SPhT-T07/021

Invariants of algebraic curves and topological expansion B. Eynard1 , N. Orantin2 Service de Physique Théorique de Saclay, F-91191 Gif-sur-Yvette Cedex, France.

Abstract:

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a tau function attached to an algebraic curve.

These invariants are constructed in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e. the (p, q) minimal models of conformal field theory.

As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV τ -function.

Introduction

Computing the topological expansion of various matrix integrals has been an interesting problem for more than 30 years. The reason for it, is that formal matrix integrals (cf [START_REF] Eynard | Formal matrix integrals and combinatorics of maps[END_REF] for a definition of formal integrals), are known to be combinatorics generating functionals. Some formal matrix integrals enumerate maps, or colored maps of given topology [START_REF] Brezin | [END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Francesco | 2D gravity, matrix models and graph combinatorics[END_REF], some count intersection numbers (the Kontsevitch integral and its generalizations [START_REF] Kontsevitch | Intersection theory on the moduli space of curves and the matrix Airy function[END_REF]), and physicists have also tried to reach the limit of continuous maps through critical limits, and thus tried to recover Liouville's field theory [START_REF] Knizhnik | [END_REF][START_REF] Di Francesco | Conformal field theory[END_REF].

Many methods have been invented to compute those formal matrix integrals, and the most successful is undoubtedly the "loop equations" method [START_REF] Kazakov | The appearance of matter fields from quantum fluctuations of 2D-garvity[END_REF][START_REF] David | Loop equations and nonperturbative effects in two-dimensional quantum gravity[END_REF], which is in fact nothing but integration by parts, or Tutte's equations [START_REF] Tutte | A census of planar triangulations[END_REF][START_REF] Tutte | A census of planar maps[END_REF], or Schwinger-Dyson equations, or Ward identities, or Virasoro constraints, or W-algebra [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF]. Until recently, those loop equations were solved only for the first few orders (mostly planar or torus), and case by case (for each matrix model). One of the most remarkable methods was obtained in [START_REF] Ambjørn | Matrix model calculations beyond the spherical limit[END_REF]. Let us also mention that other methods were invented using orthogonal polynomials [START_REF] Mehta | Random Matrices[END_REF] (only in the case were the formal integral comes from an actual convergent integral), or topological string theory methods [START_REF] Mariño | Les Houches lectures on matrix models and topological strings[END_REF][START_REF] Dijkgraaf | Matrix Models, Topological Strings, and Supersymmetric Gauge Theories[END_REF][START_REF] Bershadsky | Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes[END_REF] using the so-called holomorphic anomaly equations.

In 2004, a new method for computing the large N expansion of matrix integrals was introduced in [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF], and further developped in [START_REF] Eynard | Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula[END_REF][START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF]. The starting point of that method was not new, it was the same as in [START_REF] Ambjørn | Matrix model calculations beyond the spherical limit[END_REF], it consists in solving the loop equations recursively in powers of the expansion parameter 1/N 2 , where N is the size of the matrix. To leading order, loop equations become algebraic equations, and give rise to an algebraic curve E(x, y) = 0 where E is some polynomial in 2 variables, which we call the "classical spectral curve".

The new feature which was introduced in [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF] was to use contour integrals and functions on the curve rather than on the x-plane as in [START_REF] Ambjørn | Matrix model calculations beyond the spherical limit[END_REF]. When written on the curve, the loop equations, together with the Cauchy residue formula and the Riemann bilinear identity, simplify enormously, and take a very universal structure which can be written entirely in terms of geometric properties of the curve. In other words, the solution of loop equations of many different matrix models, depends only on the properties of the spectral curve, and not on the matrix model which gives that curve. In particular, they can be written for any arbitrary algebraic curve, even for curves which don't come from matrix models. It is thus tempting to define "free energies" for any algebraic curve. This is what we do in this article.

Therefore, in this article, for any arbitrary algebraic curve E(x, y) = 0, we define an infinite sequence of complex numbers F (g) (E), computed as residues of meromorphic forms on the curve. Out of these F (g) (E)'s, we build a formal power series:

ln Z N (E) = - ∞ g=0 N 2-2g F (g) (E) (1-1)
and we study its properties.

We compute the variations of F (g) under variations of the curve (variations of its complex structure, its moduli, and modular transformations). We show that the F (g) 's are invariant under some transformations of the curve, namely under transformations of the curve which preserve the symplectic form up to a sign ±dx ∧ dy.

We also show that Z N (E) satisfies bilinear Hirota equations, and thus Z N (E) is a formal τ -function, and we construct the associated formal Baker-Hakiezer function [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF].

We thus have a notion of a τ function associated to an algebraic curve. Such notion has already been encountered in the litterature [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF], and it is not clear whether our definition coincides with other existing definitions. What can be understood so far, is that we are defining a sort of quantum deformation of a classical τ -function whose spectral curve is E. The classical τ function being only the dispersionless limit ln Z ∞ (E) = -F (0) (E), while our Z N (E) concerns the full system.

Almost by definition, if E is the algebraic curve coming from the large N limit of the loop equations of a matrix model, then Z N (E) is the matrix integral.

What is more interesting is to see what is Z N (E) for curves not coming from the large N limit of the loop equations of a matrix model.

We study in details a few examples.

-The double scaling limit of a matrix model. It has been well known since [START_REF] Va Kazakov | Critical properties of randomly triangulated planar random surfaces[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF], that if we fine tune the parameters of a matrix model so that the algebraic curve E develops a singularity, the free energies become singular and the most singular part of the free energies form the KP-hierarchy τ -function (KdV hierarchy for the 1-matrix model). We show, by looking at the double scaling limit of matrix models, that the KP τ -function (resp. KdV τ -function), coincides with our definition for the classical limit of the (p, q) systems (resp. (p, 2)).

-It has been well known since the works of Kontsevitch [START_REF] Kontsevitch | Intersection theory on the moduli space of curves and the matrix Airy function[END_REF], that the KdV τfunction can be represented by another matrix integral called Kontsevitch integral. Kontsevitch introduced that integral as a counting function for intersection numbers, and proved that it is a KdV τ -function. One of the key features is that it depends on the eigenvalues of a diagonal matrix Λ, only through the quantities t k = tr Λ -k for odd k (cf [START_REF] Itzykson | Combinatorics of the Modular Group II: The Kontsevich integrals[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF]). Another important known property is that, if t k = 0 for k > p, it coincides with the (p, 2) τ -function found from the double scaling limit of the 1-matrix model, i.e. the (p, 2) conformal minimal model.

Here, we prove that the Kontsevitch matrix integral coincides with our Z N (E) when E is the large N limit of the Schwinger-Dyson equation of the Kontsevitch integral. The remarkable fact, is that for our Z N (E), the above properties (i.e. the fact that it depends only on odd t k 's and the fact that it gives the (p, 2) τ -function if t k = 0 for k > p) are trivial. We thus provide a new proof of those properties, and maybe a new interpretation.

Main results of this article

In this section we just sketch briefly the contents of the main body of the article.

Definitions

Given a polynomial of two variables E(x, y), we construct an infinite sequence of multilinear meromorphic forms over the curve of equation E(x, y) = 0, which we call:

W (g) k (p 1 , p 2 , . . . , p k )
, k, g ∈ N.

(2-1)

In particular W (g) 0

= -F (g) are complex numbers F (g) (E). The F (g) 's and the W (g) k 's are defined in terms of residues near the branchpoints of the curve only, i.e. they depend only on the local behavior of the curve near its branch points.

Then we show some properties:

• the W (g) k 's are symmetric in their k variables;

• there is a "loop insertion operator" which increases k → k + 1:

D B(p k+1 ,.) W (g) k (p 1 , p 2 , . . . , p k ) = W (g)
k+1 (p 1 , p 2 , . . . , p k , p k+1 );

(2-2)

• there is an inverse operator which contracts k → k -1:

Res

p k →branch points Φ(p k ) W (g) k (p 1 , p 2 , . . . , p k ) = (2g + k -3) W (g)
k-1 (p 1 , p 2 , . . . , p k-1 ).

(2-3)

The F (g) 's and the W (g) k 's are defined in a way which mimics the solution of matrix models loop equations, and almost by definition, they coincide with matrix model's free energy and correlation functions when the polynomial E is chosen as the classical large N limit of the matrix model's spectral curve:

ln dM exp -N Tr V (M) = - ∞ g=0 N 2-2g F (g) (E 1MM ) (2-4)
and

Tr dx 1 x 1 -M Tr dx 2 x 2 -M . . . Tr dx k x k -M = ∞ g=0 N 2-2g-k W (g)
k (p(x 1 ), p(x 2 ), . . . , p(x k ))

(2-5) The same construction works also for the 2-matrix model and the matrix model in an external field:

F (g) 1MM = F (g) (E 1MM ) F (g) 2MM = F (g) (E 2MM ) F (g) ext.field = F (g) (E ext.field ).
( [START_REF] Aganagic | Topological strings and integrable hierarchies[END_REF][START_REF] Akemann | Higher genus correlators for the Hermitian matrix model with multiple cuts[END_REF][START_REF] Akemann | New universal spectral correlators[END_REF][START_REF] Alexandrov | M-Theory of Matrix Models[END_REF][START_REF] Alexandrov | Instantons and Merons in Matrix Models[END_REF] in particular it works for the Kontsevitch integral

F (g) Kontsevitch = F (g) (E Kontsevitch ) (2-7)
where the LHS is the topological expansion of the corresponding matrix integral, and the RHS is the functional defined in this article, applied to the curve E(x, y) = 0 coming from the large N limit of the Schwinger-Dyson equations of the corresponding matrix model. Let us emphasize that not every curve E is the large N limit of a matrix model's spectral curve, and thus our functional F (g) (E) is defined beyond matrix models, and is really an algebro-geometric object. It has many remarkable properties, and we list below some of the most important ones: where G

Remarkable properties

(g) k+1 (p, p 1 , . . . , p k ) is a set of trivalent graphs (built on trees), and w is a Feynmanlike weight function associating values to edges and integrals (residues in fact) to vertices of the graph.

This theorem is important because it makes all formulae particularly easy to remember, and many theorems below can be proved in a diagrammatic way. Theorem 8.1 Singular limits: if the curve becomes singular, the functional F (g) commutes with the singular limit, i.e.: lim

F (g) (E) = F (g) (lim E).
(2-9)

Theorem 9.2 Integrability: the formal series:

ln Z N (E) = - ∞ g=0 N 2-2g F (g) (E) (2-10)
obeys Hirota's bilinear equations, and thus is a τ function.

Theorem 5.3 Homogeneity : F (g) (E) is homogeneous of degree 2 -2g in the moduli of the curve:

(2 -2g)F (g) = k t k ∂F (g) ∂t k . (2-11) Theorem 5.1 Deformations : if the curve E is deformed into E + δE, the differential ydx is deformed into ydx → ydx + δ(ydx)
where δ(ydx) is a meromorphic one-form which we denote δ(ydx) = -Ω, and which can be written as:

Ω = ∂Ω W (0)
2 Λ for some appropriate contour ∂Ω and some appropriate function Λ. Then we have:

δW (g) k (p 1 , p 2 , . . . , p k ) = ∂Ω Λ(p k+1 ) W (g) k+1 (p 1 , p 2 , . . . , p k , p k+1 ).
(2-12)

Theorem 7.1 Symplectic invariance : F (g) (E) is unchanged under the following changes of curve E(x, y):

y → y + R(x) , R(x)=rational fraction of x, y → cy , x → c -1 x , c=complex number, y → -y , x → x , y → x , x → y .
(2-13) all those transformations conserve the symplectic form dx ∧ dy up to the sign. Theorem 6.2 Modular transformations : the modular dependence of F (g) (E) is only in the Bergmann kernel (defined in section 3.1.5), and thus the modular transformations of F (g) (E) are derived from those of the Bergmann kernel. Under a modular transformation, the Bergmann kernel is changed into B(p, q) → B(p, q) + 2iπ du(p)κdu(q), and thus we introduce a new kernel for any arbitrary symmetric matrix κ:

B κ (p, q) → B(p, q) + 2iπ du(p)κdu(q) (2-14)
We thus define some

F (g)
κ (E), and we compute:

∂F (g) ∂κ (2-15)
We also remark that when κ = (τ -τ ) -1 , F

κ (E) is modular invariant.

Some applications, Kontsevitch's integral

(p, q) minimal models, KP and KdV Hierarchies: it is well known [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF] that some rational singular limits of matrix models correspond to (p, q) minimal models, and theorem 8.1 implies that:

F (g) (p,q) = F (g) (E (p,q) ) (2-16)
and it is well known that (p, q) minimal models are some reductions of KdV hierarchy for q = 2 and KP hierarchy for general (p, q). Notice that the x ↔ y symmetry of theorem 7.1 (i.e. Eq. (2-13)) implies the famous (p, q) ↔ (q, p) duality [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Di Francesco | Conformal field theory[END_REF][START_REF] Kharchev | On p -q duality and explicit solutions in c < 1 2d gravity models[END_REF]. Kontsevitch integral's properties: Kontsevitch's integral is defined as:

Z Kontsevitch (Λ) = dM e -N Tr M 3 3 -M Λ 2 , ln Z Kontsevitch = - ∞ g=0 N 2-2g F (g) Kontsevitch
(2-17) and we define the Kontsevitch's times:

t k = 1 N Tr Λ -k . (2-18)
It is straightforward to write the Schwinger-Dyson equations and find the classical spectral curve:

E Kontsevitch = x(z) = z + 1 2N Tr 1 Λ 1 z-Λ y(z) = z 2 + t 1 . (2-19)
According to theorem 10.3, we have:

F (g) Kontsevitch = F (g) (E Kontsevitch ). (2-20)
Using the x ↔ y invariance of eq.2-13, we see that the only branch point in y is located at z = 0, and since the F (g) 's depend only on the local behavior near the branchpoint, we may perform a Taylor expansion of x(z) near z = 0:

E Kontsevitch (t 1 , t 2 , . . .) = x(z) = z -1 2 ∞ k=0 t k+2 z k y(z) = z 2 + t 1 . (2-21)
From the symplectic invariance theorem 7.1, (i.e. eq.2-13), we may add to x any rational function of y i.e. of z 2 , thus we may substract to x its even part, and thus the following curves are related by symplectic invariance:

E Kontsevitch (t 1 , t 2 , t 3 , . . .) ∼ E Kontsevitch (t 1 , 0, t 3 , 0, t 5 , . . .). (2-22)
We thus have a very easy proof that F (g) Kontsevitch depends only on odd times. Moreover, if t k = 0 for k > p + 2, we have:

E Kontsevitch (t 1 , t 2 , . . . , t p+2 , 0, . . .) = x(z) = z -1 2 p k=0 t k+2 z k y(z) = z 2 + t 1 (2-23)
which is exactly the curve of the (p, 2) minimal model, i.e. it satisfies KdV hierarchy. We thus have a very easy proof that Z Kontsevitch is a KdV tau-function.

Those are old and classical results about the Kontsevitch integral, and we just propose a new proof, in order to illustrate the power of the tools we introduce.

Algebraic curves, reminder and notations

We begin by recalling some elements of algebraic geometry, which are used to fix the notations. We refer the reader to [START_REF] Farkas | Riemann surfaces[END_REF] or [START_REF] Fay | Theta functions on Riemann surfaces[END_REF] for further details about algebro-geometric concepts.

-→ Summary of notations E(x, y) = 0 -→ classical spectral curve.

d 1 + 1 = deg x E -→ x-degree of the polynomial E. d 2 + 1 = deg y E -→ y-degree of the polynomial E (number of sheets). a = {a i } -→ set of branch points dx(a i ) = 0. α = {α i } -→ poles of ydx. g / -→ genus of the curve. A i ∩ B j = δ ij -→ cannonical basis of non-contractible cycles. du i -→ cannonical holomorphic forms A j du i = δ ij . τ ij = B j du i -→ Riemann's matrix of periods. u i (p) = p p 0 du i -→ Abel map. A = A -κ(B -τ A) -→ κ-modified A-cycles B = B -τ A -→ κ-modified B-cycles, A i ∩ B j = δ ij dS q 1 ,q 2 (p)
-→ 3rd kind differential with simple poles q 1 and q 2 , such that Res q 1 dS q 1 ,q 2 = 1 = -Res q 2 dS q 1 ,q 2 and A i dS q 1 ,q 2 = 0. B(p, q) -→ Bergmann kernel, i.e. 2nd kind differential with double pole at p = q, no residue and vanishing A-

cycle integrals. z = n+τ m 2 -→ regular odd characterisic, i.e. g i=1 n i m i =odd. dh z = i ∂θz( v) ∂v i v=0
du i -→ Holomorphic form with only double zeroes.

E(p, q) = θz(u(p)-u(q),τ ) √ dhz(p)dhz (q)
-→ Prime form independent of z, with a simple zero at p = q. Φ(p) = p o ydx -→ some antiderivative of ydx defined on the universal covering. p, x(p) = x(p) -→ if p is near a branchpoints a, then p = p is the unique other point near a such that x(p) = x(p). p i (p), x(p i ) = x(p) -→ The p i 's, i = 0, . . . , d 2 , are the pre-images of x(p) on the curve. By convention p 0 (p) = p.

D Ω = δ Ω + tr (κ δ Ω τ κ ∂ ∂κ ) -→ Covariant variation wrt Ω = δ(ydx).
Consider an (embedded) algebraic curve given by its equation:

E(x, y) = 0 (3-1)
where E is an almost arbitrary polynomial of two variables. This is equivalent to considering a compact Riemann surface Σ and 2 meromorphic functions x and y, such

that ∀p ∈ Σ , E(x(p), y(p)) = 0. (3-2)
We only require that E(x, y) is not factorizable, and that all branchpoints (zeroes of dx) are simple, i.e. near a branchpoint a i , y behaves like a square root x -x(a i ).

a 1 a 2 y x x(p) x(q) q p (2) p (2)
q q p (3-4)

Figure 1: Example of an algebraic curve with two x-branch points a 1 and a 2 and a three sheeted structure (x has three preimages). One can see that the map p → p is not globally defined, for instance when q → p, we have q → p (2) . The notion of conjugated point depends on the branch point.

Some properties of algebraic curves

Sheets

For each complex x, there exist d 2 + 1 = deg y E solutions for y of E(x, y) = 0. This means that there are exactly d 2 +1 points on the Riemann surface Σ for which x(p) = x: Σ has a sheet structure with d 2 + 1 x-sheets. We call them:

x(p) = x ↔ p = p i (x) , i = 0, . . . , d 2 .

(3-3)

Branch points and conjugated points

Let a i , i = 1, . . . , n, a = {a 1 , . . . , a n } be the set of branch-points, solutions of dx = 0:

∀a ∈ a , dx(a) = 0. (3-5)
Since we assume that the branch-points are simple zeros of dx, we have the following property: if p is in the vicinity of a branch-point a i , there is a unique point p = p, such that x(p) = x(p), which is also in the vicinity of a i . p depends on i, and in general, p is not globaly defined (see fig. 1 for an example).

Notice that p is one of the p k defined in the previous section.

Genus, cycles, Abel map

If the curve has genus g / , there are 2g / homologicaly independent non-trivial cycles, and we may choose a (not unique) cannonical basis:

A i ∩ B j = δ ij , A i ∩ A j = 0 , B i ∩ B j = 0. (3-6) A B ⇔ B A (3-11)
Figure 2: Example of canonical cycles and the corresponding fundamental domain in the case of the torus (genus g / = 1).

The simply connected domain obtained by removing all A and B-cycles from the curve is called the "fundamental domain" (see fig.

(2) for the example of the torus). On a genus g / curve, there are g / linearly independent holomorphic forms du 1 , . . . , du g / , which we choose normalized on the A-cycles:

A j du i = δ ij . (3-7)
The Riemann matrix of period is defined by the B-cycles

τ ij = B j du i . (3-8)
They have the property that

τ ij = τ ji , Im τ > 0. (3-9)
Given a base point p 0 on the curve (we assume it is not on any A or B-cycle), we define the Abel map

u i (p) = p p 0 du i (3-10)
where the integration path is in the fundamental domain. The g / -dimensional vector u(p) = (u 1 (p), . . . , u g / (p)) maps the curve into its Jacobian.

Theta-functions and prime forms

We say that z ∈ C g / is a characteristic if there exist two vectors with integer coefficients a ∈ Z g / and b ∈ Z g / such that:

z = a + τ. b 2 . (3-12) z is called an odd characteristic if g / i=1 a i b i = odd. (3-13)
Given a characteristic z = a+τ b 2 , and given a symmetric matrix τ ij = τ ji such that Imτ is positive definite, and a vector v ∈ C g / , we define the θ z function:

θ z (v, τ ) = n∈Z g / e iπ( n-b/2) t τ ( n-b/2) e 2iπ(v+ a/2) t .( n+ b/2) . (3-14)
If z is an odd characteristic, θ z is an odd function of v, and in particular θ z (0, τ ) = 0 and we define the following holomorphic form:

dh z (p) = g / i=1 du i (p) . ∂θ z (v) ∂v i v=0 . (3-15)
All its g / -1 zeroes are double zeroes, so that it makes sense to consider its square root defined on the fundamental domain. The prime form is:

E(p, q) = θ z (u(p) -u(q)) dh z (p) dh z (q) . (3-16)
It is independent of z, and it vanishes only if p = q (and with a simple zero), and has no pole.

Bergmann kernel

We define a bilinear meromorphic form called the "Bergmann kernel" [START_REF] Bergman | Kernel functions and elliptic differential equations in mathematical physics[END_REF][START_REF] Fay | Fourrier coefficients of the resolvent for a Fuchsian group[END_REF]:

B(p, q) = Bergmann kernel (3-17)
as the unique 1-form in p, which has a double pole with no residue at p = q and no other pole, and which is normalized such that B(p, q) ∼ p→q dz(p)dz(q) (z(p) -z(q)) 2 + finite ,

A I B = 0 (3-18)
where z is any local coordinate on the curve in the vicinity of q. The Bergmann kernel depends only on the complex structure of the curve, and not on the details of E. For instance if E has genus zero, B is the Bergmann kernel of the projective complex plane (the Riemann sphere), and if the curve has genus 1, B is related to the Weierstrass function.

Properties:

B(p, q) = B(q, p) ,

q∈B i B(p, q) = 2iπ du i (p). (3-19)
For any odd characteristic z, we have:

B(p, q) = d p d q ln (θ z (u(p) -u(q))) (3-20)
If f (p) is any meromorphic function, its differential is given by

df (p) = Res q→p B(p, q)f (q).
(3-21)

3rd type differentials

Given two points q 1 and q 2 on the curve , we define the 1-form dS q 1 ,q 2 by:

dS q 1 ,q 2 (p) = q 1 q 2 B(p, q) (3-22)
where the integration path is in the fundamental domain. dS q 1 ,q 2 is the unique meromorphic form with only simple poles at q 1 and q 2 , such that:

Res

q 1 dS q 1 ,q 2 = 1 = -Res q 2 dS q 1 ,q 2 , A i dS q 1 ,q 2 = 0. (3-23)
3.1.7 Modified set of cycles:

In order to easily deal with modular properties of the objects we are going to introduce, it is convenient to define some modified cycles and kernels with an arbitrary symmetric matrix κ. When κ = 0, all those quantities reduce to the unmodified ones. The modular transformations of the modified objects, merely amount to a change of κ.

We thus choose an arbitrary g / × g / symmetric matrix κ with complex coefficients, and we define another set of cycles:

A i = A i - j κ ij (B j - l τ jl A l ) B i = B i - j τ ij A j (3-24)
They satisfy:

A i ∩ B j = δ ij , A i ∩ A j = 0 , B i ∩ B j = 0 (3-25)
and we straightforwardly have

A i du j = δ ij , B i du j = 0.
(3-26)

Modified Bergmann kernel

We also define the modified Bergmann kernel, normalized on A instead of A:

B(p, q) = B(p, q) + 2iπ i,j du i (p) κ ij du j (q). (3-27)
It is such that B(p, q) = B(q, p) ,

A I B = 0 , q∈B i B(p, q) = 2iπ du i (p) (3-28)
and if f (p) is any meromorphic function, its differential is given by:

df (p) = Res q→p B(p, q)f (q).
(3-29)

• For κ = 0 we have B = B.

• For κ = (τ -τ ) -1 , B is the Schiffer kernel [START_REF] Bergman | Kernel functions and elliptic differential equations in mathematical physics[END_REF][START_REF] Fay | Fourrier coefficients of the resolvent for a Fuchsian group[END_REF], and it is modular invariant.

Modified prime form

Similarly we define a modified prime form:

E(p, q) = E(p, q) e 2iπu t (p)κu(q) (3-30)
It vanishes only if p = q (with a simple zero), and has no pole.

Modified 3rd type differentials

In the same fashion, we define the modified 3rd type differentials dS q 1 ,q 2 by:

dS q 1 ,q 2 (p) = q 1 q 2 B(p, q) (3-31)
where the integration path is in the fundamental domain. dS q 1 ,q 2 is the unique meromorphic form with only simple poles at q 1 and q 2 , such that:

Res

q 1 dS q 1 ,q 2 = 1 = -Res q 2
dS q 1 ,q 2 , A i dS q 1 ,q 2 = 0.

(3-32)

Properties:

dS q 1 ,q 2 = -dS q 2 ,q 1 (3-33)

dS q 1 ,q 2 (p) = d p ln θ z (u(p) -u(q 1 )) θ z (u(p) -u(q 2 )) + 2iπ i,j du i (p)κ ij (u j (q 1 ) -u j (q 2 )) (3-34) B i dS q 1 ,q 2 = 2iπ(u i (q 1 ) -u i (q 2 )) (3-35) d q 1 (dS q 1 ,q 2 (p)) = B(q 1 , p) (3-36) p 2 p 1 dS q 1 ,q 2 = q 2 q 1 dS p 1 ,p 2 (3-37)
Cauchy residue formula: for any meromorphic function f (p) we have:

f (p) = -Res q 1 →p
dS q 1 ,q 2 (p)f (q 1 ).

(3-38)

Bergmann tau function

The Bergmann τ -function τ Bx was introduced and studied in [START_REF] Kokotov | Bergmann tau-function on Hurwitz spaces and its applications[END_REF][START_REF] Kokotov | Tau functions on Hurwitz spaces[END_REF][START_REF] Eynard | 1/N 2 corrections to free energy in Hermitian two-matrix model[END_REF], it is such that

∂ ln (τ Bx ) ∂x(a i ) = Res p→a i B(p, p) dx(p) (3-39)
It is well defined because the Rauch variational formula [START_REF] Rauch | Weierstrass points, branch points, and moduli of Riemann surfaces[END_REF] implies that the RHS is a closed form. Notice that τ Bx is defined only up to a multiplicative constant which will play no role in all the sequel.

3.2 Examples: genus 0 and 1

• Genus 0 If the curve E has a genus g / = 0, it is conformaly equivalent to the Riemann sphere, i.e. the complex plane with a point at ∞, and there exists a rationnal parametrization of the curve. It means that there exists two rational functions X(p) and Y (p) such that:

E(x, y) = 0 ↔ ∃p ∈ C , x = X(p) , y = Y (p) (3-40)
In this case, the Bergmann kernel is the Bergmann kernel of the Riemann sphere:

B(p, q) = B(p, q) = dpdq (p -q) 2 = d p d q ln (p -q).
(3-41)

The prime form is:

E(p, q) = E(p, q) = p -q √ dp dq . (3-42)
• Genus 1

If the curve has genus g / = 1, then it can be parametrized on a rhombus corresonding to the fundamental domain of a torus (see fig. 2). It means that there exists two elliptical functions X(p) and Y (p) such that (see [START_REF] Wang | Special Functions[END_REF] for elliptical functions):

E(x, y) = 0 ↔ ∃p ∈ C , x = X(p) , y = Y (p) X(p + 1) = X(p + τ ) = X(p) , Y (p + 1) = Y (p + τ ) = Y (p) (3-43)
Then, the Bergmann kernel is the corresponding Weierstrass function [START_REF] Wang | Special Functions[END_REF]:

B(p, q) = (℘(p -q, τ ) + π Imτ ) dpdq. (3-44)
The prime form is: -45) When κ = -1 2iImτ , the modified Bergmann kernel is the Schiffer kernel, and if g / = 1 it is the Weierstrass function:

E(p, q) = θ 1 (p -q, τ ) θ ′ 1 (0, τ ) √ dp dq . ( 3
B(p, q) = ℘(p -q, τ ) dpdq.
(3-46)

Riemann bilinear identity

If ω 1 and ω 2 are two meromorphic forms on the curve. Let p 0 be an arbitrary base point, we consider the function Φ 1 defined on the fundamental domain by

Φ 1 (p) = p p 0 ω 1 . (3-47)
We have

Res

p→all poles Φ 1 (p)ω 2 (p) = 1 2iπ g / i=1 A i ω 1 B i ω 2 - B i ω 1 A i ω 2 .
(3-48)

Note that this identity holds also for the modified cycles with any κ:

Res p→all poles Φ 1 (p)ω 2 (p) = 1 2iπ g / i=1 A i ω 1 B i ω 2 - B i ω 1 A i ω 2 .
(3-49)

In particular with ω 1 (p) = B(p, q), we have:

Res p→all poles dS p,p 0 (q) ω(p) = - g / i=1 du i (q) A i ω (3-50)
and:

ω(q) = Res p→poles of ω dS p,p 0 (q) ω(p) + g / i=1 du i (q) A i ω.
(3-51)

Moduli of the curve

The curve E(x, y) = 0 is parameterized by:

• a genus g / compact Riemann surface Σ, with periods τ ij .

• punctures α i at the poles of x and y, whose moduli are given by the negative coefficients of the Laurent series of ydx near the poles.

• the A-cycle integrals of ydx, called filling fractions.

Filling fractions

We define:

ǫ i = 1 2iπ A i ydx (3-52)
which are called "filling fractions" by analogy with matrix models ( [START_REF] Eynard | Large N expansion of the 2-matrix model, multicut case[END_REF]).

Moduli of the poles

Consider a pole α of ydx, define the "temperatures":

t α = Res α ydx. (3-53)
Notice that:

α t α = 0. (3-54)
Then consider the 3 cases:

• Either α is a pole of x of degree d α , then we define the local parameter near α as:

z α (p) = x(p) 1 dα ; (3-55)
• or α is not a pole of x neither a branchpoint (thus it is a pole of y), then we define the local parameter near α as:

z α (p) = 1 x(p) -x(α) . (3-56)
• or α is not a pole of x, and it is a branchpoint (thus it is a pole of y), then we define the local parameter near α as:

z α (p) = 1 x(p) -x(α) . (3-57)
In all cases, in the vicinity of α, we define the "potential"

V α (p) = Res q→α y(q)dx(q) ln 1 - z α (p) z α (q) (3-58)
which is a polynomial in z α (p):

V α (p) = deg Vα k=1 t α,k z k α (p). (3-59)
The t α,k are the moduli of the pole α.

We have the following properties: We have from Eq. (3-51):

dV α (p) = Res q→α y(q)dx(q) dz α (p) z α (p) -z α (q) , ( 3
y(p)dx(p) = - α Res q→α B(p, q)V α (q) + α t α dS α,o (p) + 2iπ i ǫ i du i (p). (3-63) If we introduce B α,k (p) = -Res q→α B(p, q) z α (q) k , (3-64)
we can turn this expression to

ydx = α,k t α,k B α,k + α t α dS α,o + 2iπ i ǫ i du i (p) (3-65)
in order to exhibit the moduli of the curve.

Definition of Correlation functions and free energies

In all this section, the curve E(x, y) = 0 and a symmetric matrix κ are given and fixed.

The unfamiliar reader may choose κ = 0 since most usual applications (matrix models) correspond to that case.

Notations

Consider an arbitrary point p ∈ Σ, and a point q of Σ which is in the vicinity of a branch point a i (so that q is well defined). We define: Definition 4.1 Diagrammatic rules:

vertex : ω(q) = (y(q) -y(q))dx(q) (4-1) line -propagator :

B(p, q) (4-2)
arrow -propagator :

dE q (p) = 1 2 q q B(ξ, p) (4-3)
where the integration path is a path which lies entirely in a vincinity of a i (thus it is uniquely defined) 3 .

root : Φ(q) = q o ydx (4-4)
where o is an arbitrary base point on the curve, i.e. Φ is an arbitrary antiderivative of ydx, i.e. dΦ = ydx.

The reason why we call these objects diagramatic rules and vertices, propagator or root, is explained in section 4.5 below. Notice that dE depends on i, i.e. on which branchpoint we are considering, but we omit to mention the index i in order to make the notations easier to read. In all what follows it is always clear which i is being considered.

Notation for subset of indices:

Given a set of points of the curve {p 1 , p 2 , . . . , p n }, if K = {i 1 , i 2 , . . . , i k } is any subset of {1, 2, . . . , n}, we denote:

p K = {p i 1 , p i 2 , . . . , p i k } (4-5)
4.2 Correlation functions and free energies

Correlation functions

The k-point correlation functions to order g, W

k , are meromorphic multilinear forms, defined by the following recursive triangular system:

Definition 4.2 Correlation functions W (g) k = 0 if g < 0 (4-6) W (0) 1 (p) = 0 (4-7) W (0) 2 (p 1 , p 2 ) = B(p 1 , p 2 ) (4-8)
and define recursively (remember that p K is a k-uplet of points cf eq.4-5):

W (g) k+1 (p, p K ) = Res q→a dEq(p) ω(q) g m=0 J⊂K W (m) |J|+1 (q, p J )W (g-m) k-|J|+1 (q, p K/J ) + W (g-1) k+2 (q, q, p K ) (4-9)
This system is triangular because all terms in the RHS have lower 2g + k than in the LHS and given

W (0) 1 and W (0)
2 , it has a unique solution. Notice that W (g) k+1 (p, p 1 , . . . , p k ) is a multilinear meromorphic form in each of its arguments, it is clearly symmetric in the last k-ones, and we prove below (theorem 4.6) that it is in fact symmetric in all its arguments.

More properties of W (g) k+1 are studied below in section 4.4.

Free energies

We define the free energies which are complex numbers: Definition 4.3 Free energies.

For g > 1

F (g) = 1 2 -2g i Res q→a i Φ(q)W (g) 1 (q) (4-10)
and

F (1) = - 1 2 ln (τ Bx ) - 1 24 ln i y ′ (a i ) -ln (det κ) (4-11) where y ′ (a i ) = dy(a i ) dz i (a i ) , z i (p) = x(p) -x(a i ) (4-12)
and τ Bx is the Bergmann τ -function defined in Eq. . F (0) is defined in the next section.

4.2.3

Leading order free energy F (0) .

Let us define F (0) as follows :

F (0) = 1 2 α Res α V α ydx + 1 2 α t α µ α - 1 4iπ i A i ydx B i ydx (4-13)
where µ α is given by

µ α = o α (ydx -dV α + t α dz α z α ) + V α (o) -t α ln (z α (o)) (4-14)
Notice that µ α depends on some base point o, but the sum α t α µ α does not.

Special free energies and correlation functions

All the quantities defined so far, were defined with the κ-modified cycles and modified Bergmann kernel. Let us also define them for κ = 0 (for instance F (1) obviously needs another definition). Therefore we also define the unmodified quantities corresponding to κ = 0, as:

∀k, g, W (g) k (p 1 , . . . , p k ) := W (g) k (p 1 , . . . , p k ) κ=0 , (4-15)
for g > 1,

F (g) := 1 2 -2g i Res q→a i Φ(q)W (g) 1 (q) = F (g) κ=0 , (4-16
)

F (1) = - 1 2 ln (τ Bx ) - 1 24 ln i y ′ (a i ) , (4-17)
and

F (0) = 1 2 α Res α V α ydx + 1 2 α t α µ α - 1 4iπ i A i ydx B i ydx . (4-18)
Remark 4.1 The special functions, except F (1) and F (0) , are obtained by changing B and dS by B and dS in the diagrammatic rules defined in section 4.5.

Tau function Definition 4.4

We define the tau-function as the formal power series in N -2 :

ln (Z N (E)) = - ∞ g=0 N 2-2g F (g) . (4-19)
We show in section 9 that Z N (E) is indeed a tau-function because it obeys Hirota equations, order by order in N -2 .

Properties of correlation functions

The loop functions defined in definition 4.2 satisfy the following theorems, whose proofs can be found in Appendix A: is worth:

W (0) 3 (p, p 1 , p 2 ) = Res q→a B(q, p)B(q, p 1 )B(q, p 2 ) dx(q)dy(q) (4-20)
In particular, W

is symmetric in its 3 variables.

∀i = 1, . . . , g / p∈A i W (g) k+1 (p, p 1 , . . . , p k ) = 0, (4-21) ∀i = 1, . . . , g / , ∀m = 1, . . . , k pm∈A i W (g) k+1 (p, p 1 , . . . , p k ) = 0. (4-22)
Theorem 4.4 For every k and g, we have:

i W (g) k+1 (p i , p 1 , . . . , p k ) dx(p i ) = δ k,1 δ g,0 dx(p 1 ) (x(p) -x(p 1 )) 2 (4-23) and if k ≥ 1: i W (g) k+1 (p 1 , p i , p 2 , . . . , p k ) dx(p i ) = δ k,1 δ g,0 dx(p 1 ) (x(p) -x(p 1 )) 2 (4-24)
where we recall that p i are all the points such that x(p i ) = x(p) (see section 3.1).

Theorem 4.5 For (k, g) = (0, 1),

P (g) k (x(p), p K ) = 1 dx(p) 2 i -2y(p i )dx(p)W (g) k+1 (p i , p K ) + W (g-1) k+2 (p i , p i , p K ) + g m=0 J⊂K W (m) j+1 (p i , p J )W (g-m) k-j+1 (p i , p K/J ) (4 -25)
is a rational function of x(p), with no poles at branch-points.

p→a i W (g) k+1 (p, p 1 , . . . , p k ) = 0, (4-26)
∀i, Res

p→a i x(p)W (g) k+1 (p, p 1 , . . . , p k ) = 0, (4-27)
i

Res

p→a i y(p)W (g) k+1 (p, p 1 , . . . , p k ) = 0, (4-28)
i

Res p→a i x(p)y(p)W (g) k+1 (p, p 1 , . . . , p k ) = 0. (4-29)
Theorem 4.7 For k ≥ 1 we have:

Res p k+1 →a,p 1 ,...,p k Φ(p k+1 )W (g) k+1 (p K , p k+1 ) = (2g + k -2)W (g) k (p K ) + δ g,0 δ k,1 y(p 1 )dx(p 1 ). (4-30)
Notice that for k = 0 and g ≥ 2, it holds by definition if we define W (g) 0

= -F (g) .

Diagrammatic representation

The recursive definitions of W (g) k

and F (g) can be represented graphically. We represent the multilinear form W (g) k (p 1 , . . . , p k ) as a blob-like "surface" with g holes and k legs (or punctures) labeled with the variables p 1 , . . . , p k , and F (g) with 0 legs and g holes.

W (g) k+1 (p, p 1 , . . . , p k ) := (g) p p p p 1 2 k , F (g) := (g) (4-31) We represent the Bergmann kernel B(p, q) (which is also W (0)
2 , i.e. a blob with 2 legs and no hole) as a straight non-oriented line between p and q B(p, q) := p q .

(4-32)

We represent dEq(p) ω(q) as a straight arrowed line with the arrow from p towards q, and with a tri-valent vertex whose legs are q and q dE q (p) ω(q) := q q p . (4-33)

Graphs

Definition 4.5 For any k ≥ 0 and g ≥ 0 such that k + 2g ≥ 3, we define: Let G (g) k+1 (p, p 1 , . . . , p k ) be the set of connected trivalent graphs defined as follows:

1. there are 2g + k -1 tri-valent vertices called vertices.

2. there is one 1-valent vertex labelled by p, called the root.

3. there are k 1-valent vertices labelled with p 1 , . . . , p k called the leaves.

4. There are 3g + 2k -1 edges.

5. Edges can be arrowed or non-arrowed. There are k + g non-arrowed edges and 2g + k -1 arrowed edges.

6. The edge starting at p has an arrow leaving from the root p.

7. The k edges ending at the leaves p 1 , . . . , p k are non-arrowed.

8. The arrowed edges form a "spanning4 planar5 binary skeleton6 tree" with root p.

The arrows are oriented from root towards leaves. In particular, this induces a partial ordering of all vertices.

9. There are k non-arrowed edges going from a vertex to a leaf, and g non arrowed edges joining two inner vertices. Two inner vertices can be connected by a non arrowed edge only if one is the parent of the other along the tree.

10. If an arrowed edge and a non-arrowed inner edge come out of a vertex, then the arrowed edge is the left child. This rule only applies when the non-arrowed edge links this vertex to one of its descendants (not one of its parents).

We have the following useful lemma:

Lemma 4.1 There is a 1 to 3g + 2k -1 map from G (g) k+1 (p, p K ) to G (g) k+2 (p, p K , p k+1 ).
proof:

If G is a graph in G (g)
k+2 (p, p 1 , . . . , p k , p k+1 ), remove the non-arrowed edge attached to the leaf p k+1 and remove the corresponding vertex, and merge the incoming and the other outgoing edges of that vertex. You clearly get a graph G ′ ∈ G In order to simplify the drawing, we can draw a black dot to specify the right child. This way one gets only planar graphs. Remark that without the prescriptions 9) and 10), one would get 13 different graphs whereas we only have 5.

Weight of a graph

Consider a graph G ∈ G (g) k+1 (p, p 1 , . . . , p k ). Then, to each vertex i = 1, . . . , 2g + k -1 of G, we associate a label q i , and we associate q i to the beginning of the left child edge, and q i to the right child edge. Thus, each edge (arrowed or not), links two labels which are points on the Riemann surface Σ.

• To an arrowed edge going from q ′ towards q, we associate a factor dEq(q ′ ) (y(q)-y(q))dx(q) .

• To a non arrowed edge going between q ′ and q we associate a factor B(q ′ , q).

• Following the arrows backwards (i.e. from leaves to root), for each vertex q, we take a residue at q → a, i.e. we sum over all branchpoints.

After taking all the residues, we get the weight of the graph:

w(G) (4-38)
which is a multilinear form in p, p 1 , . . . , p k . Similarly, we define weights of linear combinations of graphs by:

w(αG 1 + βG 2 ) = αw(G 1 ) + βw(G 2 ) (4-39)
and for a disconnected graph, i.e. a product of two graphs:

w(G 1 G 2 ) = w(G 1 )w(G 2 ). (4-40)
Theorem 4.8 We have: Such graphical notations are very convenient, and are a good support for intuition and even help proving some relationships. It was immediately noticed after [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF] that those diagrams look very much like Feynman graphs, and there was a hope that they could be the Feynman's graphs for the Kodaira-Spencer theory. But they ARE NOT Feynman graphs, because Feynman graphs can't have non-local restrictions like the fact that non oriented lines can join only a vertex and one of its descendent.

W (g) k+1 (p, p 1 , . . . , p k ) = G∈G (g) k+1 (p,p 1 ,...,p k ) w(G) = w    G∈G (g) k+1 (p,p 1 ,...,p k ) G    (4-41
Those graphs are merely a notation for the recursive definition 4.2.

Lemma 4.2 Symmetry factor:

The weight of two graphs differing by the exchange of the right and left children of a vertex are the same. Indeed, the distinction between right and left child is just a way of encoding symmetry factors.

proof:

This property follows directly from theorem 4.4 and the definition Eq. (4-9). Consider one term contributing to the first part of RHS of Eq. (4-9):

Res q→a dE q (p) ω(q) W (m) |J|+1 (q, P J )W (g-m) k-|J|+1 (q, P K/J ) = -Res q→a dE q (p) ω(q) W (m) |J|+1 (q, P J )W (g-m) k-|J|+1 (q, P K/J ) = Res q→a dE q (p) ω(q) W (m) |J|+1 (q, P J )W (g-m)
k-|J|+1 (q, P K/J ).

(4-43)

where the equalities are obtained by adding terms without residues at the branch points to the integrand and using theorem 4.4. One can perform the same trick for the second term in Eq. (4-9) and this proves the lemma.

Examples.

Let us present some examples of correlation functions and free energy for low order.

Correlation functions.

To leading order, one has the first correlation functions given by: dE r (q) ω(r) [B(q, p 1 )B(r, p 2 )B(r, p 3 ) +B(q, p 1 )B(r, p 2 )B(r, p 3 ) + B(q, p 2 )B(r, p 1 )B(r, p 3 ) +B(q, p 2 )B(r, p 1 )B(r, p 3 ) + B(q, p 3 )B(r, p 2 )B(r, p 1 ) +B(q, p 3 )B(r, p 2 )B(r, p 1 )]

W (0) 2 (p, q) = B(p, q). (4-44) W (0) 3 (p, p 1 , p 2 ) = p p p 1 2 + p p p 1 2 = Res q→a dE q (p) ω(q) [B(q, p 1 )B(q, p 2 ) + B(q, p 1 )B(q, p 2 )]
+ Res q→a Res r→a dE q (p) ω(q) dE r (q) ω(r) [B(q, p 1 )B(r, p 2 )B(r, p 3 )
+B(q, p 1 )B(r, p 2 )B(r, p 3 ) + B(q, p 2 )B(r, p 1 )B(r, p 3 ) +B(q, p 2 )B(r, p 1 )B(r, p 3 ) + B(q, p 3 )B(r, p 2 )B(r, p 1 ) +B(q, p 3 )B(r, p 2 )B(r, p 1 )] (4 -46)

First orders for the one point correlation function read:

W (1) 1 (p) = p = Res q→a dE q (p) ω(q) B(q, q) (4 -47) W (2) 1 (p) = p + p + p + p + p = Res q→a Res r→a Res s→a dE q (p) ω(q) dE r (q) ω(r) dE s (q) ω(s) B(r, r)B(s, s) + Res q→a Res r→a Res s→a dE q (p) ω(q) dE r (q) ω(r) dE s (r) ω(s) B(r, q)B(s, s) + Res q→a Res r→a Res s→a dE q (p) ω(q) dE r (q) ω(r) dE s (r) ω(s) [B(q, r)B(s, s) +B(s, q)B(s, r) + B(s, q)B(s, r)] = 2 p + 2 p + p (4-48)
where the last expression is obtained using lemma 4.2.

Free energy.

The second order free energy reads

-2F (2) = Res p→a Res q→a Res r→a Res s→a Φ(p)dE q (p) ω(q) dE r (q) ω(r) dE s (q) ω(s) B(r, r)B(s, s) + Res p→a Res q→a Res r→a Res s→a Φ(p)dE q (p) ω(q) dE r (q) ω(r) Φ(p)dE s (r) ω(s) B(r, q)B(s, s) + Res p→a Res q→a Res r→a Res s→a Φ(p)dE q (p) ω(q) dE r (q) ω(r) dE s (r) ω(s) [B(q, r)B(s, s) +B(s, q)B(s, r) + B(s, q)B(s, r)] (4 -49)

Remark: Teichmuller pant gluings

Every Riemann surface of genus g with k punctures can be decomposed into 2g + k pants whose boundaries are 3g+k closed geodesics (in the metric with constant negative curvature). The number of ways (in the combinatorial sense) of gluing 2g + k pants by their boundaries is clearly the same as the number of diagrams of G (g) k , and each diagram corresponds to one pant decomposition.

Example with k = 1 and g = 2:

W (2) 1 = + 2 + 2

Variations of the curve

The goal of this section is to study how the various F (g) and correlation functions change under the variations of moduli of the curve. Consider an infinitesimal variation of the curve E → E + δE. It induces a variation of the function y(x) at fixed x:

δ Ω y| x dx = -Ω.
(5-1)

If we use a local coordinate z, we may prefer to work at fixed z instead of fixed x, we have a Poisson structure:

δ Ω y| z dx -δ Ω x| z dy = -Ω (5-2)
The possible Ω's can be classified as first type (holomorphic), second type (residueless, and vanishing A-cycles) and third type (only simple poles and vanishing A-cycles), see [START_REF] Bertola | Two-matrix model with semiclassical potentials and extended Whitham hierarchy[END_REF] for this classification.

Rauch variational formula

Equation 5-2 implies that the variation of position of a branch point a i is given by:

δ Ω x(a i ) = Ω(a i ) dy(a i ) .
(5-3)

We assume here that Ω dy has no pole at branchpoints. Then, Rauch variational formula [START_REF] Rauch | Weierstrass points, branch points, and moduli of Riemann surfaces[END_REF][START_REF] Fay | Theta functions on Riemann surfaces[END_REF] implies that the change of the Bergmann kernel is

δ Ω B(p, q)| x(p),x(q) = i Ω(a i ) dy(a i ) Res r→a i B(r, p)B(r, q) dx(r) = i Res r→a i Ω(r)B(r, p)B(r, q) dx(r)dy(r) . (5-4)
In particular after integrating over a B-cycle we have:

δ Ω du(p)| x(p) = i Res r→a i Ω(r)B(r, p)du(r) dx(r)dy(r) , (5 -5)
and integrating again over a B-cycle:

δ Ω τ = 2iπ i Res r→a i Ω(r)du(r)du t (r) dx(r)dy(r) .
(5 -6)

Let us compute the variations of the κ-modified Bergmann kernel:

δ Ω B(p, q)| x(p),x(q) = δ Ω B(p, q)| x(p),x(q) + 2iπ δ Ω du t (p) x(p) κdu(q) +2iπdu t (p) κ δ Ω du(q)| x(q)
= Res r→a Ω(r)B(r, p)B(r, q) dx(r)dy(r) +2iπ Res r→a Ω(r)B(r, p)du t (r)κdu(q) dx(r)dy(r) +2iπ Res

r→a Ω(r)B(r, q)du t (p) κ du(r) dx(r)dy(r)

= Res

r→a Ω(r)B(r, p)B(r, q) dx(r)dy(r)

+ Res

r→a Ω(r)B(r, p)(B(r, q) -B(r, q)) dx(r)dy(r)

+ Res

r→a Ω(r)(B(r, p) -B(r, p))B(r, q) dx(r)dy(r)

= Res

r→a Ω(r)B(r, p)B(r, q) dx(r)dy(r)

+4π 2 Res r→a Ω(r)du t (p)κdu(r)du t (r)κdu(q) dx(r)dy(r) = Res r→a Ω(r)B(r, p)B(r, q) dx(r)dy(r) -2iπ du t (p)κ δ Ω τ κdu(q) (5-7)
i.e.

δ Ω + tr (κ δ Ω τ κ ∂ ∂κ ) x(p),x(q) B(p, q) = Res r→a Ω(r)B(r, p)B(r, q) dx(r)dy(r) = -2 Res r→a Ω(r)dE r (p)B(r, q) ω(r) .
(5 -8)

We thus define the covariant variation:

D Ω = δ Ω + tr (κ δ Ω τ κ ∂ ∂κ ).
(5-9)

It is more convenient to rewrite Eq. (5-8) as follows:

D Ω B(p, q) = -2 Res r→a Ω(r)dE r (p)B(r, q) ω(r) = -2 Res r→a Res s→r Ω(r)dE r (p)B(s, q) (y(r) -y(r))(x(s) -x(r)) = 2 Res r→a Res s→r Ω(r)dE r (p)B(s, q) (y(r) -y(r))(x(s) -x(r)) = 2 Res r→a Ω(r)dE r (p)B(r, q) ω(r) = Res r→a dE r (p) ω(r)
[Ω(r)B(r, q) + Ω(r)B(r, q)] .

(5-10) because now we recognize the propagator and vertex of def.4.1. Similarly, by integrating once with respect to q, near a branch point a j we get:

D Ω dE q (p)| x(p),x(q) = -2 Res r→a dE r (p) ω(r) Ω(r)dE q (r) = Res r→a dE r (p) ω(r)
[Ω(r)dE q (r) + Ω(r)dE q (r)]

(5 -11)

Those two relations can be depicted:

D p q + p q Ω p q Ω = = + p q Ω p q Ω and D p q + = p q Ω p q Ω .
From this last variation, one can compute the covariant variations of the correlation functions and free energies through the following lemma:

Lemma 5.1 For any symmetric bilinear form f (q, p) = f (p, q): D Ω Res q→a dE q (p) ω(q) f (q, q) x(p) = 2 i,j
Res

r→a i Res q→a j dE r (p) ω(r) Ω(r) dE q (r) ω(q) f (q, q) + j Res q→a j dE q (p) ω(q)
D Ω (f (q, q)) x(q) .

(5 -12)

D p + = p q Ω p q Ω q
Graphically, this means that taking the variation of a diagram just consists in adding a leg Ω in all possible edges of the graph. In particular if Ω can be written as:

Ω(p) = ∂Ω B(p, q)Λ(q) (5-13)
where the path ∂Ω does not intersect small circles around branch-points8 , then we have:

Theorem 5.1 Variations of correlation functions and free energies: For g + k > 1 we have

D Ω W (g) k (p 1 , . . . , p k ) x(p i ) = ∂Ω W (g) k+1 (p 1 , . . . , p k , q)Λ(q) (5-14)
and, for g ≥ 1,

D Ω F (g) = - ∂Ω W (g) 1 (p)Λ(p).
(5-15

)
This theorem is proved in appendix B. It follows directely from lemma 4.1 and lemma 5.1.

Loop insertion operator

In particular for any point q lying away from the branch-points, if we choose Ω(p) = B(p, q)

(5-16)

we call D B(.,q) the loop insertion operator, by analogy with matrix models [START_REF] Ambjørn | Matrix model calculations beyond the spherical limit[END_REF][START_REF] Eynard | An introduction to random matrices[END_REF]. It acts on the correlation functions and free energies as follows:

Theorem 5.2 D B W (g) k (p 1 , . . . , p k ) = W (g) k+1 (p 1 , . . . , p k , q) (5-17) D B F (g) = -W (g) 1 (q) (5-18)
and

D B F (0) = y(q)dx(q) + 1 4iπ κ B ydx t B B W 3,0 κ B ydx. (5-19)
Thus, the loop insertion operator, adds one leg to correlation functions.

Variations with respect to the moduli

Let us consider canonical variations of the curve corresponding to each moduli of the curve defined in section 3.4. We use Eq. (3-65)

ydx = α,k t α,k B α,k + α t α dS α,o + 2iπ i ǫ i du i (p) (5-20)
to identify the Ω's corresponding to varying only one modulus.

Variation of filling fractions

Consider the variation of the curve

Ω(p) = -2iπdu i (p) = - B i B(p, q).
(5-21)

i.e. ∂Ω = B i and Λ = -1. It is such that:

δ Ω ǫ j = δ ij , δ Ω t α = 0 , δ Ω V α = 0. (5-22)
Therefore it is equivalent to varying only the filling fraction ǫ i = 1 2iπ A i ydx:

D -2iπdu i = ∂ ∂ǫ i (5-23) Theorem 5.1 gives ∂ ∂ǫ i W (g) k (p 1 , . . . , p k ) = - B i W (g) k+1 (p 1 , . . . , p k , q), (5-24)
and

∂ ∂ǫ i F (g) = B i W (g) 1 (q), (5-25)
and

∂ ∂ǫ i F (0) = - B j ydx + 1 4iπ κ B ydx t δ -2iπdu i (τ ) κ B ydx.
(5-26)

Variation of temperatures

Let α and α ′ be two distinct poles of ydx. Consider the variation of the curve

Ω(p) = -dS α,α ′ (p) = α ′ α B(p, q) , i.e. ∂Ω = [α, α ′ ] , Λ = 1 (5-27)
It is such that:

δ Ω ǫ j = 0 , δ Ω t β = δ α,β -δ α ′ ,β , δ Ω V β = 0.
(5-28)

Therefore it is equivalent to varying only the temperatures t α and t α ′ :

D -dS α,α ′ = ∂ ∂t α - ∂ ∂t ′ α (5-29)
Notice that it is impossible to vary only one temperature t α since we have β t β = 0. Theorem 5.1 gives

∂ ∂t α - ∂ ∂t ′ α W (g) k (p 1 , . . . , p k ) = α ′ α W (g) k+1 (p 1 , . . . , p k , q), (5-30)
∂ ∂t α - ∂ ∂t ′ α F (g) = α α ′ W (g) 1 (q) (5-31) and ∂ ∂t α - ∂ ∂t ′ α F (o) = µ α -µ α ′ + 1 4iπ κ B ydx t δ -dS α,α ′ (τ ) κ B ydx.
(5-32)

Variation of the moduli of the poles

Let α be a pole of ydx. Consider the variation of the curve

Ω(p) = -B α,k = Res α B(p, q)z k α (q), (5-33)
i.e. ∂Ω is a small circle around α and Λ = 1 2iπ z k α . It is such that:

δ Ω ǫ j = 0 , δ Ω t β = 0 , δ Ω t β,k ′ = δ α,β δ k,k ′ (5-34)
Therefore it is equivalent to varying only the coefficient t α,k :

D -B α,k = ∂ ∂t α,k (5-35)
Theorem 5.1 gives

∂ ∂t α,k W (g) k (p 1 , . . . , p k ) = Res α z k α (q)W (g) k+1 (p 1 , . . . , p k , q), (5-36)
∂ ∂t α,k F (g) = -Res α z k α (q)W (g) 1 (q)
(5-37)

and ∂ ∂t α,k F (o) = Res α ydxz k α + 1 4iπ κ B ydx t δ B α,k (τ ) κ B ydx.
(5-38)

Homogeneity

Theorem 5.3 For g > 1, we have the homogeneity property:

(2 -2g)F (g) = α,k t α,k ∂ ∂t α,k F (g) + α t α ∂ t α F (g) + i ǫ i ∂ ∂ǫ i F (g) (5-39)
i.e. F (g) is homogeneous of degree 2 -2g.

The proof is given in appendix B.

5.5 Variations of F (0) with respect to the moduli.

In this section we compute the first and second deriatives of F (0) with respect to the moduli of the curve. This paragraph is only for bookkeeping since those expressions have been known for some time [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][START_REF] Bertola | Free Energy of the Two-Matrix Model/dToda Tau-Function[END_REF]. Here we set κ = 0.

First derivatives of F (0) .

∂F (0) ∂t α,k = Res α z k α ydx
(5-40)

∂F (0) ∂t α,β = ∂ ∂t α - ∂ ∂t β F (0) = µ α -µ β (5-41) ∂F (0) ∂ǫ i = - B i ydx (5-42)
Second derivatives of F (0) .

∂ 2 F (0) ∂t α,k ∂t β,l = (δ α,β -1) Res p→α Res q→β z α (p) k B(p, q)z β (q) l (5-43) ∂ 2 F (0) ∂t α,k ∂t γ,β = Res α z k α dS γ,β (5-44)
∂ 2 F (0) ∂t α,k ∂ǫ i = 2iπ Res α z k α du i = - B i B α,k (5-45)
∂ 2 F (0) ∂ǫ i ∂t α,β = 2iπ(u i (β) -u i (α)) (5-46) ∂ 2 F (0) ∂ǫ i ∂ǫ j = 0 (-2iπτ ij f or F (0) ) (5-47) ∂ 2 F (0) ∂t 2 α,β = ln (dζ α (α)dζ β (β)E(α, β) 2 ) (5-48) ∂ 2 F (0) ∂t α,β ∂t α,γ = ln dζ α (α)E(α, β)E(α, γ) E(β, γ) (5-49) ∂ 2 F (0) ∂t α,β ∂t δ,γ = ln E(δ, β)E(α, γ) E(α, δ)E(β, γ) (5-50)
where ζ α = 1 zα is a local coordinate around the pole α.

Remark 5.1 The definition of F (0) given in Eq. (4-13) is nothing but the homogeneity property since it is written in terms of the first derivatives. One can also write a formula focusing more on the second order derivatives of F (0) :

F (0) = - 1 2 α,β Res p→α Res q→β V α (p)B(p, q)V β (q) + α,β t β Res α V α dS β,o - 1 2 α,β t α t β ln (γ α,β ) - 1 2 ǫ t B ydx (5 -51)
where

ln γ α,α = - o α (dS α,o ′ + dz α z α ) + ln (z α (o)) (5-52) and ln γ α,β = ln E(α, β)E(o, o ′ ) E(α, o ′ )E(β, o) .
(5-53)

One can notice that, in these terms,

∂ 2 F (0) ∂t 2 α,β
= ln(γ α,α γ β,β ).

(5-54)

6 Variations with respect to κ and modular transformations

Variations with respect to κ

We have introduced the matrix κ in order to easily compute modular transformations of our functions. Somehow variations of κ play the role of infinitesimal modular transformations. Therefore it is important to compute ∂/∂κ, and we will use this result in section 6.2. First, notice that W (g) k (p 1 , . . . , p k ) is a polynomial in κ of degree 3g + 2k -3, and F (g) is a polynomial in κ of degree 3g -3 for g > 1 (number of propagators in a graph of G (g) k ). Theorem 6.1

2iπ ∂ ∂κ ij W (g) k (p K ) = 1 2 r∈B j s∈B i W (g-1)
k+2 (p K , r, s)

+ 1 2 h L⊂K r∈B i W (h) |L|+1 (p L , r) s∈B j W (g-h) k-|L|+1 (p K/L , s) (6-1
) and in particular for g ≥ 2:

-2iπ ∂ ∂κ ij F (g) = 1 2 r∈B j s∈B i W (g-1) 2 (r, s) + 1 2 g-1 h=1 r∈B i W (h) 1 (r) s∈B j W (g-h) 1 (s) (6-2) and ∂ ∂κ ij F (1) = 1 κ ji . (6-3)
This theorem is proved in Appendix B Notice that these equations are to be compared with the Kodaira-Spencer theory [1, 2, 11, 42].

Modular transformations

Consider a modular change of cycles:

A B = δ AA ′ δ AB ′ δ BA ′ δ BB ′ A ′ B ′ , A ′ B ′ = δ A ′ A δ A ′ B δ B ′ A δ B ′ B A B (6-4)
where

δ A ′ A = δ t BB ′ , δ A ′ B = -δ t AB ′ , δ B ′ B = δ t AA ′ , δ B ′ A = -δ t BA ′ and the matrices δ AA ′ , δ AB ′ , δ BA ′ , δ BB ′ have integer coefficients and satisfy δ AA ′ δ t BB ′ -δ BA ′ δ t AB ′ = Id.
Under this transformation of the cycle homology basis, the Abel map and the matrix of period change like:

du ′ = J du , du = J -1 du ′ (6-5) with J = (δ t AA ′ + τ ′ δ t AB ′ ) = (δ BB ′ -τ δ AB ′ ) -1 and τ ′ = (δ BB ′ -τ δ AB ′ ) -1 (-δ BA ′ + τ δ AA ′ ) , τ = (δ t AA ′ + τ ′ δ t AB ′ ) -1 (δ t BA ′ + τ ′ δ t BB ′ ).
(6-6) Let us define the following symmetric matrix:

κ = κ t = (δ BB ′ δ -1 AB ′ -τ ) -1 = δ AB ′ J (6-7)
it is such that the Bergmann kernel changes like:

B ′ = B + 2iπ du t κdu. (6-8)
The κ-modified Bergmann kernel changes like:

B ′ (p, q) = B(p, q) + 2iπ du ′t (p)κdu ′ (q) + du t (p) κdu(q) = B(p, q) + 2iπdu t (p) κ + J t κJ du(q). (6-9)
In other words, the effect of a modular change of cycles is equivalent to a change κ → κ + J t κJ in the definition of the kernel B(p, q). Thus, the modular variations of the free energies satisfy the following theorem: Theorem 6.2 For g ≥ 2 the modular transformation of the free energies F (g) consists in changing κ to κ + J t κJ in the definitions of the modified Bergmann kernel and Abelian differential. The first correction F (1) changes like:

F (1) ′ = F (1) - 1 2 ln (δ BB ′ -τ δ AB ′ ). (6-10)
An equivalent way of saying the same thing, is that: if we change the basis of cycles and change the matrix κ → J t -1 (κ -κ)J -1 , then the F (g) 's are unchanged for g > 1.

proof:

The result for g ≥ 2 comes directly from the variation of the Bergmann kernel.

F (1) depends on the cycles only through the Bergmann tau function. Since,

∂ ln (τ B ′ x ) ∂x(a i ) - ∂ ln (τ Bx ) ∂x(a i ) = Res p→a i B ′ (p, p) -B(p, p) dx(p) = 2iπ Res p→a i du t (p) κdu(p) dx(p) = -2iπ Res p→a i du t (p) κdu(p) dx(p) = -Tr κ ∂τ ∂x(a i ) = -Tr (δ BB ′ -τ δ AB ′ ) -1 ∂τ δ AB ′ ∂x(a i ) = ∂ ln det(δ BB ′ -τ δ AB ′ ) ∂x(a i ) (6 -11)
and this characterizes the Bergmann tau function totally (up to a general multiplicative factor), one obtains the second result of the theorem. Remark 6.1 The transformation of leading order F (0) is more complicated and its computation is more involved as the final result depends explicitely on the position of the poles α in the fundamental domain. Let us just mention that it depends on all the parameters of the modular transformation explicitely. Theorem 6.3 If one chooses κ = i 2 Imτ , then F (g) (κ) is modular invariant.

proof:

It is well known that for that value of κ, the modified Bergmann kernel is the Schiffer kernel and is modular invariant. Indeed, it is easy to check that if κ = i 2 Imτ one has κ + J t κJ = i 2 Imτ ′ . Since the only modular dependence of F (g) for g ≥ 1 is in the Bergmann kernel, this proves the modular invariance of the F (g) 's.

Symplectic invariance

The following theorem is mostly the reason why we call F (g) 's invariants of the curve. This theorem seems to be rather important and it has beautiful applications as we will see in section 10.4.1.

Theorem 7.1 The following transformations of E leave the F (g) 's unchanged:

• x → ax+b cx+d and y → (cx+d) 2 ad-bc y. • y → y + R(x) where R is any rational function.

• y → y and x → -x.

• y → x and x → y.

Notice that these are transformations which conserve the symplectic form |dx ∧ dy| (7-1)

In particular we have the P SL 2 (C) invariance:

x y -→ a b c d x y , (ad -bc) 2 = 1 (7-2)
That symplectic invariance seems to be a very powerful tool to see if different matrix models have the same topological expansion. For example, in section 10.4.1 we show how symplectic invariance can be used to provide a new and very easy proof of some properties of the Kontsevitch integral.

proof:

Invariance under the first 2 transformations is obvious from the definitions, because the only x and y dependance of the W (g) k 's is in ω(q) = (y(q) -y(q))dx(q) which is clearly unchanged under the first 2 transformations (notice that the transformation x → ax+b cx+d conserves the branchpoints). In fact the first 2 transformations leave W (g) k unchanged.

In the 3rd transformation the only thing which changes is the sign of ω, and it is easy to see that W (g) k is multiplied by (-1) 2g-2+k = (-1) k , and

F (g) is multiplied by (-1) 2g-2 = 1.
The 4th transformation is the difficult one. The proof consists in building some "mixed correlation functions", and seeing that their definition by inverting the roles of x and y lead to the same objetcs. Since it is long and involves new results in the framework of matrix model, it is written in a separate paper [START_REF] Eynard | Mixed traces and free energy symetry[END_REF].

The case of F (0) and F (1) are done separately in appendix C.

Singular limits

Consider a 1-parameter family of algebraic curves:

E(x, y, t) (8-1)
such that the curve at t = 0 has a singular branchpoint a with a p/q rational singularity, i.e. in some local coordinate z near a we have:

   t = 0 x(z) ∼ x(a) + (z -a) q y(z) ∼ y(a) + (z -a) p (8-2)
At t = 0, the singularity is smoothed, and we have (the local parameter is now

ζ = z t -ν ): x(z, t) ∼ x(a) + t qν Q(ζ) + o(t qν ) y(z, t) ∼ y(a) + t pν P (ζ) + o(t pν ) (8-3)
where Q is a polynomial of degree q and P is a polynomial of degree p, and where ν is some exponent which depends on the choice of the parameter t.

The curve

E sing (ξ, η) = ξ(ζ) = Q(ζ) η(ζ) = P (ζ) = Resultant(Q -ξ, P -η) (8-4)
is called the singular spectral curve. One observes that F (g) (E) is singular in the small t limit, and it behaves like

F (g) (E(t)) ∼ t γg F (g) sing + o(t γg ) , for g ≥ 2 (8-5)
F (1) (E(t)) ∼ -1 24 (p -1)(q -1)ν ln (t) + O(1) , for g = 1 (8-6)

F (g)
sing is called the double scaling limit of F (g) . The exponent γ g and F (g) sing are given by the following theorem: Theorem 8.1 Singular limits:

F (g) sing (E) = F (g) (E sing ) , for g ≥ 2 (8-7)
and

γ g = (2 -2g)(p + q)ν. (8-8)
In other words, our construction of F (g) commutes with the singular limit.

proof:

It is easy to see that the most singular term in the limit of the Bergmann kernel in that regime, behaves like t 0 , and thus dE z ′ (z) as well. The denominator ((y(z)y(z))dx(z) in the recursion behaves like t ν(p+q) (P (ζ) -P (ζ))Q ′ (ζ)dζ, and by recursion on k and g, we easily see that:

W (g) k (z 1 , . . . , z k ) ∼ t (2-2g-k)(p+q)ν ω (g) k (ζ 1 , . . . , ζ k ) (8-9)
if all z i 's are close to a, and is subdominant if some z i 's are not in the vicinity of a.

The leading contribution to W (g) k is thus obtained by taking z ′ and z ′ in the vicinity of a only in eq.4-9, i.e. ω (g) k (ζ 1 , . . . , ζ k ) obey the same recursion formula as eq.4-9, with the curve E sing . The same holds for the free energy.

Integrability

Here, we prove that Z is a tau-function, because it satisfies some Hirota equation.

Baker-Akhiezer function

Given two points ξ and η in the fundamental domain, we define the following kernel as a formal series in 1/N:

K N (ξ, η) = e -N ξ η ydx E(ξ, η) dx(ξ)dx(η) exp - ∞ g=0 ∞ l=1,2-2g-l<0 1 l! N 2-2g-l ξ η ξ η . . . ξ η W (g) l (p 1 , . . . , p l ) (9 -1)
where the integration path lies in the fundamental domain. This kernel has the following properties:

• Notice that (x(ξ) -x(η))K N (ξ, η) → 1 when η → ξ.

• We have:

lim η→ξ K N (ξ, η) - 1 (x(ξ) -x(η)) = -Ny(ξ) + W 1 (ξ) dx(ξ) (9-2)
where

W 1 = ∞ g=1 N 1-2g W (g)
1 .

• we have:

K N (ξ, η) = K -N (η, ξ) (9-3)
• One may think that K N is singular at branchpoints because ln K N has poles, however, using the singular limit theorem 8.1, we see that the leading behavior of all W (g) l 's is given by the W (g) l 's of the Airy curve y = √ x described in section 10.5. Therefore, near a branchpoint a, when ξ, η → a, to leading order K N is the Tracy-Widom kernel [START_REF] Traicy | Level-spacing distributions and the Airy kernel[END_REF]:

K N (ξ, η) ∼ Ai( ξ)Ai ′ (η) -Ai ′ ( ξ)Ai(η) ξ - η ξ = N 2/3 (x(ξ) -x(a)) , η = N 2/3 (x(η) -x(a)) (9-4)
In other words, K N is not singular near branchpoints.

• The only singularities of K N (ξ, η) are essential singularities at all the poles of ydx, with a singular part equal to exp (-N ξ η ydx), as well as a simple pole at ξ = η.

Then, given a pole α of ydx, we define for ξ in the vicinity of α:

ψ α,N (ξ) = e -N Vα(ξ) e -N ξ α (ydx-dVα+tα dzα zα ) E(ξ, α) dx(ξ)dζ α (α) (z α (ξ)) N tα exp - ∞ g=0 ∞ l=1,2-2g-l<0 1 l! N 2-2g-l ξ α ξ α . . . ξ α W (g) l (p 1 , . . . , p l ) = lim η→α K(ξ, η) dx(η) dζ α (η) e -N Vα(η) (z α (η)) N tα (9-5)
where ζ α is the local parameter near α, ζ α = 1 zα , and φ α,N (ξ) = ψ α,-N (ξ). They have the following properties:

• ψ α,N was defined only in the vicinity of α, but it can be easily analytically continued to the whole curve, by choosing an arbitrary point o in the vicinity of α and writing:

ξ α (ydx -dV α + t α dz α z α ) + V α (ξ) -t α ln (z α (ξ)) = ξ o ydx + o α (ydx -dV α + t α dz α z α ) + V α (o) -t α ln (z α (o)) (9-6)
• Using the singular limit theorem 8.1 near branchpoints, we see that the leading behavior of all W (g) l 's is given by the W (g) l 's of the Airy curve y = √ x described in section 10.5. Therefore, near a branchpoint a, when ξ, η → a we have:

ψ α,N (ξ) ∼ C Ai( ξ) , ξ = N 2/3 (x(ξ) -x(a)) (9-7)
where C is some normalization constant (C = ψ α,N (a)/Ai(0)). In other words, ψ α,N is not singular near branchpoints.

• The only singularities of ψ α,N are essential singularities at all the poles of ydx, with a singular part equal to exp (-N ξ ydx).

This is why we call those formal functions Baker-Akhiezer functions (cf [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF]).

Remark 9.1 In fact, those functions are exactly Baker-Akhiezer functions only when the curve has genus g / = 0. In the general case, the Baker-Akhiezer functions must also have the property that they take the same value after going around a non-trivial cycle. It is not difficult to multiply ψ α,N by the appropriate θ function in order to fulfill that property. However, if we do that, we destroy the 1/N 2 expansion. This is why the ψ α,N defined above can be called a "formal Baker-Akhiezer function". This definition is sufficient to find a formal Hirota equation, valid only order by order in 1/N 2 .

Sato relation

Given two points ξ and η on Σ, and a complex number r, we define the curve:

E + r[ξ, -η] = x(p), y(p) + r dS ξ,η (p) dx(p) , p ∈ Σ (9-8)
The differential ydx + rdS ξ,η has the same A-contour integrals as ydx, the same poles with the same singular part, plus two additional simple poles, one located at p = ξ with residue r, and one at p = η with residue -r.

We have Sato's relation:

Theorem 9.1 K N (ξ, η) = Z N (E + 1 N [ξ, -η]) Z N (E) , ψ α,N (ξ) = Z N (E + 1 N [ξ, α]) Z N (E) (9-9)
Indeed, the definition of K N is the formal Taylor expansion in powers of r = 1/N of the RHS.

Hirota equation

Consider two algebraic curve E(x, y) and Ẽ(x, y) with the same conformal structure (i.e. the same compact Riemann surface Σ), then we have an Hirota bilinear relation:

Theorem 9.2
We have the bilinear relation:

Res η→ζ dx(η) K N (ξ, η) K Ñ (η, ζ) = K N (ξ, ζ) (9-10)
and also:

Res ξ→α dx(ξ) ψ α,N (ξ) ψα,-Ñ (ξ) = 0 if Ñ tα > Nt α + 1 (9-11)
which takes exactly the form of the Hirota equation [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF][START_REF] Kostov | Bilinear functional equations in 2d quantum gravity[END_REF].

It is important to notice that this Hirota equation makes sense only order by order in its 1/N 2 expansion. In the way we have obtained it, it is meaningless for finite N (appart maybe from the genus zero case g / = 0, under the condition that the 1/N 2 series is convergent). Therefore, we have a "formal Hirota equation".

Application: topological expansion of matrix models

In this section, we show how the objects defined in section.4.2.4 (i.e. κ = 0) correspond to the terms of the topological expansion of the free energy and correlation functions of various matrix models when one considers appropriate curves E(x, y). Notice that in all this section we consider κ = 0.

Formal 1-matrix model

The formal 1-matrix model (cf [START_REF] Eynard | Formal matrix integrals and combinatorics of maps[END_REF]), is known to be the generating function which enumerates maps of given topology since the work of [START_REF] Brezin | [END_REF], then [START_REF] Ambjørn | Diseases of triangulated random surface models, and possible cures[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]. Its topological expansion was computed in several steps. The authors of [START_REF] Ambjørn | Matrix model calculations beyond the spherical limit[END_REF] introduced a recursive algorithm to compute the F (g) 's, only in the 1-cut case (i.e. g / = 0), and then the method was extended to other cases [START_REF] Akemann | Higher genus correlators for the Hermitian matrix model with multiple cuts[END_REF][START_REF] Akemann | New universal spectral correlators[END_REF]. The computation of the subleading term F (1) was done in general by Chekhov [START_REF] Chekhov | Genus one corrections to multi-cut matrix model solutions[END_REF]. The computation to all orders of the correlation functions was found in [START_REF] Eynard | Topological expansion for the 1-hermitian matrix model correlation functions[END_REF], and the free energies in [START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF]. Consider a "semi-classical" potential V (x), i.e. such that V ′ (x) is a rational function. Let D(x) be its denominator, i.e. D(x)V ′ (x) is a polynomial of degree d, and let X 1 , . . . , X d be its zeroes:

D(x)V ′ (x) = d i=1 (x -X i ) (10-1)
We write:

δV i (x) = V (x) -V (X i ) - 1 2 V ′′ (X i )(x -X i ) 2 (10-2)
Choose an integer n, and a d-partition of n, n = {n 1 , . . . , n d }, such that

d j=1 n j = n. (10-3)
The following gaussian integral (where each matrix M i is of size n i ) is a polynomial in T of the form:

(-1) l n l l! T l e -n T i n i V (X i ) dM 1 . . . dM d d i=1 e -nV ′′ (X i ) 2T Tr (M i -X i 1n i ) 2 i>j det(M i ⊗ 1 n j -1 n i ⊗ M j ) 2 ( i Tr δV i (M i )) l = dl/2 k=l/2 A k,l T k (10-4)
We define the formal matrix integral as the formal power series in T :

Z 1MM = ∞ k=0 T k ( 2k j=0
A k,j ). (10-5)

One can also define its formal logarithm, i.e. the free energy

F 1MM = -ln Z 1MM = ∞ k=0 T k B k (10-6)
It is a standard computation discovered by 't Hooft ( [71,[START_REF] Eynard | Formal matrix integrals and combinatorics of maps[END_REF]), that, for fixed

ǫ i = T n i n , for every k, n -2 B k is a polynomial in 1/n 2 : B k (n 1 , . . . , n d ) = gmax(k) g=0 B k,g (ǫ 1 , . . . , ǫ d ) n T 2-2g (10-7)
Thus we define the following formal power series in T :

F (g) 1MM = ∞ k=0 T k B k,g (ǫ 1 , . . . , ǫ d ) (10-8)
Remark: Here, the question of convergence of those series is not relevant. It is well known that each F (g) 1MM is a convergent series (because it is written in terms of algebraic functions of T ), but F 1MM is not.

Loop equations and classical spectral curve

It is easy to see (this property holds for each power of T because it holds for gaussian integrals) that the formal matrix integral satisfies, order by order in powers of T , the loop equations (i.e. Virasoro constraints), which can be written (see [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] David | Loop equations and nonperturbative effects in two-dimensional quantum gravity[END_REF]):

y(x) 2 + T 2 n 2 ω 2 (x, x) = V ′ (x) 2 4 - T n Tr V ′ (x) -V ′ (M) x -M (10-9)
where

y(x) = V ′ (x) 2 -T n Tr 1 x-M and ω 2 (x, x ′ ) = Tr 1 x-M Tr 1 x ′ -M c .
where the expectation value < . > is defined in a formal way similar to F .

If one identifies the coefficients of n 0 in each side, one gets an algebraic equation (here hyperelliptical), which is called the "classical spectral curve":

E 1MM (x, y) = D(x) 2 (y 2 - 1 4 V ′2 (x) + P (x)) (10-10)
where D(x)P (x) is a polynomial of degree at most deg(D(x)V ′ (x)) -1, and completely determined by the condition that the polynomial P (x) is a formal power series in powers of T such that at T = 0:

P (x, T = 0) = d i=1 ǫ i V ′ (x) x -X i (10-11)
It is such that there exist some contours A i , i = 1, . . . , d such that:

1 2iπ A i ydx = ǫ i (10-12)
Notice that the genus g / of the curve E 1MM is the number of non vanishing ǫ i 's minus one.

Most often in the litterature, V is chosen polynomial such that V ′ (0) = 0, and only the 1-cut case is considered, with only one non-vanishing filling fraction at X = 0. The resulting curve has genus g / = 0. This is the case which is relevant for enumerating polygonal surfaces.

It was proved in [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], in the case of polynomial potentials only (but it is clear that it can be extended to the semiclassical case), that one has:

Theorem 10.1 F (g) 1MM = F (g) (E 1MM ) (10-13)
This proves that F (g) (which we recall is a formal power series in T ) genericaly has a finite radius of convergence T < T c .

We also have:

Tr 1 x(p 1 ) -M . . . Tr 1 x(p k ) -M c = ∞ g=0 N 2-k-2g W (g) k (p 1 , . . . , p k ) dx(p 1 ) . . . dx(p k ) +δ k,1 N( 1 2 V ′ (x(p 1 )) -y(p 1 )) - δ k,2 (x(p 1 ) -x(p 2 )) 2 (10 -14)

2-matrix model

The formal 2-matrix model is known to be the generating function which counts bicolored maps (let us say the 2 colors are + or -, thus it is an Ising model on a random map), it was introduced by Kazakov [START_REF] Kazakov | Ising model on a dynamical planar random lattice: exact solution[END_REF]. The loop equations were first written in [START_REF] Staudacher | Combinatorial solution of the 2-matrix model[END_REF]. F (0) was computed in [START_REF] Krichever | The τ -function of the universal Whitham hierarchy, matrix models and topological field theories[END_REF][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] Eynard | Master loop equations, free energy and correlations for the chain of matrices[END_REF]. F (1) was first found in [START_REF] Eynard | Large N expansion of the 2-matrix model[END_REF] for the g / = 0 case, the in [START_REF] Eynard | Large N expansion of the 2-matrix model, multicut case[END_REF] for g / = 1, then in [START_REF] Eynard | 1/N 2 corrections to free energy in Hermitian two-matrix model[END_REF] for arbitrary g / . Then higher orders for the correlation functions were first derived in [START_REF] Eynard | Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula[END_REF], and the F (g) 's for g ≥ 2 were first found in [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF]. During the same time it became clear that matrix models topological expansion was closely related to algebraic geometry [START_REF] Kazakov | Complex Curve of the Two Matrix Model and its Tau-function[END_REF][START_REF] Dijkgraaf | Matrix Models, Topological Strings, and Supersymmetric Gauge Theories[END_REF][START_REF] Dijkgraaf | Mean field theory, topological field theory, and multimatrix models[END_REF][START_REF] Dijkgraaf | Intersection theory, integrable hierarchies and topological field theory[END_REF].

Definition

Definition 10.2 Similarly, consider V ′ 1 and V ′ 2 two rational functions with respective denominators D 1 (x) and D 2 (x). The equation:

V ′ 1 (X i ) = Y i V ′ 2 (Y i ) = X i i = 1, . . . , d (10-15
)

has d = deg(V ′ 1 D 1 ) * deg(V ′ 2 D 2 ) solutions.
We then write:

δV 1,i (x) = V 1 (x) -V 1 (X i ) -Y i (x -X i ) - V ′′ 1 (X i ) 2 (x -X i ) 2 (10-16) et δV 2,i (y) = V 2 (y) -V 2 (Y i ) -X i (y -Y i ) - V ′′ 2 (Y i ) 2 (y -Y i ) 2 (10-17)
We then choose an integer n, and a d-partition of n:

n = d i=1 n i . (10-18)
The following gaussian integral (where each matrix M i or Mi is of size n i ) is a polynomial in T of the form:

(-1) l n l l! T l e -n T i n i Tr (V 1 (X i )+V 2 (Y i )-X i Y i ) dM 1 . . . dM d d M1 . . . d Md d i=1 e -n T Tr V ′′ 1 (X i ) 2 (M i -X i 1n i ) 2 + V ′′ 2 (Y i ) 2 ( Mi -Y i 1n i ) 2 -(M i -X i 1n i )( Mi -Y i 1n i ) i>j det(M i ⊗ 1 n j -1 n i ⊗ M j ) i>j det( Mi ⊗ 1 n j -1 n i ⊗ Mj ) ( i Tr δV 1,i (M i ) + δV 2,i ( Mi )) l = dl/2 k=l/2 A k,l T k (10-19)
Similarly to the 1-matrix case, we can define the formal 2-matrix model as a formal power series in powers of T (see [START_REF] Eynard | Formal matrix integrals and combinatorics of maps[END_REF]):

Z 2MM = ∞ k=0 T k ( 2k j=0
A k,j ). [START_REF] Bergman | Kernel functions and elliptic differential equations in mathematical physics[END_REF][START_REF] Bershadsky | Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes[END_REF][START_REF] Bertola | Two-matrix model with semiclassical potentials and extended Whitham hierarchy[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 | Genus one corrections to multi-cut matrix model solutions[END_REF][START_REF] Chekhov | Matrix models with hard walls: Geometry and solutions[END_REF][START_REF] Chekhov | Hermitian matrix model free energy: Feynman graph technique for all genera[END_REF][START_REF] Chekhov | Matrix eigenvalue model: 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] One can also define its formal logarithm, i.e. the free energy

F 2MM = -ln Z 2MM = ∞ k=0 T k B k (10-21)
Again, it is a standard computation ( [71,[START_REF] Eynard | Formal matrix integrals and combinatorics of maps[END_REF]), that, for fixed

ǫ i = T n i n , for every k, n -2 B k is a polynomial in 1/n 2 : B k (n 1 , . . . , n d ) = gmax(k) g=0 B k,g (ǫ 1 , . . . , ǫ d ) n T 2-2g (10-22)
Thus we define the following formal power series in T :

F (g) 2MM = ∞ k=0 T k B k,g (ǫ 1 , . . . , ǫ d ) (10-23)

Loop equations and classical spectral curve

Again, the formal 2-matrix integral satisfies, order by order in powers of T , the loop equations (i.e. W -algebra constraints), which can be written (see [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] David | Loop equations and nonperturbative effects in two-dimensional quantum gravity[END_REF]):

(y -y(x))U(x, y) + T 2 n 2 U(x, y; x) = E(x, y) (10-24)
where

y(x) = V ′ 1 (x) -T n Tr 1 x-M 1 , U(x, y) = x -V ′ 2 (y) + T n Tr 1 x-M 1 V ′ 2 (y)-V ′ 2 (M 2 ) y-M 2 and U(x, y; x ′ ) = Tr 1 x-M 1 V ′ 2 (y)-V ′ 2 (M 2 ) y-M 2 Tr 1 x ′ -M 1 c
, and

E(x, y) = (y -V ′ 1 (x))(x - V ′ 2 (y)) -T n Tr V ′ 1 (x)-V ′ 1 (M 1 ) x-M 1 V ′ 2 (y)-V ′ 2 (M 2 ) y-M 2
+ T , and where the expectation value < . > is defined in a formal way similar to F .

If one chooses y = y(x) and identifies the coefficients of n 0 in each side, one gets an algebraic equation E 2MM (x, y) = 0, which is called the "classical spectral curve" [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]:

E 2MM (x, y) = D 1 (x) D 2 (y) ((V ′ 1 (x) -y)(V ′ 2 (y) -x) -P (x, y) + T ) (10-25)
where

D 1 (x) D 2 (y)P (x, y) is a polynomial of degree ≤ deg(D 1 V ′ 1 ) in x and a polynomial of degree ≤ deg(D 2 V ′
2 ) in y, and with fixed filling fractions:

1 2iπ A I ydx = T n I N = ǫ I , I ǫ I = T (10-26)
and such that in the limit T → 0 and ∀I ǫ I → 0, one has:

y ∼ V ′ 1 (x) - i ǫ i x -X i + O(T 2 ) , x ∼ V ′ 2 (y) - i ǫ i y -Y i + O(T 2 ). (10-27)
Then one has:

Theorem 10.2 F (g) 2MM = F (g) (E 2MM ) (10-28)
proof:

This theorem was proved in [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]. Again, this proves a posteriori, that F (g) 2MM (which we recall is a formal power series in T ) genericaly has a finite radius of convergence T < T c .

We also have:

Tr 1 x(p 1 ) -M 1 . . . Tr 1 x(p k ) -M 1 c = ∞ g=0 N 2-k-2g W (g) k (p 1 , . . . , p k ) dx(p 1 ) . . . dx(p k ) +δ k,1 N(V ′ (x(p 1 )) -y(p 1 )) - δ k,2 (x(p 1 ) -x(p 2 )) 2 . (10 -29)
Remark that the 1-Matrix model is a special case of the 2-matrix model with V 2 a quadratic polynomial.

Double scaling limits of matrix models, minimal CFT

It is well known that double scaling limits of the 1-matrix model, or 2-matrix models are in relationship with (p, q) minimal models of conformal field theory [START_REF] Kostov | Conformal field theory techniques in random matrix models[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF].

We have seen that as long as the curve is regular, all the F (g) 's can be computed. This shows that the radius of convergence in T of F (g) (T ) is reached for singular curves. So far, only rational singularities have been studied in detail.

Thus, consider the case where the potentials V 1 and V 2 are fine-tuned so that the curve E 2MM has a p/q singularity at T = T c (notice that for the one matrix model one necessarily has q = 2):

   T = T c x(z) ∼ p→a x(a) + (z -z(a)) q y(z) ∼ p→a y(a) + (z -z(a)) p (10-30)
We can use the notation of section 8 with t = T -T c , and thus at t = 0, the singularity is resolved, and we have (the local parameter is now

ζ = zt -ν ):    x(z, t) ∼ x(a) + t qν Q(ζ) + o(t qν ) y(z, t) ∼ y(a) + t pν P (ζ) + o(t pν ) ζ = zt -ν (10-31)
where Q is a polynomial of degree q and P is a polynomial of degree p.

The curve

E sing (ξ, η) = ξ = Q(ζ) η = P (ζ) = Resultant(Q -ξ, P -η) (10-32)
is called the singular spectral curve.

The dependence on T of the 2-Matrix model is such that:

d dT ydx x = dS ∞x,∞y (10-33)
where ∞ x and ∞ y are the 2 common poles of x and y. They are far away from branchpoints, and in particular from the singularity. This means that in the vicinity of the singularity we have:

d dT ydx x ∼ C t ν dζ + O(t 2ν ) (10-34)
where C is some constant of order 1. After substitution with the limit eq.10-31 this implies the Poisson relation 9 :

pP (ζ)Q ′ (ζ) -qQ(ζ)P ′ (ζ) = C ν t 1-(p+q-1)ν (10-35)
and therefore ν = 1 p + q -1 .

(10-36) Thereom 8.1 proves that:

F (g) 2MM (T ) ∼ T →Tc (T -T c ) (2-2g)(p+q)/(p+q-1) F (g) (E sing ) , for g ≥ 2 (10-37)
We also have:

F (0) 2MM (T ) ∼ T →Tc C 2 2 (p + q -1) 2 (p + q)(p + q + 1) (T -T c ) 2+2ν + reg , for g = 0 (10-38) F (1) 2MM (T ) ∼ T →Tc - 1 24 (p -1)(q -1)ν ln (T -T c ) + O(1)
, for g = 1 (10-39)

We thus have a way to compute explicitely the double scaling limit of F (g) 2MM .

(p,q) minimal models

Let us study in more details the curve E (p,q) (cf [START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF]).

As we have seen above, the curve for the (p, q) minimal model is of the form:

E (p,q) (x, y) = x = Q(ζ) y = P (ζ) = Resultant(Q -x, P -y) (10- 40
)
where P and Q are polynomials of respective degrees p and q, satisfying:

pP Q ′ -qQP ′ = t 1 ν (10-41)
The solution of which can be written [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF]:

P = (Q p/q ) + (10-42) and (Q p/q ) -= t 1 q ζ 1-q + O(ζ -q ), (10-43)
where we have used the notations f = f + + f -, with f + and f -denoting respectively the positive and the negative part of the Laurent series of f . This last equation implies q -2 equations for the coefficients of Q.

The curve E (p,q) has genus zero g / = 0, and is such that x and y have only one pole α = ∞. The Bergmann kernel is:

B(ζ 1 , ζ 2 ) = dζ 1 dζ 2 (ζ 1 -ζ 2 ) 2 .
(10-44)

The moduli (of the pole) of that curve are the Q k and P k such that:

Q(ζ) = q k=0 Q k ζ k , P (ζ) = p k=0 P k ζ k (10-45)
by a translation on ζ, we can assume that Q q-1 = 0, and by a rescaling of ζ we can assume that Q q-2 = -qQ q , and the Poisson Eq. (10-41) implies that P p-1 = 0 and P p-2 = -pP p , thus: -46) We find:

Q q-1 = P p-1 = 0 , Q q-2 Q q = -q , P p-2 P p = -p. ( 10
F (0) (E (p,q) ) = 0.
(10-47)

Other times

More generaly [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF], we can deform the (p, q) minimal model with p + q -2 times t 1 , . . . , t p+q-2 . For this purpose, one considers Q(ζ) a degree q monic polynomial,

Q(ζ) = ζ q + q-2 j=0 u q-j ζ j (10-48)
whose coefficients u 2 , . . . , u q are determined as functions of q-1 parameters, t 1 , . . . , t q-1 , by the following requirement:

(Q p/q ) -= q-2 j=1 q -j q t q-j Q -j/q + t 1 q ζ 1-q + O(ζ -q ). (10-49)
Then we define the degree p monic polynomial P (ζ) by:

P = ζ p + p-2 j=0 v p-j ζ j = Q p/q + - p-1 j=1 j + q q t q+j-1 Q j/q + (10-50)
which depends on p -1 other times t q , . . . , t q+p-2 .

The corresponding classical spectral curve is: -51) and depends on times t 1 , . . . , t p+q-2 . One can check that if t 2 = t 3 = . . . = t p+q-1 = 0, one recovers the (p, q) minimal model. It is well knwon that this curve is the spectral curve of the dispersionless Witham hierarchy [START_REF] Krichever | The τ -function of the universal Whitham hierarchy, matrix models and topological field theories[END_REF].

E (p,q) (x, y) = Resultant(x -Q, y -P ). ( 10

Examples of minimal models

• Airy curve (2, 1)

The classical spectral curve for the (2, 1) minimal model is

E (2,1) (x, y) = y 2 + t 1 -x. (10-52) Q(ζ) = ζ 2 + t 1 , P (ζ) = ζ (10-53)
It is studied with particular care in section 10.5 since it describes the behaviour of a generic curve around the branch points, and thus coincides with Tracy-Widom law [START_REF] Traicy | Level-spacing distributions and the Airy kernel[END_REF].

• Pure gravity (3, 2)

Q(ζ) = ζ 2 -2v , P (ζ) = ζ 3 -3vζ , t 1 = 3v 2 (10-54)
The classical spectral curve is:

E (3,2) (x, y) = x 3 -3v 2 x -y 2 + 2v 3 (10-55)
and is studied in details in section 10.6 .

• Ising model (4, 3) with:

Q(ζ) = ζ 3 -3vζ -3w , P (ζ) = ζ 4 -4vζ 2 -4wζ + 2v 2 - 5 3 t 5 (ζ 2 -2v)
t 1 = 4v 3 + 6w 2 , t 2 = 6vw. (10-57)
The classical spectral curve is

E (4,3) (x, y) = x 4 -y 3 -4v 3 x 2 + 3v 4 y + 2v 6 +12wv(-xy + v 2 x) + 6w 2 (-x 2 + 2vy -4v 3 ) + 8w 3 x -3w 4 . +5t 5 (-x 2 y -v 2 x 2 + 2v 3 y + 2v 5 -2wyx + 2v 2 wx + 3w 2 y -17v 2 w 2 ) + 25 3 t 2 5 (v 2 y + 2v 4 -4vwx -12vw 2 ) + 125 27 t 3 5 (-x 2 + 2v 3 -6wx -9w 2 ) (10-58)
The variations with respect to the moduli t 5 , t 2 and t 1 correspond respectively to the variations of the form ydx

Ω 5 = -dΛ 5 = d(ζ 5 -5vζ 3 + 10v 2 ζ), (10-59
) and Ω 2 = -dΛ 2 = d(ζ 2 + 5 6

Ω 1 = -dΛ 1 = d(ζ + 5 12
t 5 v 3 -w 2 (2v 2 ζ + 2vw -wζ 2 )),
t 5 v 3 -w 2 (v 2 ζ 2 -2vwζ -2w 2 )) (10-61)
in the notations of section 5.

• Unitary models (q + 1, q) E (q+1,q) (x, y) = T q+1 (x) -T q (y) (10-62)

Q(ζ) = T q (ζ) , P (ζ) = T q+1 (ζ) (10-63)
where T p is the p th Tchebychev's polynomial.

Matrix model with external field

Define the formal matrix model with external field [START_REF] Zinn-Justin | Universality of correlation functions of hermitian random matrices in an external field[END_REF]:

Z Mext = dM e -N Tr (V (M )-M Λ) (10-64)
where V ′ (x) is a rational function with denominator D(x), and Λ is a fixed N × N matrix. Consider its topological expansion (in the sense of a formal integral as in the 1-matrix case above [START_REF] Eynard | Formal matrix integrals and combinatorics of maps[END_REF]):

F Mext = -ln Z Mext = ∞ g=0 N 2-2g F (g) Mext . (10-65)
Let us assume that Λ has s distinct eigenvalues λ1 , . . . , λs with respective multiplicities m 1 , . . . , m s such that i m i = N. The minimal polynomial of Λ is:

S(y) = i (y -λi ).
(10-66)

Define the classical curve obtained by "removing" the 1/N 2 connected term in the loop equations:

E Mext (x, y) = ((V ′ (x) -y)S(y) -P (x, y))D(x) (10-67)
where

P (x, y) = 1 N Tr V ′ (x) -V ′ (M) x -M S(y) -S( Λ) y -Λ (10-68)
so that P (x, y)D(x) is a polynomial in both x and y. Then one has:

Theorem 10.3 F (g) Mext = F (g) (E Mext ) (10-69)
proof:

This theorem is proved in appendix D, and the proof is very similar to that of the 2-matrix case above, see [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF].

Application to Kontsevitch integral

The Kontsevitch integral is known to be the generating function which computes intersection numbers of moduli space of Riemann surfaces (see [START_REF] Kontsevitch | Intersection theory on the moduli space of curves and the matrix Airy function[END_REF][START_REF] Dijkgraaf | Intersection theory, integrable hierarchies and topological field theory[END_REF]). It is defined as:

Z Kontsevitch = dM e -N Tr ( M 3 3 -M (Λ 2 +t 1 )) , t 1 = 1 N Tr 1 Λ (10-70)
where Λ has eigenvalues λ 1 , . . . , λ N . Thus it is Z Mext with V (x) = x 3 /3 and Λ = Λ 2 +t 1 . Its classical curve is:

E Kontsevitch (x, y) = (x 2 -y)S(y) -xS 1 (y) -S 2 (y) (10-71)
where S 1 (y) and S 2 (y) are polynomials in y of degree at most s -1.

If we assume that the curve has genus zero (which is the case if we want the F (g) Kontsevitch to be the generating functions for intersection numbers), then we can find explicitely a rational parametrization:

E Kontsevitch (x, y) = x(z) = z + 1 2N Tr 1 Λ 1 z-Λ y(z) = z 2 + 1 N Tr 1 Λ . (10-72)
Using the symplectic invariance of theorem 7.1, we may exchange the roles of x and y.

There is a unique branch point in y, solution of y ′ (z) = 0, located at z = 0. Since the formulae for F (g) consist in taking residues of rational functions at the branch point, we may consider the Taylor expansion of x(z) near z = 0, i.e.

E Kontsevitch (x, y) = x(z) = z -1 2 ∞ k=0 t k+2 z k y(z) = z 2 + t 1 (10-73)
where we have defined the Kontsevitch times:

t k = 1 N Tr Λ -k (10-74)
so that:

F (g) Kontsevitch = F (g) (E Kontsevitch ) (10-75)
Again, using symplectic invariance of theorem 7.1, we may add to x(z) any rational function of y(z), i.e. we immediately get a 1-line proof of the following well-known theorem:

Theorem 10.4 F (g)
Kontsevitch depends only on the odd times t 2k+1 , with k ≤ 3g -2:

F (g) Kontsevitch = F (g) Kontsevitch (t 1 , t 3 , t 5 , . . . , t 6g-3 ) (10-76)
And, if we assume that t k = 0 for k ≥ p + 2, the curve is:

x(z) = z -1 2 p k=0 t k+2 z k y(z) = z 2 + t 1 (10-77)
which is identical to the curve of the (p, 2) model of section 10.3.1, and which is well known to satisfy KdV hierarchy. Thus, again we have a 1-line proof of the well known result:

Theorem 10.5 Z Kontsevitch is a KdV hierarchy tau function.

• Examples: the first few correlation functions For Kontsevitch's curve we have:

B(z, z ′ ) = dz dz ′ (z -z ′ ) 2 , dE z (z ′ ) = z dz ′ z 2 -z ′2 , ω(z) = 2z 2 dz (2 - j t 2j+3 z 2j ) (10-78)
and the only branchpoint is located at z = 0.

From the def.4.2 we easily get the first correlation functions:

W (1) 1 (z) = - dz 8(2 -t 3 ) 1 z 4 + t 5 (2 -t 3 )z 2 , (10 -79) W (0) 3 (z 1 , z 2 , z 3 ) = - 1 2 -t 3 dz 1 dz 2 dz 3 z 2 1 z 2 2 z 2 3 , (10 -80) W (1) 2
(z 1 , z 2 ) = dz 1 dz 2 8(2 -t 3 ) 4 z 6 1 z 6 2 (2 -t 3 ) 2 (5z 4 1 + 5z 4 2 + 3z 2 1 z 2 2 ) +6t 2 5 z 4 1 z 4 2 + (2 -t 3 )(6t 5 z 4 1 z 2 2 + 6t 5 z 2 1 z 4 2 + 5t 7 z 4 1 z 4
2 ) (10 -81) and The first and second order free energies are found:

W (2) 1 (z) = - dz 128(2 -t 3 ) 7 z
F (1) Kontsevitch = - 1 24 ln 1 - t 3 2 (10-83)
and

F (2) Kontsevitch = 1 1920 252 t 3 5 + 435 t 5 t 7 (2 -t 3 ) + 175 t 9 (2 -t 3 ) 2 (2 -t 3 ) 5 . (10-84)
which coincide with expressions previously found in the litterature [START_REF] Itzykson | Combinatorics of the Modular Group II: The Kontsevich integrals[END_REF].

Example: Airy curve

The curve y = √ x is particularly important, because it corresponds to the leading behaviour of any generic curve near its branch points. It is also the minimal model (1, 2) (cf [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF][START_REF] Di Francesco | Conformal field theory[END_REF]), also called Tracy-Widom law [START_REF] Traicy | Level-spacing distributions and the Airy kernel[END_REF].

Consider the curve E(x, y) = y 2 -x.

(10-85)

We chose the uniformization p = y:

x(p) = p 2 y(p) = p (10-86)
There is only one pole α = ∞, and there is only one branch point located at a = 0, the conjugated point is p = -p. The Bergmann kernel is the Bergmann kernel of the Riemann sphere:

B(p, q) = dp dq (p -q) 2 , dE q (p) = q dp q 2 -p 2 , ω(q) = 4q 2 dq (10-87)
It is easy to see that all correlation functions with 2g + k ≥ 3 are of the form:

W (g) k (p 1 , . . . , p k ) = ω (g) k (p 2 1 , . . . , p 2 k ) dp 1 . . . dp k (10-88)
Moreover, the diagrammatic rules are clearly homogenous, so that the function W (g) 1 (p) must be a homogeneous function of p. It is easy to find that:

W (g) 1 (p) = c g dp p 6g-2
(10-89) and the total 1-point function is

W 1 (p, N) = -Nydx + ∞ g=1 N 1-2g W (g) 1 (p) = W 1 (N 1 3 p, 1). (10-90)
Similarly, the total 2-point function is

W 2 (p, q, N) = ∞ g=0 N -2g W (g) 2 (p, q) = W 2 (N 1 3 p, N 1 3 q, 1) (10-91)
and in general

W k (p 1 , . . . , p k , N) = ∞ g=0 N 2-2g-k W (g) k (p 1 , . . . , p k ) = W k (N 1 3 p 1 , . . . , N 1 3 p k , 1). (10-92)
The solution of the recursion def.4.2 can be found explicitely in terms of the Airy function.

Consider g(x) = Ai ′ (x)/Ai(x) where Ai(x) is the Airy function, i.e. g ′ (x) + g 2 (x) = x = p 2 . In terms of the variable p we write:

f (p) = g(p 2 ) , f 2 + f ′ 2p = p 2 . ( 10-93)
It can be expanded for large p:

f (p) = ∞ k=0 f k p 1-3k = p - 1 4p 2 - 9 32p 5 + . . . (10-94)
where the coefficients in the expansion satisfy

4 -3k 2 f k-1 + k j=0 f j f k-j = δ k,0 . (10-95)
The solution of the recursion def.4.2 for the 1-point function is:

W 1 (p, 1) = -2 p 2 -f (p)f (-p) f (p) -f (-p) pdp = -2p 2 dp+ dp (2p) 4 + 9!! dp 3 2 (2p) 10 + 15!! 3 4 (2p) 16 . . . (10-96) W 2 (p, p ′ , 1) = -4 (f (p) -f (p ′ )) (f (-p) -f (-p ′ )) (p 2 -p ′2 ) 2 (f (p) -f (-p)) (f (p ′ ) -f (-p ′ )) pdp p ′ dp ′ (10-97)
In particular

W 2 (p, p, 1) = f ′ (p)f ′ (-p) (f (p) -f (-p)) 2 dp 2
(10-98) so that:

W 2 (p, p, 1) + W 1 (p, 1) 2 = 4p 4 dp 2 = x dx 2 (10-99)
Similarly we find for instance (with obvious cyclic conventions for the indices):

W 3 (p 1 , p 2 , p 3 , 1) = dx 1 dx 2 dx 3 (p 2 3 -p 2 2 )(p 2 3 -p 2 1 )(p 2 2 -p 2 1 ) × × 3 i=1 f (p i )f (-p i )(f (p i-1 ) + f (-p i-1 ) -f (p i+1 ) -f (-p i+1 )) (f (p 1 ) -f (-p 1 ))(f (p 2 ) -f (-p 2 ))(f (p 3 ) -f (-p 3 )) = dp 1 dp 2 dp 3 2 p 2 1 p 2 2 p 2 3 + dp 1 dp 2 dp 3 2 6 p 8 1 p 8 2 p 8 3 + . . . (10 -100)
and one can easily find similar expressions for all W k 's.

In fact all correlation functions can be written with a determinantal formula [START_REF] Dyson | correlations between the eigenvales of a random matrix[END_REF][START_REF] Eynard | Matrices coupled in a chain: eigenvalue correlations[END_REF], with the Tracy-Widom kernel [START_REF] Traicy | Level-spacing distributions and the Airy kernel[END_REF]:

K(x, x ′ ) = Ai(x)Ai ′ (x ′ ) -Ai ′ (x)Ai(x ′ ) x -x ′ (10-101)
The fact that the Baker-Akhiezer function is Ai(x) and satisfies the differential equation Ai ′′ = xAi can be seen as a consequence of the Hirota equation theorem.9.2.

Remark 10.1 To large N leading order the first term W (0) k can be written in terms of Ferrer diagrams (Young diagrams):

W (0) k (p 1 , . . . , p k ) = k -3! 2 k-2 j dp j p 2 j |λ|=k-3 M λ (1/p 2 i ) j 2λ j + 1!! λ j ! 1 n j (λ)! (10-102)
where λ is a Ferrer diagram, n i (λ) = #{j / λ j = i}, and M λ are the elementary monomial symetric polynomials:

M λ (z i ) = i 1 =i 2 =... =i k-3 z λ 1 i 1 . . . z λ k-3 i k-3 . (10-103)
For instance with k = 4 there is only one diagram (1), and M (1) (z 1 , z 2 , z 3 , z 4 ) = z 1 + z 2 + z 3 + z 4 , and: In this section we study in details the (3, 2) minimal model, also called pure gravity [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF].

W (0) 4 (p 1 , p 2 , p 3 , p 4 ) = 3 4 dp 1 dp 2 dp 3 dp 4 p 2 1 p 2 2 p 2 3 p 2 4 1 p 2 1 + 1 p 2 2 + 1 p 2 3 + 1 p 2 4 . ( 10
It corresponds to the curve

E (3,2) =    x(z) = z 2 -2v y(z) = z 3 -3vz t 1 = 3v 2 . ( 10-106)
We recognize Tchebychev's polynomials T 2 and T 3 , which satisfy the Poisson relation Eq. . Up to a rescaling z = √ v p, the curve reads:

E (3,2) =    x(p) = v(p 2 -2) y(z) = v 3/2 (p 3 -3p) t 1 = 3v 2 . (10-107)
There is only one x-branch point at p = 0, and the conjugated point is p = -p. The Bergmann kernel is the Bergmann kernel of the Riemann sphere:

B(p, q) = dp dq (p -q) 2
, dE q (p) = q dp q 2 -p 2 (10-108) ω(q) = (y(q) -y(q))dx(q) = 4v 5/2 (q 2 -3) q 2 dq (10-109)

Φ(q) = v 5/2 ( 2 q 5 5 -2q 3 ) (10-110)
Under a variation of t 1 we have:

Ω 1 (p) = - ∂y(p)dx(p) ∂t 1 x(p) = v 1/2 dp = -v 1/2 Res ∞ qB(p, q) , Λ 1 (q) = -v 1/2 q
(10-111) so the effect of ∂/∂t 1 is equivalent to:

∂ ∂t 1 W (g) k x = -v 1/2 Res q→∞ qW (g) k+1
(10-112) 10.6.1 Some correlation functions Using def.4.2, we find:

W (0) 3 (p 1 , p 2 , p 3 ) = - v -5/2 6 dp 1 dp 2 dp 3 p 2 1 p 2 2 p 2 3 (10-113) W (1) 1 (p)
= - v -5/2 (12) 2 p 2 + 3 p 4 dp (10-114) W (1)
2 (p, q) = v -5 15q 4 + 15p 4 + 6p 4 q 2 + 2p 4 q 4 + 9p 2 q 2 + 6p 2 q 4 2 5 3 3 p 6 q 6 dp dq (10-115)

W (2) 1 (p) = -v -
∂ 3 F (0) ∂t 3 1 = - 1 6v = - 1 2 √ 3t 1 -→ ∂ 2 F (0) ∂t 2 1 = - t 1/2 1 √ 3 (10-119)
and using eq.10-114:

∂F (1) ∂t 1 = - 1 (12) 2 v 2 = - 1 48t 1 -→ ∂ 2 F (1) ∂t 2 1 = 1 48 t 2 1
(10-120) as well as using eq.10-116: ∂F (2) ∂t 1 = -v -7 7 2 8 3 5 = -

7 2 8 3 3/2 t 7/2 1 -→ ∂ 2 F (2) ∂t 2 1 = 49
2 9 3 3/2 t 9/2 1
.

(10-121)

We may thus verify that the second derivative of the free energy: -122) satisfies the Painlevé equation to the first orders:

u = ∂ 2 F ∂t 2 1 = ∞ g=0 t (1-5g)/2 1 u (g) , F (g) = 4u (g) 5(1 -g)(3 -5g) . ( 10
u 2 + 1 6 u ′′ = 1 3 t 1 . (10-123)
It is well known that this equation is satisfied to all orders [START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF], and here this can be seen as a consequence of the Hirota equation theorem. 9.2.

Conclusion

In this paper, we have constructed an infinite family of invariants of algebraic curves. By construction, these invariants coincide with the topological expansion of matrix integrals in the special case where the algebraic curve is the large N part of the matrix integral's spectral curve. But we emphasize again that the construction presented here goes beyond matrix models.

Our invariants are defined only in terms of algebraic geometry, and they have many interesting properties, like homogeneity, and integrability (they obey some Hirota equation).

The problem of computing the F (g) 's for matrix models is an old problem which was addressed many times, and which found more and more elaborate answers [START_REF] Ambjørn | Matrix model calculations beyond the spherical limit[END_REF]. We claim that ours is more efficient, because it contains all multicut cases, and various types of matrix models at once. Also, even in the simplest cases (1-matrix model, 1 cut), our expressions are simpler than what existed before [START_REF] Ambjørn | Matrix model calculations beyond the spherical limit[END_REF]. Our F (g) 's are defined recursively, like those of [START_REF] Ambjørn | Matrix model calculations beyond the spherical limit[END_REF], but the recursions are much easier to handle, and it is much easier to deduce properties to any order g from our construction.

The efficiency of our method becomes striking when one wants to compare different models (Kontsevitch and KdV for instance), or when one wants to take singular limits. Another important application of our method, is to prove that our F (g) 's provide a solution to the holomorphic anomaly equations of [START_REF] Bershadsky | Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes[END_REF] in topological string theory, thus confirming the Dijkgraaf Vafa correspondence. We claim that this can be proved easily from our work, and we present it in a coming paper [START_REF] Eynard | Holomorphic anomaly and matrix models[END_REF]. It would also be interesting to compare our free energies with the D-Modules considered in [START_REF] Alexandrov | M-Theory of Matrix Models[END_REF][START_REF] Alexandrov | Instantons and Merons in Matrix Models[END_REF] as partition functions of a unified matrix M-theory by checking that they indeed satisfy the equations of [START_REF] Alexandrov | Instantons and Merons in Matrix Models[END_REF].

One of the reasons our method is very efficient also, is that it can be represented diagrammatically, without equations, and thus very easy to remember.

Perspectives and generalizations:

• The first thing one could think about is to understand what our F (g) 's compute in algebraic geometry. There has been some attempts to recognize the first few of them F (1) as the Dedekind function [START_REF] Dubrovin | Geometry of 2D topological field theories[END_REF][START_REF] Dubrovin | Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation[END_REF], F (2) as the Eisenstein series [START_REF] Huang | Topological String Theory on Compact Calabi-Yau: Modularity and Boundary Conditions[END_REF], but the answer for higher g is still obscure. Also a combinatoric interpretation is missing.

• Beyond that, it would be important to understand the combinatorics behind our diagrammatic construction. Since all diagrams are obtained from trees, it seems to be related to Schaeffer's method for counting maps [START_REF] Schaeffer | Bijective census and random generation of Eulerian planar maps[END_REF], but this issue needs to be investigated further.

• Double scaling limits of matrix models are in relationship with conformal field theory (CFT), and in particular minimal models. It would be interesting to compare our formulae with those obtained directely from CFT [START_REF] Di Francesco | Conformal field theory[END_REF][START_REF] Zamolodchikov | Structure Constants and Conformal Bootstrap in Liouville Field Theory[END_REF][START_REF] Al | On the Three-point Function in Minimal Liouville Gravity[END_REF], and, since higher genus CFT is far less known, maybe our method can bring something new to CFT.

• Also, the link between formal matrix models (i.e. those defined as combinatorial generating functions, for which it makes sense to consider a topological expansion, cf. [START_REF] Eynard | Formal matrix integrals and combinatorics of maps[END_REF]), and actual convergent matrix integrals needs to be better understood.

The difficulty lies beyond any order in perturbation, and integrability could play a key role in understanding the relationship in more details.

• It would be interesting to check that the topological expansion for the chain of matrices [START_REF] Eynard | Master loop equations, free energy and correlations for the chain of matrices[END_REF], is also given by the same F (g) 's. This is an open question, but there are strong evidences that the answer is positive. For instance it is easy to see from [START_REF] Eynard | Master loop equations, free energy and correlations for the chain of matrices[END_REF] that F (0) and some of the first few correlation functions are indeed the same.

• It would be interesting also to extend our construction to other types of matrix models, for instance non-hermitian (real symmetric or quaternionic). A first attempt was done in [START_REF] Chekhov | Matrix eigenvalue model: Feynman graph technique for all genera[END_REF]. Another possible extension is towards the O(n) matrix model, whose large N limit is known in terms of an agebraic curve [START_REF] Eynard | Exact solution of the O(n) model on a random lattice[END_REF][START_REF] Eynard | More on the exact solution of the O(n) model on a random lattice, and an investigation of the case -n-¿2[END_REF]. To leading order, the O(n) matrix model solution looks very similar to that of the 1-matrix model, except that correlation functions are no longer meromorphic functions on the curve. Instead they gain a phase shift after going around a non-trivial cycle. This would probably allow to define a twisted version of our construction.

Theorem 4.4 proof:

The case k = 1, g = 0 comes from Eq. (A-1), and by integration:

i dE q (p i ) = 0 (A-5)
which proves equation 4-23. Then, it is clear from the iterative definition of W (g) k+1 for k ≥ 1, that the dependance of W (g) k+1 (p 1 , p, p 2 , . . . , p k ) in p is a sum of integrals involving only the following quantities: j Res q→a j dE j,q (q 1 ) ω j (q) B(q, p) W (g) l+1 (q, q 2 , . . . , q l ) (A-6)

for some q 1 , . . . , q l 10 . Using equation A-1 we have to compute:

i j
Res q→a j dE j,q (q 1 ) ω j (q) B(q, p i ) W (g) l+1 (q, q 2 , . . . , q l ) = j Res q→a j dE j,q (q 1 ) ω j (q) dx(q)dx(p) (x(q) -x(p)) 2 W (g) l+1 (q, q 2 , . . . , q l ) = 1 2 j Res q→a j dE j,q (q 1 ) ω j (q) dx(q)dx(p) (x(q) -x(p)) 2 W (g) l+1 (q, q 2 , . . . , q l ) + W (g) l+1 (q, q 2 , . . . , q l ) = -1 2

q i =q,q j Res q→a j dE j,q (q 1 ) ω j (q) dx(q)dx(p) (x(q) -x(p)) 2 W (g) l+1 (q i , q 2 , . . . , q l ) = 0 (A-7)

where the second equality holds due to A-2, the third equality holds due to equation 4-23, and the last equality holds because that last expression has no poles at the branchpoints. This proves equation 4-24.

Theorem 4.5 proof: It is clearly a rational function of x(p) because it is a symmetric sum on all sheets. From theorem 4.2, the RHS may have poles at branch points, and/or at the poles of some y(p i ), and/or when x(p) = x(p l ) for some l. Let us prove that the poles at branch points actualy cancel.

Let us denote in this section:

U (g) k (q, q ′ , p K ) = g m=0 J⊂K W (m) j+1 (q, p J )W (g-m) k-j+1 (q ′ , p K/J ) + W (g-1)
k+2 (q, q ′ , p K ).

(A-8)

From theorem 4.4, we have:

i U (g)
k (q i , q i , p K ) 10 Since it may not be clear which branch point dE and ω refers to, one indicates it with an index.

=

- i l =i U (g) k (q l , q i , p K ) = -U (g) k (q, q, p K ) -U (g)
k (q, q, p K ) -

q i =q,q (U (g) k (q, q i , p K ) + U (g) k (q, q i , p K )) - q i =q,q (U (g) k (q i , q, p K ) + U (g)
k (q i , q, p K ))q i =q,q l =i,q l =,q,q U (g)

k (q l , q i , p K ) (A -9)
and from theorem 4.4 and theorem 4.2, only the first two terms have poles at branch points, and thus:

- i j Res q→a j dE j,q (p) ω j (q) U (g) k (q i , q i , p K ) = j Res q→a j dE j,q (p) ω j (q) U (g) k (q, q, p K ) + U (g) k (q, q, p K ) = 2 j Res q→a j dE j,q (p) ω j (q) U (g) k (q, q, p K ) = 2W (g) k+1 (p, p K ) (A-10)
where the last equality holds from the definition of W (g)

k+1 . Thus we have:

W (g) k+1 (p, p K ) = - 1 2 j Res q→a j dE j,q (p) ω j (q) i U (g)
k (q i , q i , p K ) .

(A -11)

Then we rewrite it in terms of P (g) k above:

W (g) k+1 (p, p K ) = - 1 2 j Res q→a j dE j,q (p) ω j (q) P (g) k (x(q), p K ) dx(q) 2 +2 i y(q i )dx(q)W (g) k+1 (q i , p K ) = - 1 2 j Res q→a j dE j,q (p) ω j (q) P (g) k (x(q), p K ) dx(q) 2 + 2y(q)dx(q)W (g) k+1 (q, p K ) +2y(q)dx(q)W (g) k+1 (q, p K ) = - 1 2 j Res q→a j dE j,q (p) ω j (q) P (g) k (x(q), p K ) dx(q) 2
+2(y(q) -y(q))dx(q)W (g)

k+1 (q, p K ) (A -12)
where we have used theorem 4.4 again. That gives

W (g) k+1 (p, p K ) = - 1 2 j Res q→a j dE j,q (p) ω j (q) P (g) k (x(q), p K ) dx(q) 2 - j Res q→a j dE j,q (p) W (g) k+1 (q, p K ). (A -13)
Let us compute that last integral:

j Res q→a j dE j,q (p) W (g) k+1 (q, p K ) = -Res q→p dE j,q (p) W (g) k+1 (q, p K ) + 1 2iπ i q ′ ∈B i B(p, q ′ ) A i W (g) k+1 (q, p K ) - 1 2iπ i q ′ ∈A i B(p, q ′ ) B i W (g) k+1 (q, p K ) = -Res q→p dE j,q (p) W (g) k+1 (q, p K ) + i du i (p) A i W (g) k+1 (q, p K ) = -Res q→p dE j,q (p) W (g) k+1 (q, p K ) = -W (g) k+1 (p, p K ) (A-14)
where we have deformed the contour of integration using Riemann bilinear identity, and then we have used theorem 4.3. Thus we have: j Res q→a j dE j,q (p) ω j (q) P (g) k (x(q), p K ) dx(q) 2 = 0. (A-15)

Since this holds for any p, we can write for any m ≥ 0:

0 = Res p→a i (y(p) -y(a i )) (x(p) -x(a i )) m j Res q→a j dE j,q (p) ω j (q) P (g) k (x(q), p K ) dx(q) 2 = -Res q→a i Res p→q,q dE j,q (p) ω j (q) (y(p) -y(a i )) (x(p) -x(a i )) m P (g) k (x(q), p K ) dx(q) 2 = -Res q→a i dx(q) (x(q) -x(a i )) m P (g) k (x(q), p K ) (A -16)
which proves that P (g) k (x(q), p K ) can have no pole at q = a i . Theorem 4.6 proof: Assume this is proved for h < g, and at g, it is proved for l < k. Then we have:

W (g) 2 (p 1 , p 2 ) = i Res q→a i dE q (p 1 ) ω(q) 2 g m=0 W (m) 2 (q, p 2 )W (g-m) 1 (q) + W (g-1) 3 (q, q, p 2 ) = i Res q→a i dE q (p 1 ) ω(q) B(q, p 2 ) W (g) 1 (q) + i Res q→a i i ′ Res q ′ →a i ′ dE q (p 1 ) ω(q) dE i ′ ,q ′ (p 2 ) ω i ′ (q ′ ) 2 g-1 m ′ =0 W (m ′ ) 2 (q ′ , q)W (g-m ′ -1) 2 (q ′ , q) +2 g m=1 W (g-m) 1 (q)W (m-1) 3 (q ′ , q ′ , q) + 2 g-1 m ′ =0 W (m ′ ) 3 (q ′ , q, q)W (g-m ′ -1) 1 (q ′ ) +4 g m=1 m m ′ =0 W (m ′ ) 2 (q ′ , q)W (m-m ′ ) 1 (q ′ )W (g-m) 1 (q) + W (g-2) 4 (q, q ′ , q, q ′ ) (A -17)
and W (g) 2 (p 2 , p 1 ) is given by the same integral except that the order for computing residues is reversed, the residue in q is computed before q ′ . The difference is thus obtained by pushing the contour of q ′ through the contour of q, and is obtained as the residue at q = q ′ . Notice that only W (0) 2 (q, q ′ ) has a pole at q = q ′ , all the other W (g) k have no poles at q = q ′ from theorem 4.2. Thus we have:

W (g) 2 (p 1 , p 2 ) -W (g) 2 (p 2 , p 1 ) -i Res q→a i dEq(p 1 ) B(q,p 2 )-dEq(p 2 ) B(q,p 1 ) ω(q) W (g) 1 (q) = i Res q→a i Res q ′ →q dEq(p 1 ) ω(q) dE q ′ (p 2 ) ω i (q ′ ) B(q, q ′ ) 4 g m=0 W (m) 1 (q ′ )W (g-m) 1 (q) + 2W (g-1) 2 (q ′ , q) = 2 i Res q→a i Res q ′ →q dEq(p 1 ) ω(q) dE q ′ (p 2 ) ω i (q ′ ) B(q, q ′ ) U (g) 0 (q, q ′ ) = i Res q→a i Res q ′ →q dEq(p 1 ) ω(q) dE q ′ (p 2 ) ω i (q ′ ) B(q, q ′ ) U (g) 0 (q, q ′ ) + U (g) 0 (q, q ′ ) (A-18)
where we have used the notation of theorem 4.5.

Similarly, for higher values of k, we find:

W (g) 2+k (p 1 , p 2 , p K ) -W (g) 2+k (p 2 , p 1 , p K ) - i Res q→a i dE q (p 1 ) B(q, p 2 ) -dE q (p 2 ) B(q, p 1 ) ω(q) W (g) k+1 (q, p K ) = i Res q→a i Res q ′ →q dE q (p 1 ) ω(q) dE q ′ (p 2 ) ω i (q ′ ) B(q, q ′ ) U (g) k (q, q ′ , p K ) + U (g) k (q, q ′ , p K ) (A -19)
Then it gives:

W (g) 2+k (p 1 , p 2 , p K ) -W (g) 2+k (p 2 , p 1 , p K ) - i Res q→a i dE q (p 1 ) B(q, p 2 ) -dE q (p 2 ) B(q, p 1 ) ω(q) W (g) k+1 (q, p K ) = i Res q→a i dE q (p 1 ) ω(q) d q ′ dE q ′ (p 2 ) ω i (q ′ ) U (g) k (q, q ′ , p K ) + U (g) k (q, q ′ , p K ) q ′ =q (A -20)
and by integrating half of it by parts we get:

W (g) 2+k (p 1 , p 2 , p K ) -W (g) 2+k (p 2 , p 1 , p K ) = -i Res q→a i B(q,p 1 )dEq(p 2 )-B(q,p 2 )dEq(p 1 ) ω(q) 2 U (g) k (q, q, p K ) + U (g) k (q, q, p K ) + i Res q→a i dEq(p 1 ) B(q,p 2 )-dEq(p 2 ) B(q,p 1 ) ω(q) W (g) k+1 (q, p K ) (A-21)
Now we use the following Lemma:

Lemma A.1 If f (q, q ′
) is localy a bilinear differential near a branchpoint a i , with no poles, and symmetric in q and q, and in q ′ and q ′ , then:

Res q→a i B(q, p 1 )dE q (p 2 ) -B(q, p 2 )dE q (p 1 ) ω(q) 2 f (q, q) = 0. (A-22)
Proof of the Lemma:

The residue is a simple pole, and we can use formula A-3, that gives:

Res q→a i B(q, p 1 )dE q (p 2 ) -B(q, p 2 )dE q (p 1 ) ω(q) 2 f (q, q) = Res q→a i
B(q, p 1 )B(q, p 2 ) -B(q, p 2 )B(q, p 1 ) ω(q)dy(q)dx(q) f (q, q) = 0 (A-23)

.

Using this Lemma, as well as theorem 4.5, we get:

W (g) 2+k (p 1 , p 2 , p K ) -W (g) 2+k (p 2 , p 1 , p K ) = -i Res q→a i B(q,p 1 )dEq(p 2 )-B(q,p 2 )dEq(p 1 ) ω(q) 2
(y(q) -y(q))dx(q)W (g) k+1 (q, p K ) + i Res q→a i dEq(p 1 ) B(q,p 2 )-dEq(p 2 ) B(q,p 1 ) ω(q) W (g) k+1 (q, p K ) = 0.

(A-24) Corollary 4.1 proof: For any rational function R(x) with no pole at x(a i ) we have:

Res a i R(x(p))W (g) k+1 (p, p 1 , . . . , p k ) = 1 2 Res a i R(x(p)) W (
k+1 (p, p K ) = - 1 2 α Res p→α x(p) m dx(p) -2y(p)dx(p)W (g) k+1 (p, p K )+ + g h=0 I⊂K W (h) |I|+1 (p, p I )W (g-h) k-|I|+1 (p, p K/I ) + W (g-1) k+2 (p, p, p K ) = 1 2 Res p→a i ,p K x(p) m dx(p) -2y(p)dx(p)W (g) k+1 (p, p K )+ + g h=0 I⊂K W (h) |I|+1 (p, p I )W (g-h) k-|I|+1 (p, p K/I ) + W (g-1) k+2 (p, p, p K ) = 1 2 k j=1 Res p→p j x(p) m dx(p) B(p, p j )W (g) k (p, p K/{j} ) + 1 2 i Res p→a i x(p) m dx(p) -2y(p)dx(p)W (g) k+1 (p, p K )+ + g h=0 I⊂K W (h) |I|+1 (p, p I )W (g-h) k-|I|+1 (p, p K/I ) + W (g-1) k+2 (p, p, p K ) = 1 2 k j=1 d p j x(p j ) m W (g) k (p K ) dx(p j ) + 1 4 i Res p→a i x(p) m dx(p) -2y(p)dx(p)W (g) k+1 (p, p K )+ + g h=0 I⊂K W (h) |I|+1 (p, p I )W (g-h) k-|I|+1 (p, p K/I ) + W (g-1) k+2 (p, p, p K ) + 1 4 i Res p→a i x(p) m dx(p) -2y(p)dx(p)W (g) k+1 (p, p K )+ + g h=0 I⊂K W (h) |I|+1 (p, p I )W (g-h) k-|I|+1 (p, p K/I ) + W (g-1) k+2 (p, p, p K ) = 1 2 k j=1 d p j x(p j ) m W (g) k (p K ) dx(p j ) + 1 4 i Res p→a i x(p) m dx(p) P (g) k (x(p), p K )dx 2 (p) = 1 2 k j=1 d p j x(p j ) m W (g) k (p K ) dx(p j ) (A -26)
due to theorem 4.5.

Theorem 4.7 proof:

The case k = 1, g = 0 is easy:

Res p 2 →p 1 Φ(p)B(p 1 , p 2 ) = dΦ(p 2 ) = y(p 1 )dx(p 1 ). (A-27)
We prove the theorem by recursion on g and k. Suppose it is proved for all k ′ for g ′ ≤ g-1, and for k ′ ≤ k-1 if g ′ = g. We write K = {1, . . . , k} and K ′ = {1, . . . , k-1}.

Then we have from Eq. (4-9):

Res

p k →a Φ(p k )W (g) k+1 (p, p 1 , . . . , p k ) = Res p k →a Res q→a Φ(p k ) dE q (p) ω(q) g m=0 J⊂K ′ W (m) j+2 (q, p J , p k )W (g-m) k-j (q, p K ′ /J ) + g m=0 J⊂K ′ W (m) j+1 (q, p J )W (g-m) k-j+1 (q, p K ′ /J , p k ) + W (g-1)
k+2 (q, q, p K ′ , p k ) .

(A -28)

Then we exchange the contours of integration: Res .

(A-29)

Thus

Res

p k →a Φ(p k )W (g) k+1 (p, p 1 , . . . , p k ) = Res q→a Res p k →a Φ(p k ) dE q (p) ω(q) g m=0 J⊂K ′ W (m) j+2 (q, p J , p k )W (g-m) k-j (q, p K ′ /J ) + g m=0 J⊂K ′ W (m) j+1 (q, p J )W (g-m) k-j+1 (q, p K ′ /J , p k ) + W (g-1) k+2 (q, q, p K ′ , p k ) + Res q→a Res p k →q,q Φ(p k ) dE q (p) ω(q) g m=0 J⊂K ′ W (m) j+2 (q, p J , p k )W (g-m) k-j (q, p K ′ /J ) + g m=0 J⊂K ′ W (m) j+1 (q, p J )W (g-m) k-j+1 (q, p K ′ /J , p k ) + W (g-1)
k+2 (q, q, p K ′ , p k ) .

(A -30)

The first term is computed from the recursion hypothesis, and the second term can exist only if the correlation function containing p k has poles at p k = q or p k = q, and from theorem 4.2, this can happen only if the correlation function containing p k is a Bergmann kernel. That gives:

Res p k →a Φ(p k )W (g) k+1 (p, p 1 , . . . , p k ) = Res q→a dEq(p) ω(q) g m=0 J⊂K ′ (2m + (j + 1) -2)W (m) j+1 (q, p J )W (g-m) k-j (q, p K ′ /J ) + g m=0 J⊂K ′ (2(g -m) + (k -j) -2)W (m) j+1 (q, p J )W (g-m) k-j (q, p K ′ /J ) +(2(g -1) + k + 1 -2)W (g-1) k+1 (q, q, p K ′ ) + Res q→a Res p k →q,q Φ(p k ) dEq(p) ω(q) B(q, p k )W (g) k+1 (q, p K ′ ) + W (g) k (q, p K ′ )B(q, p k ) = (2g + k -3) Res q→a dEq(p) ω(q) g m=0 J⊂K ′ W (m) j+1 (q, p J )W (g-m) k-j (q, p K ′ /J ) +W (g-1)
k+1 (q, q, p K ′ ) + Res q→a dEq(p) y(q)-y(q) y(q)W (g)

k+1 (q, p K ′ ) + y(q)W (g) k (q, p K ′ ) = (2g + k -3) W k (p, p 1 . . . , p k-1 ) + Res q→a dEq(p) y(q)-y(q) y(q)(W (g) k+1 (q, p K ′ ) + W (g) k (q, p K ′ )) + (y(q) -y(q))W (g) k (q, p K ′ ) = (2g + k -3) W k (p, p 1 . . . , p k-1 ) + Res q→a dEq(p) y(q)-y(q) (y(q) -y(q))W (g) k (q, p K ′ ) = (2g + k -3) W k (p, p 1 . . . , p k-1 ) -Res q→a dE q (p) W (g) k (q, p K ′ ) = (2g + k -3) W k (p, p 1 . . . , p k-1 ) + 1 2 Res q→a dS q,o (p) W (g) k (q, p K ′ ) -1 2 Res q→a dS q,o (p) W (g) k (q, p K ′ ) = (2g + k -3) W k (p, p 1 . . . , p k-1 ) + 1 2 Res q→a dS q,o (p) W (g) k (q, p K ′ ) -1 2 Res q→a dS q,o (p) W (g) k (q, p K ′ ) = (2g + k -3) W k (p, p 1 . . . , p k-1 ) + Res q→a dS q,o (p) W (g) k (q, p K ′ ) = (2g + k -3) W k (p, p 1 . . . , p k-1 ) -Res q→p dS q,o (p) W (g) k (q, p K ′ ) -1 2iπ j B j B(q, p) A j W (g) k (q, p K ′ ) -A j B(q, p) B j W (g) k (q, p K ′ ) = (2g + k -3) W k (p, p 1 . . . , p k-1 ) -Res q→p dS q,o (p) W (g) k (q, p K ′ ) -du t (p) (1 + κτ ) A W (g) k (q, p K ′ ) -κ B W (g) k (q, p K ′ ) = (2g + k -3) W k (p, p 1 . . . , p k-1 ) -Res q→p dS q,o (p) W (g) k (q, p K ′ ) -du t (p) A W (g) k (q, p K ′ ) = (2g + k -3) W k (p, p 1 . . . , p k-1 ) -Res q→p dS q,o (p) W (g) k (q, p K ′ ) = (2g + k -2) W k (p, p 1 . . . , p k-1 ) (A-31)

B Variation of the curve

In this appendix, we proove the theorems stated in sections ( 5) and section (6.1). Lemma 5.1 proof:

D Ω j Res q→a j dEq(p)
ω(q) f (q, q)

x(p)

= j Res q→a j dEq(p) ω(q) D Ω (f (q, q)) x(q)j Res q→a j dEq(p)

(ω(q)) 2 (Ω(q) -Ω(q)) f (q, q) +2 j Res q→a j i Res r→a i dEr(p) ω(r) Ω(r) dEq(r) ω(q) f (q, q) = j Res q→a j dEq(p) ω(q) D Ω (f (q, q)) x(q) -2 j Res q→a j dEq(p) (ω(q)) 2 Ω(q) f (q, q) +2 j Res q→a j i Res r→a i dEr(p) ω(r) Ω(r) dEq(r) ω(q) f (q, q) = j Res q→a j dEq(p) ω(q) D Ω (f (q, q)) x(q) -2 j Res q→a j Res r→q dEq(r)dEr (p) ω(q)ω(r) Ω(r) f (q, q) +2 j Res q→a j i Res r→a i dEr(p) ω(r) Ω(r) dEq(r) ω(q) f (q, q) = j Res q→a j dEq(p) ω(q) D Ω (f (q, q)) x(q) +2 i Res r→a i j Res q→a j dEr(p) ω(r) Ω(r) dEq(r) ω(q) f (q, q) (B-1)

Theorem 5.1 proof: This theorem straightforwardly comes from the diagrammatic rules described in the preceding paragraph except for the variation of F (1) .

Let us prove it for F (1) with Ω(p) = C B(p, q)Λ(q). One has

-C W (g) 1 (p)Λ(p) = -Res q→a C dEq(p)Λ(p) ω(q)
B(q, q)

= -Res q→a C dEq(p)Λ(p)dz i (q)dz i (q) ω(q)

1 (z(q)-z(q)) 2 + 1 6 S B (q) (B-2)
where z i is a local variable near the branch point a i and S B is the corresponding Bergmann projective connection. Since the last term has a simple pole at the branch point a i , one can write -Res q→a C dE q (p)Λ(p)dz i (q)dz i (q) ω(q)

1 6 S B (q) = - 1 2 i
Ω(a i ) dy(a i ) Res

q→a i B(q, q) dx(q) = - 1 2 δ Ω lnτ Bx . (B -3)
On the other hand, one can express the first term thanks to the local variables z i and compute -Res q→a C dE q (p)Λ(p)dz i (q)dz i (q) ω(q)(z(q) -z(q)) 2 = -1 24 δ Ω (y ′ (a i ))

(B -4) provided that z i (q) -z i (q) = 2z i (q). Theorem (5.3) proof:

Using theorem 4.7, we have: (B-7) Since the poles α and the branch points a i do not coincide, one can exchange the order of integration. Then, one can move the integration contours for p in order to integrate only around the last pole p → q:

(2 -2g)F (g) = α t α α q=o Res p→q dS p,o ′ (q)W (g) 1 (p)α Res q→α Res p→q V α (q)dS p,o ′ (q)W (g) 1 (p) + i ǫ i q∈B i Res p→q dS p,o ′ (q)W (g) 1 (p) = α Res q→α V α (q)W (g) 1 (q)α t α α q=o W (g) 1 (q)i ǫ i q∈B i W (B -9)

Thus the theorem holds for k = 2 and g = 0. And by integration we get -2iπ ∂ ∂κ ij dE q (p) = 2(iπ) 2 (du i (p)(u j (q) -u j (q)) + du j (p)(u i (q) -u i (q))) = -1 2 r∈B i s∈B j (B(p, r)dE q (s) + B(p, s)dE q (r)). (y(q)-y(q))dx(q) W (g-1)

k+2 (q, q, p K ) + W (h) j+1 (q, p J )W (g-h)

k-j+1 (q, p K/J ) = 1 2 r∈B i s∈B j Res q→a B(p,r)dEq(s)+B(p,s)dEq(r) (y(q)-y(q))dx(q) W (g-1)

k+2 (q, q, p K ) + W (h) j+1 (q, p J )W (g-h)

k-j+1 (q, p K/J ) + Res q→a dEq(p) (y(q)-y(q))dx(q) 2iπ ∂ ∂κ ij W (g-1)

k+2 (q, q, p K ) + Res q→a dEq(p) (y(q)-y(q))dx(q) 2iπ ∂ ∂κ ij W (h) j+1 (q, p J )W (g-h)

k-j+1 (q, p K/J ) + Res q→a dEq(p) (y(q)-y(q))dx(q) W (h)

j+1 (q, p J )2iπ ∂ ∂κ ij W (g-h)
k-j+1 (q, p K/J ) = 1 2 r∈B i s∈B j B(p, r)W (y(q)-y(q))dx(q) W (g-2)

k+4 (q, q, p K , r, s) +2W (h) j+3 (q, q, p J , r)W (g-1-h) k-j+1 (p K/J , s) + 2W (h) j+2 (q, p J , r)W (g-1-h) k-j+2 (q, p K/J , s) + 1 2 r∈B i s∈B j Res q→a dEq(p) (y(q)-y(q))dx(q) J W (g-h) k-j+1 (q, p K/J )× × W (h-1) j+3 (q, p J , r, s) + 2 L W (m) l+2 (q, p L , r)W (h-m) k-j-l+1 (p K/(J∪L) , s) + 1 2 r∈B i s∈B j Res q→a dEq(p) (y(q)-y(q))dx(q) J W (g-h) k-j+1 (q, p K/J )× × W (h-1) j+3 (q, p J , r, s) + 2 L W (m) l+2 (q, p L , r)W k+1 (r, p K ) + r∈B i s∈B j Res q→a dEq(p) (y(q)-y(q))dx(q) J W (h-1) j+3 (q, p J , r, s)W (g-h) k-j+1 (q, p K/J ) + r∈B i s∈B j Res q→a dEq(p) (y(q)-y(q))dx(q) W (h) j+2 (q, p J , r)W (g-1-h) k-j+2 (q, p K/J , s) + 1 2 r∈B i s∈B j Res q→a dEq(p) (y(q)-y(q))dx(q) W (g-2)

k+4 (q, q, p K , r, s) + r∈B i s∈B j Res q→a dEq(p) (y(q)-y(q))dx(q) × × J L W (m) l+2 (q, p L , r)W (h-m) k-j-l+1 (p K/(J∪L) , s)W (g-h) k-j+1 (q, p K/J ) + r∈B i s∈B j Res q→a dEq(p) (y(q)-y(q))dx(q) × × J L W (m) l+2 (q, p L , r)W (h-m) k-j-l+1 (p K/(J∪L) , s)W (g-h) k-j+1 (q, p K/J ) + 1 2 r∈B i s∈B j Res q→a dEq(p) (y(q)-y(q))dx(q) 2W (h) j+3 (q, q, p J , r)W (g-1-h) k-j+1 (p K/J , s) (B-12) We recognize the recursion relation eq.4-9 in lines 2 to 5, and in lines 6 to 8, and this gives the theorem.

The theorem for the free energies, is easy obtained using theorem 4.7.

D Matrix model with an external field

We consider the matrix model with external field defined in section (10.4):

Z(Λ) :=

Hn dM e -N T r(V (M )-M Λ) (D-1)

where we assume that Λ is the diagonal matrix: .

Λ = diag (
In particular, the polynomial S(y) := s i=1 (y -λ i ) is the minimal polynomial of Λ. We define the correlation functions w k (x 1 , . . . , x k ) := N k-2 k i=1 tr 1 Notice that u k, (x, y; x K ) is a polynomial in y of degree s -1, and D(x)P k (x, y; x K ) is a polynomial in x of degree d -1 and in y of degree s -1 (note that P 0 corresponds to P in Eq. (10-68)).

It is convenient to renormalize those functions, and define: which defines an algebraic curve.

D.2 Leading order algebraic curve

Let us study the curve E M ext (x, y) = E (0)

M ext (x, y) = 0 defining a compact Riemann surface Σ and two functions x and y defined on it.

Because y is a solution of a degree s + 1 equation, E M ext (x, y) has s + 1 x-sheets. The sheets can be identified by their large x behavior:

• in the physical sheet, we have Y (x) ∼ V ′ (x) -1/x + O(1/x 2 )

• in the other sheets, Y (x) ∼ λ i + n i N 1

x + O(1/x 2 ) Let us note by p i ∈ Σ with i = 0 . . . s the different points of the curve whose x-projection are x(p), i.e. ∀i, j x(p i ) = x(p j ).

(D-16)

The superscript 0 corresponds to the point in the physical sheet. From the correlation functions previously defined on the x and y projections, one defines the corresponding meromorphic 1-forms on the curve as follows: where E (h) (x, y) is the h'th term in the 2 -expansion of the spectral curve.

D.3 Diagrammatic rules for the correlation functions and the free energy

In this section, one proves that the correlation functions' and the free energy's topological expansion of this model do coincide with the W It implies that dx(p)dx(q) (x(p) -x(q)) 2 = i W (0) 2 (p i , q) and ∀h > 1 , i W (h) 2 (p i , q) = 0, (D-30)

i.e. The coefficient of y s-1 is:

i<j Y (p i )Y (p j ) + 1 N 2 ω 2 (p i , p j ) = V ′ (x) i Λ i + i<j Λ i Λ j + 1 N tr V ′ (x) -V ′ (M 1 )
x -M 1 .

(D-32) Notice that:

i<j Y (p i )Y (p j ) + 1 N 2 ω 2 (p i , p j ) = 1 2 i Y (p i )(V ′ (x) + j Λ j -Y (p i )) - 1 N 2 ω 2 (p i , p i ) = 1 2 (V ′ (x) + j Λ j ) 2 - 1 2 i Y (p i ) 2 + 1 N 2 ω 2 (p i , p i ) .
(D -33)

Thus:

V ′ (x) 2 + i Λ 2 i - 2 N tr V ′ (x) -V ′ (M 1 ) x -M 1 = i Y (p i ) 2 + 1 N 2 ω 2 (p i , p i ) . (D-34)
Notice that the LHS is the ratio of a polynomial in x and D(x):

Q(x) D(x) = V ′ (x) 2 + i Λ 2 i -2 N tr V ′ (x)-V ′ (M 1 )
x-M 1 . The topological expansion of this equation reads for h ≥ 1 .

(D-35) From now on, following the lines of [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF], one multiplies these equations by 1 2 dE p,p (q) y(p)-y(p) , takes the residues when p → µ α and sums over all the branch points and obtains:

Theorem 4 . 8

 48 Diagrammatic representation: W (g) k+1 (p, p 1 , . . . , p k ) = G∈G (g) k+1 (p,p 1 ,...,p k )

Theorem 4 . 1

 41 The correlation function W (0) 3

Theorem 4 . 2

 42 For (k, g) = (1, 0), the loop function W (g) k+1 (p, p 1 , . . . , p k ) has poles (in any of its variables p, p 1 , . . . , p k ) only at the branch points.

Theorem 4 . 3

 43 For every A cycle we have:

  , p 1 , . . . , p k ). It is clear that the same graph is obtained 3g + 2k -1 times (the number of edges of G ′ . And it is clear that from any G ′ ∈ G (g) k+1 (p, p 1 , . . . , p k ), you can obtain 3g + 2k -1 graphs G ∈ G (g) k+2 (p, p 1 , . . . , p k , p k+1 ) by adding a new vertex on any edge, and linking this new vertex to the leaf p k+1 .Example of G(2) 1 (p) As an example, let us build step by step all the graphs of G

  ), i.e. g = 2 and k = 0.We first draw all planar binary skeleton trees with one root p and 2g + kdraw g + k = 2 non-arrowed edges in all possible ways such that every vertex is trivalent, also satisfying rule 9) of definition.4.5. There is only one possibility for the first graph and two for the second one: to specify the left and right children for each vertex. The only possibilities in accordance with rule 10) of def.4.5 are 7 :

  This is precisely what the recursion equations 4-9 of def.4.2 are doing. Indeed, one can represent them diagrammatically by

( 4 -

 4 [START_REF] Eynard | More on the exact solution of the O(n) model on a random lattice, and an investigation of the case -n-¿2[END_REF])W (0) 4 (p, p 1 , p 2 , p 3 ) =

10. 1 . 1 DefinitionDefinition 10 . 1

 11101 Formal 1-matrix model.

Remark 10 . 2

 102 -104) The free energies are all vanishing for that curve: ∀g F (g) = 0. (10-105) 10.6 Example: pure gravity (3,2)

  g) k+1 (p, p 1 , . . . , p k ) + W (g) k+1 (p, p 1 , . . . , p k ) = 0 (A-25) due to theorem 4.4. For m = 0, 1 compute α Res p→α x(p) m y(p)W (g)

( 2 -ΦW (g) 1 (B- 5 )( 2 -) 1 =α

 21521 2g)F (g) = -Res a and using eq.3-63, we can choose for any arbitrary o′ : 2g)F (g) = -Res a ΦW (gRes p→a Res q→α V α (q)dS p,o ′ (q)W (g) 1 (p)α t α Res p→a α q=o dS p,o ′ (q)W (g) 1 (p) i ǫ i Res p→a q∈B i dS p,o ′ (q)W (g) 1 (p).

2 ( 1 2

 21 du i (p 1 )du j (p 2 ) + du i (p 2 )du j (p 1 )) = r∈B j s∈B i (B(p 1 , r)B(p 2 , s) + B(p 2 , r)B(p 1 , s)).

(B - 10

 10

  , p K ) + B(p, s)W (g) k+1 (r, p K ) + 1 2 r∈B i s∈B j Res q→a dEq(p)

  (h-m) k-j-l+1 (p K/(J∪L) , s) (B-11)We regroup together all terms with 2 W 's and all terms with 3 W 's:2iπ ∂ ∂κ ij W (g) k+1 (p, p K ) = 1 2 r∈B i s∈B j B(p, r)W (g) k+1 (s, p K ) + B(p, s)W (g)

n 1 λ 1 2 λ 2

 1122 , . . . , λ 1 , n , . . . , λ 2 , . . . , ns λ s , . . . , λ s ) (D-2) and V ′ (x) is a rational fraction with denominator D(x): V ′ (x) = d k=0 g k x k D(x)

3 )

 3 We also define the auxiliary functionsu k (x, y; x K ) := N |K|-P k (x, y; x K ) := N |K|-1 tr V ′ (x) -V ′ (M 1 )x -M S(y) -S(Λ) y -

u

  k (x, y; x K ) := u k (x, y; x K ) -δ k,0 S(y) (D-6) andw k (x K ) := w k (x K ) + δ k,2 (x 1 -x 2 ) 2 .(D-7)D.1 Loop equationsConsider the change of variablesδM = 1 x -M S(y) -S(Λ) y -Λ . (D-8)You get the loop equationw 1 (x)u 0 (x, y)+ 1 N 2 u 1 (x, y; x) = V ′ (x)u 0 (x, y)-P 0 (x, y)-yu 0 (x, y)+S(y)w 1 (x) (D-9)i.e.(y+w 1 (x)-V ′ (x))(u 0 (x, y)-S(y))+ 1 N 2 u 1 (x, y; x) = (V ′ (x)-y)S(y)-P 0 (x, y). (D-10)We define the polynomial both in x and yE M ext (x, y) := ((V ′ (x) -y)S(y) -P 0 (x, y)) D(x) (D-11) and Y (x) := V ′ (x) -w 1 (x). (D-12)The loop equation thus implies:(y -Y (x))u 0 (x, y)D(x) + 1 N 2 u 1 (x, y; x)D(x) = E M ext (x, y) (D-13)and in particularE M ext (x, Y (x)) = 1 N 2 u 1 (x, Y (x); x)D(x). (D-14)The leading order of the topological expansion reads E (0) M ext (x, Y (x)) = 0 (D-15)

WD. 2 . 1

 21 k (p K ) := w k (x(p K )) dx(p 1 ) . . . dx(p k ) (D-17) and U k (p, y; p K ) := u k (x(p), y; x(p K )) dx(p)dx(p 1 ) . . . dx(p k ) (D-18)as well as there topological expansionsW k (p K ) = ∞ h=0 N 2-2h W (h) k (p K ) and U k (p, y; p K ) = Fillingfractions and genusThe curve has a genus g ≤ ds -1 and we work with fixed filling fractionsǫ I := 1 2iπ A I ydx. (D-20)D.2.2 Subleading loop equationsConsider the topological expansion of the loop equation Eq. (D-14). It reads, for h ≥ 1:E (h) (x, y) = D(x)(y -Y (x))u

k

  's and F (h) 's defined following the definitions of Eq. (4-15) and Eq. (4-16) for the classical spectral curve E M ext (x, y) = 0.D.3.1 The semi-classical spectral curveLet us reexpress the semi-classical spectral curve (i.e. the whole formal series E M ext (x, y)) in terms of the classical one E (0) M ext (x, y). Theorem D.1 E M ext (x, y) = -D(x)" s i=0 (y -V ′ (x(p)) + 1 N Tr 1 x(p (i) )-M ) ′′ = D(x) [(V ′ (x) -y)S(y) -P 0 (x, y)] (D-22)

ω 2 (p, q) + s i=1 ω 2

 22 (p i , q) = 0. (D-31)

2 D

 2 , p i ) + 2 Q (h) (x(p))dx(p) (x(p))

W

1 2

 1 dE p,p (q)(W(h-1) 2,0 (p, p) + h-1 m=1 W (m) 1,0 (p)W (h-m) 1,0(p)) (y(p) -y(p))dx(p) . (D-36)

  10 252 t 4 5 z 8 + 12 t 2 5 z 6 (2 -t 3 )(50 t 7 z 2 + 21 t 5 ) +z 4 (2 -t 3 ) 2 (252 t 2 5 + 348 t 5 t 7 z 2 + 145 t 2 7 z 4 + 308 t 5 t 9 z 4 ) +z 2 (2 -t 3 )(203 t 5 + 145 z 2 t 7 + 105 z 4 t 9 + 105 z 6 t 11 ) +105 (2 -t 3 ) 4 . (10 -82)

  , p 2 , p 3 , p 4 , p 5 ) dp 1 dp 2 dp 3 dp 4 dp 5

	15/2 7	135 + 87p 2 + 36p 4 + 12p 6 + 4p 8 2 10 3 5 p 10	dp		(10-116)
	W 4 (p 1 , p 2 , p 3 , p 4 ) = (0)	v -5 9p 2 1 p 2 2 p 2 3 p 2 4	1 + 3	i		1 i p 2	dp 1 dp 2 dp 3 dp 4	(10-117)
	W 5 (p 1 = (0)	v -15/2 9p 2 1 p 2 2 p 2 3 p 2 4 p 2 5	1 + 3	i	1 p 2 i	+ 6	i<j	1 p 2 i p 2 j	+ 5	i	1 p 4 i
												(10-118)
	etc...
	Using def.4.3, and eq.10-112, we find from eq.10-113:

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

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

This definition is the opposite of the notation used in[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] since the integral goes from q to q instead of going from q to q.

It goes through all vertices.

planar tree means that the left child and right child are not equivalent. The right child is marked by a black disk on the outgoing edge.

a binary skeleton tree is a binary tree from which we have removed the leaves, i.e. a tree with vertices of valence 1, 2 or 3.

Note that the graphs are not necessarily planar.

This excludes the case where Ω corresponds to the variation of an hard edge, cf[START_REF] Eynard | Loop equations for the semiclassical 2-matrix model with hard edges[END_REF][START_REF] Chekhov | Matrix models with hard walls: Geometry and solutions[END_REF][START_REF] Bertola | Two-matrix model with semiclassical potentials and extended Whitham hierarchy[END_REF].

This Poisson relation is well known and can be found in[START_REF] Daul | Rational Theories of 2D Gravity from the Two-Matrix Model[END_REF][START_REF] Di Francesco | 2D Gravity and Random Matrices[END_REF]

Acknowledgements

We would like to thank Mina Aganagic, Michel Bergère, Leonid Chekhov, Philippe Di Francesco, Robert Dijkgraaf, Aleix Prats Ferrer, Ivan Kostov, Amir Kashani-Poor, Marcos Mariño, Andrei Okounkov, Pierre Vanhove, Jean-Bernard Zuber, 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 ANR-05-BLAN-0029-01, by the Enrage European network MRTN-CT-2004-005616, by the European Science foundation through the Misgam program, by the French and Japaneese governments through PAI Sakurav, by the Quebec government with the FQRNT.

A Properties of correlation functions

In this section we prove the theorems stated in section 4. [START_REF] Akemann | New universal spectral correlators[END_REF].

We use very much the following obvious properties: i B(p i , q) = dx(p)dx(q) (x(p) -x(q)) 2 , (A-1)

and lim q→a i dE q (p) ω(q) dx(q) = -1 2 lim q→a i B(p, q) dy(q) dx(q). (A-3)

The third one is nothing but De L'Hôpital's rule.

Theorem 4.1 proof:

From the definition 4.2, we have:

3 (p, p 1 , p 2 ) = Res q→a dE q (p) ω(q) B(q, p 1 )B(q, p 2 ) + B(q, p 2 )B(q, p 1 ) = 2 Res q→a dE q (p) ω(q) B(q, p 1 )B(q, p 2 ) = 2 Res q→a Res r→q dE q (p) (y(q) -y(q))(x(r) -x(q)) B(q, p 1 )B(r, p 2 ) = -2 Res q→a Res r→q dE q (p) (y(q) -y(q))(x(r) -x(q)) B(q, p 1 )B(r, p 2 ) = -2 Res q→a dE q (p) ω(q) B(q, p 1 )B(q, p 2 ) = Res q→a B(q, p)B(q, p 1 )B(q, p 2 ) dx(q)dy(q) (A-4)

where we have used Eq. (A-2), Eq. (A-1) and Eq. (A-3).

Theorem 4.2 proof:

It is obvious from the definition that if p is away from branchpoints, the residues are finite integrals, and W (g) k+1 is finite. The only poles can be obtained when p pinches an integration contour, i.e. at branch points.

Then, it is easy to see by recursion on k and g that it holds for any p i .

Theorem 4.3 proof:

The first one is a property of dE q (p), and the second one follows from recursion on k and g, and it holds for

C Proof of the symplectic invariance of F (0) and F (1) C.1 F (1) Let us study how F (1) is changed under the exchange of the roles of x and y. For this purpose, we define the images of F (1) and W q B(p, ξ) 2(x(q) -x(q))dy(q) B(q, q) (C-1)

and

where b denotes the set of y-branch points, q is the only point satisfying y(q) = y(q) and approaching a branch point b j when q → b j , τ By is the Bergmann tau function associated to y and x ′ (b j ) = dx(b j ) dz j (b j ) . According to theorem 5.1, for any variation Ω of the curve, the variation of the free energy reads

Thus, the variation of the difference between the two "free energies" reads

1 (p) + W

(1)

(p) Λ(p). (C-4)

In order to evaluate this quantity, one needs the following lemma:

Lemma C.1 For any choice of variable z:

where the derivatives are taken with respect to the variable z and S Bz denotes the Bergmann projective connection associated to z.

proof:

From the definitions , one easily derives that

Let us now expand the integrand in terms of an arbitrary local variable z when p → q. The different factors read

Inserting it altogether inside Eq. (C-6), one can explicitly compute the residues and one recognizes the formula C-5.

We can now prove the following theorem stating the symmetry of F (1) under the exchange of x ↔ y:

The two free energies transform in the same way under any variation of the moduli of the curve:

proof:

We already know that

(C-10) for an arbitrary variable z.

We just have to check that this quantity vanishes for the transformations corresponding to varying the moduli of the curve. Because the function inside the differential wrt p vanishes at the poles of ydx, one can check that it is indeed the case.

Thus the first correction to the free energy satisfies the following variation under symplectic transformations: Theorem C.1 F (1) does not change under the following transformations of E:

• y → y + R(x) where R is a rational function.

• y → cy and x → 1 c x where c is a non-zero complex number. • x → x + R(y) where R is a rational function.

• y → x and x → y.

F (1) is shifted by a multiple of iπ/12 when E is changed by x → -x.

proof:

The first transformation is obvious from the definition since neither lnτ Bx nor y ′ (a i ) changes.

The second one follows from theorem 2 in [START_REF] Eynard | 1/N 2 corrections to free energy in Hermitian two-matrix model[END_REF] for f = x and g = x c which shows that the variations of ln(τ Bx ) and y ′ (a i ) compensate.

The fourth one is nothing but lemma (C.2) and it gives the third one when combined with the first.

The last transformation holds because lnτ Bx is left unchanged and y ′ (a i ) changes sign.

C.2 F (0)

Theorem C.2 F (0) does not change under the following transformations of E:

• y → y + P(x) where P is a polynomial.

• y → y and x → -x.

• y → cy and x → 1 c x where c is a non-zero complex number. • x → x + P(y) where P is a polynomial.

• y → x and x → y.

proof:

The second and third transformations come from the structure of F (0) which is bilinear in objects proportionnal to ydx.

One can obtain the last one following the same method as for F (1) : we compute the variation of the difference of the two free energies under the changes of the moduli of E and show that they vanish.

Let us now show the first invariance. In this case, the variation of the free energy is δ P(x)dx F (0) . Since P(x) is a polynomial one can write it

where Q(x) is also a polynomial in x. Then, theorem (5.1) implies the invariance of F (0) under this transformation. The fourth transformation is a combination of the first and the last one.

and

where " < . > ′′ means that one replace w 2 by w 2 in the expansion.

proof:

One proves that the 1 N 2 -expansions of

and

coincide with the expansion of E M ext (x, y) and U 0 (p, y).

Let the topological expansions be

Expanding the expressions of E(x, y) and U (p, y) into cumulants, one recovers

which coincides with Eq. (D-21).

One easily proves that this system of equations admits a unique solution thanks to the polynomial properties of U (p, y) and that the leading orders h = 0 coincide. The proof is extremely similar to that for the 2-matrix model (cf. theorem 1 in [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF]).

D.3.2 Diagrammatic solution

One has:

where B is the Bergmann kernel of the algebraic curve E M ext . The coefficient of y s of Eq. (D-22), divided by D(x), is:

Differentiating wrt the potential V (x i ), one can finally write down an expression for the correlation functions:

k+1,0 (q, p K ) = α Res p→µα (D-37)

This coincides with the reccursive definition 4-15 and ensures the equality of the correlation functions with the former defined special "loop functions".

Keeping on following [START_REF] Chekhov | Free energy topological expansion for the 2-matrix model[END_REF], one finds that the topological expansion of the free energy also coincides with the special free energies defined on 4-16, that is the τ -function of the algebraic curve.