exp−cε (|t| + |t1|)|ξ| + |λ|1/2 1 + |ξ| + |λ|1/2−1−k−(|q|+2p)−(r+l)+m+n(3.59)with positive constants C1(ε) and cε independent of ν ∈R2 \ (0, 0).Proposition 3.7.

The analytic continuation to the whole complex plane C ∋s ofthe integral over Mj of ps,εj,∂(x) (which is defined for Re s > n/2) is estimated byZMjps,εj,∂(x) < c1,ε maxρ−Re s, 1exp (ε| Im s|)X|s −sk|−1 ++ c2,ε exp (ε| Im s|)2c2(Re s−1)εΓ2(1 −Re s), cερ1/2+ πρ maxρ−Re s, 1,(3.60)where the sum is over 1 ≤k ≤m + 1 such that (−sk) /∈Z+ ∪0.Proof. The trace of the operator (3.58) for Re s > n/2 is given by the integral(2π)−n XlZdx′ dtϕl(x′, t)Zdξ′ i2πZΓ(ε)λ−sdλθ1(ξ′, λ)mXk=0d−−2−k(x′, t, ξ′, t, λ)!.74The whole symbol of ∆• does not depend on t in the neighborhood of N.

114S.M. VISHIK(We suppose that ϕl(x′, t) are independent of t for 0 ≤|t| ≤1.) It follows from theestimate (3.59) that the density on N corresponding to the integralZR+dt (ϕl(x′, t) −ϕl(x′, 0))Zdξ′ tr i2πZΓ(ε)λ−sdλ θ1Xd−−2−k!has the analytic continuation which is an entire function of s ∈C and which satisfiesthe estimateCε,m,n1 exp (ε| Im s|) hρ(cε, Re s)Z(1 + |ξ|)−n1 exp(−cε|ξ′|)dξ′,(3.61)hρ(cε, Re s) :=Z ∞ρexp−cεt1/2t−Re sdt + πρ maxρ−Re s, 1== 2c2(Re s−1)εΓ2(1 −Re s), cερ1/2+ πρ maxρ−Re s, 1,where n1 ∈Z+ is sufficiently large.The density on N is defined by the integralps,εl,j,∂(x′) := (2π)−n ϕl(x′, 0)ZR+dtZdξ′ tr i2πZΓ(ε)λ−sdλ θ1mXk=0d−−2−k!

(3.62)which is absolutely convergent for Re s > (n−1)/2. Hence it is analytic in s for suchs.

The integral over t ∈R+ of d−−2−k is a positive homogeneous of degree (−2 −k) in(ξ′, λ1/2) symbol, which is smooth in (x′, ξ′, λ) and analytic in λ for c|λ| < |ξ′|2 (wherec is the same as in (3.43), and (3.44)) and in λ ∈Λε/2 := {λ: ε/2 < arg λ < 2π−ε/2}for (ξ′, λ) ̸= (0, 0) ([Se1], Lemma 2, [Se2], Lemma 2). So the proof of Proposition 3.6is valid also for the density (3.62) (where (x, ξ, n) are replaced by (x′, ξ′, n −1)).

Weconclude that this density has a meromorphic continuation ps,εl,j,∂(x′) with no more thansimple poles at the points s1, . .

. , sm+1.

This proof provides us with the estimateps,εl,j,∂(x′) < c1,ε maxρ−Re s, 1exp (ε| Im s|)X|s −sk|−1 ,(3.63)where the sum is over 1 ≤k ≤m+1. The analytic continuation of the density on N,defined by the sum (over l) of the integrals (3.62) for the interior integrals over thecontour Γ(π) (i.e., with ε = π) is regular at s = sk for (−sk) ∈Z+ ∪0.

(It is proved inTheorem 3.1.) For |ξ′|2 > 1 −ρ the integral over Γ(ε) in (3.62) is equal to the integralover Γ(π).

Hence the estimate (3.63) is satisfied if the sum is over k, 1 ≤k ≤m + 1,such that (−sk) /∈Z+ ∪0. The estimate (3.60) follows from (3.63), (3.61).□The estimate (3.40) follows from Propositions 3.6, 3.7, and from (3.46).ThusProposition 3.4 is proved.□Proof of Proposition 3.5.

The estimate (3.46) in the proof of Proposition 3.4 issatisfied by the integral of (rmλ )x1,x2. So it is enough to obtain the estimate (3.41) for

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY 115the kernel ofi2πZΓ(ε)λ−sP mλ dλ,where Re s > n/2.The term ps,εint(x1, x2) in this kernel has the same form as in(3.49) but with the addition factor exp (iξ(x1 −x2)) under the integral sign (wherex and ϕj(x) are replaced by x1 and by ψj(x1)ϕj(x2)).The integration over thedomains (3.50) in the integral corresponding to (3.49) represents this kernel as thesumJs0,ε + Js1,ε + Js2,ε(x1, x2), where Jsl,ε corresponds to the integration over theappropriate domain in (3.50).The term Js0,ε(x1, x2) satisfies the estimate (3.51)if Re s > n/2.

(In this estimate maxρ−Re s, 1can be replaced by ρ−Re s sinceRe s > 0 and since 0 < ρ < 1. )The contour Γ(ε) of the interior integral in (3.49) for Js2,ε can be replaced by thecontour Γ(ε),|ξ| defined by (3.52).

The sum of the integrals over the straight line piecesof Γ(ε) and of Γ(ε),|ξ| in the kernelJs1,ε + Js2,ε(x1, x2) has the same form as (3.53)(but with the factor exp (i(x1 −x2)ξ) under the integral sign). The integral over thecircle part of Γ(ε),|ξ| for the kernel Js2,ε(x1, x2) is also completely analogous to (3.55).This provides us with the estimate for the kernel ps,εint(x1, x2) (where Re s > n/2):|ps,εint(x1, x2)| < Cε exp (ε| Im s|) ρ−Re s (Re s −n/2)−1 .

(3.64)The proof of (3.41) for an arbitrary closed manifold M follows from the estimate(3.64) together with the estimate (3.46) for (rmλ )x1,x2. (They also give us the proofof (3.41) for a part ps,εint(x1, x2) of the kernel (T−s)x1,x2 defined by a local parametrixPj ψjP mλ,intϕj.) If (M, gM) is mirror symmetric with respect to (N, gN) (and the ν-transmission interior boundary conditions are given on N) then the kernel (T−s)x1,x2for Re s > −n/2 can be represented by the formulas (analogous to (2.54), (2.55),and to (2.118)), where ν = (α, β) ̸= (0, 0) and T M−s corresponds to a closed manifold(M, gM) (or to ν0 = (1, 1) that is the equivalent according to Proposition 1.1):(T−s)x1,x2 =T M−sx1,x2 + β2 −α2α2 + β2σ∗1T M−sx1,x2for x1, x2 ∈M1,(T−s)x1,x2 =T M−sx1,x2 + α2 −β2α2 + β2σ∗1T M−sx1,x2for x1, x2 ∈M2,(3.65)(T−s)x1,x2 =2αβα2 + β2T M−sx1,x2for x1, x2 from different Mk,where σ1 is the mirror symmetry on M with respect to N, acting on the variable x1.The kernelT M−sx1,x2 can be analytically (meromorphically) continued to the wholecomplex plane C ∋s (separately on the diagonal x1 = x2 and offthe diagonal).

Itfollows from [Se2], Theorem 1 or from the proof of Theorem 3.1. Hence (3.65) is true

116S.M. VISHIKfor all s ∈C.

So the estimate (3.41) is satisfied also in the case of the ν-transmissioninterior boundary conditions on N if (M, gM) is mirror-symmetric with respect to N.The boundary term75 P ψjDm,Uϕj of the parametrix P m can be identified with thesame term in the mirror-symmetric case (as it is defined in a neighborhood N × I ofN = N × 0, I = [−2, 2]). The estimate (3.46) for the integral over Γ(ε) of (rmλ )x1,x2is satisfied for the mirror-symmetric case also.

So the estimate (3.41) is satisfied bythe kernel p∂,εx1,x2(s) of the operatorP ψjDm,Uϕj. This estimate for p∂,εx1,x2(s) togetherwith the estimates (3.64), (3.46) for ps,εint(x1, x2), and with the estimate (3.46) of theintegral of (rmλ )x1,x2over Γ(ε) provides us with the estimate (3.41).□3.5.

Appendix.Trace class operators and their traces. A bounded linearoperator A acting in a separable Hilbert space H is a trace class operator if the seriesof its singular numbers (i.e., of the arithmetic square roots of the eigenvalues for theself-adjoint operator A∗A) is absolutely convergent.

If A is a trace class operatorthen its matrix trace exists for any orthonormal basis (ei) in H:X(Aei, ei) =: Sp Aand this sum is independent of the orthonormal basis ([Kr]). It is called the matrixtrace of A.

The Lidskii theorem ([Li]) claims that if A is a trace class operator thenthe series of its eigenvalues is absolutely convergent: P |λj(A)| < ∞and its traceTr A := P λj(A) is equal to its matrix trace: Tr A = Sp A. (Here the sums are overall the eigenvalues λj(A) of A including their algebraic multiplicities, [Ka], Ch.

1,§ 5.4. )Proposition 3.8.

For t > 0 the operators exp (−t∆ν0,j) and p1 exp (−t∆ν0,j) aretrace class operators in the L2-completion (DRj(M))2 of DRj(M)76 and their tracesare equal to the integrals of the densities defined by the restrictions to the diagonalsof their kernels:Tr exp(−t∆ν0,j) =Xr=1,2ZMrtr∗x2i∗MrEjt,x1,x2(ν0),(3.66)Tr (p1 exp(−t∆ν0,j)) =ZM1tr∗x2i∗M1Ejt,x1,x2(ν0). (3.67)Here pk : (DRj(M))2 →(DRj (Mk))2 ֒→(DRj(M))2 is the composition of the re-striction to Mk of differential forms and of their prolongation to M by zero on anotherpiece of M, iMr : M r ֒→M r × M r is an immersion of the diagonal, and the exteriorproduct of the double forms (restricted to the diagonal) is implied.75The operator Dm = Dm,λ(ν) is defined in the coordinate chart Rn−1 × R1 ∋(x′, t) by (3.12),(3.11), (3.10).76DRj M 1⊕DRj M 2⊂DRj(M)2

GENERALIZED RAY-SINGER CONJECTURE.I. A MANIFOLD WITH BOUNDARY 117Proof.

The operator At := exp(−t∆ν0,j) is positive definite on (DRj(M))2 and foran arbitrary f ∈(DRj(M))2, f ̸= 0, it holds (Atf, f) > 0 (where the scalar producton (DRj(M))2 corresponds to (1.23)).The operator Bt := p1 exp (−t∆ν0,j) is positive definite on the subspace (DRj(M1))2of (DRj(M))2. Namely (Btm, m) > 0 for m ∈(DRj(M1))2, m ̸= 0, and it is a non-negative operator on (DR•(M))2 : (Btf, f) ≥0 for f ∈(DRj(M))2.The operator exp(−t∆ν0,j) is self-adjoint on (DRj(M))2 by Theorem 3.2.

Its kernelA(x1, x2) is smooth on M r1 × M r2 (as it is proved in Proposition 3.2) and its trace isequal toTrexp(−t∆ν0,j)=Xk=1,2Tr(pkexp(−t∆ν0,j))=Xk=1,2Tr(pkexp(−t∆ν0,j)pk) . (3.68)(The matrix trace for exp (−t∆ν0,j) in (DRj(M))2 =: H can be computed with thehelp of an orthonormal basis (ei(1)) , (ei(2)) in H, where (ei(k)) is an orthonormalbasis in Hk := (DRj (Mk))2.

)The operator Ak := pk exp (−t∆ν0,j) pk acting in H = H1 ⊕H2 has a continuouskernel Ak(x1, x2) = Ejt,x1,x2(ν0) on M k ×M k (and it has the zero kernel on M k1 ×M k2for k1 ̸= k2). The operator Ak is a self-adjoint operator acting in the Hilbert spaceHk and it is positive definite, (Akf, f) > 0 for f ∈Hk, f ̸= 0.

So according to theMercer theorem [GG], IV.3, [RiN], § 98, the Fourier series for the kernel of Ak by theeigenforms of AkAk(x1, x2) =Xµjωj(x1) ⊗ωj(x2),(3.69)(where µj > 0 are the eigenvalues of Ak) converges absolutely and uniformly withrespect to M k × Mk. Hence integrating this series over the diagonals in M k × M k(for k = 1, 2) we obtain the equality (3.66):Tr exp (−t∆ν0,j) =Xk=1,2Tr Ak =Xk=1,2ZMktri∗MkAk(x1, x2).The equality (3.67) is obtained similarlyTr (p1 exp (−t∆ν0,t)) = Tr (p1 exp (−t∆ν0,t) p1) =ZM1tri∗M1A1(x1, x2).The proposition is proved.□Proposition 3.9.

For Re s > n/2 the operator T−s defined by the integral77 ( 3.18)and the operators pj (∆•ν)−s are trace class operators (n := dim M). The traces of77The operator T−s for such s is defined on (DR•(M))2 and it is equal to the direct sum of theoperator (∆•ν)−s on the orthogonal complement to Ker (∆•ν) and of the zero operator on Ker (∆•ν),by Theorem 3.1.

118S.M. VISHIKthese operators for Re s > n/2 are equal to the integrals over the diagonals of thedenisities, defined by the restrictions of their kernels to these diagonals.Tr(∆•ν)−s=Xr=1,2ZMrtr∗x2i∗MrT−s(x1, x2),(3.70)Trpj (∆•ν)−s=ZMjtr∗x2i∗MjT−s(x1, x2).(3.71)Proof.

The kernel T−s(x1, x2) for Re s > n/2 is continuous on M j1 × M j2 (Theo-rem 3.1). The operator T−s for such s is nonnegative, (T−sf, f) ≥0, and self-adjoint.It is a trace class operator (Theorem 3.1).

For Re s > n/2 the equality holds (anal-ogous to (3.68):Tr T−s =Xj=1,2Tr(pjT−spj).The operator pjT−spj is self-adjoint in the Hilbert subspace (DR•(Mj))2 of (DR•(M))2and its kernel coincides with the kernel K−s,j of T−s on M j×M j. So for Re s > n/2 itskernel is continuous and, according to the Mercer theorem, the series on M j ×M j forK−s,j by the eigenforms of pjT−spj (analogous to (3.69)) is absolutely and uniformlyconvergent on M j × M j.Hence for such s the integral over the diagonal of thedensity, defined by the restriction of the kernel K−s,j, is equal to Tr (pjT−spj).

Thusthe equalities (3.71) and (3.70) are proved.□Acknowledgements.I express my gratitude to R. Bott, W. M¨uller, C. Taubes,M. Wodzicki, and S.-T. Yau for stimulating discussions.

I am indebted to J. Bern-stein, A. Beilinson, D. Kazhdan, and I.M. Singer for a series of conversations on thetheme of this paper.

I am very grateful to the Department of Mathematics of HarvardUniversity and to the Max-Planck-Institut f¨ur Mathematik for their hospitality andfor financial support, which made it possible for me to work in the excellent intellec-tual atmosphere of these mathematical centers. The results and the methods of thispaper were reported in the seminars of R. Bott, C. Taubes, and S.-T. Yau in Harvardduring the Fall of 1992, in Oberseminar Max-Planck-Institut f¨ur Mathematik in Mayof 1993, and to H. McKean and L. Nirenberg in the Courant Institute in Decemberof 1992.

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