Guillaume Noyel

Michel Jourlin

Spatio-colour Asplünd's metric and Logarithmic Image Processing for Colour Images (LIPC)

Keywords: Asplünd's metric, spatio-colour metric, colour Logarithmic Image Processing, double-sided probing, colour pattern recognition

Asplünd's metric, which is useful for pattern matching, consists in a double-sided probing, i.e. the over-graph and the sub-graph of a function are probed jointly. This paper extends the Asplünd's metric we previously defined for colour and multivariate images using a marginal approach (i.e. component by component) to the first spatiocolour Asplünd's metric based on the vectorial colour LIP model (LIPC). LIPC is a non-linear model with operations between colour images which are consistent with the human visual system. The defined colour metric is insensitive to lighting variations and a variant which is robust to noise is used for colour pattern matching.

Introduction

The Asplünd's metric initially defined for binary shapes [START_REF] Asplünd | Comparison between plane symmetric convex bodies and parallelograms[END_REF][START_REF] Grünbaum | Measures of symmetry for convex sets[END_REF] has been extended to grey-scale images by Jourlin et al. [START_REF] Jourlin | Chapter 7 -Logarithmic image processing: Additive contrast, multiplicative contrast, and associated metrics[END_REF][START_REF] Jourlin | Asplünd's metric defined in the logarithmic image processing (LIP) framework: A new way to perform double-sided image probing for non-linear grayscale pattern matching[END_REF] and to colour and multivariate images in the LIP framework by Noyel et al. [START_REF] Noyel | Asplünd's metric defined in the logarithmic image processing (LIP) framework for colour and multivariate images[END_REF]. It consists in probing a function by two homothetic template functions, i.e. the probes which are computed by the LIP multiplication.

The Logarithmic Image Processing (LIP) model initially defined for grey level images by Jourlin et al. [START_REF] Jourlin | A model for logarithmic image processing[END_REF][START_REF] Jourlin | Logarithmic image processing: The mathematical and physical framework for the representation and processing of transmitted images[END_REF] is perfectly suited for images acquired by transmitted light (i.e. when the observed object is located between the source and the sensor) and by reflected light because of its consistency with the Human Vision [START_REF] Brailean | Evaluating the EM algorithm for image processing using a human visual fidelity criterion[END_REF]. The necessity to analyse together the channels of the colour images (i.e. by a vectorial analysis) has led to the introduction of the Logarithmic Image Processing for Colour images (LIPC) by Jourlin et al. [START_REF] Jourlin | Chapter 2 -Logarithmic image processing for color images[END_REF].

The LIP Asplünd's metric was defined in [START_REF] Noyel | Asplünd's metric defined in the logarithmic image processing (LIP) framework for colour and multivariate images[END_REF] in a marginal way (i.e. channel by channel). In this paper, our contribution is to extend this metric by using the spatio-colour properties [START_REF] Noyel | Morphological segmentation of hyperspectral images[END_REF][START_REF] Noyel | Multivariate mathematical morphology for DCE-MRI image analysis in angiogenesis studies[END_REF] of the colour LIPC framework.

After some prerequisites about the colour LIPC model and about the marginal LIP Asplünd's metric, we will define a spatio-colour Asplünd's metric in the LIPC framework. Then we will perform spatio-colour pattern matching which is robust to noise. Examples will illustrate the definitions.

Prerequisites

LIPC model

A colour image f , defined on a domain D ⊂ R N , with values in T 3 = [0, M [ 3 , M ∈ R, is written: f : D → T 3 = [0, M [ 3 x → f (x) = (f R (x), f G (x), f B (x)) (1)
f R , f G , f B are the red, green and blue channels (i.e. components) of f , f (x) is a vector-pixel and x is the spatial coordinate of the vector-pixel. The real value M is equal to 2 8 = 256 for 8 bits images. Given P the number of pixels, the matrix F of E → T , E = 3 × P , associated to the image f is written:

F =   f R (x 1 ) f R (x 2 ) ... f R (x P ) f G (x 1 ) f G (x 2 ) ... f G (x P ) f B (x 1 ) f B (x 2 ) ... f B (x P )   (2)
To make the comments easier, the word "image" designates both the matrix F and the image f . The image space for 24-bits images F is written I 3 . A colour image is a particular case of a multivariate image defined as f λ : D → T L , where L ∈ N is the number of channels [START_REF] Noyel | Morphological segmentation of hyperspectral images[END_REF][START_REF] Noyel | Multivariate mathematical morphology for DCE-MRI image analysis in angiogenesis studies[END_REF].

As for the grey-level LIP, the colour LIPC framework is based on colour transmittance [START_REF] Jourlin | Chapter 2 -Logarithmic image processing for color images[END_REF]. It is valid for transmitted and reflected images [START_REF] Brailean | Evaluating the EM algorithm for image processing using a human visual fidelity criterion[END_REF]. It models the human perceptual system approach by taking into account: i) the sensitivity of the human eye in the visible domain characterised by colour matching functions of [START_REF] Stiles | colour-matching investigation: Final report[END_REF] [START_REF] Stiles | colour-matching investigation: Final report[END_REF] and ii) the spectral distribution of light with the D65 illuminant [START_REF] Schanda | Colorimetry: Understanding the CIE System[END_REF].

In the LIPC framework, the transmittance of the sum of two images T F + c G is equal to the product of their transmittances T F and T G : T F + c G = T F * T G . The symbol of the LIPC addition is + c and * represents the element-wise multiplication [START_REF] Jourlin | Chapter 2 -Logarithmic image processing for color images[END_REF]. The addition of two images F, G ∈ I 3 is:

F + c G = Ḱ-1 Ú( Ú-1 ḰF * Ú-1 ḰG). (3)
Ḱ and Ú are real matrices of size 3 × 3 corresponding to the LIPC mixing model 3 . From the LIPC addition, a multiplication by a scalar α ∈ R has been defined:

α × c F = Ḱ-1 Ú( Ú-1 ḰF) α . ( 5
)
3 With colour matching functions of [START_REF] Stiles | colour-matching investigation: Final report[END_REF] and D65 illuminant [START_REF] Jourlin | Chapter 2 -Logarithmic image processing for color images[END_REF], matrices Ḱ and Ú equal to: The space (I 3 , + c , × c ) is the positive cone of a vector space with robust mathematical properties.

Ú =   25
Physical interpretation [START_REF] Jourlin | Chapter 2 -Logarithmic image processing for color images[END_REF]: the LIPC addition corresponds to the superposition of two semi-transparent layers. A LIPC multiplication by a scalar α ∈]0, 1[ brightens the result by suppressing layers, while a scalar α ∈]1, +∞[ darkens the result by superimposing α times the image on itself.

Marginal Asplünd's metric for colour and multivariate images

In [START_REF] Noyel | Asplünd's metric defined in the logarithmic image processing (LIP) framework for colour and multivariate images[END_REF], an Asplünd's metric between colour images was defined with the LIP model by using a marginal approach (i.e. channel by channel) [START_REF] Noyel | Morphological segmentation of hyperspectral images[END_REF][START_REF] Noyel | Multivariate mathematical morphology for DCE-MRI image analysis in angiogenesis studies[END_REF] . Definition 1. The Asplünd's metric (with LIP multiplication) between two colour images f and g on a region

Z ⊂ D is d × As,Z (f , g) = ln(λ/µ) ( 6
)
with λ = inf {k, ∀x ∈ Z, k × g R (x) ≥ f R (x), k × g G (x) ≥ f G (x), k × g B (x) ≥ f B (x)} and µ = sup {k, ∀x ∈ Z, k × g R (x) ≤ f R (x), k × g G (x) ≤ f G (x), k × g B (x) ≤ f B (x)}.
In particular, by the property of the distance

d × As,Z (f , g) = d × As,Z (g, f ).

Asplünd's metric defined in the Logarithmic Image Processing Colour (LIPC) framework

Given two colours

C 1 = (r 1 , g 1 , b 1 ), C 2 = (r 2 , g 2 , b
2 ) ∈ T 3 , as we are only looking for lower and upper bounds, a marginal order [START_REF] Barnett | The ordering of multivariate data[END_REF] is used:

C 1 ≥ C 2 ⇔ {r 1 ≥ r 2 and g 1 ≥ g 2 and b 1 ≥ b 2 }.
Definition 2. Given two colours C 1 , C 2 ∈ T 3 , their Asplünd's distance (with LIPC multiplication) is equal to:

d × c As (C 1 , C 2 ) = ln(µ/λ) (7) λ = inf k {k × c C 2 ≥ C 1 } and µ = sup k {k × c C 2 ≤ C 1 }. Strictly speaking, d × c
As is a metric if the colours C n are replaced by their equivalence classes Cn

= C ∈ T 3 /∃α ∈ R + , α × c C = C n }.
Comment: in eq. 7 contrary to the Asplünd's distance (with LIP multiplication) defined in [START_REF] Noyel | Asplünd's metric defined in the logarithmic image processing (LIP) framework for colour and multivariate images[END_REF] (eq. 6), we have λ ≤ µ because, by definition of the colour LIPC model the scales are inverted as compared to the grey LIP model [START_REF] Jourlin | Chapter 2 -Logarithmic image processing for color images[END_REF].

Colour metrics (with LIPC multiplication) between two colour images f and g may be defined as the sum (d

1 metric) or the supremum (d ∞ ) of d × c As (C 1 , C 2 ) on the region of interest Z ⊂ D of cardinal #Z d × c 1,Z (f , g) = 1 #Z x∈Z d × c As (f (x), g(x)) d × c ∞,Z (f , g) = sup x∈Z d × c As (f (x), g(x)) (8)
The Asplünd's metric can be extended to colour functions. Definition 3. The colour Asplünd's metric (with LIPC multiplication) between two colour images f and g on a region Z ⊂ D is

d × c As,Z (f , g) = ln(µ/λ) (9) λ = inf k {∀x ∈ Z, k × c g(x) ≥ f (x)} and µ = sup k {∀x ∈ Z, k × c g(x) ≤ f (x)}.
In fig. 1, the Asplünd's metric has been computed between the colour probe g and the colour function f on their definition domain D. As,D (f , g) = 0.43. Each colour channel is represented by a line of the same colour.

Comment: the lower (resp. upper) bound µ × c g (resp. λ × c g) may not be equal to any point of the function f but strictly less (or greater) than the function. Indeed, one can demonstrate that the following assertion is verified:

"given C 0 ∈ T 3 , ∀C ∈ T 3 , ∃λ ∈ R + /λ × c C 0 = C". The metric d × c
As,Z can be adapted to local processing with a colour template image (i.e. the probe) t defined on a spatial support D t ⊂ D. For each point x ∈ D, the distance d

× c
As,Dt (f |Dt(x) , t) is computed on the neighbourhood D t (x) centred in x where f |Dt(x) is the restriction of f to D t (x). Definition 4. Given a colour image f defined on D with values in T 3 , T 3 D , a colour probe t defined on D t with values in T 3 , T 3 Dt , and D t (x) the neighbourhood D t centred in x ∈ D, the map of Asplünd's distances (with × c ) is:

As × c t f : T 3 D × T 3 Dt → (R + ) D (f , t) → As × c t f (x) = d × c
As,Dt (f |Dt(x) , t)

In figure 2, the map of Asplünd's distances is computed between a colour function and a colour probe. The minima of the map corresponds to the location of a pattern which is similar to the probe.

Asplünd's distance is sensitive to noise because the probe lays on regional extrema that may be caused by noise (Figure 1). In [START_REF] Jourlin | Asplünd's metric defined in the logarithmic image processing (LIP) framework: A new way to perform double-sided image probing for non-linear grayscale pattern matching[END_REF][START_REF] Noyel | Asplünd's metric defined in the logarithmic image processing (LIP) framework for colour and multivariate images[END_REF], definitions of Asplünd's distance with a tolerance on the extrema have been introduced. In this paper, we extend this definition for colour images with LIPC model. To reduce the sensitivity of Asplünd's distance to the noise, the "Measure metric" or "M-metric" has been defined in the context of "Measure Theory". The image being digitized, the number of pixels of D is finite and the "measure" of a subset of D is linked to the cardinal of this subset, e.g. the percentage P of its elements with respect to D. We are looking for a subset D of D, such that f |D and g |D are neighbours for Asplünd's metric and the complementary set D \ D of D into D is of small size when compared to D. This last condition is written as:

P (D \ D ) = #(D\D ) #D
≤ p, where p is an acceptable percentage and #D is the number of elements in D.

Given a small positive real number, the neighbourhood of function f is

N P,d As , ,p (f ) = g \ ∃D ⊂ D, d × c As,D (f |D , g |D ) < and #(D \ D ) #D ≤ p (11)
The closest points of the probe to the function are discarded as in [START_REF] Jourlin | Chapter 2 -Logarithmic image processing for color images[END_REF][START_REF] Noyel | Asplünd's metric defined in the logarithmic image processing (LIP) framework for colour and multivariate images[END_REF].

Definition 5. Given two constant vector-pixels c µ , c λ ∈ T 3 , a percentage p of points to be discarded. The colour Asplünd's metric (with LIPC multiplication) with tolerance between two colour images f and g on a region Z ⊂ D is c λ are increased such as a percentage p of points is discarded.

d × c As,Z,p (f , g) = ln(µ /λ ) (12)
λ = inf k {∀x ∈ Z, k × c g(x) ≥ f (x) -c λ } and µ = sup k {∀x ∈ Z, k × c g(x) ≤ f (x) + c µ }. c µ and
In figure 3, a tolerance of p = 20% is used to discard two points. The Asplünd's distance decreases from 0.43 to 0.21.

A map of Asplünd's distances (with × c ) can now be defined.

Definition 6. Given a colour image f of T 3 D , a colour probe t of T 3 Dt and a tolerance p ∈ [0, 1], the map of Asplünd's distances with a tolerance is:

As × c
t,p f :

T 3 D × T 3 Dt → (R + ) D (f , t) → As × c t,p f (x) = d × c
As,Dt,p (f |Dt(x) , t) In figure 4, we look for the bricks of a wall, similar to a colour probe. A blue brick has been added to the wall. In the image without noise f , the regional minima of the map As × c t f (dark points in fig. 4b) correspond to the centre of the bricks similar to the probe (according to the Asplünd's distance). The white rectangle corresponds to the maxima of the distances between the blue brick and the probe. Therefore, the distance is sensitive to colour (i.e. the hue). In the image with noise f , the map without tolerance As × c t f is more sensitive to noise (fig. 4e) than the map with tolerance As × c t,p f (fig 4c). Indeed, the minima are preserved into the map with tolerance (fig. 4c) compared to the map without (fig. 4e). The minima can be extracted using mathematical morphology [START_REF] Matheron | Eléments pour une théorie des milieux poreux[END_REF][START_REF] Serra | Image analysis and mathematical morphology[END_REF]. Importantly, all the maps of Asplünd's distances are insensitive to the vertical lighting drift. Moreover, a correlation map is useless to find the location of the bricks (fig. 4f).

D t (x) is the neighbourhood D t centred in x ∈ D. (13)
In figure 5

Conclusion and perspectives

A new spatio-colour Asplünd's distance based on colour LIPC model has been defined. It is a true colour (i.e. vectorial) metric based on a colour model consistent with the human visual system. It is also consistent with the previous properties given in [START_REF] Jourlin | Asplünd's metric defined in the logarithmic image processing (LIP) framework: A new way to perform double-sided image probing for non-linear grayscale pattern matching[END_REF][START_REF] Noyel | Asplünd's metric defined in the logarithmic image processing (LIP) framework for colour and multivariate images[END_REF]. An extension of this metric robust to noise has been presented and illustrated on pattern recognition examples. This double-sided probing distance is efficient for colour pattern matching and performs better than traditional correlation methods. In future work, we will evaluate in details the properties of this colour distance on practical applications (e.g. in medical, remote sensing or industrial images). We will compare it to the marginal colour Asplünd's distance and we will study the links between Asplünd's probing and mathematical morphology.

Fig. 1 .

 1 Fig. 1. Computation of the Asplünd's distance between two colour functions d × c

Fig. 2 .

 2 Fig. 2. (c) Map of the Asplünd's distances As × c t f between a colour function and a probe. (a) and (b) Each colour channel is represented by a line of the same colour.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Colour Asplünd's distance with a tolerance of p = 20%. (µ, λ) are the scalars multiplying the probe without tolerance. (µ , λ ) are the scalars multiplying the probe with tolerance. d × c As,D (f , g) = 0.43 and d × cAs,D,p=20% (f , g) = 0.21

  , two images of the same scene, a bright image f and a dark image f , are acquired with two different exposure times. The probe t is extracted in the bright image and used to compute the map of Asplünd's distance As × c t f in the darker image. By finding the minima of the map, all the balls are detected and their contours are added to the image in figure 5 (b). One can notice that the Asplünd's distance is very robust to the lighting variations.

Fig. 5 .

 5 Fig. 5. Detection of coloured balls on a dark image f with a probe t extracted in the bright image f . (a) The border of the probe t is coloured in white.


B Eynard

N Orantin

Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F g algebraic invariants

.

Introduction

Formal matrix integrals can be regarded as an efficient toy model to explore the link between algebraic geometry and integrable systems [START_REF] Krichever | The τ -function of the universal Whitham hierarchy, matrix models and topological field theories[END_REF][START_REF] Bertola | Two-matrix model with semiclassical potentials and extended Whitham hierarchy[END_REF]. The theory of quantum gravity [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] David | Planar diagrams, two-dimensional lattice gravity and surface models[END_REF][START_REF] Kazakov | Bilocal regularization of models of random surfaces[END_REF] is based on the idea that matrix models provide a generating function to measure "volumes" of moduli spaces of Riemann surfaces, and random matrix models were introduced in the 80's [START_REF] Brezin | [END_REF] as a discretized version of 2d quantum gravity, i.e. conformal field theory coupled to gravity.

The formal matrix integral is at the same time a tau-function of some integrable hierarchy [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF], and it has a 't Hoof topological expansion [33,[START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Ambjørn | Matrix model calculations beyond the spherical limit[END_REF]:

ln formal dMe -N Tr V (M ) = ∞ g=0 N 2-2g F (g) (1-1)
which is related to algebraic geometry (see [START_REF] Bertola | Free Energy of the Two-Matrix Model/dToda Tau-Function[END_REF][START_REF] Bertola | Two-matrix model with semiclassical potentials and extended Whitham hierarchy[END_REF][START_REF] Kazakov | Complex Curve of the Two Matrix Model and its Tau-function[END_REF][START_REF] Eynard | Large N expansion of the 2-matrix model[END_REF]).

In a recent work [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF][START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF][START_REF] Eynard | Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula[END_REF][START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF][START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], we have developped a method to compute the F (g) 's for various formal hermitian matrix models (1-matrix model, 2-matrix model, matrix model with an external field, double scaling limits of 2-matrix model) out of the data of an algebraic equation (called the classical spectral curve):

E(x, y) = 0 , E = polynomial. (1-2)
The construction of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] extends beyond matrix models, and the F (g) 's can be computed for any algebraic equation of the type E(x, y) = 0. However the construction of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] assumes an embedding of the curve into C 2 , i.e. the choice of 2 meromorphic functions x and y on the curve. It was claimed in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] that F (g) is invariant under the exchange x ↔ y, and the proof was announced to be published separately.

This is what we do in the present paper, together with additional results.

Mixed correlations

In order to prove this claim, we first explore the case where the F (g) 's come from a formal 2-matrix model (the symmetry x ↔ y holds almost by definition in that case, see [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF]). We write the loop equation relations (W-algebra) [START_REF] Staudacher | Combinatorial solution of the 2-matrix model[END_REF][START_REF] Eynard | Master loop equations, free energy and correlations for the chain of matrices[END_REF], which we solve, and we are led to define new mixed correlation functions (W k,l and H k,l below), which did not appear in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF].

In the application of the 2-matrix model to quantum gravity and conformal field theory, those mixed correlation functions were known to play an important role in the understanding of boundary operators. But their explicit computation has been a challenge until recently. The main reason is that they don't reduce to eigenvalues of the matrices, and could not be computed by standard methods. The first explicit computations were obtained in [START_REF] Bertola | Mixed Correlation functions of the 2-Matrix Model[END_REF] and [START_REF] Eynard | 2-matrix versus complex matrix model, integrals over the unitary group as triangular integrals[END_REF]. Here in this paper, we show how to compute the topological expansion of a family of mixed correlation functions of the 2-matrix model. In a coming work [START_REF] Eyner | Whole topological expnasion of any correlation function in the two matrix model[END_REF], we shall show how to compute all mixed correlations, and introduce a link with group theory and Bethe ansatz (this is a generalization of [START_REF] Eynard | Mixed correlation functions in the 2-matrix model, and the Bethe ansatz[END_REF]).

Then, for the general case (i.e. if E was not obtained from a matrix model), we mimic those mixed correlation functions and that allows to prove the x ↔ y symmetry of F (g) .

Mixed traces of matrix models

Consider the formal 2-matrix integral3 :

Z = dM 1 dM 2 e -N tr (V 1 (M 1 )+V 2 (M 2 )-M 1 M 2 ) (2-1)
where we assume in this section that V 1 is a polynomial of degree d 1 + 1 and V 2 is a polynomial of degree d 2 + 1.

Our goal is to compute the following connected expectation values:

W k,l (x 1 , . . . , x k |y 1 , . . . , y l )

= tr 1 x 1 -M 1 tr 1 x 2 -M 1 . . . tr 1 x k -M 1 tr 1 y 1 -M 2 tr 1 y 2 -M 2 . . . tr 1 y l -M 2 c = ∞ g=0 N 2-2g-k-l W (g)
k,l (x 1 , . . . , x k |y 1 , . . . , y l ).

(2-2) and H k,l (x, y; x 1 , . . . , x k |y 1 , . . . , y l )

= tr 1 x -M 1 1 y -M 2 tr 1 x 1 -M 1 . . . tr 1 x k -M 1 tr 1 y 1 -M 2 . . . tr 1 y l -M 2 c = ∞ g=0 N 2-2g-k-l-1 H (g)
k,l (x, y; x 1 , . . . , x k |y 1 , . . . , y l )

(2-3)

W (g)
k,l is the generating function which counts connected genus g bi-colored discrete surfaces with k boundaries of the first color, and l boundaries of the second color. H (g) k,l is the generating function which counts genus g bi-colored discrete surfaces with k boundaries of the first color, and l boundaries of the second color, and one additional boundary which carries the 2 colors. The power of N in both cases is the Euler characteristic of such surfaces. The 2-matrix model was introduced in [START_REF] Kazakov | Ising model on a dynamical planar random lattice: exact solution[END_REF] as a discrete version of the Ising model on a random surface.

Notice that in H (g) k,l , the first trace contains both matrices M 1 and M 2 , we call it a mixed trace because it cannot be expressed in terms of eigenvalues of M 1 and M 2 . In applications of matrix models to conformal field theories, such objects correspond to the insertion of a pair of boundary operators, and are thus very interesting. H (0) 0,0 was computed in many works [START_REF] Eynard | Master loop equations, free energy and correlations for the chain of matrices[END_REF][START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF], and in the context of convergent integrals (instead of formal integrals), H 0,0 was computed in [START_REF] Bertola | Mixed Correlation functions of the 2-Matrix Model[END_REF][START_REF] Eynard | 2-matrix versus complex matrix model, integrals over the unitary group as triangular integrals[END_REF][START_REF] Bergère | Mixed correlation function and spectral curve for the 2-matrix model[END_REF].

The W (g) k,0 's were already computed in [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF][START_REF] Eynard | Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula[END_REF][START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF], and are given by the algebraic invariants defined in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], they are the non mixed traces.

It is known (see for instance [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF]) that all those functions are multivalued functions of their x or y variables, and they are in fact functions living on a Riemann surface called the spectral curve of equation:

E(x, y) = 0.
(2-4)

On this curve, we chose a canonical basis of cycles 4 A i ∩ B j = δ i,j , i, j = 1, . . . G, where G denotes the genus of the curve E. We will note by p i (resp. pj ) the different points of E whose projection in the complex plane by the meromorphic function x (resp. y) are equal:

∀i = 1 . . . d 2 , x(p i ) = x(p 0 ) , ∀i = 1 . . . d 1 , y(p i ) = x(p 0 ), (2-5)
where the superscript 0 refers to the x-and y-physical sheets.

It is thus more convenient to redefine W (g) k,l and H (g) k,l in terms of meromorphic forms on the curve:

W (g) k,l (p 1 , . . . , p k |q 1 , . . . , q l ) = W (g)
k,l (x(p 1 ), . . . , x(p k )|y(q 1 ), . . . , y(q l )) dx(p 1 ) . . . dx(p k )dy(q 1 ) . . . dy(q l ) +δ g,0 δ k,1 δ l,0 (y(p 1 ) -

V ′ 1 (x(p 1 )))dx(p 1 ) + δ g,0 δ k,0 δ l,1 (x(q 1 ) -V ′ 2 (y(q 1 )))dy(q 1 ) + δ g,0 δ k,2 δ l,0 dx(p 1 )dx(p 2 ) (x(p 1 ) -x(p 2 )) 2
+ δ g,0 δ k,0 δ l,2 dy(q 1 )dy(q 2 ) (y(q 1 ) -y(q 2 )) 2 (2 -6) 4 All required definitions relative to algebraic geometry can be found in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] or more generally in [START_REF] Fay | Theta functions on Riemann surfaces[END_REF][START_REF] Farkas | Riemann surfaces[END_REF]. We will use all along these notes the notations of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]. The A and B-cycles may be the modified cycles of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF].

where the p i 's and q j 's are now points on the curve E, instead of points in the complex plane. We have also "renormalized the unstable functions" with 2 -2g -k -l ≥ 0.

With those notations we have [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF][START_REF] Bertola | Free Energy of the Two-Matrix Model/dToda Tau-Function[END_REF]:

W (0) 1,0 = W (0) 0,1 = 0, (2-7)
W (0) 2,0 (p, q) = -W (0) 1,1 (p, q) = W (0) 0,2 (p, q) = B(p, q) (2-8)
where B is the Bergmann kernel, i.e. the unique bilinear form on E with a double pole at p = q and no other pole, with vanishing residue, and normalized on A-cycles:

B(p, q) ∼ p→q dz(p)dz(q) (z(p) -z(q)) 2 + finite , ∀i = 1 . . . G , A B = 0. (2-9)
We also define the differentials corresponding to the mixed correlation functions:

H (g) k,l (p, q; p 1 , . . . , p k |q 1 , . . . , q l ) = H (g)
k,l (x(p), y(q); x(p 1 ), . . . , x(p k )|y(q 1 ), . . . , y(q l )) dx(p 1 ) . . . dx(p k )dy(q 1 ) . . . dy(q l ) +δ g,0 δ k,0 δ l,0

and we normalize them by the leading order of the simplest mixed correlation function:

h (g)
k,l (p, q; p 1 , . . . , p k |q 1 , . . . , q l ) =

H (g)
k,l (p, q; p 1 , . . . , p k |q 1 , . . . , q l ) H (0) 0,0 (p, q)

.

(2-11)

It is well known [START_REF] Eynard | Large N expansion of the 2-matrix model[END_REF][START_REF] Eynard | Master loop equations, free energy and correlations for the chain of matrices[END_REF][START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF] (and it can be rederived from Eq. (2-18) and Eq. (2-21) below) that: -12) We also need to introduce:

H (0) 0,0 (p, q) = E(x(p), y(q)) (x(p) -x(q))(y(p) -y(q)) . ( 2
U k,l (p, y; p 1 , . . . , p k |q 1 , . . . , q l ) = tr 1 x(p) -M 1 V ′ 2 (y) -V ′ 2 (M 2 ) y -M 2 tr dx(p 1 ) x(p 1 ) -M 1 . . . tr dx(p k ) x(p k ) -M 1 tr dy(q 1 ) y(q 1 ) -M 2 . . . tr dy(q l ) y(q l ) -M 2 c +δ g,0 δ k,0 δ l,0 (V ′ 2 (y) -x(p)) = ∞ g=0 N 2-2g-k-l-1 U (g)
k,l (p, y; p 1 , . . . , p k |q 1 , . . . , q l ), [START_REF] Bergère | Mixed correlation function and spectral curve for the 2-matrix model[END_REF][START_REF] Bertola | Two-matrix model with semiclassical potentials and extended Whitham hierarchy[END_REF][START_REF] Bertola | Mixed Correlation functions of the 2-Matrix Model[END_REF][START_REF] Bertola | Free Energy of the Two-Matrix Model/dToda Tau-Function[END_REF][START_REF] Brezin | [END_REF][START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF][START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF][START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF][START_REF] David | Loop equations and nonperturbative effects in two-dimensional quantum gravity[END_REF][START_REF] David | Planar diagrams, two-dimensional lattice gravity and surface models[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF] which is a polynomial of y of degree at most d 2 -1, Ũk,l (x, q; p 1 , . . . , p k |q 1 , . . . , q l ) = tr

V ′ 1 (x) -V ′ 1 (M 1 ) x -M 1 1 y(q) -M 2 tr dx(p 1 ) x(p 1 ) -M 1 . . . tr dx(p k ) x(p k ) -M 1 tr dy(q 1 ) y(q 1 ) -M 2 . . . tr dy(q l ) y(q l ) -M 2 c +δ g,0 δ k,0 δ l,0 (V ′ 1 (x) -y(p)) = ∞ g=0 N 2-2g-k-l-1 Ũ(g)
k,l (x, q; p 1 , . . . , p k |q 1 , . . . , q l ), [START_REF] Bergère | Mixed correlation function and spectral curve for the 2-matrix model[END_REF][START_REF] Bertola | Two-matrix model with semiclassical potentials and extended Whitham hierarchy[END_REF][START_REF] Bertola | Mixed Correlation functions of the 2-Matrix Model[END_REF][START_REF] Bertola | Free Energy of the Two-Matrix Model/dToda Tau-Function[END_REF][START_REF] Brezin | [END_REF][START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF][START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF][START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF][START_REF] David | Loop equations and nonperturbative effects in two-dimensional quantum gravity[END_REF][START_REF] David | Planar diagrams, two-dimensional lattice gravity and surface models[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF][START_REF] Eynard | Large N expansion of the 2-matrix model[END_REF] which is a polynomial of x of degree at most d 1 -1 and

-E k,l (x, y; p 1 , . . . , p k |q 1 , . . . , q l ) = tr V ′ 1 (x) -V ′ 1 (M 1 ) x -M 1 V ′ 2 (y) -V ′ 2 (M 2 ) y -M 2 tr dx(p 1 ) x(p 1 ) -M 1 . . . tr dx(p k ) x(p k ) -M 1 tr dy(q 1 ) y(q 1 ) -M 2 . . . tr dy(q l ) y(q l ) -M 2 c +δ g,0 δ k,0 δ l,0 ((V ′ 1 (x) -y(p))(V ′ 2 (y) -x(p)) -1) = - ∞ g=0 N 2-2g-k-l-1 E (g)
k,l (x, y; p 1 , . . . , p k |q 1 , . . . , q l ), [START_REF] Bergère | Mixed correlation function and spectral curve for the 2-matrix model[END_REF][START_REF] Bertola | Two-matrix model with semiclassical potentials and extended Whitham hierarchy[END_REF][START_REF] Bertola | Mixed Correlation functions of the 2-Matrix Model[END_REF][START_REF] Bertola | Free Energy of the Two-Matrix Model/dToda Tau-Function[END_REF][START_REF] Brezin | [END_REF][START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF][START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF][START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF][START_REF] David | Loop equations and nonperturbative effects in two-dimensional quantum gravity[END_REF][START_REF] David | Planar diagrams, two-dimensional lattice gravity and surface models[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF][START_REF] Eynard | Large N expansion of the 2-matrix model[END_REF][START_REF] Eynard | Large N expansion of the 2-matrix model, multicut case[END_REF] which is a polynomial of x of degree d 1 -1 and of y of degree d 2 -1.

We have:

E (0) 0,0 (x, y) = E(x, y) , U (0) 0
,0 (p, y) = E(x(p), y) y -y(p) , Ũ(0) 0,0 (x, q) = E(x, y(q)) x -x(q) ,
(2-16) and P (0) 0,0 (x, y) = -E(x, y).

(2-17)

Loop equations

In order to obtain a closed set of equations computing these mixed correlation functions, we consider 4 families of loop equations [START_REF] Staudacher | Combinatorial solution of the 2-matrix model[END_REF][START_REF] Eynard | Master loop equations, free energy and correlations for the chain of matrices[END_REF][START_REF] Eynard | Loop equations for the semiclassical 2-matrix model with hard edges[END_REF] corresponding to different infinitesimal changes of variables M i → M i + ǫδM i in the matrix integral.

δM 2 = 1 x(p)-M 1 1 y(q)-M 2 k i=1 tr 1 x(p i )-M 1 l j=1 tr 1 y(q j )-M 2 gives: -U (g) k,l (p, y(q); p K |q L ) = (x(p) -x(q))H (g) k,l (p, q; p K |q L ) + h I,J W (h) i,j+1 (p I |q J , q)H (g-h) k-i,l-j (p, q; p K/I |q L/J ) dy(q) + H (g-1)
k,l+1 (p, q; p K |q L , q) dy(q)

- n d qn H (g) k,l-1 (p, q n ; p K |q L/{n} ) y(q) -y(q n ) (2-18) δM 1 = 1 x(p)-M 1 1 y(q)-M 2 k i=1 tr 1 x(p i )-M 1 l j=1 tr 1 y(q j )-M 2 gives: - Ũ(g) k,l (x(p), q; p K |q L ) = (y(q) -y(p))H (g) k,l (p, q; p K |q L ) + h I,J W (h) i+1,j (p, p I |q J )H (g-h)
k-i,l-j (p, q; p K/I |q L/J ) dx(p)

+ H (g-1)
k+1,l (p, q; p,

p K |q L ) dx(p) - m d pm H (g) k-1,l (p m , q; p K/{m} |q L ) x(p) -x(p m ) (2-19) δM 2 = V ′ 1 (x(p))-V ′ 1 (M 1 ) x(p)-M 1 1 y(q)-M 2 k i=1 tr 1 x(p i )-M 1 l j=1 tr 1 y(q j )-M 2 gives: E (g) k,l (x(p), y(q); p K |q L ) = (x(p) -x(q)) Ũ(g) k,l (x(p), q; p K |q L ) + h I,J W (h) i,j+1 (p I |q J , q) Ũ(g-h) k-i,l-j (x(p), q; p K/I |q L/J ) dy(q) + Ũ(g-1) k,l+1 (x(p), q; p K |q L , q) dy(q) - m d qm Ũ(g) k,l-1 (x(p), q m ; p K |q L/{m} ) y(q) -y(q m ) - m d pm H (g) k-1,l (p m , q; p K/{m} |q L ) (2-20)
and

δM 1 = 1 x(p)-M 1 V ′ 2 (y(q))-V ′ 2 (M 2 ) y(q)-M 2 k i=1 tr 1 x(p i )-M 1 l j=1 tr 1 y(q j )-M 2 gives: E (g) k,l (x(p), y(q); p K |q L ) = (y(q) -y(p))U (g) k,l (p, y(q); p K |q L ) + h I,J W (h) i+1,j (p, p I |q J )U (g-h) k-i,l-j (p, y(q); p K/I |q L/J ) dx(p) + U (g-1)
k+1,l (p, y(q); p,

p K |q L ) dx(p) - m d pm U (g) k-1,l (p m , y(q); p K/{m} |q L ) x(p) -x(p m ) - m d qm H (g) k,l-1 (p, q m ; p K |q L/{m} ). (2-21)
Those loop equations can be seen to be equivalent to W-algebra constraints [START_REF] David | Loop equations and nonperturbative effects in two-dimensional quantum gravity[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF], or to a generalization of Tutte's equations for the combinatorics of discrete surfaces [START_REF] Tutte | A census of planar triangulations[END_REF][START_REF] Tutte | A census of planar maps[END_REF].

Solution of loop equations

Theorem 2.1 The solution of loop equations is such that:

h (g) k,l (p, q; p K |q L ) = Res r→q j ,p,p K 1 (x(p) -x(r))(y(r) -y(q)) h (g-1)
k+1,l (r, q; r,

p K |q L ) + h I⊂K J⊂L W (h) i+1,j (r, p I |q J )h (g-h)
k-i,l-j (r, q; p K/I |q L/J ) ,

(2 -22) W (g) k,l+1 (p K |q L , q) = Res r→q j ,p K dy(q) (y(r) -y(q)) h (g-1)
k+1,l (r, q; r,

p K |q L ) + h I⊂K J⊂L W (h) i+1,j (r, p I |q J )h (g-h)
k-i,l-j (r, q; p K/I |q L/J ) .

(2 -23)

where Res

r→q j
means that one takes the residues around all the points qj = q such that y(q j ) = y(q).

Given the initial conditions:

h (0) 0,0 = 1 , W (g) k,0 (p 1 , . . . , p k ) = W (g) k (p 1 , . . . , p k ) E (2-24)
where

W (g) k (p 1 , . . . , p k ) E
is the function defined in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], the above system is triangular and computes univocally any h

(g) k,l and W (g)
k,l in at most k + l + g 2 2 steps. One easily proves by recursion on 2g + k + l that:

H (g) k,l (p, q; p K |q L ) has poles        in p = a, q, q L in q = b, p, p K in p j = a, q, q L in q j = b, p, p K (2-25) and W (g) k,l (p K |q L ) has poles in p j = a, q L in q j = b, p K (2-26) proof: Since Ũ(g) k,l (x(p), q; p K |q L ) is a polynomial in x(p) of degree at most d 1 -2,
it is given by the Lagrange interpolation formula:

Ũ(g) k,l (x(p), q; p K |q L ) = Ũ(0) 0,0 (x(p), q) d 1 j=1 Ũ(g) k,l (x(q j ), q; p K |q L ) (x(p) -x(q j )) Ũ(0) 0,0 x (x(q j ), q) = Ũ(0) 0,0 (x(p), q) d 1 j=1 Res r→q j Ũ(g) k,l (x(q j ), q; p K |q L ) dx(r) (x(p) -x(r)) Ũ(0) 0,0 (x(r), q) . (2 -27)
Then we replace Ũ(g) k,l (x(q j ), q; p K |q L ) by its value from the loop equation 2-19:

Ũ(g) k,l (x(p), q; p K |q L ) = - d 1 j=1
Res r→q j

Ũ (0) 0,0 (x(p),q) (x(p)-x(r)) Ũ (0) 0,0 (x(r),q) h I,J W (h) i+1,j (r, p I |q J )H (g-h) k-i,l-j (r, q; p K/I |q L/J ) +H (g-1)
k+1,l (r, q; r,

p K |q L ) - m d pm H (g) k-1,l (pm,q;p K/{m} |q L ) dx(r) x(r)-x(pm)
(2-28) Notice that the same residue computed at r → p gives the terms in the RHS of the loop equation 2-19, and therefore:

(y(q) -y(p))H (g) k,l (p, q; p K |q L ) = Res r→p,q j Ũ(0) 0,0 (x(p), q) (x(p) -x(r)) Ũ(0) 0,0 (x(r), q) h I,J W (h) i+1,j (r, p I |q J )H (g-h) k-i,l-j (r, q; p K/I |q L/J ) +H (g-1)
k+1,l (r, q; r,

p K |q L ) - m d pm H (g) k-1,l (p m , q; p K/{m} |q L ) dx(r) x(r) -x(p m ) .
(2 -29)

Moreover the last term d pm

H (g) k-1,l (pm,q;p K/{m} |q L ) dx(r) x(r)-x(pm)
can be computed explicitely:

d pm Res r→p,q j Ũ (0) 0,0 (x(p),q) (x(p)-x(r)) Ũ (0) 0,0 (x(r),q) H (g) k-1,l (pm,q;p K/{m} |q L ) dx(r) x(r)-x(pm) = d pm Res r→p,q j E(x(p),y(q))(x(r)-x(q)) (x(p)-x(r))(x(p)-x(q)) E(x(r),y(q)) H (g) k-1,l (pm,q;p K/{m} |q L ) dx(r) x(r)-x(pm)
.

(2-30)

Under this form, one can see that the integrant is a rational function of x(r). Thus, the residue can be computed on the complex plane obtained by the projection x and we can move the integration contours on the complex plane instead of the curve E itself. This term is then equal to:

d pm Res x→x(p),x(q j ) E(x(p), y(q))(x -x(q)) (x(p) -x)(x(p) -x(q)) E(x, y(q)) H (g) k-1,l (p m , q; p K/{m} |q L ) dx x -x(p m ) = -d pm Res x→x(pm) E(x(p), y(q))(x -x(q)) (x(p) -x)(x(p) -x(q)) E(x, y(q)) H (g) k-1,l (p m , q; p K/{m} |q L ) dx x -x(p m ) = -d pm E(x(p), y(q))(x(p m ) -x(q)) (x(p) -x(p m ))(x(p) -x(q)) E(x(p m ), y(q)) H (g) k-1,l (p m , q; p K/{m} |q L ) = -Res r→pm Ũ(0) 0,0 (x(p), q) (x(p) -x(r)) Ũ(0) 0,0 (x(r), q) H (g) k-1,l (r, q; p K/{m} |q L )W (0) 2,0 (r, p m ) = -Res r→pm Ũ(0) 0,0 (x(p), q) (x(p) -x(r)) Ũ(0) 0,0 (x(r), q) h,I,J W (h) i+1,j (r, p I |q J )H (g-h) k-i,l-j (r, q; p K/I |q L/J ) +H (g-1)
k+1,l (r, q; r,

p K |q L ) , (2-31)
where the last equality holds thanks to the loop equation Eq. [START_REF] Bergère | Mixed correlation function and spectral curve for the 2-matrix model[END_REF][START_REF] Bertola | Two-matrix model with semiclassical potentials and extended Whitham hierarchy[END_REF][START_REF] Bertola | Mixed Correlation functions of the 2-Matrix Model[END_REF][START_REF] Bertola | Free Energy of the Two-Matrix Model/dToda Tau-Function[END_REF][START_REF] Brezin | [END_REF][START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF][START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF][START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF][START_REF] David | Loop equations and nonperturbative effects in two-dimensional quantum gravity[END_REF][START_REF] David | Planar diagrams, two-dimensional lattice gravity and surface models[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF][START_REF] Eynard | Large N expansion of the 2-matrix model[END_REF][START_REF] Eynard | Large N expansion of the 2-matrix model, multicut case[END_REF][START_REF] Eynard | Loop equations for the semiclassical 2-matrix model with hard edges[END_REF][START_REF] Eynard | 2-matrix versus complex matrix model, integrals over the unitary group as triangular integrals[END_REF][START_REF] Eynard | Master loop equations, free energy and correlations for the chain of matrices[END_REF][START_REF] Eynard | Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula[END_REF]. Therefore:

(y(q) -y(p))H (g) k,l (p, q; p K |q L ) = Res r→p,q j ,p K Ũ(0) 0,0 (x(p), q) (x(p) -x(r)) Ũ(0) 0,0 (x(r), q) h I,J W (h) i+1,j (r, p I |q J )H (g-h) k-i,l-j (r, q; p K/I |q L/J ) +H (g-1)
k+1,l (r, q; r, p K |q L ) .

(2-32)

If we divide by Ũ(0) 0,0 (x(p), q) we obtain:

-h

(g) k,l (p, q; p K |q L ) = Res r→p,q j ,p K 1 (x(p) -x(r)) (y(r) -y(q)) h I,J W (h) i+1,j (r, p I |q J )h (g-h) k-i,l-j (r, q; p K/I |q L/J ) +h (g-1)
k+1,l (r, q; r, p K |q L ) .

(2-33)

The other half of the theorem is obtained from the fact that for large x:

tr 1 x -M 1 1 y -M 2 → 1 x tr 1 y -M 2 (2-34)
and thus:

H (g) k,l (p, q; p K |q L ) → 1 x(p) W (g) k,l+1 (p K |q L , q) dy(q) (2-35) when p → ∞ x 5 .

Examples, first few terms

Let us solve the recursive definition and give explicit formulae for the simplest functions.

Example W (0)

1,1 : In particular, definitions Eq. and Eq. give:

W (0) 1,1 (p 1 |q) = Res r→q j ,p 1 dy(q) B(r, p 1 ) (y(r) -y(q))
= -Res r→q dy(q) B(r, p 1 ) (y(r) -y(q)) = -B(q, p 1 ).

(2-36)

Therefore we recover:

W (0) 2,0 (p 1 , q) + W (0) 1,1 (p 1 |q) = 0. (2-37) Example H (0) 1,0 : h (0)
1,0 (p, q; p 1 ) = Res

r→q j ,p,p 1 B(r, p 1 ) (x(p) -x(r))(y(r) -y(q)) = -Res r→p i ,q B(r, p 1 ) (x(p) -x(r))(y(r) -y(q)) . (2-38) Example H (0) 0,1 : h (0) 0,1 (p, q; p 1 ) = Res r→q j ,p W (0) 1,1 (r|p 1 ) (x(p) -x(r))(y(r) -y(q)) = -Res r→q j ,p B(r, p 1 ) (x(p) -x(r))(y(r) -y(q)) = Res r→p i ,q,p 1 B(r, p 1 ) (x(p) -x(r))(y(r) -y(q))
.

(2-39)

Moreover we have:

h (0) 1,0 (p, q; p 1 ) + h (0) 0,1 (p, q; p 1 ) = Res r→p 1 B(r, p 1 ) (x(p) -x(r))(y(r) -y(q)) = d p 1 1 (x(p) -x(p 1 ))(y(p 1 ) -y(q)) . (2-40) Example W (0)
2,1 :

W (0) 2,1 (p 1 ,p 2 |q) dy(q) = Res r→q j ,p 1 ,p 2 B(r,p 1 )h (0) 1,0 (r,q;p 2 )+B(r,p 2 )h (0)
1,0 (r,q;p 1 )+W

(0) 3,0 (r,p 1 ,p 2 ) (y(r)-y(q)) = -Res r→q,a B(r,p 1 )h (0) 1,0 (r,q;p 2 )+B(r,p 2 )h (0) 1,0 (r,q;p 1 )+W (0) 3,0 (r,p 1 ,p 2 ) (y(r)-y(q))
.

(2-41)

Conclusion of section 2

Therefore, through theorem 2.1, we have an effective explicit method to compute any H (g) k,l and any W (g) k,l for the 2-matrix model. This is an interesting result in itself, since none of those quantities were computed before, and those quantities are of importance in applications of random matrices to combinatorics of maps with colored boundaries, i.e. boundary conformal field theory.

An important remark, is that we have chosen to emphasize the role of the loop equation 2-19, rather than equation 2-18, i.e. we have used the Lagrange interpolation formula for a polynomial in x, whereas we could have done the same thing with a polynomial in y. In other words, we have chosen the x-representation rather than the y-representation, although both methods must give the same answer. In particular, given W k,0 , theorem 2.1 allows to compute W 0,l . W k,0 can be computed with the method of [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF][START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] using the x-representation, while W 0,l can be computed with the method of [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF][START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] using the y-representation, i.e. under the exchange x ↔ y .

(2-42)

Therefore, in the following section, we improve the result of theorem 2.1, in order to prove that the diagrammatic rules of [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF][START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] are indeed symmetric under the exchange of x and y. In other words we prove theorem 7.1 of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], as announced in that article.

3 Proof of the symmetry x-y of the algebraic invariants F (g) (E)

Consider the two algebraic curves:

Ê(x, y) = E(x, y) and Ě(x, y) = E(y, x) (3-1)
In [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], for any curve E an infinite sequence of invariants F (g) was defined. Here we consider those invariants for the 2 curves Ê and Ě.

In this section we prove the following theorem (which was announced in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]):

Theorem 3.1 Symmetry under the exchange x ↔ y:

F (g) ( Ê) = F (g) ( Ě) (3-2)
where the functional F (g) (E) is defined for any curve E(x, y) in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF].

Preliminaries

For the curve Ê(x, y) = 0, we have defined in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] an infinite sequence of meromorphic forms:

Ŵ (g) k (p 1 , . . . , p k ) = W (g) k (p 1 , . . . , p k ) Ê (3-3)
with poles only at the zeroes a = {a i } of dx, and some free energies

F (g) = 1 2 -2g Res p→a Φ(p) Ŵ (g) 1 (p) (3-4)
where Φ is any antiderivative of ydx, dΦ = ydx and Res p→a stands for i Res p→a i .

And likewise, for the curve Ě(x, y) = 0, we have defined an infinite sequence of meromorphic forms:

W (g) k (q 1 , . . . , q k ) = W (g) k (q 1 , . . . , q k ) Ě (3-5)
with poles only at the zeroes b = {b i } of dy, and some free energies

F (g) = 1 2 -2g Res q→b Ψ(q) W (g) 1 (q) (3-6)
where dΨ = xdy.

Our first step is to extend those forms into two families of multilinear meromorphic forms similar to those of section 2 (i.e. mimicking the mixed traces of matrix models):

Ŵ (g) k,l (p 1 , . . . , p k |q 1 , . . . , q l ) and W (g) k,l (p 1 , . . . , p k |q 1 , . . . , q l ) (3-7) such that: Ŵ (g) k,0 = Ŵ (g) k , W (g) 0,l = W (g) l . (3-8)
Our second step, is to prove that:

Ŵ (g) k,l = W (g) k,l . (3-9)
Our third step, is to prove that:

Ŵ (g) k+1,l (p K , p|q L ) + W (g) k,l+1 (p K |p, q L ) = d p A (g) k,l (p; p K |q L ) dx(p)dy(p) (3-10)
where

A (g)
k,l (p; p K |q L ) has poles of degree at most 2 at the poles of ydx, so that in particular for k = l = 0 we have:

Ŵ (g) 1,0 (p) + W (g) 0,1 (p) = d p A (g) 0,0 (p) dx(p)dy(p) (3-11)
where A (g) 0,0 has poles of degree at most 2 at the poles of ydx. This last step is sufficent to prove that F (g) = F (g) .

(3-12)

Definitions of mixed correlators

Ŵ (g) k,l and W (g) k,l
We define the initial terms:

Ê(0) 0,0 (x, y) = Ě(0) 0,0 (x, y) = E(x, y), (3-13)
Ĥ(0) 0,0 (p, q) = Ȟ(0) 0,0 (p, q) = E(x(p), y(q)) (x(p) -x(q))(y(p) -y(q)) ,

Ŵ (0) 1,0 (p) = Ŵ (0) 0,1 (p) = W (0) 1,0 (p) = W (0) 0,1 (p) = 0, (3-14)
Ŵ (0) 2,0 (p, q) = Ŵ (0) 0,2 (p, q) = - Ŵ (0) 1,1 (p, q) = B(p, q), (3-15)
and

W (0) 2,0 (p, q) = W (0) 0,2 (p, q) = - W (0) 1,1 (p, q) = B(p, q). (3-17)
Let us define recursively the following quantities for any g, k, l ≥ 0:

J (g) k,l (p, q; p K |q L ) := k m 1 ,m 2 =0 l n 1 ,n 2 =0 g h,h ′ =1 Ŵ (h) m 1 +1,n 1 (p, p M 1 |q N 1 )× × W (h ′ ) m 2 ,n 2 +1 (p M 2 |q N 2 , q)H (g-h-h ′ ) k-m 1 -m 2 ,l-n 1 -n 2 (p, q; p K/{M 1 M 2 } |q L/{N 1 N 2 } ) + g-1 h=1
(x(p) -x(q)) Ŵ (h) 1,0 (p)dy(q) + (y(q) -y(p))

W (h) 0,1 (q)dx(p) H (g-h) k,l (p, q; p K |q L ) + k m=0 l n=0;mn =kl g h=0 H (g-h) k-m,l-n (p, q; p K/M |q L/N )× × (x(p) -x(q)) Ŵ (h) m+1,n (p, p M |q N )dy(q) + (y(q) -y(p)) W (h) m,n+1 (p M |q N , q)dx(p) +(x(p) -x(q))H (g-1)
k+1,l (p, q; p, p K |q L )dy(q) + (y(q) -y(p))H

(g-1)
k,l+1 (p, q; p K |q L , q)dx(p)

+ k m=0 l n=0 g-1 h=0 Ŵ (h) m+1,n (p, p M |q N )H (g-h-1)
k-m,l-n+1 (p, q; p K/M |q L/N , q)

+ 1 2 Ŵ (h) m+1,n+1 (p, p M |q N , q) + W (h) m+1,n+1 (p, p M |q N , q) H (g-h-1) k-m,l-n (p, q; p K/M |q L/N ) + W (h) m,n+1 (p M |q N , q)H (g-h-1) k-m+1,l-n (p, q; p, p K/M |q L/N ) + H (g-2)
k+1,l+1 (p, q; p, p K |q L , q) (3-18)

and

J (g) k,l (p, q; p K |q L ) := J (g) k,l (p, q; p K |q L ) - k α=1 d pα dx(p) x(p)-x(pα) (x(p α ) -x(q))dy(q) H (g) k-1,l (p α , q; p K-{α} |q L ) + g h=1 W (h) 0,1 (q)H (g-h) k-1,l (p α , q; p K-{α} |q L ) + h <kl i,j=0 H (g-h) i-1,j (p α , q; p I-{α} |q J ) W (h) k-i,l-j+1 (p K-I |q L-J , q) - l β=1 d q β dy(q) y(q)-y(q β ) (y(q β ) -y(p))dx(p) H k,l-1 (p, q β ; p K |q L-{β} ) + g h=1 Ŵ (h) 1,0 (p)H (g-h) k,l-1 (p, q β ; p K |q L-{β} ) + <kl i,j=0 h H (g-h) i,j-1 (p, q β ; p I |q J-{β} ) Ŵ h k-i+1,l-j (p, p K-I |q L-J ) + k α=1 l β=1 d pα d q β dx(p) x(p)-x(pα) dy(q) y(q)-y(q β ) H k-1,l-1 (p α , q β ; p K-{α} |q L-{β} ) - k α=1 d pα H (g-1)
k-1,l+1 (pα,q;p K-{α} |q L ,q)dx(p)

x(p)-x(pα)

- l β=1 d q β H (g-1)
k+1,l-1 (p,q β ;p,p K |q L-{β} )dy(q) y(q)-y(q β ) .

( Now we define:

Ŵ (g) k+1,l (p, p K |q L ) := Res s→a,q L dS s,o (p) 1 d 1 d 1 j=1 J (g) k,l (s, sj ; p K |q L ) U (0)
0,0 (s, y(s))dy(s)

+ 1 d 2 d 2 i=1 J (g) k,l (s i , s; p K |q L ) U (0) 0,0 (x(s), s)dx(s) , (3 -20) Ŵ (g) k,l+1 (p K |q L , q) := Res s→b,p K dS s,o (q) 1 d 1 d 1 j=1 J (g) k,l (s, sj ; p K |q L ) U (0)
0,0 (s, y(s))dy(s)

+ 1 d 2 d 2 i=1 J (g) k,l (s i , s; p K |q L ) U (0) 0,0 (x(s), s)dx(s) , (3 -21) and -U (g) k,l (p, q; p K |q L ) := (x(p) -x(q))H (g) k,l (p, q; p K |q L ) + h I,J W (h) i,j+1 (p I |q J , q)H (g-h)
k-i,l-j (p, q; p K/I |q L/J ) dy(q)

+ H (g-1)
k,l+1 (p, q; p K |q L , q) dy(q

) 2 - n d qn H (g)
k,l-1 (p, q n ; p K |q L/{n} ) y(q) -y(q n ) .

(3 -30)

Those definitions form a triangular system of definitions, and each term is well defined in a unique recursive way. Remark 3.2 Definitions eq.3-29 and eq.3-30 coincide with loop equation 2-19 and 2-18 in the matrix model case, i.e. when E is the classical spectral curve of the 2 matrix model.

Theorems

Theorem 3.2 For 2g + k + l ≥ 3, one has the following properties:

• Ŵ (g) k,l (p K |q L ) (resp. W (g) k,l (p K |q L )
) has poles only when p i → a, q L and q j → b, p K ;

• in any of the k +l variables, the A-cycle integrals vanish:

A Ŵ (g) k,l = A W (g) k,l = 0; • H (g)
k,l (p, q; p K |q L ) = Ȟ(g) k,l (p, q; p K |q L ) has poles only when p → q, a, q L and q → p, b, p K , and

E (g) k,l (x(p), q; p K |q L ) = Ě(g) k,l (p, y(q); p K |q L ) := E (g) (x(p), y(q); p K |q L ) (3-31) is a polynomial of degree d 1 -1 in x(p) and d 2 -1 in y(q); • U (g) k,l (p, y(q); p K |q L ) (resp. U (g) k,l (x(p), q; p K ; q L )) is a polynomial in y(q) (resp. x(p)) of degree d 2 -1 (resp. d 1 -1).

proof:

Let us proceed by induction on 2g + k + l. Suppose that the properties are satisfied for any g ′ , k ′ , l ′ such that 2g ′ + k ′ + l ′ < 2g + k + l. Let us prove that they are true for g, k, l. In order to make the proof more readable, we split it into pieces. Nevertheless, for every step, the global recursion hypothesis is needed.

We need the following lemma:

Lemma 3.1 The quantity f (g) k,l (s; p K ; q L ) := J (g) k,l (s, sj ; p K ; q L ) U (0) 0,0 (s, y(s))dy(s) (3-32)
is independent of j = 0, it is a meromorphic one-form in the variable s, with poles at s = a, q L , and it vanishes to order at least deg(ydx) -1 near the poles of ydx.

Similarly, the quantity

f (g) k,l (s; p K ; q L ) := J (g) k,l (s i , s; p K ; q L ) U (0) 0,0 (x(s), s)dx(s) . (3-33)
is independent of i = 0, it is a meromorphic one-form in the variable s, with poles at s = b, q L , and it vanishes to order at least deg(xdy) -1 near the poles of xdy. Moreover one has:

A f (g) k,l (s; p K ; q L ) + f (g) k,l (s; p K ; q L ) = 0, (3-34)
B f (g) k,l (s; p K ; q L ) + f (g) k,l (s; p K ; q L ) = 0 (3-35)
and:

f (g) k,l (s; p K ; q L ) + f (g) k,l (s; p K ; q L ) = Res q→a,b,p K ,q L dS q,o (s) f (g)
k,l (q; p K ; q L ) + f (g) k,l (q; p K ; q L ) .

(3-36)

Proof of the lemma: First of all, One can remark that the definition of J (g) k,l involves only quantities whose properties are known by the recursion hypothesis. One can note that it can be written under the following forms:

J (g) k,l (p, q; p K ; q L ) := - g-1 h=1 k m=0 l n=0 W (h) m,n+1 (p M ; q, q N ) U g-h k-m,l-n (x(p), q; p K/M , q L/N )dx(p) - k m=0 l n=0,(m,n) =(0,0) W (0) m,n+1 (p M ; q, q N ) U g k-m,l-n (x(p), q; p K/M , q L/N )dx(p) - k m=0 l n=0,(m,n) =(k,l) W (g) m,n+1 (p M ; q, q N ) U 0 k-m,l-n (x(p), q; p K/M , q L/N )dx(p) -U (g-1)
k,l+1 (x(p), q; p K ; q, q L )dx(p)

+ g-1 h=1 k m=0 l n=0 d pα W (h) m,n+1 (p M ;q,q N )H g-h k-m-1,l-n (pα,q;p K/M/{α} ,q L/N )dx(p) x(p)-x(pα) + k m=0 l n=0,(m,n) =(0,0) d pα W (0) m,n+1 (p M ;q,q N )H g k-m-1,l-n (pα,q;p K/M/{α} ,q L/N )dx(p) x(p)-x(pα) + k m=0 l n=0,(m,n) =(k,l) d pα W (g) m,n+1 (p M ;q,q N )H 0 k-m-1,l-n (pα,q;p K/M/{α} ,q L/N )dx(p) x(p)-x(pα) +d pα H (g-1)
k-1,l+1 (pα,q;p K/{α} ;q,q L )dx(p)

x(p)-x(pα)

+(x(p) -x(q)) g-1 h=1 k m=0 l n=0 Ŵ (h) m+1,n (p, p M ; q N )dy(q)H (g-h)
k-m,l-n (p, q; p K/M ; q L/N )

+ k m=0 l n=0,(m,n) =(k,l) Ŵ (0) m+1,n (p, p M ; q N )dy(q)H (g) k-m,l-n (p, q; p K/M ; q L/N ) + k m=0 l n=0,(m,n) =(0,0) Ŵ (g) m+1,n (p, p M ; q N )dy(q)H (0) k-m,l-n (p, q; p K/M ; q L/N ) +H (g-1)
k+1,l (p, q; p, p K ; q L )dy(q) (3-37) and

J (g) k,l (p, q; p K ; q L ) = - g-1 h=1 k m=0 l n=0 W (h) m,n+1 (p M ; q, q N ) U g-h k-m,l-n (x(p), q; p K/m , q L/N )dx(p) - k m=0 l n=0,(m,n) =(0,0) W (0) m,n+1 (p M ; q, q N ) U g k-m,l-n (x(p), q; p K/m , q L/N )dx(p) - k m=0 l n=0,(m,n) =(k,l) W (g) m,n+1 (p M ; q, q N ) U 0 k-m,l-n (x(p), q; p K/m , q L/N )dx(p) -U (g-1)
k,l+1 (x(p), q; p K ; q, q L )dx(p) +d q β U (g) k,l-1 (p,q β ;p K ;q L/{β} ) y(q)-y(q β ) dx(p)dy(q)

-d pα x(pα)-x(q) x(p)-x(pα) H (g) k-1,l (p α , q; p K:{α} , q L )dx(p)dy(q) +(x(p) -x(q)) g-1 h=1 k m=0 l n=0 Ŵ (h) m+1,n (p, p M ; q N )dy(q)H (g-h) k-m,l-n (p, q; p K/M ; q L/N ) + k m=0 l n=0,(m,n) =(k,l) Ŵ (0) m+1,n (p, p M ; q N )dy(q)H (g) k-m,l-n (p, q; p K/M ; q L/N ) + k m=0 l n=0,(m,n) =(0,0) Ŵ (g) m+1,n (p, p M ; q N )dy(q)H (0) k-m,l-n (p, q; p K/M ; q L/N ) +H (g-1)
k+1,l (p, q; p, p K ; q L ) dy(q). (3-38) Thanks to the properties implied by the recursion hypothesis (U and Ũ are polynomials), one has:

J (g) k,l (q i , q; p K ; q L ) = - g-1 h=1 k m=0 l n=0 W (h) m,n+1 (p M ; q, q N ) U g-h k-m,l-n (x(q), q; p K/M , q L/N )dx(q) - k m=0 l n=0,(m,n) =(0,0) W (0) m,n+1 (p M ; q, q N ) U g k-m,l-n (x(q), q; p K/M , q L/N )dx(q) - k m=0 l n=0,(m,n) =(k,l) W (g) m,n+1 (p M ; q, q N ) U 0 k-m,l-n (x(q), q; p K/M , q L/N )dx(q) -U (g-1)
k,l+1 (x(q), q; p K ; q, q L )dx(q) +d q β U (g) k,l-1 (x(q), q β ; p K ; q L/{β} ) y(q) -y(q β ) dx(q)dy(q)

+d pα H (g)
k-1,l (p α , q; p K/{α} , q L ) dx(q)dy(q) (3 -39) for any non vanishing i. Thus this quantity does not depend on i, and f is clearly a meromorphic 1-form, whose poles can be easily seen on this expression using the recursion hypothesis.

The same considerations give the equivalent through the exchange of x ↔ y:

J (g) k,l (p, pj ; p K ; q L ) = - g-1 h=1 k m=0 l n=0 Ŵ (h) m+1,n (p, p M ; q N )U g-h k-m,l-n (p, y(p); p K/M , q L/N )dy(p) - k m=0 l n=0,(m,n) =(0,0) Ŵ (0) m+1,n (p, p M ; q N )U g k-m,l-n (p, y(p); p K/M , q L/N )dy(p) - k m=0 l n=0,(m,n) =(k,l) Ŵ (g) m+1,n (p, p M ; q N )U 0 k-m,l-n (p, y(p); p K/M , q L/N )dy(p) -U (g-1)
k+1,l (p, y(p); p, p K ; q L )dy(p)

+d pα U (g) k-1,l (p α , y(p); p K/{α} ; q L ) x(p) -x(p α ) dx(p)dy(p) +d q β H (g) k,l-1 (p, q β ; p K/{α} , q L/{β} ) dx(p)dy(p) (3 -40)
This quantity does not depend on j, and f is clearly a meromorphic 1-form, whose poles can be easily seen on this expression using the recursion hypothesis.

The fact that the A and B cycle integrals vanish comes from the symmetry x ↔ y. Indeed under the symmetry x ↔ y, f is changed to f and f is changed to f . At the same time the A-cycles are changed to -A because 2iπǫ = A ydx = -A xdy, and the B-cycles are changed to -B in order to form a canonical basis. Therefore, the A and B cycle integrals of f + f vanish. Equation 3-36 simply comes from Cauchy residue formula and Riemann's bilinear identity.

The fact that f vanishes to order at least deg(ydx) -1 near a pole α of ydx follows from the definition of J :

J (g) k,l (p, pj ; p K |q L ) dx(p)dy(p) ∼ p→α ∼ p→α x(p) -x(p j ) dx(p) k m=0 l n=0 g h=0 Ŵ (h) m+1,n (p, p M |q N )H (g-h) k-m,l-n (p, pj ; p K/M |q L/N ) +H (g-1)
k+1,l (p, pj ; p, p K |q L )

- k α=1 d pα (x(p α ) -x(p j )) x(p) -x(p α ) H (g) k-1,l (p α , pj ; p K-{α} |q L )
where the last equality holds because the integrant has no pole when s → qj . Then

H (g) k,l (p,q;p K ;q L ) E(x(p),y(q)) = = Res r→q,p i Res s→p,q j G (g) k,l (s,r;p K ;q L ) (y(q)-y(r))(x(s)-x(p))H (0) 0,0 (s,r) 1 (y(r)-y(s))(x(p)-x(q)) + 1 (x(s)-x(r))(y(q)-y(p)) - 1 (y(r)-y(s))(x(p)-x(q)) = Res r→q,p i Res s→p,q j G (g) k,l (s,r;p K ;q L ) (y(q)-y(r))(x(s)-x(p))(y(r)-y(s))(x(p)-x(q))H (0) 0,0 (s,r) + d 2 i=1 G (g) k,l (p,p i ;p K ;q L ) (y(q)-y(p))(y(q)-y(p i ))H (0) 0,0 (p,p i )dx(p) 2 .
(3-44) Note that the first term corresponds exactly to

Ȟ(g) k,l (p,q;p K ;q L ) E(x(p),y(q))
with the integration contours for r and s exchanged. However, the poles of the integrand are known and thus: (3 -45)

The last term does not contribute because the integrant is regular when r i → s, thus

H (g) k,l (p,q;p K ;q L ) E(x(p),y(q)) so that E (g) k,l = Ě(g) k,l . Moreover, one can see from Eq. (3-27) that E (g) k,l (p, q; p K |q L ) is a polynomial of y(q) while Ě(g) k,l (p, q; p K |q L ) is a polynomial of x(p), therefore E (g) k,l (x(p), y(q); p K |q L ) = E (g) k,l (p, q; p K |q L ) = Ě(g) k,l (p, q; p K |q L ) (3-51)
is a polynomial in two variables.

•U

k,l and Ũ (g) k,l are polynomials.

Eq. (3-49), Eq. (3-50), Eq. (3-37) and Eq. imply that

E (g) k,l (x(p), y(q); p K |q L ) = (x(p) -x(q)) Ũ(g) k,l (x(p), q; p K |q L ) + h I,J W (h) i,j+1 (p I ; q J , q) Ũ(g-h) k-i,l-j (x(p), q; p K/I |q L/J ) dy(q) + Ũ(g-1) k,l+1 (x(p), q; p K |q L , q) dy(q) - m d qm Ũ(g) k,l-1 (x(p), q m ; p K |q L/{m} ) y(q) -y(q m ) - m d pm H (g) k-1,l (p m , q; p K/{m} |q L ) (3-52) and E (g) k,l (x(p), y(q)
; p K |q L ) = (y(q) -y(p))U (g) k,l (p, y(q); p K |q L ) + h I,J Ŵ (h) i+1,j (p, p I ; q J )U (g-h) k-i,l-j (p, y(q); p K/I |q L/J ) dx(p) + U (g-1)
k+1,l (p, y(q); p,

p K |q L ) dx(p) - m d pm U (g) k-1,l (p m , y(q); p K/{m} |q L ) x(p) -x(p m ) - m d qm H (g) k,l-1 (p, q m ; p K |q L/{m} ) (3-53)
from which (together with the recursion hypothesis), we deduce that U (g) k,l and Ũ(g) k,l are polynomials.

This proves the theorem 3.2.

Theorem 3.3 Symmetry of the W (g) k,l . For any k, l, g we have:

Ŵ (g-1) k+1,l+1 (p, p K |q L , q) = W (g-1) k+1,l+1 (p, p K |q L , q) (3-54) proof:
Let us prove it by recursion on 2g + k + l. Assume we have already proved it for any g ′ , k ′ , l ′ such that 2g

′ + k ′ + l ′ < 2g + k + l.
Insert Eq. (3-30) into Eq. (3-53) in order to eliminate the U's, and then insert the result into Eq. . Most of the terms cancel (in fact the definitions of J

(g) k,l , J (g) k,l , G (g)
k,l were designed for that purpose), and using the recursion hypothesis, the only term left is:

W (g-1) k+1,l+1 (p, p K |q L , q) = 1 2 W (g-1) k+1,l+1 (p, p K |q L , q) + Ŵ (g-1) k+1,l+1 (p, p K |q L , q) (3-55)
which proves the theorem.

Corollary 3.1 Ŵ (g) k,l (p K |q L ) = W (g) k,l (p K |q L
) is a symmetric function of its variables p 1 , . . . , p k , and a symmetric function of its variables q 1 , . . . , q l .

proof:

It is clear from the definitions that W (g) k,l (p K |q L ) is a symmetric function of its variables p 1 , . . . , p k , and that Ŵ (g) k,l (p K |q L ) is a symmetric function of its variables q 1 , . . . , q l . Now, we prove the following theorem:

Theorem 3.4 Ŵ (g) k,0 (p K |) = Ŵ (g) k (p K ) (3-56) and W (g) 0,l (|q L ) = W (g) l (q L ). (3-57)
proof: Write Eq. (3-53) for l=0:

E (g) k,0 (x(p), y(q); p K ) = (y(q) -y(p))U (g) k,0 (p, y(q); p K ) + h I Ŵ (h) i+1,0 (p, p I )U (g-h)
k-i,0 (p, y(q); p K/I ) dx(p)

+ U (g-1)
k+1,0 (p, y(q); p, p K ) dx(p) -

m d pm U (g)
k-1,0 (p m , y(q); p K/{m} ) x(p) -x(p m ) .

(3 -58)

Using Lemma B.2, we obtain:

Ŵ (g) k,0 (p K |) = Ŵ (g) k (p K ) (3-59)
The other equality is obtained by writing Eq. (3-52) for k = 0 and exchanging the roles of x and y in the Lemma B.2.

Theorem 3.5

Ŵ (g) k+1,l (p, p K |q L ) + W (g) k,l+1 (p K |q L , p) = d p A (g) k,
Ŵ (g) k+1,l (p, p K |q L ) = Res s→a,q L dS s,o (p) f (g) k,l (s; p K |q L ) (3-63)
and we have:

Res p→α x(p)y(p) Ŵ (g) k+1,l (p, p K |q L ) = Res p→α Res s→a,q L x(p)y(p)dS s,o (p) f (g) k,l (s; p K |q L ) = Res s→a,q L Res p→α x(p)y(p)dS s,o (p) f (g) k,l (s; p K |q L ) = -Res s→a,q L (x(s)y(s) -x(o)y(o)) f (g) k,l (s; p K |q L ) (3 -64) since f (g)
k,l vanishes near the poles of ydx to order at least deg ydx -1, the expression above has no other poles than a, q L , and thus the total residue is zero. Theorem 3.7 For any k,l,g such that k + l + g ≤ 1, one has

Res p→a,q L Φ(p) Ŵ (g) k+1,l (p, p K |q L ) = Res q→b,p K Ψ(q) W (g) k,l+1 (p K |q L , q) = (2 -2g -k -l) Ŵ (g) k,l (p K |q L ).
(3-65)

proof:

We have:

Res p→a,q L Φ(p) Ŵ (g) k+1,l (p, p K |q L ) -Res p→b,p K Ψ(p) W (g) k,l+1 (p K |q L , p) = Res p→a,q L x(p)y(p) Ŵ (g) k+1,l (p, p K |q L ) -Res p→a,q L Ψ(p) Ŵ (g) k+1,l (p, p K |q L ) -Res p→b,p K Ψ(p) W (g) k,l+1 (p K |q L , p) = - Res p→a,b,p K ,q L Ψ(p)( Ŵ (g) k+1,l (p, p K |q L ) + W (g) k,l+1 (p K |q L , p)) = Res p→a,b,p K ,q L
x(p)dy(p) A (3-66)

The fact that Res p→a,q L Φ(p)

Ŵ (g) k+1,l (p, p K |q L ) = (2 -2g -k -l) Ŵ (g) k,l (p K |q L )
, can be proved by recursion on 2g + k + l and using corolary 3.1. This allows to prove our main theorem: Theorem 3.8 The F (g) 's are symmetric under the exchange x ↔ y:

F (g) = F (g) (3-67)
proof: Indeed, we have:

(2 -2g) F (g) = Res a Φ(p) Ŵ (g) 1,0 (p) , (2 -2g) F (g) = Res b Ψ(p) W (g) 0,1 (p). (3-68)

Additional properties

The following theorem relates H and W : Theorem 3.9 We have:

Ŵ (g) k+1,l (p, p K |q L ) = Res q→α H (g) k,l (p, q; p K |q L ) H (0) 0,0 (p, q) dy(q) (3-69) W (g) k,l+1 (p K |q L , q) = Res p→α H (g) k,l (p, q; p K |q L ) H (0) 0,0 (p, q)
dx(p).

(3-70)

proof:

Multiply equation 3-30 by dx(p)dy(q)/(y(q) -y(p))H (0) 0,0 (p, q) and take the residues at q → α.

Remark 3.3 This theorem was expected from the matrix model property that

tr 1 x -M 1 1 y -M 2 → 1 x tr 1 y -M 2 (3-71) when x → ∞.

Conclusion

In this article, we have proved the x ↔ y symmetry which was announced in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]. This symmetry has many applications, for instance in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] it was used to recover the (p, q) ↔ (q, p) duality of minimal models [START_REF] Kharchev | On p -q duality and explicit solutions in c < 1 2d gravity models[END_REF], or to give a very short proof that Kontsevitch integral indeed depends only on odd times and satisfies KdV hierarchy [START_REF] Itzykson | Combinatorics of the Modular Group II: The Kontsevich integrals[END_REF].

In addition we have shown how to compute some family of mixed correlation functions of the 2-matrix model.

This could open the route to some matrix model approach to the understanding of boundary conformal field theory in higher genus. In a forthcoming article, we shall introduce a similar algebraic geometry method to compute all possible mixed correlation functions [START_REF] Eyner | Whole topological expnasion of any correlation function in the two matrix model[END_REF]. This work also raises many questions, and calls the following prospects: • It would be interesting to see what the H k,l and W k,l correspond to for other matrix models (e.g. Kontsevitch's integral, chain of matrices), although we may guess that they also correspond to mixed traces expectation values in those cases.

• More interesting would be to understand what the H (g) k,l and W (g) k,l compute in algebraic geometry. Those should correspond to "volume" or "intersection numbers of some moduli spaces" ?

Appendix A Spectral curve

We recall that the curve E(x, y), called the classical spectral curve, is given by a polynomial of the form:

E(x, y) = d 2 +1 j=0 E j (x)y j (1-1)
We define the "quantum spectral curve" as the formal power series:

E N (x, y) = g N -2g E (g) (x, y) (1-2)
where

E (g) (x, y) = E d 2 +1 (x) d 2 r=1 J 1 ∪...∪Jr=K g 1 ,...,gr δ l (g l +|J l |-1),g r l=1 W (g l ) |J l | (p J l ) (1-3) with K = {1, . . . , d 2 } (1-4) and W (g) k (p K ) := W (g) k (p K ) + δ k,1 δ g,0 (y -Y (p 1 )) (1-5)
where W (g) k (p K ) is the meromorphic form defined in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] for the curve E(x, y).

Lemma A.1 For any g, E (g) (x, y) is a polynomial in x and y, whose degrees are at most those of E.

proof:

It is clear that E N (x, y) is a polynomial in y, and a rational function of x. Let us prove that E (g) (x, y) is indeed a polynomial in x for g ≥ 1. The coefficient of y k in E (g) (x, y) is:

E (g) k (x) E d 2 +1 (x) = J 0 ⊂K,|J 0 |=k j∈J 0 y(p j ) d 2 -k r=1 J 1 ∪...∪Jr=K/J 0 g 1 ,...,gr δ l (g l +|J l |-1),g r l=1 W |J l | (g l ) (p J l ) (1-6)
First, notice that the product of W 's can have poles only at branch-points, and the product of y's can have poles only at poles of y. The poles of y which are not poles of x, are killed by the prefactor E d 2 +1 (x), as they are in the classical curve E(x, y). Let us consider the poles at a branch-point a. The only terms which might diverge at p → a are of either of the following forms

• (W (h) 1+|J| (p, p J ) + W (h) 1+|J| (p, p J )) × (reg)
where reg means a term with no poles at p → a. This term is regular because of theorem 4.4 in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF].

• or (W

(g 1 ) 1+|J 1 | (p, p J 1 )W (h-g 1 ) 1+|J|-|J 1 | (p, p J/J 1 ) + W (h-1)
2+|J| (p, p, p J )) × (reg) again, this expression is regular when p → a, because of theorems 4.4 and 4.5 in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF].

Thus, we have proved that E (g) k (x) is a rational function of x whose only poles are the poles of x, i.e. it is a polynomial in x.

Consider a pole ∞ x of x, the behavior of E (g) (x(p), y(p)) when p → ∞ x is at most that of J 0 ⊂K j∈J 0 y(p j ). Notice that J 0 cannot be equal to K itself, because the product of the corresponding W 's vanishes (it contains no term), and |J 0 | cannot be equal to |K| -1, because the prefactor vanishes due to theorem 4.4 in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]. Thus, |J 0 | ≤ |K| -2, which implies that E (g) (x(p), y(p))dx(p) has a pole of degree at most that of E y (x(p), y(p)), i.e. E (g) (x(p), y(p)) is contained in the Newton's polytope of E(x, y). This means that

E (g) (x(p), y(p)) E y (x(p), y(p)) dx(p) (1-7)
is a holomorphic differential.

Appendix B Lemma: unicity of the solution of loop equations

Lemma B. [START_REF] Bergère | Mixed correlation function and spectral curve for the 2-matrix model[END_REF] The system of equations:

E (g) k (x(p), y(q); p K ) = (y(q) -y(p))U (g) k (p, y(q); p K ) + h I W (h) i+1 (p, p I )U (g-h)
k-i (p, y(q); p K/I ) dx(p)

+ U (g-1)
k+1 (p, y(q); p, p K ) dx(p) - where:

• if 2g + k > 2, W (g)
k+1 (p, p K ) has poles only at branchpoints in any of its variables, and vanishing A-cycle integrals,

• E (g) k (x(p), y(q); p K ) is a polynomial in x(p) of degree at most d 1 -1, and a polynomials in y(q) of degree at most d 2 -1,

• U (g) k (p, y(q); p K ) is a polynomials in y(q) of degree at most d 2 -1, has a unique solution.

This solution is such that

W (g) k (p K ) = Ŵ (g) k (p K ). (2-2)
Proof of the Lemma: Unicity:

We prove it by recursion on 2g + k. Assume it is already proved for any g ′ , k ′ such that 2g ′ + k < 2g + k.

At p = q, Eq. (2-1) gives: has no pole at the branchpoints and does not contribute to the residue, and similarly the las term of Eq. (2-3) does not contribute to the residue. We get: W k-i (r, y(r); p K/I ) .

(2-6)

Since all the terms in the RHS are already known from the recursion hypothesis, this determines W (g) k+1 (p, p K ) uniquely. Then, we write Eq. (2-1) for p = qj with j = 1, . . . , d 1 :

E (g) k (x(q j ), y(q); p K ) = h I W (h)
i+1 (q j , p I )U (g-h)

k-i (q j , y(q); p K/I ) dx(q j ) + U (g-1)

k+1 (q j , y(q); qj , p K ) dx(q j ) -

m d pm U (g)
k-1 (p m , y(q); p K/{m} ) x(q j ) -x(p m ) (2 -7) since all terms in the RHS are uniquely determined, so is the LHS. And since E (g) k (x(p), y(q); p K ) is a polynomial in x(p) of degree d 1 -1 and we know its value in d 1 points, then E (g) k (x(p), y(q); p K ) is uniquely determined. Then, using Eq. (2-1) once again, we uniquely determine U (g) k (p, y(q); p K ). This proves the unicity for g and k.

Existence: Start from the meromorphic form W (g) k (p K ) defined in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] for the curve E(x, y), and define: W Therefore, the functions E k (p, y; p K ). Thus, we have found an explicit solution of the system of lemma B.2, which proves the existence.

  (p; p K |q L ) dx(p)dy(p) = -Res p→α x(p)dy(p) A (g) k,l (p; p K |q L ) dx(p)dy(p) = 0.

k- 1

 1 (p m , y(q); p K/{m} ) x(p) -x(p m ) (2 -1)

Wk- 1

 1 (g)k+1 (p, p K ) = E (g) k (x(p), y(p); p K )dx(p) U i+1 (p, p I )U (g-h) k-i (p, y(p); p K/I ) U (p m , y(p); p K/{m} )dx(p) (x(p) -x(p m ))U k+1 (p, p K ) = -Res r→p dS r,o (p) W (g) k+1 (r, p K ).(2-4)Since we know the poles of W (g) k+1 (p, p K ) and its A-cycle integrals, we may move the integration contour using Riemann's bilinear identity and get:W (g) k+1 (p, p K ) = Res r→a dS r,o (p) W (g) k+1 (r, p K ).(2-5)Now, we replace W (g) k+1 (r, p K ) by its value in Eq. (2-3). We see that the termE (g)k (x(r),y(r);p K )dx(r) U

  (g) k (p K ) := W (g) k (p K )/dx(p K ) + δ k,1 δ g,0 (y -y(p 1 )) (2-8)Then, let K 0 = {0, 1, . . . , d 2 } ∪ K and K 1 = {1, . . . , d 2 } ∪ K, and define:E (g) k (x(p 0 ), y; p K ) = E d 2 +1 (x) d 2 +1+kr=1 J 1 ∪...∪Jr=K 0 g 1 ,...,grδ l (g l +|J l |-1),g r l=1 W (g l ) |J l | (p J l ) k (p 0 , y; p K ) = E d 2 +1 (x) d 2 +k r=1 J 1 ∪...∪Jr=K 1 g 1 ,...,gr δ l (g l +|J l |-1),g r l=1 W (g l ) |J l | (p J l ). (2-10) It is clear that both E (g) k (x, y; p K ) and U(g)k (p, y; p K ) are polynomials in y of degree at most d 2 -1. Following the same line as in lemma A.1, it is easy to get that E (g) k (x, y; p K ) is also a polynomial in x of degree at most d 1 -1.

k

  (x, y; p K ), U (g) k (p, y; p K ) and W (g) k (p K ) obey the requirements of lemma B.2, and eq.2-1 is clearly satisfied from the definitions of E (g) k (x, y; p K ) and U (g)

  (p; p K |q L ) has at most simple poles when p → α. K |q L , p) are vanishing, and thus it is the differential of some function. The fact that A (p; p K |q L ) has at most simple poles when p → α, follows from lemma 3.1.

			l (p; p K |q L ) dx(p)dy(p)	(3-60)
	(g) where A k,l proof:
	From Eq. (3-42), it is easy to see that all contour integrals of W (g) k,l+1 (p (g)	Ŵ (g) k+1,l (p, p K |q L ) +
	k,l Theorem 3.6
	Res p→α	x(p)y(p)	Ŵ (g) k+1,l (p, p K |q L ) = 0,		(3-61)
	Res p→α	x(p)y(p)	Ŵ (g) k,l+1 (p K |q L , p) = 0.		(3-62)
	proof:
	By definition:

E-mail: eynard@spht.saclay.cea.fr

E-mail: orantin@spht.saclay.cea.fr

A formal integral is defined as a formal power series in some expansion parameter t, as explained in[START_REF] Eynard | Formal matrix integrals and combinatorics of maps[END_REF] or[START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]. Formal matrix integrals always have a 1/N 2 expansion order by order in t, called the topological expansion.

∞ x is the only point on the curve where the meromorphic function x has a simple pole (see[START_REF] Eynard | Large N expansion of the 2-matrix model, multicut case[END_REF] for further details).

Acknowledgements

We would like to thank Michel Bergère and Aleix Prats Ferrer for fruitful discussions on this subject. This work is partly supported by the Enigma European network MRT-CT-2004-5652, by the ANR project Géométrie et intégrabilité en physique mathématique

G (g)

k,l (p, q; p K |q L ) := J (g) k,l (p, q; p K |q L ) + H (0) 0,0 (p, q) (x(p) -x(q)) Ŵ (g) k+1,l (p, p K |q L )dy(q) + (y(q) -y(p)) W (g) k,l+1 (p K |q L , q)dx(p) ,

k,l (p, q; p K |q L ) := J (g) k,l (p, q; p K |q L ) + H (0) 0,0 (p, q) (x(p) -x(q)) Ŵ (g) k+1,l (p, p K |q L )dy(q) + (y(q) -y(p))

k,l (p, q; p K |q L ) E(x(p), y(q)) := Res

(we prove below that Ĥ(g)

k,l ) as well as

k+1,l (p, q; p,

which is at most finite if p approaches a pole α of ydx. Then it implies that

vanishes at order at least deg(ydx) -1.

The same holds for f.

k,l has poles only when p i → a, q L and q j → b, p K , and

From the definition eq.3-20, it is clear that Ŵ (g) k+1,l (p, p 1 , . . . , p k |q 1 , . . . , q l ) is finite when p is not close to a branch point or to one of the q j 's, and becomes infinite only if the integration contour is pinched. Thus in the variable p, the only poles of Ŵ (g) k+1,l (p, p 1 , . . . , p k |q 1 , . . . , q l ) are at p = a, q L . The poles of Ŵ (g) k+1,l (p, p 1 , . . . , p k |q 1 , . . . , q l ) in any other variable, follow from the recursion hypothesis, and thus they are at p i = a, q L , and at q j = b, p, p K .

The fact that A Ŵ (g) k+1,l = 0 when one integrates over the first variable comes from the fact that this is a property of dS, and in the other variables it comes from the recursion hypothesis.

By a symmetric argument, the same holds for W (g) k,l+1 (p 1 , . . . , p k |q 1 , . . . , q l , p), and we see that Ŵ (g) k,l and W (g) k,l have the same poles. We have (from the Cauchy residue formula and Riemann bilinear identity):

One has:

k,l (p,r;p K ;q L ) (y(q)-y(p))(y(q)-y(r))(x(p)-x(r))H k,l (s,r;p K ;q L ) (y(q)-y(p))(y(q)-y(r))(x(s)-x(r))(x(s)-x(p))H (0) 0,0 (s,r) = Res r→q,p i Res s→p,q j G (g) k,l (s,r;p K ;q L ) (y(q)-y(p))(y(q)-y(r))(x(s)-x(r))(x(s)-x(p))H (0) 0,0 (s,r)

k,l (q j ,q;p K ;q L ) (x(p)-x(q))(x(q j )-x(p))H (0) 0,0 (q j ,q)dy(q) 2 = Ȟ(g) (p,y(q)) E(x(p),y(q))

is a holomorphic 1-form in s, i.e. it has no poles. We have:

(y(p) -y(p i )) (y(q) -y(p))(y(q) -y(p i ))

(x(q) -x(q j )) (x(p) -x(q))(x(p) -x(q j ))

Res s→p i (y(p) -y(s)) (y(q) -y(p))(y(q) -y(s))

k,l (p, q; p K |q L ) = Ȟ(g) k,l (p, q; p K |q L ) = H (g) k,l (p, q; p K |q L ).

• E (g) k,l (p, q; p K |q L ) = Ě(g) k,l (p, q; p K |q L ).

We have from Eq. (3-27)

k,l (p, q, p K |q L ) = (x(p) -x(q))(y(p) -y(q)) H k,l (p, q, p K |q L ) = (x(p) -x(q))(y(p) -y(q)) Ȟ(g) k,l (p, q, p K |q L ) -G (g) k,l (p, q; p K |q L ) dx(p)dy(q) , (3-50)