Luc Hillairet
email: luc.hillairet@univ-orleans.fr

Victor Kalvin
email: vkalvin@gmail.com

Alexey Kokotov
email: alexey.kokotov@concordia.ca

Moduli spaces of meromorphic functions and determinant of Laplacian

The Hurwitz space is the moduli space of pairs (X, f ) where X is a compact Riemann surface and f is a meromorphic function on X. We study the Laplace operator ∆ |df | 2 of the flat singular Riemannian manifold (X, |df | 2 ). We define a regularized determinant for ∆ |df | 2 and study it as a functional on the Hurwitz space. We prove that this functional is related to a system of PDE which admits explicit integration. This leads to an explicit expression for the determinant of the Laplace operator in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic differential df. The proof has several parts that can be of independent interest. As an important intermediate result we prove a decomposition formula of the type of Burghelea-Friedlander-Kappeler for the determinant of the Laplace operator on flat surfaces with conical singularities and Euclidean or conical ends. We introduce and study the S-matrix, S(λ), of a surface with conical singularities as a function of the spectral parameter λ and relate its behavior at λ = 0 with the Schiffer projective connection on the Riemann surface X. We also prove variational formulas for eigenvalues of the Laplace operator of a compact surface with conical singularities when the latter move.

Introduction 1.General part

Studying the determinants of Laplacians on Riemann surfaces is motivated by needs of quantum field theory (in connection with various partition functions) and geometric analysis (in particular, in connection with Sarnak program, [START_REF] Sarnak | Extremal geometries[END_REF]). The explicit expressions for the determinant of the Laplacian in the metric of constant negative curvature ( [START_REF] D'hoker | On Determinants of Laplacians on Riemann Surfaces[END_REF]) and in the Arakelov metric (obtained in [START_REF] Alvarez-Gaumé | Bosonization on higher genus Riemann surfaces[END_REF] in relation to so-called bosonization formulas from the string theory) for compact Riemann surfaces of genus g > 1 are among the most spectacular results of the subject. According to Sarnak program, these determinants (which are functions on the moduli space of Riemann surfaces) can be used to study the geometry of the moduli space via methods of Morse theory. In particular, their behavior at the boundary of moduli space is of great importance and was intensively studied (see, e. g., [START_REF] Wolpert | Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces[END_REF], [START_REF] Wentworth | Precise constants in bosonization formulas on Riemann surfaces[END_REF]).

It seems very interesting to consider the case which is in a certain sense opposite to the case of the metric of constant curvature: instead of distributing the curvature uniformly along the Riemann surface X one can concentrate it at a finite set {P 1 , . . . , P M } ⊂ X. This leads to a flat metric m on X with singularities (e. g. conical) at P k . Determinants of Laplacians for various classes of singular flat metrics were introduced and studied at least on the formal level (via path integrals) by physicists ( [START_REF] Sonoda | Functional determinants on punctured Riemann surfaces and their application to string theory[END_REF], [START_REF] Al | Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation function[END_REF], [START_REF] Knizhnik | Analytic fields on Riemann surfaces[END_REF], [START_REF] Bershadsky | g-loop amplitudes in bosonic string theory in terms of branch points[END_REF]) and certain explicit expressions for them were produced (see, e. g. , [START_REF] Sonoda | Functional determinants on punctured Riemann surfaces and their application to string theory[END_REF], [START_REF] Al | Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation function[END_REF]).

One of the main challenges is to study such determinants from the spectral theory of self-adjoint operators and perturbation theory point of view. This was done in the mathematical literature for the determinants of the Laplacians of smooth metrics, in particular, those two mentioned above (see, e. g., Fay's book [START_REF] Fay | Kernel functions, analytic torsion and moduli spaces[END_REF] for complete compendium and consistent exposition). The standard definition of the determinant uses the ζ-function of the corresponding Laplace operator

ln det∆ m = -ζ ′ ∆ m (0) , (1.1)
ζ ∆ m (s) = j 1 λ s j , (1.2)
where in the latter expression the sum is extended over all non-zero eigenvalues of ∆ m . This definition makes sense when the metric m have conical singularities provided a regular self-adjoint extension is considered, for instance the Friedrichs one (see [START_REF] Hillairet | Krein formula and S-matrix for Euclidean surfaces with conical singularities[END_REF] for the definition of regular) . Indeed, the Laplace operator ∆ m (with natural domain consisting of smooth functions on X supported outside the conical points P k ) is not essentially self-adjoint. This fact is never mentioned by the physicists and it is not clear whether this issue has been addressed. Comparing the determinants of the different self-adjoint extensions of ∆ m leads to a nice application of Birman-Krein theory and is done in [START_REF] Hillairet | Krein formula and S-matrix for Euclidean surfaces with conical singularities[END_REF] (see also the references therein). In what follows we consider the Friedrichs extension of ∆ m , our results refer to this self-adjoint extension only.

In [START_REF] Kokotov | Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray-SInger formula[END_REF] it was found an explicit expression for the determinant of (the Friedrichs extension of) the Laplace operator corresponding to a flat conical metric m with trivial holonomy. Any metric of this type can be written |ω| 2 , where ω is a holomorphic one-form on X, zeros of ω of multiplicity ℓ are the conical points of the metric |ω| 2 with conical angle 2π(ℓ + 1). The moduli space of pairs (X, ω), where X is a compact Riemann surface and ω is a holomorphic one-form on X, is stratified according to the multiplicities of ω (see [START_REF] Kontsevich | Connected components of the moduli spaces of Abelian differentials with prescribed singularities[END_REF]). In [START_REF] Kokotov | Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray-SInger formula[END_REF] it was proved that on each stratum of the moduli space of holomorphic differentials the ratio det∆ |ω| 2 Area(X)det ℑB where B is the matrix of b-periods of the Riemann surface X, coincides with the modulus square of a holomorphic function τ on the stratum. This holomorphic function τ (the socalled Bergman tau-function on the space of holomorphic differentials) admits explicit expression through theta-functions, prime-forms, and the divisor of the holomorphic one-form ω. In the case g = 1 the holomorphic one-forms have no zeroes, the metric |ω| 2 is smooth, and the corresponding result coincides with the classical Ray-Singer formula for the determinant of the Laplacian on an elliptic curve with flat conformal metric.

In [START_REF] Kokotov | Polyhedral surfaces and determinant of Laplacian[END_REF] it was found a comparison formula (an analog of classical Polyakov formula) relating determinants of the Laplacians in two conformally equivalent flat conical metrics. This lead to the generalization of the results of [START_REF] Kokotov | Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray-SInger formula[END_REF] to the case of arbitrary flat conformal metrics with conical singularities.

Together with determinant of the Laplacians in flat conical metrics given by the modulus square of the holomorphic one form (these metrics have finite volume and the spectra of the corresponding self-adjoint Laplacians are discrete) in physical literature appear determinants of the Laplacians corresponding to flat metrics |ω| 2 , where ω is now a meromorphic one form on X. Depending on the order of the poles of ω, the corresponding non compact Riemannian manifold (X, |ω| 2 ) of the infinite volume has cylindrical, Euclidean, or conical ends. The spectrum of the corresponding Laplace operator is continuous (with possible embedded eigenvalues, say, in case of cylindrical ends) and the Ray-Singer regularization of the determinant (1.1, 1.2) is no longer applicable. The way to regularize such determinants is, in principle, also well-known (see, e. g., [START_REF] Müller | Relative zeta functions, relative determinants and scattering theory[END_REF]): considering the Laplacian ∆ as a perturbation of some properly chosen "free" operator ∆, one introduces a relative determinant det(∆, ∆) in terms of the relative ζ-function

ζ(s; ∆, ∆) = 1 Γ(s) ∞ 0
Tr(e -∆t -e -∆t )t s-1 dt,

where a suitable regularization of the integral is made (being understood in the conventional sense the integral is usually divergent for any value of s). Following this approach, in [START_REF] Hillairet | Spectral Determinants on Mandelstam Diagrams[END_REF] we studied the regularized determinants det(∆, ∆) = e -ζ ′ (0;∆, ∆)

of the Laplacians on the so-called Mandelstam diagrams -the flat surfaces with cylindrical ends (more precisely, Riemann surfaces X with the metric |ω| 2 , where ω is a meromorphic one-form on X having only simple poles and such that all the periods of ω are pure imaginary and all the residues of ω at the poles are real).

In the present paper we consider determinants of the Laplacian corresponding to flat metrics with even wilder singularities: the corresponding Riemannian manifolds have Euclidean (i. e. isometric to a vicinity of the point at infinity of the Euclidean plane) and/or conical ends (i. e. isometric to a vicinity of the point at infinity of a straight cone). These metrics are given as the modulus square of the differential of an arbitrary meromorphic function f on a compact Riemann surface X. The moduli space of pairs (X, f ) is called the Hurwitz space H. We define and study the regularized determinant of the Laplace operator corresponding to the metric |df | 2 as a functional on H. The main result of this work is an explicit formula for the determinant.

It should be mentioned that such determinants appeared for the first time in [START_REF] Al | Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation function[END_REF], [START_REF] Bershadsky | g-loop amplitudes in bosonic string theory in terms of branch points[END_REF] (see also [START_REF] Knizhnik | Analytic fields on Riemann surfaces[END_REF]), although no attempt was made to define them rigorously.

Results and organization of the paper

Let X be a Riemann surface and f be a meromorphic function f : X → P 1 . The metric |df | 2 gives to X a structure of a (non-compact) flat Riemannian manifold with conical singularities and conical (or Euclidean) ends. The conical singularities are located at the critical points P 1 , . . . , P M of f (or, equivalently, at the zeros of the meromorphic oneform df ), the ends of X are located at the poles of df . The moduli space of (equivalence classes) of such pairs (X, f ) is known as Hurwitz space H and the critical values z m , z m = f (P m ), m = 1, . . . , M , locally parametrize H.

Given such a Riemannian manifold (X, |df | 2 ), we introduce the reference manifold ( X, m) as the disjoint union of the complete cones corresponding to the ends of (X, |df | 2 ). By ∆ and ∆ we denote the Laplace operators on (X, |df | 2 ) and ( X, m) correspondingly.

The first part of the paper aims at defining the relative zeta-regularized determinant det ζ (∆, ∆) and proving a version of the Burghelea-Friedlander-Kappeler (BFK in what follows) gluing formula (see [START_REF] Burghelea | Meyer-Vietoris type formula for determinants of elliptic differential operators[END_REF]). This new BFK type formula is a generalization of the Hassell-Zelditch formula for the determinant of the Laplacian in exterior domains [START_REF] Hassell | Determinants of Laplacians in exterior domains[END_REF]; here we rely on ideas from [START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF][START_REF] Carron | Le saut en zéro de la function de décalage spectral[END_REF][START_REF] Hassell | Determinants of Laplacians in exterior domains[END_REF].

In order to obtain the gluing formula, we cut X along some hypersurface Σ. This decomposes X into a compact part X -and the union of conical/Euclidean ends X + . The latter is isometric to the reference surface X with a compact part Xremoved. There is some latitude in choosing the initial Σ. In order to choose Σ we first specify some large R and then in each end of (X, |df | 2 ) we take a circle whose radius depends on R and on the cone angle of the end, see Definition 3. As expected the gluing formula then involves the Neumann jump operator N on Σ and the Dirichlet Laplacian ∆ D -on X -, see Theorem 1 below.

Theorem 1. For R large enough we have the BFK gluing formula

det ζ (∆, ∆) = C det * ζ N • det ζ ∆ D -,
where N, ∆ D -depend on R. The constant C depends on R but not on the moduli parameters z 1 , . . . z M as long as the corresponding critical points P m do not approach Σ.

Note that the proof of the gluing formula also holds for a more general class of metric (see Remark 1).

Let us now sketch some steps leading to this Theorem. First we start from the BFK gluing formula for det ζ (∆ -λ, ∆ -λ) obtained in [START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF] for negative (regular) values of the spectral parameter λ. In order to obtain a gluing formula for det ζ (∆, ∆), we study the behaviour of all ingredients in the gluing formula for det ζ (∆ -λ, ∆ -λ) as λ → 0-(i.e. at the bottom of the continuous spectrum of ∆ and ∆) and then pass to the limit. As usual, this essentially reduces to derivation of asymptotics as λ → 0-for the zeta regularized determinant of the Neumann jump operator and for the spectral shift function of the pair (∆, ∆). In principle both asymptotics were obtained in [START_REF] Carron | Le saut en zéro de la function de décalage spectral[END_REF] for Schrödinger type operators on manifolds with conical ends. Unfortunately those asymptotics cannot be used for our purposes because the asymptotic for the Neumann jump operator contains an unspecified constant and the asymptotic for the spectral shift function is not sufficiently sharp. We demonstrate that at least in our setting (no potential) the methods of [START_REF] Carron | Le saut en zéro de la function de décalage spectral[END_REF] can be improved to specify the constant and to obtain a sufficiently sharp asymptotic of the spectral shift function as needed for the proof of our BFK formula. Once these asymptotics are obtained, we follow the lines of [START_REF] Hassell | Determinants of Laplacians in exterior domains[END_REF] in our study of the behaviour of det ζ (∆ -λ, ∆ -λ) as λ → 0-and also in definition of det ζ (∆, ∆).

Using this BFK formula, we prove (as it was done in similar situations in [START_REF] Kokotov | Polyhedral surfaces and determinant of Laplacian[END_REF], [START_REF] Hillairet | Spectral Determinants on Mandelstam Diagrams[END_REF]) that the variations of the determinant of the Laplacian with respect to the moduli parameters z k remain the same if we replace the metric m = |df | 2 of infinite volume by a metric m of finite volume, where m coincides with m outside vicinities of the poles of f and with some standard nonsingular metric of finite volume inside these vicinities. The aim of the second part of the paper is thus to study the zeta-regularized determinant of this new metric m and its variation with respect to moduli parameters.

We show that these variations can be conveniently expressed using the so-called S-matrix, so we start the second part of the paper by introducing this object and deriving some of its properties. We think that the S-matrix is an important characteristic of a compact Riemann surface X equipped with a conformal metric m with conical singularities. It is introduced in analogy with scattering problems and the general theory of boundary triples (see [START_REF] Grubb | Distributions and operators[END_REF]). Basically, the S-matrix is a meromorphic matrix function of the spectral parameter λ. It serves as a formal analog of the scattering matrix on complete non-compact manifolds (say, on manifolds with cylindrical ends) in the sense that the elements of S are coefficients in asymptotics of certain eigenfunctions (all growing terms in the asymptotics near conical singularities of m are interpreted as incoming waves and all decaying terms -as outgoing). Let us also note that there is no true scattering on the (incomplete) manifold (X, m) (corresponding operators have no continuous spectrum) and that S(λ) is a non-unitary matrix with poles at the eigenvalues λ of the Friedrichs self-adjoint Laplacian on (X, m) [START_REF] Hillairet | Krein formula and S-matrix for Euclidean surfaces with conical singularities[END_REF].

It turns out that the value, S(0), of the S-matrix can be found explicitly: it depends only on the conical angle at the conical point, the conformal class of the surface X, and the choice of the holomorphic local parameter near the conical point (the so-called distinguished local parameter for the metric m). For instance, when the conical angle is 4π, we express the matrix elements of S(0) through the Bergman reproducing kernel for holomorphic differentials and a certain special projective connection on X (the so-called Schiffer projective connection).

In the general case we prove that a certain linear combination of the matrix elements of S(0) (actually, the one that appears later in the variational formulas for the determinant) can be expressed as the derivative (of the order depending on the conical angle) of the Schiffer projective connection.

We continue by studying the moduli variations of the zeta-regularized determinant of ∆ m. We use the Kato-Rellich perturbation theory to compute the variation of individual eigenvalue branches and then a contour argument similar to the one from [START_REF] Hillairet | Krein formula and S-matrix for Euclidean surfaces with conical singularities[END_REF] to get the variational formula for the determinant. This formula involves a combination of the matrix elements of S(0) and hence the Schiffer projective connection. Writing it into an invariant form, we obtain the following theorem.

Theorem 2. Let P m be a zero of the meromorphic differential df of multiplicity ℓ m and let z m = f (P m ) be the corresponding critical value of f . Let the metric m be obtained from the metric |df | 2 via smoothing of the conical ends. Then

∂ zm ln det * ζ (∆ m) detℑB = - 1 12πi Pm S B -S f df (1.4)
where S B is the Bergman projective connection, S f =

f ′′′ f ′ -3 2 (f ′′ ) 2 (f ′ ) 2
is the Schwarzian derivative, and B is the matrix of b-periods.

In this theorem we can replace det * ζ (∆ m) by det(∆, ∆) since we proved before that the moduli variations of both functions coincide.

The system of PDE for det ∆ m that appears in Theorem 2 is the governing system for the Bergman tau-function on the Hurwitz space (introduced and studied in [START_REF] Kokotov | Tau-functions on Hurwitz spaces[END_REF], [START_REF] Kokotov | Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray-SInger formula[END_REF], [START_REF] Kokotov | Isomonodtromic tau function on the space of admissible covers[END_REF], [START_REF] Kokotov | On the isomonodromic tau-function for the Hurwitz spaces of branhed coverings of genus zero and one[END_REF]). The latter system was explicitly integrated in [START_REF] Kokotov | Isomonodromic tau-function of Hurwitz Frobenius manifolds and its appplications[END_REF] and in §5 we remind this result (unfortunately, technically involved). This leads to the following explicit formula for det ζ (∆, ∆). Theorem 3. Let (X, f ) be an element of the Hurwitz space H(M, N ) and let τ (X, f ) be given by expressions (6.10, 6.9, 6.8). There is the following explicit expression for the regularized relative determinant of the Laplacian ∆ on the Riemann surface X:

det ζ (∆, ∆) = C detℑB |τ | 2 , (1.5)
where C is a constant that depends only on the connected component of the space H(M, N ) containing the element (X, f ).

We finish the paper with two illustrating examples in genus 0, deriving the formulas for the determinant of the Laplacian on the space of polynomials of degree N and on the space of rational functions with three simple poles.

Acknowledgements. This work was finished during the stay of the third author (A. K.) at the Max Planck Institute of Mathematics (Bonn). A. K. thanks the Institute for hospitality and excellent working conditions. The work of L.H is partly supported by the ANR program Gerasic-ANR-13-BS01-0007-0. The work of A. K. is supported by NSERC and ANR program Gerasic-ANR-13-BS01-0007-0.

The regularized determinant as a functional on Hurwitz space and a BFK gluing formula

To a pair (X, f ), where X is a compact Riemann surface and f is a meromorphic function on X (i. e. to an element of the Hurwitz space), there corresponds a Riemannian manifold (X, |df | 2 ). Our aim is to define a regularized determinant of the corrresponding Laplacian and to prove a BFK-type gluing formula. Since the metric m = |df | 2 has conical singularities and non-compact conical ends, this is not that straightforward and requires several steps. First, we consider regular values of the spectral parameter λ 2 , i.e.

λ 2 ∈ C \ [0, ∞).
In that case, the definition of the relative determinant and the BFKgluing formula are the same as in [START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF]. Then we derive estimates for the determinant of the Dirichlet-to-Neumann operator as λ approaches 0; our methods here are closely related to those of [START_REF] Carron | Le saut en zéro de la function de décalage spectral[END_REF]. These estimates allow us to define a zeta-regularized determinant for λ = 0 similarly to [START_REF] Hassell | Determinants of Laplacians in exterior domains[END_REF]. Then we prove the gluing formula for thus defined determinant. At the end of this section we use the gluing formula to compactify (X, |df | 2 ) in such a way, that locally, the moduli variations remain the same.

The flat Laplacian of an element in Hurwitz space

We will be dealing with conical singularities and conical ends. These are defined in the following way.

Definition 1.

• For any ℓ ∈ N the Euclidean cone of total angle 2ℓπ is the Riemannian manifold (C, |ℓy ℓ-1 dy| 2 ).

• A point P in a Riemannian manifold will be a conical singularity of angle 2ℓπ if there is a neighbourhood of P that is isometric to the set {|y| < ε}, |ℓy ℓ-1 dy| 2 for some positive ε.

• A open set Ω ⊂ X of a Riemannian manifold (X, m) such that (Ω, m) is isometric to {|y| > R}, |ℓy ℓ-1 dy| 2 for some positive R will be called a conical end of angle 2ℓπ (Euclidean end if ℓ = 1).

Let (X, m) be a Riemannian manifold such that the metric is flat with a finite number of conical singularities and conical ends. Denote by ∆ the self-adjoint Friedrichs extension of the (non-negative) symmetric Laplace operator defined on smooth compactly supported functions that vanish near the conical singularities.

Let f be a meromorphic function on a compact Riemann surface X of genus g ≥ 0 or, what is the same, a ramified covering of the Riemann sphere f : X → P 1 .

(2.1)

Two coverings f 1 : X 1 → P 1 and f 2 : X 2 → P 1 are called equivalent if there exists a biholomorhic map g :

X 1 → X 2 such that f 1 = f 2 • g.
The following constructions are standard, we recall them for convenience of the reader.

The critical points, P m , m = 1, . . . , M, of the function f (i. e. those points for which df (P m ) = 0) are the ramification points of the covering, the points z m = f (P m ) are called the critical values. The ramification index of the covering at the point P m equals to ℓ m + 1, where ℓ m is the order of the zero of the one-form df at P m . Denote by ∞ 1 , . . . , ∞ K the poles of f , and let k 1 , . . . , k K be their multiplicities.

Then the covering (2.1) has degree N = k 1 + . . . + k K and the following Riemann-Hurwitz formula holds:

M m=1 ℓ m - K j=1 (k j + 1) = 2g -2 ,
where g is the genus of X.

Pick some regular value z 0 ∈ P 1 and draw on P 1 the segments I 0 := [z 0 , ∞], I m = [z 0 , z m ], m = 1, . . . , M . It may happen that some segment is repeated several times if different critical points take the same critical value. We may also choose z 0 such that all these segments have pairwise disjoint interiors. Denote by L := M m=0 I m the union of these segments and observe that P 1 \ L contains only regular values of f . It follows that X \ f -1 (L), the complement of the preimage of L by f in X has N connected components. By construction f is a biholomorphic map from each of these connected components onto P 1 \ L. We denote these connected components by C n , n = 1 . . . N and call them the sheets of the covering. Each C n can be seen as a copy of the complex plane equipped with the cuts provided by L.

On each sheet, the metric |dz| 2 lifted from the base P 1 to the covering space coincides with the metric |df | 2 . The Riemannian manifold (X, |df | 2 ) is thus obtained by gluing N copies of a Euclidean plane (C, |dz| 2 ) with a system of non-intersecting cuts, one of which extends to infinity.

For each critical point P m of ramification index ℓ m + 1, we obtain a (ℓ m + 1)-cycle γ j obtained by looking in which order the sheets are following one another when making a small loop around P m . It follows that P m is a conical singularity of angle 2π(ℓ m + 1).

For each z m , m = 0, we obtain a permutation in S N by composing the cycles for each critical point in f -1 (z m ). We thus obtain M ′ permutations σ m ′ , m ′ = 1 . . . M ′ , where M ′ is the number of different critical values. The critical value z m is thus associated with one cycle in one of the permutations σ m ′ , m ′ = 1 . . . M ′ .

In the same manner we obtain a permutation σ 0 by looking at the preimage of a large loop that surrounds z 0 (or equivalently, a small loop around ∞ in the base P 1 ). This permutation describes the structure at infinity of the Riemannian manifold (X, |df | 2 ) : each fixed point of σ 0 corresponds to a flat Euclidean end and a cycle of length k to a conical end of angle 2kπ. A pole in f of order k corresponds to a conical end of angle 2kπ (and therefore a Euclidean end for a simple pole).

The flat structure on (X, |df | 2 ) is completely characterized by the positions of the critical values z m , m = 1 . . . M , and by the permutations σ m ′ , m ′ = 0 . . . M ′ . Conversely, starting from M ′ + 1 permutations of S N , and M ′ distinct points w 1 , . . . w m ′ in C, we construct the sequence z 0 , . . . z M by choosing a distinct point z 0 and, for m > 0, by repeating w m ′ as many times as there are disjoint cycles in σ m ′ . We then glue N sheets according to the scheme prescribed by the permutations and obtain a (not necessarily connected) flat surface (X, m) with conical singularities and conical ends.

It turns out that it is always possible to find a meromorphic function f from

X to P 1 such that (X, m) is isometric to (X, |df | 2 ).
Introduce the Hurwitz space H(N, M ) of equivalence classes of coverings f : [START_REF]Hurwitz schemes and irreducibility of moduli of algebraic curves[END_REF], we notice here that it may have more than one connected components) and the critical values z 1 , . . . , z M can be taken as local coordinates on H(N, M ).

X → P 1 of degree N with M ramification points of (fixed) indices ℓ 1 + 1, . . . , ℓ M + 1 and K poles of (fixed) multiplicities k 1 , . . . , k ℓ ; k 1 + . . . , k K = N . The space H(N, M ) is a complex manifold of dimension M (see
If all the critical points of the maps f are simple, then the corresponding Hurwitz space is usually denoted by H g,N (k 1 , . . . , k K ) and is known to be connected (see [START_REF] Natanzon | Topology of 2-dimensional coverings and meromorphic functions on real and complex algebraic curves[END_REF]). Definition 2. We will refer to the coordinates z 1 , . . . , z M as moduli.

From the flat metric point of view, moving z m can be easily realized by cutting a small ball around P m , then move P m inside this ball. Since the boundary of the ball does not change we can glue the new ball back into the surface.

For such a Riemannian manifold (X, |df | 2 ) we define a reference manifold ( X, m) which is obtained in the following way. Take those N sheets with cuts that correspond to X, and in the gluing scheme of X, keep σ 0 and replace all the permutations σ m ′ , m ′ > 0 by the identity. It can be easily seen that ( X, m) is the disjoint union of cones, those cones correspond to the conical ends of X and the tip of each cone is now located above z 0 .

The Laplacian ∆ can be considered as a perturbation of the free Laplacian ∆ := ∆ m acting in L 2 ( X). The perturbation is basically reduced to the change of the domain of the unbounded operator: when we make slits on X and glue them according to a certain gluing scheme, it induces boundary conditions on the sides of the cuts. The determinant of ∆ will then be defined in terms of the relative zeta function (1.3) as a regularized relative determinant det ζ (∆, ∆).

The main goal of this work is to study the relative determinant det ζ (∆, ∆) as a functional on the space H(N, M ).

Relative Determinant and BFK gluing formula for negative energies

Let X be a compact Riemann surface and let f be a nonconstant meromorphic function on X. Introduce the flat singular metric m = |df | 2 on X. As it is explained in the previous section, the flat singular Riemannian manifold (X, m) has conical points (at the zeros, P 1 , . . . , P M , of the differential df ) and conical ends of angle 2πk j at the poles, ∞ 1 , . . . , ∞ K , of f , where k j is the order of the corresponding pole. Let ∆ be the (Friedrichs) Laplacian on (X, m). Let ( X, m) be the reference unperturbed manifold and let ∆ be the associated Laplace operator. We recall that ( X, m) = K j=1 (C, |k j y k j -1 j dy j | 2 ). Since (X, m) and ( X, m) are isometric outside a compact region the methods and results of [START_REF] Carron | Le saut en zéro de la function de décalage spectral[END_REF] apply.

For R > 0 large enough, there is a subset X + (R) ⊂ X that is isometric to

∪ K j=1 {y j ∈ C : |y j | ≥ R 1/k j } ⊂ X. (2.2)
Definition 3. We denote by Σ R the boundary of the region

X + (R). It is the union of K circles {y ∈ C : |y| = R 1/k j } on X.
Note that R will be chosen at the very beginning of construction and will then be fixed. In what follows we omit the reference to R and simply write Σ, X + .

We represent X in the form

X = X -∪ Σ X + ,
where

X -= X \ (X + ∪ Σ).
Following [START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF] we first define the external Dirichlet-to-Neumann operator. We consider each conical end {y j ∈ C j : |y j | ≥ R 1/k j } of X + separately and omit the subscript j for brevity of notation. We introduce the coordinates (r, ϕ) where

r = |y| k ∈ [R, ∞) and ϕ = arg y ∈ (-π, π]. We have g = dr 2 + k 2 r 2 dϕ 2 , ∆ = r -2 (r∂ r ) 2 + k -2 ∂ 2 ϕ .

Separation of variables shows that for

λ ∈ C \ {0} with ℑλ ≥ 0 the exterior Dirichlet problem (∆ -λ 2 )u(λ) = 0 on X + , u(λ) = f on Σ, (2.3)
has a unique solution of the form

u(r, ϕ; λ) = ∞ n=-∞ C n H (1) νn (λr) H (1) νn (λR) e inϕ , ν n = |n| kR , where f ∈ C ∞ (Σ), C n = (2π) -1 π -π f (ϕ)e -inϕ dϕ, and H (1)
n is the Hankel function. This solution is in L 2 (X + ) if ℑλ > 0. If ℑλ = 0, it is the unique outgoing solution that satisfies the Sommerfeld radiation condition

√ r ∂ r u(λ) -iλru(λ) → 0 as r → ∞.
The external Dirichlet-to-Neumann operator on Σ acts by the formula

N + (λ)f = -∂ r u(λ) ↾ r=R . (2.4) Thus ψ n (ϕ) = (Vol Σ) -1/2 e inϕ are eigenfunctions of N + (λ) with ψ n L 2 (Σ) = 1, and
µ n (λ) = µ -n (λ) = - ∂ r H (1)
νn (λr) ↾ r=R H

(2.5)

are the corresponding eigenvalues (if ℑλ ≥ 0, λ = 0). We can also consider Σ and X + as subsets of X. Then in the same manner we have X = X-∪ Σ X+ .

Let ∆ D ± be the Friedrichs extensions of the Dirichlet Laplacians in L 2 (X ± ). We denote by ∆ D := ∆ D -⊕ ∆ D + the Friedrichs Laplace operator on L 2 (X) with Dirichlet boundary condition on Σ. Similarly, we define ∆D ± and ∆D . The spectrum spec(∆ D -) of the positive self-adjoint operator ∆ D -is discrete. For any

λ 2 ∈ C \ σ(∆ D -) and f ∈ H 1 (Σ) there exists a unique solution u(λ) ∈ H 3/2 (X -) to the Dirichlet problem (∆ -λ 2 )u(λ) = 0 on X -\ Σ, u(λ) = f on Σ, (2.6) such that u(λ) = f -(∆ D --λ 2 ) -1 (∆ -λ 2 ) f , (2.7
)

where f ∈ H 3/2 (X -) is a continuation of f and (∆ D --λ 2 ) -1 : H -1/2 (X -) → H 3/2 (X -) is a holomorphic function of λ 2 ∈ C \ σ(∆ D -); here u; H s (X -) = (∆ D -) s/2 u; L 2 (X -) . The Dirichlet-to-Neumann operator N -(λ) on Σ acts by the formula N -(λ)f = ∂ r u(λ) ↾ r=R ,
where f is the right hand side in (2.6) and u(λ) is defined by (2.7). The function

λ 2 → N -(λ) ∈ B(H 1 (Σ), L 2 (Σ)) is holomorphic in C \ σ(∆ D
-); here and elsewhere B(X, Y) stands for the space of bounded operators from X to Y.

Finally, we introduce the Neumann jump operator

N(λ) = N + (λ) + N -(λ),
which is a first order elliptic classical pseudodifferential operator on Σ with the principal symbol 2|ξ|. For λ 2 ≤ 0 the operator N(λ) is formally self-adjoint and nonnegative, it is positive if λ 2 < 0, and ker N(0) = {c ∈ C} (see e.g. [START_REF] Carron | Le saut en zéro de la function de décalage spectral[END_REF]Sec. 3.3]

• ; ∆, ∆ D ) ∈ L 1 (R + , (1 + λ 2 ) -2 λ dλ) such that Tr (∆ + 1) -1 -(∆ D + 1) -1 = - ∞ 0 ξ(λ ; ∆, ∆ D )(1 + λ 2 ) -2 2λ dλ. (2.8)
Moreover, the following representation is valid

Tr e -t∆ -e -t∆ D = -t ∞ 0 e -tλ 2 ξ(λ ; ∆, ∆ D )2λ dλ, (2.9)
which implies that the left hand side in (2.9) is absolutely bounded uniformly in t > ǫ > 0. The heat trace asymptotic Tr e -t∆ -e -t∆ D ∼ j≥-2

a j t j/2 , t → 0+, (2.10)
can be obtained in a usual way, see e.g. [START_REF] Hillairet | Spectral Determinants on Mandelstam Diagrams[END_REF]Lemma 4]. Thus for λ 2 < 0 the relative zeta function given by

ζ(s; ∆ -λ 2 , ∆ D -λ 2 ) = 1 Γ(s) ∞ 0 t s-1 e λ 2 t
Tr(e -t∆ -e -t∆ D ) dt is defined for ℜs > 1 and continues meromorphically to the complex plane with no pole at s = 0 by the usual argument. The relative determinant is defined to be

det ζ (∆ -λ 2 , ∆ D -λ 2 ) = e -ζ ′ (0;∆-λ 2 ,∆ D -λ 2 ) .
By [START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF]Theorem 4.2] we have the gluing formula

det ζ (∆ -λ 2 , ∆ D -λ 2 ) = det ζ N(λ), λ 2 < 0.
(2.11) (Although only smooth manifolds are considered in [START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF], it is fairly straightforward to see that the argument in [START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF] remains valid for (2.11) as far as we consider only Friederichs extensions and there are no conical points on Σ.) All the constructions above can also be done for ( X, m). Thus similarly to (2.11) we have

det ζ ( ∆ -λ 2 , ∆D -λ 2 ) = det Nζ (λ), λ 2 < 0. (2.12)
Observe that since all operators can be seen as acting on L 2 ( X) we have

e -t∆ -e -t ∆ = e -t∆ -e -t∆ D -e -t ∆ -e -t ∆D + e -t∆ D -e -t ∆D = e -t∆ -e -t∆ D -e -t ∆ -e -t ∆D + e -t∆ D --e -t ∆D - ,
where for the last line we have used that

∆ D = ∆ D -⊕ ∆ D + , ∆D = ∆D -⊕ ∆ D + since X + and X+ are isometric.
It follows that we can take the trace of both sides and thus define the following relative zeta function for ℜs > 1

ζ(s; ∆ -λ 2 , ∆ -λ 2 ) = 1 Γ(s) ∞ 0 t s-1 e λ 2 t
Tr(e -t∆ -e -t ∆) dt, λ 2 < 0. Moreover, we obtain the relation

ζ(s; ∆ -λ 2 , ∆ -λ 2 ) = ζ(s; ∆ -λ 2 , ∆D -λ 2 ) -ζ(s; ∆ -λ 2 , ∆D -λ 2 ) + ζ(s, ∆ D --λ 2 ) -ζ(s, ∆D --λ 2
). All the functions continue meromorphically to the complex plane with no pole at 0. Passing to the determinant, we obtain

det ζ (∆ -λ 2 , ∆ D -λ 2 ) det ζ ( ∆ -λ 2 , ∆D -λ 2 ) = det ζ (∆ -λ 2 , ∆ -λ 2 ) det ζ ( ∆D --λ 2 ) det ζ (∆ D --λ 2 )
.

Thus dividing (2.11) by (2.12) we obtain

det ζ (∆ -λ 2 , ∆ -λ 2 ) det ζ ( ∆D --λ 2 ) det ζ (∆ D --λ 2 ) = det ζ N(λ) det ζ N(λ) , λ 2 < 0, (2.13)
where N(λ) and ∆D -are moduli independent. In order to take the limit λ 2 → 0-in (2.13), we will need the asymptotic behavior of all the ingredients in the latter equation. We start with det ζ N(λ).

Asymptotic of det ζ N(λ) as |λ| → 0+, ℑλ ≥ 0

In this section we follow [START_REF] Carron | Le saut en zéro de la function de décalage spectral[END_REF], where a similar problem is studied. Our purpose here is to refine the asymptotic of det ζ N(λ) from [START_REF] Carron | Le saut en zéro de la function de décalage spectral[END_REF] as needed for the proof of our gluing formula.

First we need to understand the behavior of the internal and external Dirichlet-to-Neumann operators. Since the internal Dirichlet Laplacian ∆ D -is positive, there is no problem in letting λ go to 0 in the definition of N -(λ).

Concerning the external Dirichlet-to-Neumann operator, by separation of variables in each conical end, we see that in the case λ = 0 the exterior Dirichlet problem (2.3) has a unique solution of the form

u(r, ϕ; 0) = ∞ n=-∞ C n R r νn e inϕ , (2.14)
where f = C n e inϕ . We recall that ν n = |n| kR and ψ n (φ) = (2π) -1 2 e inφ . The external Dirichlet-to-Neumann N + (0) is obtained by applying -∂ r to this solution.Clearly, {|n|/(kR 2 ), ψ n } ∞ n=-∞ is a complete set of the eigenvalues and orthonormal eigenfunctions of the operator N + (0). Remark 1. Note that thanks to the special choice of the lower bound on y in (2.2) the eigenvalue µ 0 (λ) of N + (λ) corresponding to the constant eigenfunction ψ 0 does not depend on k. This will be important in our proof of the BFK gluing formula in the case K ≥ 1 if k j = k i for some i, j = 1, . . . , K.

It is convenient to present the argument in the case where K = 1 so that X has only one conical end. We will explain afterwards how the proof is modified for K > 1.

The case K = 1

In the series (2.14) only the terms with ν n > 1 are in L 2 (X + ). As a result, in a neighborhood of zero, properties of λ → N + (λ) on the eigenspaces of N + (0) corresponding to the eigenvalues |n|/(kR 2 ) > 1/R and |n|/(kR 2 ) ≤ 1/R are essentially different. Consider the spectral projector P = 0≤n≤kR P n of N + (0) on the interval [0, 1/R]; here

P 0 = (•, ψ 0 ) L 2 (Σ) ; P n = (•, ψ n ) L 2 (Σ) ψ n + (•, ψ -n ) L 2 (Σ) ψ -n .
Lemma 1 (see [START_REF] Carron | Le saut en zéro de la function de décalage spectral[END_REF]Prop 4.5]). We have

N + (λ) Id -P = Ψ(λ 2 ) + L(λ 2 ),
where Ψ(z) is an elliptic pseudo-differential operator of order 1 which is a holomorphic function of z in a neighbourhood of zero and L(z) is an operator with smooth integral kernel which is a C 1 function of z in a neighbourhood of zero with ℑz ≥ 0.

Recall that the eigenfunctions ψ n of N + (λ) do not depend on λ and we have

N + (λ)Pf = 0≤n≤kR µ n (λ)P n
where the µ n have been defined in (2.5).

The eigenvalues of N + (0) on [0, 1/R] are the limits of µ n (λ). As |λ| → 0+, ℑλ ≥ 0, the formula (2.5) and properties of the Hankel functions (see [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]) imply that

µ 0 (λ) = - 1 R ln λ 1 -ln R 2 + πγ 2 -i π 2 1 ln λ + O 1 (ln λ) 2 , (2.15)
where γ is the Euler's constant, and

µ n (λ) = |n|/(kR 2 ) + O(λ ǫ ), 0 < |n| < kR, (2.16)
with some ǫ > 0.

We show the following proposition.

Proposition 1. Assume (X, |df | 2 ) has only one conical end then, for any R large enough we have, as |λ| → 0+, ℑλ ≥ 0,

det ζ N(λ) = - 1 R ln λ det * ζ N(0) 1 -ln R 2 + πγ 2 -i π 2 1 ln λ + O 1 (ln λ) 2 ,
where det ζ N(λ) is the zeta regularized determinant of N(λ), and det * ζ N(0) is the zeta regularized determinant of N(0) with zero eigenvalue excluded.

Proof. Due to the representation L 2 (Σ) = ker P 0 ⊕ ker(P -P 0 ) ⊕ ker(Id -P) we have

N(λ) =   N 0,0 (λ) N 0,1 (λ) N 0,2 (λ) N 1,0 (λ) N 1,1 (λ) N 1,2 (λ) N 2,0 (λ) N 2,1 (λ) N 2,2 (λ)   , (2.17)
where N i,j (λ) = P i N(λ)P j with P 0 = P 0 , P 1 = P -P 0 , and P 2 = Id -P. The operator N 2,2 (0) is invertible and therefore det ζ N 2,2 (0) = 0. Note that N 2,2 (λ) = P 2 N -(λ)P 2 + P 2 N + (λ)P 2 , where N + (λ)P 2 = N + (λ)(Id -P) is the same as in Lemma 1 and N -(λ) is a holomorphic function of λ 2 in a small neighbourhood of zero. This implies that

det ζ N 2,2 (λ) -det ζ N 2,2 (0) = o(1), |λ| → 0+, ℑλ ≥ 0.
(2.18)

Thanks to Lemma 1 we also have

N 2,2 (λ) -N 2,2 (0); B(H 1 (Σ), L 2 (Σ)) = O(λ 2 ), ∂ λ N 2,2 (λ) -∂ λ N 2,2 (0); B(H 1 (Σ), L 2 (Σ)) = O(λ). (2.19)
In order to refine (2.18), we estimate the absolute value of

∂ λ ln det ζ N 2,2 (λ). Since ∂ λ N 2,2 (λ) and N -1
2,2 (λ) are pseudodifferential operators of order -1, the operator

N -1 2,2 (λ)∂ λ N 2,2 (λ)
is in the trace class, and hence

∂ λ ln det ζ N 2,2 (λ) = Tr N -1 2,2 (λ)∂ λ N 2,2 (λ) , (2.20)
see [START_REF] Burghelea | Meyer-Vietoris type formula for determinants of elliptic differential operators[END_REF][START_REF] Forman | Functional determinants and geometry[END_REF]. The first estimate in (2. [START_REF] Knizhnik | Analytic fields on Riemann surfaces[END_REF]) and the Neumann series for N -1 2,2 (λ) give

N -1 2,2 (λ) = Id +L(λ) N -1 2,2 (0), L(λ); B(H 1 (Σ)) = O(λ 2 ); N -1 2,2 (λ) = N -1 2,2 (0) Id +R(λ) , R(λ); B(L 2 (Σ)) = O(λ 2 ). (2.21)
As a consequence of (2.20), (2.21), and (2.19) we get

|∂ λ ln det ζ N 2,2 (λ)| = | Tr N -1 2,2 (λ)∂ λ N 2,2 (λ) | ≤ N 2,2 (λ)(Id +L(λ))N -2 2,2 (0)(Id +R(λ))∂ λ N 2,2 (λ) 1 ≤ N 2,2 (λ); B(H 1 (Σ), L 2 (Σ)) Id +L(λ); B(H 1 (Σ)) N -2 2,2 (0) 1 × Id -R(λ); B(L 2 (Σ)) ∂ λ N 2,2 (λ); B(H 1 (Σ), L 2 (Σ)) = O(1),
where • 1 is the trace norm. This together with (2.18) implies |∂ λ det ζ N 2,2 (λ)| = O(1). Now, as a refinement of (2.18), we obtain

det ζ N 2,2 (λ) -det ζ N 2,2 (0) = O(λ).
This together with (2.17) implies

det ζ N(λ) = det Fr   N 0,0 (λ) N 0,1 (λ) N 0,2 (λ)N 2,2 (λ) -1 N 1,0 (λ) N 1,1 (λ) N 1,2 (λ)N 2,2 (λ) -1 N 2,0 (λ) N 2,1 (λ) Id   det ζ   Id 0 0 0 Id 0 0 0 N 2,2 (λ)   = det Fr   N 0,0 (λ) N 0,1 (λ) N 0,2 (λ)N 2,2 (λ) -1 N 1,0 (λ) N 1,1 (λ) N 1,2 (λ)N 2,2 (λ) -1 N 2,0 (λ) N 2,1 (λ) Id   det ζ N 2,2 (0) 1 + O(λ) ;
(2.22) see [START_REF] Kontsevich | Geometry of determinants of elliptic operators[END_REF] for the first equality. On the next step we rely on the estimate

| det Fr (Id +A) -det Fr (Id +B)| ≤ A -B 1 e A 1 + B 1 +1 , (2.23)
see [35, f-la (3.7), and references therein], for

Id +A =   N 0,0 (λ) 0 0 0 N 1,1 (0) N 1,2 (0)N -1 2,2 (0) 0 N 2,1 (0) Id   , Id +B =   N 0,0 (λ) N 0,1 (λ) N 0,2 (λ)N 2,2 (λ) -1 N 1,0 (λ) N 1,1 (λ) N 1,2 (λ)N 2,2 (λ) -1 N 2,0 (λ) N 2,1 (λ) Id   .
Since N -(0) is a selfadjoint operator in L 2 (Σ) and ker N -(0) = {c ∈ C}, we have N -(0)P 0 = P 0 N -(0) = 0. Then thanks to

P i N(λ) -N(0) P j = δ ij µ j (λ) -µ j (0) + P i N -(λ) -N -(0) P j , i, j ∈ [0, 1/R],
where δ ij is the Kronecker delta function, and

P j N(λ) -N(0) (Id -P) = P j N -(λ) -N -(0) (Id -P), j ∈ [0, 1/R],
together with (2.16) and (2.21), we obtain A -B 1 = O(λ ǫ ) with some ǫ > 0. From (2.22) and (2.23) we get

det ζ N(λ) = det Fr   N 0,0 (λ) 0 0 0 N 1,1 (0) N 1,2 (0)N 2,2 (0) -1 0 N 2,1 (0) Id   det ζ N 2,2 (0)(1 + O(λ ǫ )).
(2.24) It remains to note that

N 0,0 (λ) = (µ 0 (λ) + P 0 (N -(λ) -N -(0))P 0 = (µ 0 (λ) + O(λ 2 ))P 0 , det * ζ N(0) = det Fr N 1,1 (0) N 1,2 (0)N 2,2 (0) -1 N 2,1 (0) Id det ζ N 2,2 (0).
This together with (2.24) and (2.15) completes the proof.

Corollary 1. The spectral shift function ξ in (2.9) satisfies + (λ) is defined on the circle {y ∈ C j : |y| = R 1/k j } as in (2.4). The first eigenvalue of N (j) + (λ) is µ 0 (λ) and the corresponding eigenspace consists of constant functions on the circle. As a consequence, in the estimate (2.24) the eigenvectors in N + with eigenvalue µ 0 (λ) contribute at the order O( 1 ln λ ) instead of O(λ ǫ ) and this is not good enough for our purpose.

ξ(λ ; ∆, ∆ D ) = (ln λ 2 ) -1 + O (ln λ) -2 , λ → 0 + . ( 2
We thus introduce P 0 the orthogonal projection onto the eigenspace of N + (λ) corresponding to µ 0 (λ) (note that P 0 does not depend on λ, and that rank P 0 = K). Observe that we have ker N(0) ⊂ ker(Id -P 0 ). We repeat the argument of Proposition 1, where P 0 is now the orthogonal projection onto ker N(0) = {c ∈ C}, P 1 = (Id -P 0 )P where P is the spectral projection of N + (0) on the interval [0, 1/R], and P 2 = (Id -P). Clearly, N + (λ)P 0 = µ 0 (λ)P 0 and N + (λ)(Id -P 0 ) = (Id -P 0 )N + (λ).

The same argument as in the case K = 1 leads to

det ζ N(λ) = - 1 R ln λ det Fr N 1,1 (0) + µ 0 (λ)P 0 (Id -P 0 ) N 1,2 (0)N 2,2 (0) -1 N 2,1 (0) Id × det ζ N 2,2 (0) 1 -ln R 2 + πγ 2 -i π 2 1 ln λ + O 1 (ln λ) 2 .
(Note that in the case K = 1 we have P 0 = P 0 and the term µ 0 (λ)P 0 (Id -P 0 ) does not appear.) This together with (2.15) and (2.23) gives

det ζ N(λ) = - 1 R ln λ det Fr N 1,1 (0) -1 R ln λ P 0 (Id -P 0 ) N 1,2 (0)N 2,2 (0) -1 N 2,1 (0) Id × det ζ N 2,2 (0) 1 -ln R 2 + πγ 2 -i π 2 1 ln λ + O 1 (ln λ) 2 .
Here the Fredholm determinant is a holomorphic function of the parameter τ := 1 R ln λ , we have

det ζ N(λ) = - 1 R ln λ det * ζ N(0) 1 -C + ln R 2 + πγ 2 -i π 2 1 ln λ + O 1 (ln λ) 2 (2.26)
with some constant C = C(R).

We observe that C must be real since det ζ N(λ) is positive for λ ∈ C, Arg λ = π/2 (as N(λ) is a positive self-adjoint operator for those values of λ). Thus C does not influence the calculation of the argument in the asymptotic of det ζ N(λ) and Corollary 1 remains valid for K > 1. We will use Corollary 1 to define a relative determinant of (∆, ∆) at λ = 0.

The relative determinant and the gluing formula at λ = 0

In this section we prove the following Theorem. Observe that this theorem first requires a definition for the left-hand side of the equality. Once this is done, we will let λ go to zero in (2.13) and study the limit of both sides.

As before the case K = 1 is simpler than the general one. We will present the proof for this case first. The case K > 1 is more technically involved but the arguments we need can be adapted from [START_REF] Hassell | Determinants of Laplacians in exterior domains[END_REF].

2.4.1

The case K = 1

In this case, the definition of det ζ (∆, ∆) is rather straightforward, since, for K = 1, a conventional regularization (see, e.g., [START_REF] Hillairet | Spectral Determinants on Mandelstam Diagrams[END_REF] and references therein) for the relative zeta function makes sense. Indeed, the first integral in the representation

ζ(s; ∆, ∆) = 1 0 + ∞ 1 t s-1 Γ(s)
Tr e -t∆ -e -t ∆ dt (2.27) defines an analytic in ℜs > 1 function that has a meromorphic continuation to C with no pole at zero by the usual argument based on short time heat trace asymptotic Tr e -t∆ -e -t ∆ ∼ j≥-2 a j t j/2 , t → 0 + .

(2.28)

For the second integral we need the long time heat trace behaviour given by the following lemma. where δ > 0 is the smallest eigenvalue in the spectra of ∆ D -and ∆D -. (In (2.29) we also used (2.9) for ∆ and ∆.) As a consequence of Corollary 1 (which is also valid for ξ in the case K = 1) we have

ξ(λ ; ∆, ∆ D ) -ξ(λ, ∆, ∆D ) = O (ln λ) -2 , λ → 0 + .
This together with (2.29) implies the assertion; see e.g. [17, Theorem 1.7] for details.

As a consequence, the second integral in (2.27) defines a holomorphic in ℜs < 0 function that has a continuous in ℜs ≤ 0 derivative. Thus ζ(s; ∆, ∆) is a meromorphic function in ℜs < 0 and ζ ′ (s; ∆, ∆) tends to a certain limit ζ ′ (0; ∆, ∆) as s → 0-. The relative zeta regularized determinant is defined to be det ζ (∆, ∆) = e -ζ ′ (0;∆, ∆) .

(2.30)

We now prove the gluing formula in the case K = 1. First observe that by Proposition 1 (applied also to N(λ)) we have

det ζ N(λ) det ζ N(λ) → det * ζ N(0) det * ζ N(0) as λ 2 → 0-, K = 1.
(2.31)

The limit λ → 0 is then addressed by the Proposition 2. In the case K = 1 we have

det ζ (∆ -λ 2 , ∆ -λ 2 ) → det ζ (∆, ∆) as λ 2 → 0-,
where the determinant det ζ (∆, ∆) has been defined in (2.30).

Proof. Let us write the relative zeta function in the form

ζ(s; ∆ -λ 2 , ∆ -λ 2 ) = 1 0 + ∞ 1 t s-1 e tλ 2 Γ(s)
Tr e -t∆ -e -t ∆ dt.

Thanks to (2.10) the first integral converges for ℜs > 1 uniformly in λ ≤ 0 and has a meromorphic continuation to C with no pole at zero (by the usual argument based on the short time heat trace asymptotic (2.28)). Due to Lemma 2 the second integral defines a holomorphic in ℜs < 0 and continuous in λ 2 ≤ 0 and ℜs ≤ 0 function. Moreover, as 1/Γ(s) has a first order zero at s = 0, Lemma 2 also implies that the first derivative with respect to s of the second integral is also continuous in λ 2 ≤ 0 and ℜs ≤ 0. Thus we obtain

ζ ′ (0; ∆ -λ 2 , ∆ -λ 2 ) → ζ ′ (0; ∆, ∆), λ 2 → 0 -.
where ζ ′ (0; ∆, ∆) is defined using (2.27).

Proof of Theorem 4 in the case K = 1. We pass to the limit as

λ 2 → 0-in (2.13). Since ∆ D -is positive, we have det ζ (∆ D --λ 2 ) → det ζ ∆ D -as λ 2 → 0-,
C = det ζ ∆D -det * ζ N(0) -1
is moduli independent.

The case K > 1

In the case K > 1 we have N(λ) = ⊕ K j=1 N(j) (λ), where N(j) (λ) is the Neumann jump operator on the circle {y ∈ C j : |y| = R 1/k j } located on the infinite cone (C j , |dy k j | 2 ). We have

det ζ N(λ) = K j=1 det ζ N(j) (λ), det * ζ N(0) = K j=1 det * ζ N(j) (0).
We apply Proposition 1 to each det ζ N(j) (λ), j = 1, . . . , K and get

det ζ N(λ) = (-R ln λ) -K det * ζ N(0) 1 -ln R 2 + πγ 2 -i π 2 K ln λ + O 1 (ln λ) 2 , (2.32) as |λ| → 0+, ℑλ ≥ 0.
Thanks to the relation ξ(λ ; ∆, ∆D ) = π -1 Arg det N( √ λ 2 + i0) as λ 2 → 0+, calculation of the argument in (2.32) leads to

ξ(λ ; ∆, ∆D ) = K(ln λ 2 ) -1 + O((ln λ) -2 ), λ → 0+, where ξ(• ; ∆, ∆D ) ∈ L 1 (R + , (1 + λ 2 ) -2 dλ 2 ) is the spectral shift function satisfying Tr ( ∆ + 1) -1 -( ∆D + 1) -1 = - ∞ 0 ξ(λ ; ∆, ∆D )(1 + λ 2 ) -2 dλ 2 ;
cf. Corollary 1. This together with Corollary 1 gives

ξ(λ, ∆, ∆ D ) -ξ(λ, ∆, ∆D ) = -(K -1)(ln λ 2 ) -1 + O((ln λ) -2 ), λ → 0 + . (2.33)
Besides, Proposition 1 together with (2.32) implies that ln i λ

1-K det ζ N(λ) det ζ N(λ) → R K-1 det * ζ N(0) det * ζ N(0) as λ 2 → 0-, K ≥ 1. (2.34)
Recall that for λ 2 < 0 the relative zeta function is defined as the meromorphic continuation of (Now the short time asymptotic (2.28) is a consequence of (2.10) and similar short time asymptotics for Tr e -t ∆ -e -t ∆D , Tr e -t∆ D -, and Tr e -t ∆D -.) Let

ζ(s; ∆ -λ 2 , ∆ -λ 2 ) = ∞ 0 t s-1 e tλ 2 Γ(s) Tr e -t∆ -e -t ∆ dt (2.
N (λ) = j:λ 2 j ≤λ 2 dim ker(∆ D --λ 2 j ),
where λ 2 j are the eigenvalues of ∆ D -, be the counting function of ∆ D -. Similarly, let N (λ) be the counting function of ∆D -. Then

Tr(e -t∆ D --e -t ∆D -) = t ∞ 0 e -tλ 2 N (λ) -N (λ) 2λ dλ and ξ(λ; ∆ ; ∆) = ξ(λ ; ∆, ∆ D ) -ξ(λ ; ∆, ∆D ) -N (λ) + N (λ)
is the spectral shift function for the pair (∆, ∆) such that Tr e -t∆ -e -t ∆ = -t ∞ 0 e -tλ 2 ξ(λ; ∆, ∆)2λ dλ.

Since the operators ∆ D -and ∆D -are positive, from (2.33) it follows the asymptotic

ξ(λ; ∆, ∆) = -(K -1)(ln λ 2 ) -1 + O((ln λ) -2 ), λ → 0 + . (2.36)
Introduce a cutoff function χ ∈ C ∞ (R) such that χ(µ) = 1 for µ < 1/2 and χ(µ) = 0 for µ > 3/4. Following the scheme in [START_REF] Hassell | Determinants of Laplacians in exterior domains[END_REF] we write Tr e -t∆ -e -t ∆ = e 1 (t) + e 2 (t), where

e 1 (t) = -t ∞ 0 e -tµ 2 χ(µ)ξ(µ; ∆, ∆)2µ dµ, e 2 (t) = -t ∞ 0 e -tµ 2 (1 -χ(µ))ξ(µ; ∆, ∆)2µ dµ;
cf. (2.9). Note that e 2 is exponentially decreasing as t → +∞. Thanks to the short time asymptotic (2.28) and smoothness of e 1 at t = 0, we see that e 2 (t) has a short time asymptotic of the same form. Therefore for λ 2 ≤ 0 the holomorphic in ℜs > 1 zeta function

ζ 2 (s; λ 2 ) = ∞ 0 t s-1 e tλ 2 Γ(s) e 2 (t) dt
continues as a meromorphic function to C with no pole at zero and ζ ′ 2 (0, λ 2 ) → ζ ′ 2 (0, 0) as λ 2 → 0-by the usual argument.

We are now in position to define the regularized determinant det ζ (∆, ∆) We start from ζ(s; ∆, ∆) = ζ 1 (s; 0) + ζ 2 (s; 0), where ζ 1 (s; 0) is defined by

ζ 1 (s; 0) = ∞ 0 t s-1 Γ(s) e 1 (t) dt = s ∞ 0 (-µ 2 ) -s-1 χ(µ)ξ(λ; ∆, ∆)2µ dµ. (2.37)
From this expression and (2.33) one can easily see that ζ 1 (s; 0) is a holomorphic function in ℜs < 0, and we already know that ζ 2 (s; 0)is a meromorphic function of s ∈ C with no pole at 0. The asymptotic behaviour of ζ(s; 0) near s = 0 is given by the following proposition.

Proposition 3. Set Z ′ 0 :=ζ ′ 2 (0; 0) + (K -1) lim δ→0+ ∞ δ χ(µ)dµ µ ln µ + ln ln 1 δ -2 ∞ 0 µ -1 χ(µ) ξ(µ; ∆, ∆) -(K -1)(ln µ 2 ) -1 dµ + (K -1)(γ + ln 2).
When s → 0-we have

ζ(s; ∆, ∆) = ζ 2 (0; 0) + s(K -1) ln(-s) + sZ ′ 0 + o(s).
This proposition gives way to the following definition of the regularized relative determinant of ∆ in case K > 1. (-µ 2 ) -s-1 χ(µ) ξ(µ; ∆, ∆) + (K -1)(ln µ 2 ) -1 2µ dµ converges uniformly in s ≤ 0 and thus gives the contribution

-s ∞ 0 2µ -1 χ(µ) ξ(µ; ∆, ∆) + (K -1)(ln µ 2 ) -1 dµ
into the expansion of ζ(s; ∆, ∆). For the second integral we have

s(1 -K) ∞ 0 (-µ 2 ) -s-1 χ(µ)(ln µ 2 ) -1 2µ dµ = s(1 -K) -ln(-s) + γ + ln 2 -lim δ→0+ ∞ δ χ(µ)dµ µ ln µ + ln ln 1 δ + o(1)
; see [14, p.13].

The proof of the gluing formula will also require that we understand the limit when λ goes to 0. At this stage we have

ζ(s; ∆ -λ 2 , ∆ -λ 2 ) = ζ 1 (s; λ 2 ) + ζ 2 (s; λ 2 ),
where only properties of the zeta function

ζ 1 (s; λ 2 ) = ∞ 0 t s-1 e tλ 2 Γ(s) e 1 (t) dt = s ∞ 0 (λ 2 -µ 2 ) -s-1 χ(µ)ξ(λ; ∆, ∆)2µ dµ (2.38)
remain unknown. Notice that the last integrand is compactly supported and therefore the integral converges uniformly near s = 0 for fixed λ 2 < 0 due to (2.36). We get

ζ ′ 1 (0; λ 2 ) = ∞ 0 (λ 2 -µ 2 ) -1 χ(µ)ξ(µ; ∆, ∆)2µ dµ.
(2.39) Proposition 4. As λ 2 → 0-we have

ζ ′ (0; ∆ -λ 2 , ∆ -λ 2 ) = ln ln i λ 1-K + (K -1) lim δ→0+ ∞ δ χ(µ) dµ µ ln µ + ln ln 1 δ -2 ∞ 0 µ -1 χ(µ) ξ(µ; ∆, ∆) + (K -1)(ln µ 2 ) -1 dµ + ζ ′ 2 (0; 0) + o(1).
Proof. We only need to study the behaviour of ζ ′ (0; λ 2 ) in (2.39) as λ 2 → 0-. Thanks to (2.36) the integral

∞ 0 (λ 2 -µ 2 ) -1 χ(µ) ξ(µ; ∆, ∆) + (K -1)(ln µ 2 ) -1 2µ dµ
converges uniformly in λ 2 ≤ 0 and thus tends to

- ∞ 0 2µ -1 χ(µ) ξ(µ; ∆, ∆) + (K -1)(ln µ 2 ) -1 dµ as λ 2 → 0-. It remains to note that ∞ 0 (λ 2 -µ 2 ) -1 χ(µ)(ln µ 2 ) -1 2µ dµ = ln ln i λ -lim δ→0+ ∞ δ χ(µ) dµ µ ln µ + ln ln 1 δ + o(1);
as λ 2 → 0-; see [14, p. 12 and appendix].

Note that by definition of det ζ (∆, ∆) = e -Z ′ 0 we have

ζ ′ (0; ∆ -λ 2 , ∆ -λ 2 ) = ln ln i λ 1-K --Z ′ 0 -(K -1)(γ + ln 2) + o(1), λ 2 → 0-, see Prop. 4.
Proof of Theorem 4 in the general case. From Propositions 4 and 3 we immediately get ln i λ

1-K det ζ (∆ -λ 2 ; ∆ -λ 2 ) → e (1-K)(γ+ln 2) det * ζ (∆, ∆), λ 2 → 0 -. (2.40)
We pass in (2.13) to the limit as λ

det ζ ∆ D - = Re γ+ln 2 K-1 det * ζ N(0) det * ζ N(0)
. This proves Theorem 4, where N ≡ N(0) and the constant

C = Re γ+ln 2 K-1 / det ζ ∆D -det * ζ N(0) is moduli independent.
Remark 1. The proof of the gluing formula holds verbatim for a more general class of metrics under the following two assumptions. First, the structure at infinity should be given by a finite union of conical/Euclidean ends. Second, we have to assume that nothing bad happens with the Laplace operator ∆ D -of the compact part. In particular, it should have a well-defined zeta-function that extends to the complex plane with no pole at 0. This works for instance if the metric is smooth in the compact part or, if it is flat with conical singularities and the Friedrichs extension is chosen.

Closing the Euclidean (conical) ends with the help of gluing formulas

Let R be a sufficiently large positive number such that all the critical values of the meromorphic function f lie in the ball {|z| < R}.

In the holomorphic local parameter η j = y -1/k j in a vicinity U j (|y j | > R) of the j-th conical end of the angle 2πk j (k j ≥ 1) of the Riemannian manifold (X, |df | 2 ) (i. e. a pole of f of order k j ) the metric m = |df | 2 takes the form

m = k 2 j |dη j | 2 |η j | 2k j +2 . Let χ j be a smooth function on C such that χ j (η) = χ j (|η|), |χ j (η)| ≤ 1, χ j (η) = 0 if |η| > (R + 1) -1/k j , χ j (η) = 1 if |η| < (R + 2) -1/k j . Introduce the metric m on X such that m = m for |z| < R [1 + (|η j | 2k j +2 -1)χ j (η j )]m in U j .
Since the (Friedrichs extension of) the Laplace operator ∆ m has discrete spectrum and the corresponding operator ζ-function is regular at s = 0 (see e. g. [START_REF] Kokotov | On the spectral theory of the Laplacian on compact polyhedral surfaces of arbitrary genus[END_REF]and references therein), one can define the determinant det * ∆ m via usual Ray-Singer zeta regularization. Moreover, for this determinant the usual BFK gluing formula ( [START_REF] Burghelea | Meyer-Vietoris type formula for determinants of elliptic differential operators[END_REF], Theorem B * ) holds (under the condition that the contour cutting the surface X does not pass through the conical singularities of the metric m). Applying this standard BFK gluing formula, we get ln det

* ζ ∆ m = ln C 0 + ln det ζ ∆ D -+ ln det * ζ N + ln det∆ m ext , (2.41)
where ∆ m ext is the operator of the Dirichlet problem for ∆ m in the union ∪ j U j . Using conformal invariance we see that N is the same as in Theorem 1 and C 0 is a moduli independent constant (C 0 = Area(X, m) length(Σ) ). Now equation (2.41) and Theorem 1 imply the following proposition.

∂ z k ln det ζ (∆, ∆) = ∂ z k ln det * ζ ∆ m (2.42) for k = 1, . . . , M .
Thus, the relative determinant of the Laplacian on a noncompact surface (X, m) with conical points and conical/Euclidean ends can be studied via consideration of the zeta-regularized determinant of Laplacian on a compact surface (X, m) with conical points. The latter surface is flat everywhere except the conical singularities (whose positions vary when one changes the moduli z 1 , . . . , z M ) and smooth ends of nonzero curvature which remain unchanged.

In the next two sections we study some spectral properties of compact surfaces with conical points. The final goal is to derive the variational formulas for ln det * ζ ∆ m.

S-matrix

In this section we introduce the so-called S-matrix and relate its behavior at λ = 0 with the Schiffer projective connection. The definition of the S-matrix originates in the general theory of boundary triplets (see [START_REF] Grubb | Distributions and operators[END_REF] sect. 13) and, more specifically, the general theory of self-adjoint extensions of elliptic operators in singular settings (see [START_REF] Nazarov | Elliptic Problems in Domains with Piecewise Smooth Boundaries[END_REF]). Here we will follow closely [START_REF] Hillairet | Krein formula and S-matrix for Euclidean surfaces with conical singularities[END_REF]. However, we should point out that some normalization constants in the latter reference are erroneous and that the normalization we use here is slightly different.

It is convenient to introduce the S-matric in the following general setting.

General Setting and normalizations

Let (X, m) be a compact singular 2d Riemannian manifold (possibly with boundary). Let P be an interior point of X such that in a neighborhood V of it X is isometric to a neighborhood of the tip of the Euclidean cone of angle 2ℓπ. We set X 0 := X\{P } and X ε := X\B(p, ε). We also denote by γ r the circle of radius r centered at P .

We will occasionnally use several different ways of parametrizing V.

• Polar coordinates (r, θ) ∈ (0, r max ) × R/2ℓπZ,

• Local complex coordinate z. The 1-form dz is well-defined on V \{P } and extends to V as a holomorphic one form α with a zero of order ℓ -1 at P . Note that it may not have a global holomorphic extension to X.

• Distinguished complex parameter y such that α = ℓy ℓ-1 dy near P .

We now want to consider the Laplace operator that is associated with m. We assume that the set of singularities of m consists of a finite number of conical points (in particular, it may consist of a single point P ). Let ∆ be the Friedrichs extension of the Laplace operator on X with domain consisting of smooth functions that vanish near the singularities.

Remark 2. Actually the choice of extension (that is the prescription of a certain asymptotics at singular points to functions from the domain of the self-adjoint extension) should be made at each singularity (see [START_REF] Hillairet | Krein formula and S-matrix for Euclidean surfaces with conical singularities[END_REF]). Here we really care only about the choice of extension at P , where we are choosing the Friedrichs extension (all the functions from the domain of this extension are bounded near P ). At other singularities one can chose any other extension, not necessarily the Friedrichs one.

By definition we set H 2 (X) to be the domain of ∆ and by H 1 (X) to be its form domain. We denote by ∆ 0 the restriction of ∆ to functions in H 2 (X) that vanish near p and by ∆ * 0 its formal adjoint. By choice, the self-adjoint extension ∆ corresponds to the Friedrichs extension of ∆ 0 . We will also denote by H 2 0 (X) := dom ∆ 0 . Near P , we have

∆ * 0 = -4∂ z ∂ z = -4 ℓ 2 |y| 2(ℓ-1) -1 ∂ y ∂ y .
Introduce a cut-off function ρ such that ρ has support in r ≤ r max and equals 1 near r = 0. Define the functions F 0 , F a ν and F h ν via

F 0 (z) = c 0 ln(zz)ρ(z) = c 0 ln(r 2 )ρ(z), F a ν (z) = c ν z -ν ρ(z), ν = k ℓ , 0 < k < ℓ F h ν (z) = c ν z -ν ρ(z), ν = k ℓ , 0 < k < ℓ , where c 0 = 1 2 √ ℓπ , c ν = 1 2 √ νℓπ . (3.1)
Remark 3. The indices a and h correspond to "antiholomorphic" and "holomorphic" behavior of the corresponding functions F at 0. In all the formulas below the index ν runs through the set {k/ℓ} k=1,...,ℓ-1 , where 2πℓ is the conical angle at P .

Separating variables near P , it can be shown that any function in dom(∆ * 0 ) admits the following expression (cf. [START_REF] Kokotov | On the spectral theory of the Laplacian on compact polyhedral surfaces of arbitrary genus[END_REF], [START_REF] Mooers | Heat kernel asymptotics on manifolds with conic singularities[END_REF])

u = c 0 Λ 0 (u) + c 0 Λ 0,-(u)F 0 (z) + ν c ν Λ h,- ν (u)z -ν + c ν Λ a,- ν (u)z -ν + ν c ν Λ h ν (u)z ν + c ν Λ a ν (u)z -ν + u 0 , (3.2)
Remark 4. The functions F, f, g depend on the initial choice of ρ but the linear functionals Λ and the functions G are cut-off independent.

The S-matrix is defined by blocks :

S :=   S 00 S 0h 0ν S 0a 0ν S h0 µ0 S hh µν S ha µν S a0 µ0 S ah µν S aa µν   . (3.5)
Remark 5. If there are several conical points on the surface, then there are several ways to define the S-matrix depending on how many points we want to take into account. The S-matrix defined in [START_REF] Hillairet | Krein formula and S-matrix for Euclidean surfaces with conical singularities[END_REF] takes into account all the conical points whereas the one defined here deals only with the conical point P (even if there are other conical points on the surface). Thus the S-matrix that is constructed here is only a part of the one from [START_REF] Hillairet | Krein formula and S-matrix for Euclidean surfaces with conical singularities[END_REF].

Applying Green's formula, we get

Λ ♯ ν (u) = G(u, F ♯ ) = (∆ -λ)u • F ♯ ν -u • f ♯ ν (•; λ)dS = (∆ -λ)u • F ♯ ν + u • (∆ -λ)g ♯ ν (•; λ)dS = (∆ -λ)u • G ♯ ν dS (3.6)
for any test function u ∈ H 2 (X) (here dS is the area element on X). We have used here that F 0,a,h = F 0,h,a ∈ dom(∆ * 0 ), u and g are in H 2 (X), and that ∆ is real (i.e. commutes with complex conjugation) and self-adjoint.

Applying (3.6) to the g-functions gives the following alternative expressions for the S-matrix entries (we omit the dependence on λ):

S ♯♭ µν = G ♯ µ (∆ -λ) g ♭ ν dS = -G ♯ µ f ♭ ν dS. (3.7)
Remark 6. The S-matrix allows the description of the elements of ker(∆ * 0 -λ) in the following way. For any element u ∈ dom(∆ * 0 ), denote by L ± the column of its coefficients Λ ♯,± ν that describe the singular behaviour of u near P . We have

(∆ * 0 -λ)u = 0 ⇔ L + = S(λ)L -.
This gives a (pure) formal analogy with a typical scattering situation. In our setting, any solution to the equation (∆ * 0 -λ)u = 0 plays the role of scattered field, L ± is the "incoming" and "outgoing" parts and the S-matrix is the "scattering" matrix.

4 Basic properties of the S-matrix

Analyticity and complex conjugation

From the analyticity of the resolvent we see that the S-matrix depends analytically on λ. Besides, the expression

S ♯♭ µν (λ) = -G ♯ µ (•; λ)f ♭ ν (•; λ) dS
and the fact that

f a,h ν (•, λ) = f h,a ν (•; λ), G a,h (•; λ) = G h,a ν (•; λ)
, lead to the following identities :

S hh (λ) = S aa (λ), S ah (λ) = S ha (λ). ( 4.1)
4.0.2 Behavior for λ going to -∞ For r > 0 consider the equation

-u ′′ (r) - 1 r u ′ (r) + ν 2 r 2 u(r) = λu(r)
. Since ν = 0, any solution to this equation has the following asymptotic behaviour near zero :

u(r) = a -r -ν + a + r ν + o(r ν ).
and the vector space of solutions that belongs to L 2 (rdr) is one-dimensional. We set k ν (r; λ) to be the unique solution to this equation which is in L 2 (rdr) and that is normalized in such a way that

F h ν (r, θ) -k ν (r; λ) exp(-iνθ) = O(r ν ), F a ν (r, θ) -k ν (r; λ) exp(iνθ) = O(r ν ) (i.e.
we adjust the coefficient of r -ν in k ♯ ν so that it coincides with the coefficient of F ♯ ν ). By inserting a cut-off ρ, we define

K h ν (r, θ; λ) := k ν (r; λ) exp(-iνθ)ρ(r), K a ν (r, θ; λ) := k ν (r; λ) exp(iνθ)ρ(r)
as functions on X. We compute

R ♯ ν ( • ; λ) = (∆ * 0 -λ) K ♯ ν , where ♯ = a, h. Lemma 1. For ♯ = a, h we have G ♯ ν ( • ; λ) = K ♯ ν -[∆ -λ] -1 R ♯ ν . Proof.
By construction it is straightforward that both sides of the equation are in ker(∆ * 0 -λ) and by choice of normalization, both share the same singular behaviour.

We define by κ ν (λ) the coefficient of r ν in the asymptotic expansion of k ν ( • ; λ).

Corollary 1.

As ℜλ goes to -∞ we have

S hh (λ) = O(|λ| -∞ ), S aa (λ) = O(|λ| -∞ ), S ah (λ) = diag(κ ν (λ)) + O(|λ| -∞ ), S ha (λ) = diag(κ ν (λ)) + O(|λ| -∞ ).
Proof. The asymptotic expansion of Bessel functions implies that

R ♭ ν L 2 = O(|λ| -∞ ). Therefore Λ ♯ ν [∆ -λ] -1 R ♭ ν = O(|λ| -∞
). Thus all entries of the S-matrix are given by Λ ♯ µ (K ♭ ν ) up to O(|λ| -∞ ). The first term is seen to be 0 except for the diagonal terms in S ah or S ha for which it is κ ν (λ).

Remark 7. A different proof is given in [START_REF] Hillairet | Krein formula and S-matrix for Euclidean surfaces with conical singularities[END_REF] using the heat kernel.

S(0) and the Schiffer projective connection

Chose a marking for the Riemann surface X, i. e. the canonical basis a 1 , b 1 , . . . , a g , b g of H 1 (X, Z). Let {v 1 , . . . , v g } be the basis of holomorphic differentials on X normalized via

a i v j = δ ij .
Then the matrix of b-periods of the marked Riemann surface X is defined via

B = || b i v j || .
Let W ( • , • ) be the canonical meromorphic bidifferential on X × X, with properties The bidifferential W has the only double pole along the diagonal P = Q. In any holomorphic local parameter x(P ) one has the asymptotics

W (P, Q) = W (Q, P ), a i W ( • , P ) = 0,
W (x(P ), x(Q)) = 1 (x(P ) -x(Q)) 2 + H(x(P ), x(Q)) dx(P )dx(Q), (4.4)
S(P, Q) = W (P, Q) -π i,j (ℑB) -1 ij v i (P )v j (Q).
The Schiffer projective connection, S Sch , is defined via the asymptotic expansion

S(x(P ), x(Q)) = 1 (x(P ) -x(Q)) 2 + 1 6 S Sch (x(P )) + O(x(P ) -x(Q)) dx(P )dx(Q)
One has the equality

S Sch (x) = S B (x) -6π i,j (ℑB) -1 ij v i (x)v j (x) . (4.5)
In contrast to the canonical meromorphic differential and the Bergman projective connection, the Schiffer bidifferential and the Schiffer projective connection are independent of the marking of the Riemann surface X.

Introduce also the so-called Bergman kernel (which is in fact the Bergman reproducing kernel for holomorphic differentials on X) as

B(x, x) = ij (ℑB) -1 ij v i (x)v j (x) .
Proposition 6. Let X be a Riemann surface and let m be a conformal metric on X, suppose that m has a conical singularity of angle 2ℓπ at p. Let also x be the distinguished local parameter for m near p. Then there is the following relation between the entries of the holomorphic-holomorphic part, S hh (0), of the S-matrix :

ℓ-1 k=1 k(ℓ -k) ℓ S hh ℓ ℓ ℓ-k ℓ (0) = - 1 6ℓ(ℓ -2)! d dx ℓ-2 S Sch (x) x=0 .
(4.6)

Remark 8. The same would hold true for a conical singularity of angle β with 2π(ℓ -1) < β ≤ 2πℓ.

Remark 9. Observe that using the indices µ, ν = k ℓ the left-hand-side of (4.6) can be written as

µ+ν=1 √ µ √ νS hh µν (0).
Proof. Introduce the following one forms Ω k and Σ k on X:

Ω k = - 1 (k -1)! d dx k-1 W ( • , x) dx x=0 + 2πi (k -1)! α,β (ℑB) -1 αβ ℑv (k-1) β (0) v α (•) Σ k = -i 1 (k -1)! d dx k-1 W ( • , x) dx x=0 + 2πi (k -1)! α,β (ℑB) -1 αβ ℜv (k-1) β (0) v α (•) , where v (k-1) β (0) := d dx k-1 v β (x) dx x=0 .
All the periods of the differentials Ω k and Σ k are pure imaginary, therefore, one can correctly define the function f k on X via

f k (Q) = ℜ Q P 0 Ω k -iℜ Q P 0 Σ k
where P 0 is an arbitrary base point not coinciding with P . Clearly, f k is harmonic in X \ {P } and

f k (x) = 1 x k + const + ∞ j=1 (c j x j + d j xj ) (4.7)
in a vicinity of P . One gets

c l = - 1 l!(k -1)! ∂ l-1 x ∂ k-1 y H(x, y) x=y=0 + π l!(k -1)! α,β (ℑB) -1 αβ v (k-1) β (0)v (l-1) α (0)
and

S hh k ℓ l ℓ (0) = l k c l .
This implies that

ℓ-1 k=1 k(ℓ -k) ℓ S hh k ℓ ℓ-k ℓ (0) = - 1 ℓ ℓ-2 k=0 1 k!(ℓ -2 -k)! ∂ ℓ-2-k x ∂ k y H(x, y) x=y=0 + π ℓ(ℓ -2)! d dx ℓ-2 α,β (ℑB) -1 αβ v α (x)v β (x) (dx) 2 x=0 . Since 1 6 S B (x) = H(x, x) = ∞ n=0 1 n! (∂ x + ∂ y ) n H(x, y) x=y x n ,
we have 1 6 S

(n)

B (0) = n p=0 n! p!(n -p)! ∂ p x ∂ n-p y H(0, 0),
which implies the proposition.

Remark 10. From (4.7) with k = 1 it follows that for conical angles 2π < β ≤ 4π we have S hh (0) S ha (0)

S ah (0) S aa (0) = -1 6 S Sch (0) B(0, 0) B(0, 0) -1 6 S Sch (0)
, where the Schiffer projective connection and the Bergman kernel are calculated in the distinguished local parameter at P .

Variational formulas with respect to moduli

In this section we derive the variational formulas for ln det∆ m. This derivation goes as follows. First, using Kato-Rellich theory (see [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF]), we prove variational formulas for the individual eigenvalues of the operator ∆ m. Using these formulas and the contour integral representation of the zeta-function of ∆ m, we express the variations of the value ζ ′ ∆ m (0) with respect to the critical value z k through a combination of the matrix elements of the S-matrix at the conical point P k (the zero of the meromorphic differential df ) of the metric m. The latter combination is the one appearing in Proposition 6 and can be expressed through the Schiffer projective connection.

Variational formula for eigenvalues of ∆ m

Remark 11. In this section we will use w for the moduli parameter and on the surface we will use the complex parameter z and (x, y) for the associated local cartesian coordinates (so that z = x + iy). We warn the reader that in the rest of the paper we use z i as the moduli parameters and x as a local complex parameter on X.

Moving conical points

Let m be a metric as constructed in section 2.5. Let P be one of its conical points. We wish to define a metric mw corresponding to the shift of P by w ∈ C. The following makes this construction precise.

Let C be the complex plane with pointed origin. We set Xw to be the ℓ-fold covering of C with one ramification point at w so that Xw can be identified with the Euclidean cone of total angle 2ℓπ.

Fix a cutoff function ρ and define a map φ w from C to itself by φ w (z) = z + ρ(|z|)w.

For w small enough, this defines a family of smooth selfdiffeomorphisms of C. The cone X0 can be obtained by gluing together ℓ copies of the plane after cutting along a fixed half-line d that emanates from the origin. The cone Xw can then be obtained by gluing ℓ copies of C after cutting it along φ w (d).

The function φ w thus defines a family of smooth diffeomorphisms from X0 onto Xw . Let the metric g w on X0 be the pull-back of the Euclidean metric on Xw by φ w .

We write w = a + ib and use the local cartesian coordinates x + iy = z near P . For the metric g w = A(x, y; w)dx 2 + 2B(x, y; w)dxdy + C(x, y; w)dy 2 we obtain the following expressions :

A B B C = t Dφ w Dφ w , Dφ w = 1 + ax r ρ ′ 1 (r) ay r ρ ′ 1 (r) bx r ρ ′ 1 (r) 1 + by r ρ ′ 1 (r)
.

(5.1)

It follows by direct verification that the coefficients of g w are polynomials in a, b. Observe that g w coincide with g 0 outside a ball centered at P so that g w can be smoothly extended by any Riemannian metric that coincides with the Euclidean one in an annulus centered at p. This allows us to define a metric mw on our given setting X ≡ X 0 that corresponds to some X w that is obtained by fixing the exterior of a small ball centered at P ∈ X and, in an even smaller ball, by shifting the conical point by w.

We denote by J w the jacobian determinant of the metric mw on X, by q w the Dirichlet energy quadratic form associated with mw , and by n w the Riemannian L 2 (X) scalar product on (X, mw ).

We thus have the following expressions (for a real u that is supported near P )

q w (u) = X 0 C(∂ x u) 2 -2B∂ x u∂ y v + A(∂ y u) 2 J -1 2 w dxdy, n w (u) = u 2 J 1 2
w dxdy.

(5.2)

Observe that q w (u) and n w (u) do not depend on w for u supported away of P . In order to apply spectral perturbation theory, we will need the first order variations of q w (u) and n w (u). We prove the relevant lemma below. Lemma 2. For any λ ∈ C and any u ∈ H 1 (X) we have

[-∂ w q + λ∂ w n] w=0 (u) = 2 X 0 (∂ z u) 2 zρ ′ (r) r dxdy + λ 2 X 0 u 2 zρ ′ (r) r dxdy, [-∂ w q + λ∂ w n] w=0 (u) = 2 X 0 (∂ z u) 2 zρ ′ (r) r dxdy + λ 2 X 0 u 2 zρ ′ (r) r dxdy.
(5.3)

Proof. Denote by

G w := C -B -B A so that we have q w (u) := X 0 t ∇uG∇u • J -1 2
w dxdy.

Differentiating at w = 0, we obtain

∂ w q w (u) = X 0 t ∇u • ∂ w G - 1 2 ∂ w JI • ∇udS.
A straightforward computation yields

∂ a G - 1 2 ∂ a JI = -xρ ′ r -yρ ′ r -yρ ′ r xρ ′ r ∂ b G - 1 2 ∂ b JI = yρ ′ r -xρ ′ r -xρ ′ r -yρ ′ r
From this we find

t ∇u • ∂ w G - 1 2 ∂ w JI • ∇u = t ∇u • -zρ ′ 2r izρ ′ 2r izρ ′ 2r zρ ′ 2r • ∇u = - zρ ′ 2r (∂ x u) 2 -(∂ y u) 2 -2i∂ x u∂ y u = - 2zρ ′ r (∂ z u) 2 .
The other terms proceed in the same way.

Variational formulas for eigenvalues of m

In this section, we compute variational formulas for the eigenvalues of mw . In order to do so we use the Kato-Rellich analytic perturbation theory [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF]. It should be noticed that the family of metrics g w is smooth in w but not analytic, see (5.1). We thus fix w and introduce q t = q tw and n t = n tw , which are analytic in t. In this way we may only consider directional derivatives. The eigenvalue equation that gives the spectrum of q t relatively to n t is

q t (u t , v) = λ t n t (u t , v). (5.4)
This problem is analytic in t so that the eigenvalues are organized into real-analytic branches, see [18, Chapter VII.6.5] for details. The first-order variation for the eigenbranch (λ t , u t ) is given by the following Feynman-Hellmann formula

dλ dt = dq dt (u) -λ dn dt (u) (5.5)
which is obtained by differentiating eq. (5.4) with v fixed and then evaluating at v = u t .

Proposition 3. Let r be small enough, then for λ ∈ spec(∆ m) we have

∂ w λ = 2 i γr (∂ z u) 2 dz - λ 4 u 2 dz ∂ w λ = - 2 i γr (∂ z u) 2 dz - λ 4 u 2 dz.
(5.6) Lemma 3. Let A = [a µν ] be the matrix defined by

a µν = 4πµc µ • νc ν if µ + ν = 1, a µν = 0 otherwise.
We have the alternative expressions

∂ w λ = µ+ν=1 Λ h ν (u)a µν Λ h µ (u), ∂ w λ = µ+ν=1 Λ a ν (u)a µν Λ a µ (u). ( 5
(∂ z (u -u 0 )) 2 dz.
By definition we have

u -u 0 = c 0 Λ 0 (u) + ν c ν Λ h ν (u)z ν + c ν Λ a (u)z ν ,
so that the claim follows by a direct computation.

Using the definition of Λ h ν and the fact that u is an eigenfunction, we obtain

Corollary 2. For any λ ∈ C\[0, ∞) the series λn∈spec(∆ m ) ∂ w λ n (λ n -λ) -2 is absolutely convergent and λn∈spec(∆ m ) ∂λ n (λ n -λ) 2 = Tr A ∂S hh ∂λ (λ) .
Proof. To prove the absolute convergence it suffices to show that for any ν we have

λn∈spec(∆ m ) Λ h ν (u n ) λ n -λ 2 < ∞. Since Λ h ν (u n ) = (∆ -λ)u n G h ν = (λ n -λ) G h ν , u ,
the claim follows by remarking that the eigenfunctions u n form an orthonormal basis. By Plancherel formula, we then obtain

λn∈spec(∆ m ) ∂ w λ n (λ n -λ) 2 = µ,ν a µν X G h µ (x ; λ)G h ν (x ; λ) dS.
We now remark that using (4.2)

X G h µ (x ; λ)G h ν (x ; λ) dS = ∂ λ S hh µν ∂λ ,
and A is a symmetric matrix.

Remark 13. For λn∈spec(∆ m ) ∂λ n (λ n -λ) -2 we also have a similar formula involving S aa .

Variational formula for ζ

′ (0 ; ∆ m)
We prove the following proposition.

Proposition 4. Let mw be the family of metrics defined above, S hh be the holomorphicholomorphic part of the corresponding S-matrix and A be the matrix defined in Lemma 3. We have

∂ w ζ ′ (0 ; ∆ m) = Tr AS hh (0) = µ+ν=1 √ µ √ νS hh µν (0). (5.9)
Proof. We start from the following integral representation of the zeta-function of the operator ∆ m -λ through the trace of the second power of the resolvent:

sζ(s + 1; ∆ m -λ) = 1 2πi Γ λ (z -λ) -s Tr (∆ m -z) -2 dz, (5.10)
where Γ λ is a contour connecting -∞ + iǫ with -∞ -iǫ and following the cut (-∞, λ) at the (sufficiently small) distance ǫ > 0. Using Corollary 2, differentiation under the integral sign is legitimate and we obtain

s∂ w ζ(s + 1, ∆ m -λ) = 1 2πi Γ λ (z -λ) -s λn∈spec(∆ m ) -2∂ w λ n (λ n -z) 3 dz. (5.11)
Using again Corollary 2, it is legitimate to make an integration by parts under the integral sign to get

s∂ w ζ(s + 1, ∆ m -λ) = -s 2πi Γ λ (z -λ) -s-1 λn∈spec(∆ m ∂ w λ n (λ n -z) 2 dz.
(5.12)

We can now divide by s, use Corollary 2 once again, and replace s + 1 by s to finally obtain

∂ w ζ(s, ∆ m -λ) = -1 2πi Γ λ (z -λ) -s Tr A∂ z S hh (z) dz. (5.13)
Using the behaviour of S hh at infinity we can make an integration by parts again and obtain

∂ w ζ(s, ∆ m -λ) = -s 2πi Γ λ (z -λ) -s-1 Tr AS hh (z) dz. (5.14)
It should be noted that the right hand side of (6.2) depends holomorphically on moduli z 1 , . . . , z M and, therefore, one has

∂ 2 zm zn ln det ζ (∆, ∆) detℑB = 0 . This implies the relation det ζ (∆, ∆) = C detℑB |τ | 2 , (6.3)
where τ is a holomorphic function of moduli z 1 , . . . , z M (actually, τ is a holomorphic section of some holomorphic line bundle over the Hurwitz space, see [START_REF] Kokotov | Isomonodtromic tau function on the space of admissible covers[END_REF] for further information; here we restrict ourselves to local considerations: the reader may assume for simplicity that all happens in a small vicinity of the covering f : X → CP 1 in the Hurwitz space H(M, N )) subject to the system of PDE

∂ zm ln τ = - 1 12πi Pm S B -S f df (6.4)
and C is a moduli independent constant. System of PDE (6.4) first appeared in the context of the theory of isomonodromic deformations and Frobenius manifolds in [START_REF] Kokotov | Tau-functions on Hurwitz spaces[END_REF] and [START_REF] Kokotov | Isomonodromic tau-function of Hurwitz Frobenius manifolds and its appplications[END_REF], where, in particular, it was explicitly integrated. We remind these results in the next subsection.

Explicit expressions for τ

In this section we recall explicit formulas for the holomorphic solution, τ , of the system (6.4) derived in [START_REF] Kokotov | Isomonodromic tau-function of Hurwitz Frobenius manifolds and its appplications[END_REF], [START_REF] Kokotov | Tau-functions on Hurwitz spaces[END_REF] (see also [START_REF] Kokotov | On the isomonodromic tau-function for the Hurwitz spaces of branhed coverings of genus zero and one[END_REF] and [START_REF] Kokotov | Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray-SInger formula[END_REF] for alternative and more straightforward proofs). The result should be formulated separately for low genera g = 0, 1 and for higher genus g > 1. We start with the higher genus situation.

Let g > 1. Take a nonsingular odd theta characteristic δ and consider the corresponding theta function θ[δ](t; B), where t = (t 1 , . . . , t g ) ∈ C g . Put

ω δ = g i=1 ∂θ[δ] ∂t i (0; B) ω i .
All zeroes of the holomorphic 1-differential ω δ have even multiplicities, and √ ω δ is a well-defined holomorphic spinor on X. Following Fay [START_REF] Fay | Kernel functions, analytic torsion and moduli spaces[END_REF], consider the prime form

E(x, y) = θ[δ] y x v 1 , . . . , y x v g ; B √ ω δ (x)
√ ω δ (y) . (6.5)

To make the integrals uniquely defined, we fix 2g simple closed loops in the homology classes a i , b i that cut X into a connected domain, and pick the integration paths that do not intersect the cuts. The sign of the square root is chosen so that We introduce local coordinates on X that we call distinguished with respect to f . Consider the divisor (df ) = k d k p k , p k ∈ X, d k ∈ Z, d k = 0, of the meromorphic differential df . We take z = f (x) as a local coordinate on Xk p k and

E
x k =    (f (x) -f (p k )) 1 d k +1 if d k > 0, f (x) 1 d k +1 if d k < 0, (6.6)
near p k ∈ X. In terms of these coordinates we have E(x, y) = E(z(x),z(y))

√ dz(x) √ dz(y)

, and we define E(z, p k ) = lim Let A x be the Abel map with the basepoint x, and let K x = (K x 1 , . . . , K x g ) be the vector of Riemann constants

K x i = 1 2 + 1 2 B ii - j =i a i v i (y) y x v j dy (6.7)
(as above, we assume that the integration paths do not intersect the cuts on X). Then we have A x ((df )) + 2K x = ΩZ + Z ′ for some Z, Z ′ ∈ Z g . One has the following expression for the holomorphic solution to (6.4)(see [START_REF] Kokotov | Isomonodromic tau-function of Hurwitz Frobenius manifolds and its appplications[END_REF], here we follow the presentation of this result in [START_REF] Kokotov | Isomonodtromic tau function on the space of admissible covers[END_REF]): Here θ(t; B) = θ[0](t; B) is the Riemann theta function, t = (t 1 , . . . , t g ) ∈ C g , and W is the Wronskian of the normalized holomorphic differentials v 1 , . . . , v g on X; the expression in (6.8) is independent of ζ ∈ X. Let g = 1. Then the function τ (X, f ) is given (see [START_REF] Kokotov | Tau-functions on Hurwitz spaces[END_REF]) by the equation

τ (X, f ) = g i=1 v i (ζ) ∂
τ (X, f ) = [θ 1 ′ (0 | B)] 2/3 K k=1 h (k j +1)/12 j M m=1 f ℓm/12 m , (6.9)
where v(P ) is the normalized Abelian differential on the elliptic Torelli marked curve X; v(P ) = f m (x m )dx m near P m , where x m = (z -z m ) 1/(ℓm+1) is the distinguished local parameter near the zero, P m of the differential df ; f m ≡ f m (0); v(P ) = h j (ζ j )dζ j as P → ∞ j , ζ j = z -1/k j , where k j is the multiplicity of the pole ∞ j of f , h k ≡ h k (0); θ 1 is the Jacobi theta-function. Let g = 0 and let U : X → P 1 be a biholomorphic map such that U (∞ 1 ) = ∞ and U (P ) = (f (P )) 1/k 1 + o(1) as P → ∞ 1 . Then (see [START_REF] Kokotov | On the isomonodromic tau-function for the Hurwitz spaces of branhed coverings of genus zero and one[END_REF]) τ (X, f ) = K j=2 ( dU dζ j ζ j =0 ) (k j +1)/12 M m=1 ( dU dxm xm=0 ) lm/12 . (6.10) Summarizing (6.3) and (6.10,6.9 6.8), we get the main result of the present paper.

Theorem 6. Let (X, f ) be an element of the Hurwitz space H(M, N ) and let τ (X, f ) be given by expressions (6.10,6.9, 6.8). There is the following explicit expression for the regularized relative determinant of the Laplacian ∆ on the Riemann surface X:

det ζ (∆, ∆) = C detℑB |τ | 2 , (6.11)
where C is a constant dependent only on the connected component of the space H(M, N ) containing the element (X, f ).

Examples in genus 0

We finish the paper with two simple and especially instructive examples of the calculation of the determinant of the Laplacian ∆ in genus zero. Example 1. Let p be a polynomial with N -1 simple critical points w 1 , . . . , w N -1 and let the corresponding critical values be z 1 , . . . , z N -1 (or, what is the same, a ramified covering with N -1 simple branch points and one branch point of multiplicity N over the point at infinity of the base. In other words p is an element of the Hurwitz space H 0,N ([1] N ) of meromorphic functions of degree N on the Riemann sphere P 1 with a single pole of multiplicity N .

Let also w be the holomorphic coordinate on the cover P 1 (more precisely, on P 1 \ {∞}) and z be the holomorphic coordinate on the base P 1 . One can assume that the leading coefficient of the polynomial p(w) is equal to one.

  for details). (Note that in Theorem 4 the operator N(0) is denoted by N.) Let λ 2 < 0. The function ζ(s) = Tr N(λ) -s is holomorphic in {s ∈ C : ℜs > 1} and admits a meromorphic continuation to C with no pole at zero; we set det ζ N(λ) = e -ζ ′ (0) .It is known (see[START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF] Theorem 2.2]) that the difference(∆ + 1) -1 -(∆ D + 1) -1is in the trace class. By the Krein theorem, see e.g.[START_REF] Yafaev | Mathematical scattering theory[END_REF] Chapter 8.9] or [5, Theorem 3.3], there exists a spectral shift function ξ(

. 25 )

 25 Proof. By[START_REF] Carron | Déterminant relatif et la fonction Xi[END_REF] Theorem 3.5] we have ξ(λ) = π -1 Arg det N( √ λ 2 + i0) as λ 2 → 0+, where Arg z ∈ (-π, π], and ξ(λ) = 0 if λ 2 < 0. Calculation of the argument in the asymptotic obtained in Propositon 1 gives (2.25).

2. 3 . 2

 32 The case K > 1 Let us outline the changes in Proposition 1 and Corollary 1 needed in the case K > 1. Now we have N + (λ) = ⊕ K j=1 N (j) + (λ), where each N (j)

Theorem 4 .

 4 The gluing formuladet ζ (∆, ∆) = C det * ζ N • det ζ ∆ D is valid, where N, ∆ D -depend on R.The constant C depends on R but not on the moduli parameters z 1 , . . . z M .

Lemma 2 .e

 2 Assume that K = 1. Then Tr e -t∆ -e -t ∆ = O (ln t) -2 as t → +∞. Proof. Since ∆ D + ≡ ∆D + and the operators ∆ D -, ∆D -are Dirichlet Laplacians on compact manifolds, we have Tr e -t∆ -e -t ∆ = Tr e -t∆ -e -t∆ D -⊕∆ D + -Tr e -t ∆ -e -t ∆D -⊕ ∆D + + Tr e -t∆ D --Tr e -t ∆D --tλ 2 (ξ(λ ; ∆, ∆ D ) -ξ(λ ; ∆, ∆D ))2λ dλ + O(e -tδ ), t → +∞, (2.29)

  and the same is true for ∆D -. Thanks to (2.31) and Proposition 2 we obtain det ζ (∆, ∆) det ζ ∆D -4, where N(0) is denoted by N and the constant

35 )from

 35 ℜs > 1. We have Tr e -t∆ -e -t ∆ = Tr e -t∆ -e -t∆ D -⊕∆ D + -Tr e -t ∆ -e -t ∆D -⊕ ∆D + +Tr e -t∆ D --Tr e -t ∆D -.

Definition 4 .

 4 det ζ (∆, ∆) := e -Z ′ 0 . Proof. Since ζ 2 (s; 0) is a meromorphic function of s with no pole at zero, as s → 0-we have ζ(s; ∆, ∆) = ζ 1 (s; 0) + ζ 2 (0; 0) + sζ ′ 2 (0; 0) + O(s 2 ). It remains to study the behaviour of ζ 1 (s; 0) = s ∞ 0 (-µ 2 ) -s-1 χ(µ)ξ(µ; ∆, ∆)2µ dµ as s → 0-. We represent the last integral as a sum of two integrals. Due to (2.36) the first integral s ∞ 0

Proposition 1 .

 1 The relative zeta regularized determinant det ζ (∆, ∆) and the zetaregularized determinant det * ζ ∆ m has the same variations with respect to moduli i. e. one has

  and b j W ( • , P ) = 2πiv j (P ).

H

  (x(P ), x(Q)) = 1 6 S(x(P )) + O(x(P ) -x(Q)), as Q → P , where S B (•) is the Bergman projective connection. Consider the Schiffer bidifferential

  (x, y) = ζ(y) -ζ(x) √ dζ(x) √ dζ(y) (1 + O((ζ(y) -ζ(x)) 2 ))as y → x, where ζ is a local parameter such that dζ = ω δ .

  k , p l ) = lim x→p k y→p l E(z(x), z(y)) dx k dζ (x) dx l dζ (y) .

1

 1 BZ+4K ζ ,Z W (ζ) 2/3 k<l E(p k , p l ) 6 -1 d k d l k E(ζ, p k ) 3 -1 (g-1)d k . (6.8)

12 . 2 . 4 k=1 4 k=1 1 r

 122441 Introduce the distinguished local parameter x k = √ z -z k at P k . Then for x k = 0 one hasdw dx k = 1 z ′ (w) 2x k = w -w k p ′ (w) -p ′ (w k ) 2x k w -w kPassing to the limit x k → 0, one gets 2 w ′ (x k ) R (f, g) is the resultant of polynomials f and g (since the τ -function is defined up to multiplicative constant, the power of 2 is omitted) anddet ζ (∆ |dp| 2 , ∆) = C|R (p ′ , p ′′ )| 1Example Let r is a rational function with three simple poles (which can be assumed to coincide with ∞, 0, 1,r(w) = aw -b w -c w -1 + d;i. e., r is an element of the Hurwitz space H 0,3 (1, 1, 1) of meromorphic functions on P 1 of degree three with three simple poles.Introducing the local parameter ζ = 1 z in vicinities of the poles w = 0 and w = 1 of the cover, one gets for ζ = 0w ′ (ζ) = -1 r ′ (w) r 2 (w)and, say, for w = 1w ′ (0) = -lim w→1 1 r ′ (w) r 2 (w) = -c .Analogously, w ′ (0) = -b at the pole w = 0. For the local parameter w = w/a (for which w(P ) = z(P ) + o(1) as P tends to the pole w = ∞ of the cover) one has w′ (0) = -b/a at the pole w = 0 and w′ (0) = -c/a at the pole w = 1. On the other hand writing r ′ (w) = f g , where f and g are two polynomials and introducing the local parameterx k = √ z -z k near the critical point w k of r (k = 1, 2, 3,[START_REF] Burghelea | Meyer-Vietoris type formula for determinants of elliptic differential operators[END_REF], one gets similarly to (6.12)w′ (x k ) = a -1 w ′ (x k ), C [w ′ (x k )| x k =0 ] 2 = ′′ (w k ) = R(f, g) R(f, f ′ )Calculating the resultants, one gets τ 24 = a 3 b 3 c 3 M(a, b, c) where M(a, b, c) = a 3 + b 3 + c 3 + 3a 2 b + 3a 2 c + 3b 2 a + 3b 2 c + 3c 2 a + 3c 2 b -21abc and det ζ (∆ |dr| 2 , ∆) = C|abc| 1/4 |M(a, b, c)| 1/12 .

  .8)Proof. We prove the formula for ∂ w λ, the proof is the same for ∂ w λ. First observe that since u is bounded we have lim Br ∆v∂ z u 0 + ∂ z v∆u 0 dz ∧ dz.By continuity, this equality persists for u 0 ∈ H 2 0 . It follows that for any u ∈ H 2 and any u 0 ∈ H 2 0 we have lim r→0 γr ∂ z u∂ z u 0 dz = 0.

	r→0 γr	u 2 dz = 0.
	Now, if u 0 is smooth and compactly supported away of p, using Stokes' formula, we have that for any v ∈ H 2
	γr 4 It follows that 1 ∂ z v∂ z u 0 dz =
	∂ w λ = lim r→0 γr

where the Λ are linear functionals on dom(∆ * 0 ) that vanish on H 2 0 and u 0 is in H 2 0 . Moreover, one has Λ a ν (u) = Λ h ν (u). For u, v ∈ dom(∆ * ) we define

The Green formula implies

The domain of the Friedrichs extension of ∆ is characterized by requiring that all the coefficients with the superscript "-" vanish; it follows from (3.1) that the linear functionals Λ a,h,0 ν are continuous over H 2 (X) and supported at P . It can be proved that any linear functional Λ that is continuous on H 2 (X) and supported at P can be written as a linear combination of the linear functionals Λ 0 , Λ a ν and Λ h ν . Finally, we notice that one has the following representation for the space H 2 0 (X):

Definition of the S-matrix

We follow [START_REF] Hillairet | Krein formula and S-matrix for Euclidean surfaces with conical singularities[END_REF], paying special attention to conjugations and normalizing constants.

In the following the symbols ♯ and ♭ are to be substituted by 0, h or a. When the superscript is 0 the subscript ν is 0, when it is a or h, ν = j ℓ where j ranges from 1 to ℓ -1.

We define

Observe that by definition the g-functions belong to H 2 , which makes the latter definition consistent when seeing Λ ♯ µ as a linear functional over H 2 . Since Λ ♯ µ also makes sense as a linear functional over ker(∆ * -λ), we may also write

Differentiation with respect to λ

We denote by a dot the differentiation with respect to λ. Differentiating the defining equation for g ♯ ν , we find that

From this relation we deduce the following proposition.

Proposition 5. For any λ, S aa (λ) and S hh (λ) are symmetric matrices and

Proof. The expression for Ṡ♯♭ µν yields that Ṡaa and Ṡhh are symmetric matrices. Since both tend to symmetric matrices (actually 0) as λ goes to -∞, the first part of the claim follows. In the same way we obtain Ṡah µν -Ṡha νµ = 0.

Since S ah and t S ha tend to the same diagonal matrix as λ goes to -∞, the second part of the claim also follows.

Combining the identities (4.1) and ( 4.3) we conclude that S ah is hermitian for λ real; actually it is an analytic family of hermitian matrices, meaning that S ah (λ) = S ah (λ) * .

Behavior for λ going to 0

The matrix S(λ) is well defined a priori only for λ in the resolvent set of ∆. However it is always possible to define the function f ♯ ν ( • ; λ = 0). Whenever ♯ = 0 the latter function is in the range of ∆. We can thus find solutions g ♯ ν (•; 0) to the equation

The latter solutions are defined only up to the addition of a constant. It follows that the definition of S ♯♭ µν (0) makes sense for ♯ = 0, ♭ = 0 and the following Proposition holds.

Proposition 2. For ♯ = 0 and ♭ = 0 the matrix-valued function λ → S ♯♭ (λ) extends holomorphically to a neighbourhood of 0. Moreover S ♯♭ (0) depends only on the conformal class of m.

Proof. For λ close to 0 we have

where X g ♯,⊥ ν = 0. Since λ → X f ♯ ν (λ; 0) = 0 is holomorphic and vanish at 0, we obtain that λ → G * ν can be holomorphically continued to a neighbourhood of 0. The first statement follows. The second statement follows by remarking that G ♯ ν is a function in dom(∆ * 0 ) such that ∆ * 0 G ♯ ν = 0 and the singular behaviour near P is prescribed. Both conditions are conformally invariant so that if we change the metric in its conformal class, we may only change G by adding a constant. This will not affect the coefficients in the S-matrix we are considering here.

Let (λ t , u t ) be an eigenbranch of q t relatively to n t , then λ ′ = d dt |t=0 λ is given by

where in the expression of ∂ w λ and ∂ w λ, u = u 0 is the eigenvector of the eigenbranch (λ t , u t ) at t = 0 .

Remark 12. We remind the reader of one subtlety of perturbation theory (see [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF], [START_REF] Kriegl | Differentiable perturbation of unbounded operators[END_REF]).

In case of a multiple eigenvalue λ 0 , for any family q t there are several eigenbranches emanating from λ 0 , and the initial corresponding eigenvectors may actually depend of the chosen family. In particular the expressions ∂ w λ and ∂ w λ also depend on the initial w that defines q t := q tw . In other terms, for any direction w it is possible to organize the spectrum into eigenvalues branches but it may not be possible to organize the eigenvalues as functions that are differentiable with respect to w varying in the ball.

Proof. We start with the one form

Since (λ, u) is an eigenpair of the Laplace operator we compute

We now use Stokes formula to obtain

γr

On the other hand, in (5.5) we use the formulas provided by Lemma 2 to obtain :

Comparing the two yields the first formula in (5.6). The second one follows either from the same computation or by complex conjugation.

Since ω u is closed in B(p, r 0 ) \ {p} we may let tend r to 0 in the preceding formulas. We thus obtained a formula for ∂ w λ that is expressed only through the asymptotic expansion of u near p.

Recall that by definition of the linear functionals Λ ♯ ν , we have in the local coordinate z the following expansion near p u(z)

with u 0 ∈ H 2 0 . By letting r go to zero we obtain the following lemma.

Differentiating with respect to s and setting s = 0 gives

The claim follows by applying Cauchy's theorem. Now, using Proposition 1, the preceding Proposition and Proposition 6 we arrive at the following corollary.

Corollary 3. Let P m be a zero of the meromorphic differential df of multiplicity ℓ m and let z m = f (P m ) be the corresponding critical value of f . Let also x m = (z -z m ) 1 ℓm+1 be the distinguished local parameter in a vicinity of P m . Then

(5.15)

6 Integration of the equations for ln Det and explicit expressions for the τ -function

Let, as before, B be the matrix of b-periods of the Torelli marked Riemann surface X and let {v α } α=1,...,g be the basis of the normalized holomorphic differentials on X. Using the Rauch formulas (see, e. g., [START_REF] Kokotov | Isomonodromic tau-function of Hurwitz Frobenius manifolds and its appplications[END_REF], [START_REF] Kokotov | Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray-SInger formula[END_REF], [START_REF] Kokotov | Tau-functions on Hurwitz spaces[END_REF]),

one immediately gets the relation

where the contour integrals are taken over a small contour on X encircling the point P m (in the positive direction). Now, using the relation (4.5), the equations (6.1) and (5.15) together with elementary properties of the Schwarzian derivative (see, e. g. [START_REF] Tyurin | Periods of quadratic differentials[END_REF]), we arrive at the following version of Corollary 3 rewritten in the invariant form. where S B is the Bergman projective connection and S f =

is the Schwarzian derivative.

Remark 14. Notice that the difference S B -S f is a quadratic differential and, therefore, the integrand in (6.2) is a meromorphic one form.