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Hardy Spaces on Compact Riemann Surfaces with Boundary | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Hardy Spaces on Compact Riemann Surfaces with Boundary

Zuevsky A*

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

Corresponding Author:
Zuevsky A
School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, Galway, Ireland
Tel: 3531439 2424
E-mail: [email protected]

Received date July 21, 2015; Accepted date August 02, 2015; Published date August 31, 2015

Citation: Zuevsky A (2015) Hardy Spaces on Compact Riemann Surfaces with Boundary. J Generalized Lie Theory Appl S1: 005. doi:10.4172/1736-4337.S1-005

Copyright: © 2015 Zuevsky A. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

We consider the holomorphic unramified mapping of two arbitrary finite bordered Riemann surfaces. Extending the map to the doubles X1 and X2 of Riemann surfaces we define the vector bundle on the second double as a direct image of the vector bundle on first double. We choose line bundles of half-order differentials Δ1 and Δ2 so that the vector bundle 2 χ2 2 V X ⊗Δ on X2 would be the direct image of the vector bundle 1 1 χ ⊗Δ2 V X . We then show that the Hardy spaces 2, 1( ) ( 1, χ1 1) H J p S V ⊗Δ and 2, 2 ( ) ( 2, χ2 2) H J p S V ⊗Δ are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation χ2 of the fundamental group π1(X2, p0) given a matrix representation χ1 of the fundamental group π1(X1, p'0). On the basis of the results of Alpay et al. and Theorem 3.1 proven in the present work we then conjecture that there exists a covariant functor from the category ï��ï�� of finite bordered surfaces with vector bundle and signature matrices to the category of KreÄ­n spaces and isomorphisms which are ramified covering of Riemann surfaces.

Keywords

Half-order differentials; Hardy spaces; Riemann surfaces

Introduction

It is well known how to study Hardy spaces defined on a finite bordered Riemann surface [1-5]. For domains with more than one boundary component it is natural to introduce, besides the usual positive definite inner product on H2, indefinite inner products. Those products may be introduced by picking up different signature matrices when integrating over different components of the boundary of the Riemann surface. In the paper Alpay et al. [1] a necessary and sufficient condition for such an indefinite inner product to be non-degenerate was obtained. It was shown that when this condition is satisfied one actually gets a Kreĭn space. The result was obtained by using a covering map of the surface to the unit disk to construct an isomorphism to a Hardy-Kreĭn space over the unit disk. Furthermore, each holomorphic mapping of the finite bordered Riemann surface onto the unit disk (which maps boundary to boundary) determines an explicit isometric isomorphism between this space and a usual vector-valued Hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix. The mapping to the unit disk in Alpay et al. [1] serves as a tool to study of the Hardy-Kreĭn space over the finite bordered Riemann surface which in turn has motivation from the point of view of the study of commuting tuples of non-self adjoint operators. As it is usual when studying Hardy spaces on a multiply connected domain, the elements of the space are sections of a vector bundle rather than functions. The main point of the paper by Alpay et al. [1] was to construct an appropriate extension of this bundle to the double of the finite bordered Riemann surface and to use Cauchy kernels for certain vector bundles on a compact Riemann surface. Hardy spaces on a finite bordered Riemann surface, including indefinite Hardy spaces, are important in the model theory for commuting non-self adjoint operators [6].

Half-order differentials play a very important role in the vertex opera-tor algebra approach to construction of partition and n-point functions for conformal field theories defined on Riemann surfaces [7-9]. In particular, the Szegö kernel [10] turned out to be key object in construction of correlation functions in free fermion conformal field theories/chiral algebras on a genus two Riemann surface sewed from two genus one Riemann surfaces [11].

In this work we replace a holomorphic mapping of a finite bordered Riemann surface onto the unit disk by a holomorphic mapping of two arbitrary finite bordered Riemann surfaces S1 and S2, which we assume however to be unramified. In the spirit of Alpay et al. [1,12] one can introduce the extension of the vector bundles on a finite bordered Riemann surfaces to the respective doubles. Extending the map F to the doubles X1 and X2 of Riemann surfaces S1 and S2 we define the vector bundle on X2 as a direct image of the vector bundle image over X1. We choose line bundles of half-order differentials (i.e., square roots of the canonical bundles image, i = 1, 2) Δ1 and Δ2 so that the vector bundle image on X2 would be the direct image of the vector bundle image . We then show that the Hardy spaces image bundle and image are isometrically isomorphic. Proving that we construct a.) an explicit isometric isomorphism; b.) a matrix representation χ2 of the fundamental group π1(X2, p0) given a matrix representation χ1 of the fundamental group π1(X1, p′0). Using results of Alpay et al. [1] and Theorem 3.1 proven in the present work we then conjecture that there exists a covariant functor from the category RH of finite bordered surfaces with vector bundle and signature matrices to the category of Kreĭn spaces and isomorphisms which are ramified covering of Riemann surfaces.

The isomorphism established in this work has also an operator theoretical interpretation, namely, a (ramified) covering F allows us to construct a pair of commuting non-self adjoint operators with the model space on S2 given a pair of commuting non-self adjoint operators with the model space on S1. More generally, one might expect also possible connections with vessels construction and Bezoutians [13].

Preliminaries

As we mentioned in Introduction indefinite Hardy spaces [14-16] on a finite bordered Riemann surface were considered in Alpay et al. [1].

Definition. Let J be an m x m unitary self-adjoint matrix. Such a matrix is usually called a signature matrix. (In fact one may take J to be any non-singular self-adjoint matrix). The Hardy space image on the unit disk D endowed with the indefinite inner product

image

is a Kreĭn space denoted image, J .

This space plays an important role in interpolation theory [17] and in model theory [18]. For the general theory of Kreĭn spaces[19-21].

Suppose now that we have an open Riemann surface S such that S ∪ ∂S is a finite bordered Riemann surface (i.e., a compact Riemann surface with boundary), with the boundary ∂S consisting of k ≥ 1 componentsimage . We consider analytic sections of a rank m flat unitary vector bundle Vχ on S corresponding to a homomorphism χ from the fundamental group π1(S, p0) into the group U(m) of m × m unitary matrices. An analytic section f of Vχ over S is an analytic Cm valued function on the universal covering image of S satisfying

image

for all image and all deck transformations T of image over S, which we identify with elements of the fundamental group π1(S, p); f can be thought of as a multiplicative multivalued function on S. We consider also multiplicative half-order differentials [1], i.e., sections of a vector bundle of the form image ,where Vχ is a flat unitary vector bundle on S as above and Δ is a square root of the canonical bundle onimage

Definition. The Hardy space image on a Riemann surface S is the set of sections imageof a vector bundle image analytic in S satisfying

image(1)

for some ϵ > 0. In (1) image(r) denotes smooth simple closed curves in S approximating image, i = 0, ..., k - 1 ([1]): if zi is a boundary uniformizer near the boundary component image then image (r) is given by |zi(p)| = r.

Note that since image is a section of image, the expression image is a section of |KS|, where |KS| is the line bundle with transition functions the absolute values of the transition functions of KS ; sections of |KS| can be represented locally as η(t)|dt(p)| where t(p) is a local parameter. Therefore one can integrate image over curves in S and (1) makes sense.

The space image is a Hilbert space with the inner product

image

For a relation between image and Hardy spaces of functions on S with respect to a harmonic measure on ∂S, see Alpay et al. [1].

Definition. Denote by image the analogue of the Kreĭn space image for S which is the Hardy space image endowed with the indefinite inner product

image(2)

where image is the non-tangential boundary values of image which exists again almost everywhere on ∂S (Alpay et al.[1]) and image is a locally constant matrix function on image whose values are m x m signature matrices, satisfying

image (3)

for all image and all image. The expression image in (2) means image where image is over image. It is a welldefined section of |KS| because of the transformation property of image. There exists certain freedom in the choice of image for the given Vχ . Indeed, choose points image. Let Ci be a path on S linking p0 to pi. Set image (see Appendix). Then the (homotopy class of) Ci determines a component image of image lying over image , and the constant value Ji of image on image can be an arbitrary m x m signature matrix satisfying

image

Any other component of image lying over image can be obtained from image by some deck transformation R. The value of imageon this component is image. For the case of the line bundles (i.e., m = 1), the choice of image amounts to an arbitrary choice of a sign ±1 for each image . We will often assume the choice of components image has been made and denote image by image The space image is a natural example of an indefinite inner product space. It is related to the model theory of pairs of commuting non-selfadjoint operators and interpolation theory on multiply connected domains.

In the paper Alpay et al. [1] an appropriate extension of Vχ on S to the double X of the Riemann surface S was constructed. Given a flat unitary vector bundle on a finite bordered Riemann surface, together with a collection of signature matrices, it can be uniquely extended to a flat unitary vector bundle on the double satisfying certain symmetry properties. Let us recall that construction.

Due to the identification of the boundaries the complex structures on two copies of S constituting X are mirror images of each other, i.e., there exists an anti-holomorphic involution τ : X → X that maps image to image . Thus X is a compact real Riemann surface, or equivalently Riemann surface of a real algebraic curve. The genus g of the double of X of S is g = 2s + k − 1, where s is the genus of S. The set Xf of fixed points of τ (real points of X) coincides with the boundary ∂S of S. Furthermore X is a real Riemann surface of dividing type: the complement X \ Xf consists of two connected components X+ = S and X− = S′ interchanged by τ. The converse is also true: any real Riemann surface of dividing type is the double of a finite bordered Riemann surface. The anti-holomorphic involution τ acts both on the fundamental group π1(X, p0) and on the universal coveringimage of X (recall that the fundamental group π1(X, p0) is isomorphic to the group of deck transformations Deck ( image /X). It also acts naturally on complex holomorphic vector bundles on X: the transition functions for the vector bundle V τ complex conjugates of the transition functions for V at the point conjugate under τ.

Consider a vector bundle H on X of rank m with deg H = m(g-1) satisfying the condition h0(H) = 0. Such a vector bundle is necessarily of the form image where Vχ is a rank m flat vector bundle on X and Δ is a square root of KX [22]. These vector bundles are closely related to determinantal representations of algebraic curves and play an important role in the theory of commuting non-self adjoint operators and related theory of 2D systems [22-26].

Let H be such that there exists a non-degenerate bilinear pairing H × Hτ → KX which is parahermitian. The parahermitian property means that

image

for all local holomorphic sections image and image of H near p and pτ respectively. We assume also that the line bundle Δ has been chosen so that image and that the transition functions of Δ are symmetric with respect to τ [27]; Then

we obtain a parahermitian non-degenerate bilinear pairing image or more explicitly an everywhere nonsingular holomorphic m x m matrix–valued function G on the universal covering image with the property

image(4)

satisfying the relation

image(5)

where T ∈ π1(X, p0). The pairing H × Hτ → KX is then given explicitly by

image

Now let us introduce the (in general) indefinite inner product

image(6)

where image and image are measurable sections of H over Xf . Here and in similar expressions, the integral is computed on X and the integrand does not depend on the choice of image above p ∈ X Since in (6) image lies over a point of Xf there exists image such that image . Therefore (6) can be rewritten as

image(7)

Where

image

Note that image and

image(8)

for all image over Xf and all R ∈ π1(X, p0). Thus the vector bundle image on X defines an indefinite inner product on the sections of its restriction to Xf = ∂S.

For image lying over a point of image , we have image for i = 0 and image for i = 1, . . . , k − 1 where imageare part of the generators of the fundamental group π1(X, p0) of X (refer Appendix for the relation between generators of the fundamental groups of a Riemann surface S and the corresponding double), and R depends only on the component of the inverse image of image in image that image belongs to. Restricting image in (7) to belong to a specific component we may write

image

for i = 0 and

image for i = 1, . . . , k − 1 (refer Appendix). (The specific component depends on the choice of the generators Bi, i.e., on the homotopy classes of the paths Ci.)

It follows from the conditions deg H = m(g −1) and h0(H) = 0 that H is a semi-stable vector bundle. By a theorem of Narasimhan and Seshadri [28] H is a direct sum of stable bundles if and only if the flat vector bundle Vχ (in image) can be taken to be unitary flat. Since G is an isomorphism from Vχ to the dual of image it follows in this case that G is constant and unitary. Since it is also selfadjoint, it is a constant signature matrix. Thus for analytic sections image and image of image on S that belong to image we can rewrite the inner product (7) as

image(9)

where

J0 = G, Ji = χ(Bi)*G,

for i = 1, . . . , k−1 and p is restricted to belong to a specific component of the inverse image of χi in image as explained above. We then obtain for the vector bundle H on X the inner product (2) on the Hardy space image. Conversely, every unitary flat vector bundle on S with signature matrices J0, . . . , Jk−1 can be obtained from a vector bundle H on X as above. Let Ji, i = 0,..., k − 1 be self adjoint matrices and let image be a homomorphism satisfying

image

Then by Proposition 2.1 from Alpay et al. [1] there exists a unique extension (still denoted by χ) of χ to a homomorphism from π1(X, p0) into GL(m, image) satisfying

image

where G = J0 (refer Appendix).

If the original flat vector bundle Vχ on S is unitary flat and all the matrices Ji are unitary then the extended vector bundle is also unitary flat as it is follows form the proof of Proposition 2.1 of Alpay et al. [1]. The extension need not satisfy image; i.e., the unitary case, this condition will be satisfied ”generically” since flat unitary vector bundles Vχ on X with image form a divisor in the moduli space of flat unitary vector bundles (the generalized theta divisor [10,29]). It was proven in Proposition 2.2 from Alpay et al. [1] that if the indefinite inner product space imageis nondegenerate, then

image(10)

It follows that the condition (10) is satisfied automatically in the positive definite case (i.e., when Ji > 0 for i = 0, . . . , k − 1); for line bundles, this has been obtained in Fay et al. [27,30].

Summing up, we see that the above extension procedure establishes a one-to-one correspondence between unitary flat vector bundles on S together with a choice of signature matrices satisfying (3) and unitary flat vector bundle on X satisfying the symmetry condition (4), (5). Given a unitary flat vector bundle on S, the various choices of extension to the double X correspond to the various choices of signature matrices. We shall occasionally denote the corresponding unitary flat vector bundles on S and X by image and image respectively.

Under the condition image (i.e., that image has no global holomorphic sections), it turns out that image on X admits a certain kernel function (which is called the Cauchy kernel) which is an analogue of image for the trivial bundle on the complex plane. The Cauchy kernel is the reproducing kernel for image. In the case of line bundles the Cauchy kernel can be given explicitly in terms of theta functions [27,31]. In Alpay et al. [1] the Cauchy kernel was used to construct for any given holomorphic mapping z : S → D an explicit isometric isomorphism between image and image for appropriate M and J. In particular this implies that image is indeed non-degenerate (under the condition image) and actually a Kreĭn space.

Statement of the Main Result

Suppose that we have two finite bordered Riemann surfaces S1 and S2. Let F: S1 → S2 be an analytic mapping continuous up to the boundary.

Equivalently we may take F to be a complex analytic mapping F: X1 → X2 between the doubles of S1 and S2 equivariant with respect to the action of the anti-holomorphic involutions, i.e., such that the diagram

image

is commutative. Notice that F: S1 → S2 is unramified if and only if F: X1 → X2 is unramified.

We identify as usual a complex holomorphic vector bundle on a complex manifold with a locally free sheaf of its analytic sections. It is easily seen that if VX is a complex holomorphic vector bundle of rank m on a complex manifold X and F is a n-sheeted unramified covering, then the direct image VY = F*VX is a complex holomorphic vector bundle of rank nm on Y and the fiber of VY at a given point of Y is the direct sum of the fibers of VX at the preimages of this point on X.

The main statement of this work is the following

Theorem 3.1 Let F: S1 → S2 be a map of finite bordered Riemann surfaces which is a finite n-sheeted unramified covering (F ; S1, S2), and let image be signature matrices for a unitary flat vector bundle imageon S1 of rank m. Consider the corresponding extension of image to the double X1 of S1 satisfying the symmetry condition

image

for all R ∈ π1(X1, p′0) and all image . Choose the bundles Δ1 and Δ2 of half-order differentials on X1 and X2 respectively, such that

a.) the bundles Δi, i = 1, 2 are invariant with respect to the corresponding anti-holomorphic involutions, i.e., image and the transition functions of Δ1 and Δ2 are symmetric with respect to τ1 and τ2;

b.) the pull–back of Δ2 is equal to Δ1, i.e., Δ1 = F*Δ2.

Then

1) the direct image image is a unitary flat holomorpnic vector bundle of rank nm satisfying the symmetry condition

image

for all T ∈ π1(X2, p0) and all image , appropriate matrix function image , and representation χ2 of π1(X2, p0); furthermore image;

2) there exists a canonical isometric isomorphism

image

between Hardy spaces on S1 and S2.

Now some remarks are in order. By definition the anti-holomorphic involutions τ1 and τ2 are related by

F ○ τ1 = τ2 ○ F,

and therefore if we have a line bundle L2 on X2 then its pull-back satisfies

image

We fix Δ2 such that image and image . Then it follows that image . We choose Δ2 such that its transition functions are symmetrical. Then since F is equivariant with respect to the antiholomorphic involution then transition functions of Δ1 are also symmetrical.

The isomorphism of the spaces image and image implies that they are degenerate or nondegenerate simultaneously, i.e., image which is obvious from the definition of the direct image vector bundle.

We assume that the map F: S1 → S2 is a n-sheeted unramified covering (F; S1, S2) of the Riemann surface S2 by S1. On the other hand, a result of Alpay et al. [1] mentioned in Introduction is a construction of an isometric isomorphism between Hardy spaces when S2 = D but F is (usually) ramified (assumingimage is not degenerate, i.e., image. The next natural step would be to consider the case when S2 is an arbitrary finite bordered Riemann surface and F is a ramified covering. That will be a point of some further publication.

We have introduced the vector bundle image on the double X2 as the direct image of the vector bundle image on X1 defining the vector bundle image on S2 and the signature matrices image . On the other hand one can define the vector bundle image to be the direct image image of the vector bundle image with signature matrices defined naturally in terms of image (as direct sums). Though the main claim of Theorem 3.1 is formulated for finite bordered Riemann surfaces it seems to us that the consideration of the structures involved in its proof is more natural (in the sense of the theory of compact Riemann surfaces) on the doubles. Furthermore, this approach allows us to construct a matrix representation image of the fundamental group π1(X2, p0) given a representation of π1(X1, p′0), and the matrix function image. We will prove that signature matrices image calculated with the help of the representation χ2 do coincide with the signature matrices constructed directly from the signature matrices image. This shows the equivalence of those two approaches. From the use of Cauchy kernels in Alpay et al. [1] it seems however that in the ramified case the approach via the doubles is the only one possible.

Speaking in more abstract terms we deal in Theorem 3.1 with a category which we will denote by image. Objects of image are finite bordered Riemann surfaces S together with a unitary flat vector bundle Vχ and signature matrices image (or equivalently, compact real Riemann surfaces X of dividing type with a vector bundle image imageon X and a matrix functionon X and a matrix function image satisfying (4) and (5)) such that the space image is non degenerate, i.e., image . A morphism between the objects image and image of image is an analytic map F : X1→X2 of Riemann surfaces which is equivariant with respect to anti-holomorphic involutions τ1 and τ2, such that image (and image is correspondingly induced by image).

We conjecture that there exists a covariant functor from the above men-tioned category image to the category of Krein spaces and isomorphisms, asso-ciating to image the Hardy space image.

Theorem 3.1 proves the conjecture for a subcategory of image whose morphisms are unramified coverings. The isometric isomorphism established in Alpay et al. [1] proves another special case of the conjecture namely for a subcategory whose morphisms are restricted to have the unit disk D as a range. Somewhat related considerations of categories of functional spaces on Riemann surfaces are contained in Alling et al. [32].

Sections of the Vector Bundle image

In this section we give an explicit construction of a holomorphic section image of the bundle imageon X2 in terms of a holomorphic section image of imageon X1. When (F; S1, S2) is an unramified covering the doubles X1 and X2 possess the common universal covering image , i.e., one has a diagram

image

where π1 and π2 are the covering maps from image to X1 and X2 respectively.

Let U′ ⊂ X1 be an open set in X1. Suppose f1 is an analytic section of the holomorphic vector bundle image over U′, i.e., an analytic image valued function on image satisfying the relation

image(11)

Similarly, a section imageof the vector bundle image over U′ satisfies

image

for all image, imagewhere t1 is a local parameter on X1 lifted to image . The fundamental group image, is a subgroup of π1(X2, p0) of index n (here p0′ is a preimage of p0 ∈ X2). Enumerate fixed representatives gi, i = 1,..., n of the left cosets {π1(X1, p0′ )gi} of the group π1(X2, p0) with respect to its subgroup π1(X1, p′). We define a sheaf on X2 whose sections over an open set U⊂ X2 are analytic image valued functions on image of the vector form

image

i = 1,…, n, where f1 is a section of the bundle image over F-1(U), i.e.,image is an analytic image-valued function on image satisfying (11). It easy to see from the definition that this sheaf on X2 is isomorphic to the direct image sheaf of a sheaf on X2 of analytic sections of image , i.e., (12) defines the sheaf of analytic sections of F*image.

Now let p ∈ X2 and p′1, ..., p′n ∈ X1 be preimages of p. Let t2 and t1,i be local parameters near p and p′i, i = 1, ..., n lifted to the common universal covering image . Denote by φi the composition image . Then a section imageof a vector bundle imageis given by

image (13)

where image is a section of the vector bundle image and t1,i, i = 1, ..., n are local parameters in the vicinity of image . Since we have chosen the bundles Δ1 and Δ2 of half-order differentials in (13), the ambiguity in the sign of the square roots of image in (13) is global and since we have assumed that Δ1 = F *Δ2, the expression (13) does not depend on the choice of local parameters.

Representation χ2 of π1(X2, p0)

In this section we give an explicit formula for a unitary representation χ2 of 2 π1(X2, p0) such that image . It follows from the previous section that we have to define χ2 so that

image (14)

for every f2 given by (12). Let g ∈ π1(X2, p0). Fix a preimage p′0 ∈ X1 of p0. The element g belongs to a coset of the fundamental group π1(X2, p0) with respect to its subgroup π1(X1, p′0). Then there exist elements h ∈ π1(X1, p′0) and image such that

image

i.e., g defines a permutation σg of the preimages of p0. We take this as a definition of σg. We define the matrix representation χ2 as follows:

image (15)

It is immediate that (14) is verified. Taking into account the unitarity of χ1(g) , it can be seen from (15) that the matrices defining the representation of π1(X2, p0) are unitary, i.e.,

image

Now we check that (15) provides a representation π1(X2, p0), i.e.,

image

for all image Proving this we used the fact that χ1 is homomorphism and σg is an anti-homomorphism, i.e.,

image

which can be easily verified. In general, the matrix χ2 is given by the formula

image

where g belongs to i-th coset.

Construction of Pairing and Inner Product

Suppose that H1 = image is such that there exists a non-degenerate bilinear pairing imagewhich is parahermitian, i.e.,

image

We assume that the line bundle Δ1 is such that imageand the transition functions of Δ are symmetric with respect to τ1. Then we have a parahermitian non-degenerate bilinear pairing image and the matrix function image satisfying (4) and (5). One can define a bilinear non-degenerate pairing image where image introducing an everywhere nonsingular holomorphic mn × mn matrix-valued function G2 on the universal covering image of X1 and X2. The matrix image should have the property

image (16)

and satisfy the symmetry condition, T ∈ π1(X2, p0) ,

image (17)

Then the pairing is given by

image

Taking into account the explicit form (15) of χ2(g) one can check that the following expression for image does satisfies (17)

image (18)

where ν(k) is defined as follows. Consider the action of τ on an element g ∈ π1(X2, p0). By definition we have gτ = τgτ−1. For any gk that belongs to k-th coset of π1(X2, p) with respect to π1(X1, p′0) there exist hk ∈ π1(X1, p′0) and gν(k) ∈ π1(X2, p) such that

image (19)

We define ν(k) by (19). One can check directly that (18) does satisfy conditions (16) and (17).

We saw in Introduction how to define an indefinite inner product (2), (9) on the Hardy space image using signature matrices image. Suppose that we have such an inner product on image

image

Then we define an indefinite inner product on image

image(20)

By the same reasons as in Introduction we can rewrite (20) as

image(21)

Where

image

and introduce the matrices

image(22)

where image(refer Appendix). As in Alpay et al. [1] the extension of the bundle imageon the Riemann surface S2 to the double X2 depends on the choice of the signature matrices J2,0, ..., J2,k−1 given by (22) and which satisfies the symmetry condition (8). On the other hand, one can define the signature matrix image using the signature matrix image . One should have

image(23)

for all T ∈ π1(X2, p) and

image(24)

for all image over p ∈ X2,f. The matrix image in the form

image (25)

satisfies (23) and (24). Then we check the commutativity of the diagram

image

Where ext, Ji means the extension of the vector bundle image on Si to the double Xi . I.e., we will show that the matrix image defined by (22) coincides with (25). It easy to check that

image

for all R ∈ π1(X2, p), p ∈ X2,f and imagesuch that image where image lies over p. Using that we arrive at

image

Proof of the Isometricity

We have constructed explicitly a section image of the bundle image in terms of a section image of the bundle image. Now we will prove that the map image is an isometric isomorphism of the space image on the space image . First let us show that image if and only if image.

Suppose image is a section of the bundle image and image. That means that image , i.e., image is an analytic in X1

And

image

for some ϵ. Here image are smooth simple curves in X1 approximating the i-th boundary of the X1. The space image is the space image endowed with the indefinite inner product (2)

image

Let image be a boundary component of X2 and imagej = 1, ..., ni be corresponding preimages on X1. The boundary uniformizer z2 near the boundary component is such that z2p0′ = z2°Fp0′ . Then the approximating curves image are mapped to the approximating curves image. Due to the construction given by the formula (13) we see that image is an analytic and

image(26)

The summation in (26) with upper limits ni is performed over the components image that are preimages of image. Thus we infer that image belongs to the space image In the previous section we have introduced an indefinite inner product in the space image Thus we see that a section imageof xconstructed by the formula (13) belongs to the space image

Finally, it remains to show that the inner product (21) is isometric, i.e., that

image(27)

where imageand image are sections of the vector bundles image and image respectively. Indeed, consider the inner product of two sections of the bundle image

image

By the same argument that were used in the formulae (26) the last integral is equal to

image

where we use the invariance of sections of Δ1 with respect to deck transformations and the symmetry of the their transition functions. Hence we see that (27) holds. That completes the proof of the isometricity.

Appendix: Fundamental groups of S and Double X

Let us describe explicitly [1] the action of τ on the generators of π1(X, p0). Choose points image and let Ci be a path on S linking p0 to pi. Then π1(S, p0) is generated by

image(28)

where image for j = 1,…,k - 1 imagerepresent a canonical homology basis on S with the intersection matrix

image. The generators of π1(S, p0) satisfy a single relation

image

Now consider the fundamental group π1(X, p0). It is generated by

image

The generators Aj , A′i, Bi′ are the same as in (28)

image

for j = 1,..., k − 1 and

image

The generators of π1(X, p0) satisfy a single relation by Natanzon et al. [33-41]

image

Note that

image

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