Geometry of Spectral Curves and All Order Dispersive Integrable System

We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between"correlators", the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.


Introduction to integrability
Integrable systems play a particularly important role in physics, because they are "exactly solvable", at least in principle. Under a suitable change of variables, they can be brought into the form of a linear constant motion with constant velocity in a multidimensional torus. However, there is a difference between exactly solvable and exactly solved, and the study of integrable systems is still a very active branch of theoretical physics and mathematical physics. See the books [DJM00,BBT02] for an introduction.
There are several definition of what is an integrable system. In this article, we speak of an integrable system whenever some Hirota equations [Hir71] are present. An important class of integrable systems arise from Lax systems. The locus of eigenvalues of the Lax operator is an algebraic curve called the spectral curve. Here, conversely, we shall deal with the ways to build an integrable system (and in fact a Lax pair) starting from an arbitrary algebraic spectral curve, the so-called reconstruction problem. This has been addressed, in a dispersionless limit, in several earlier works (especially [Kri77] and [BG07]), on which we will comment along this article. Here, we propose a way to add the dispersionful part, using the algebraic invariants introduced in [EO07a].
Let us recall first generalities about Lax pairs and integrable systems.

Hamiltonian system and Liouville integrability
Consider a classical Hamiltonian system, i.e. a set of coordinates q i (t) and their conjugated momenta p i (t), obeying the Hamilton equations of motions: where H is the Hamiltonian. This can be written in vector notation: Q = (q 1 , . . . , q n , p 1 . . . . , p n ), where {, } is the Poisson bracket such that {p i , q j } = δ i,j . The system is "Liouville integrable" if there exist n independent conserved quantities H 1 , H 2 , . . . , H n (by convention H 1 = H): In this case, it is possible to find a symplectic change of coordinates in the phase space, called "action-angle" variables, in which the equations of motions reduce to linear constant motions in a torus (the Jacobian, i.e. u i is a periodic variable): The main problem is to find explicitly the conserved quantities H i , the action variables v i , and the angle variables u i as functions of the initial system of coordinates q i , p i . A way to achieve this, was found in terms of a Lax pair, which we review below, and all the quantities above can be expressed in terms of algebraic geometry.

Lax matrix formulation
Imagine that one can group the coordinates q i , p j together, using d 2 formal series L i,j (x, Q), i, j = 1, . . . , d of a formal variable x, which form a d × d matrix L(x, Q). This means that the coefficients of the expansion of L i,j (x, Q) in powers of x, are functions of the coordinates q, p. Imagine that there exists a matrix M(x, Q), such that the equations of motion Eqn. 1.2 can be rewritten as a matrix equation: It may not seem obvious that the Hamilton equations of motion Eqn. 1.2 of a Liouville integrable system can always be written in this way, but indeed a Lax pair 3 has been found for all classical integrable systems (see for example [BBT02] and references thereof).

The spectral curve
The Lax equation (Eqn. 1.5) implies that the eigenvalues of L(x, Q(t)) do not depend on the time t, they are conserved. In particular, this means that the characteristic polynomial E(x, y) is independent of time: E(x, y) = det(y id − L(x, Q(t))) (1.6) The locus of zeroes of E(x, y), i.e. the eigenvalues of L, is called the "spectral curve" S: S = {(x, y) | E(x, y) = 0} (1.7) By expanding E(x, y) into powers of x and y, one can find all the conserved quantities, and thus, knowing the spectral curve, one knows the conserved quantities. The nonconserved quantities, i.e. the angle variables, are contained in the eigenvectors of L. It was shown [Kri77,BBT02], that the eigenvectors of L, must be Baker-Akhiezer functions on the spectral curve. Before explaining what it means, we need to describe more geometry of the spectral curve, i.e. more algebraic geometry.

Outline of the article
• In Section 2, we present the algebro-geometric background needed for our purposes.
• We first review in Sections 3-4 the construction of an integrable system associated to a given compact Riemann surface. Its times are the moduli of a meromorphic 1-form defined on the curve with fixed cycle integrals. The conserved quantities are the moduli of the curve. This construction dates back to Krichever [Kri77]. It encompasses the finite gap approach when the spectral curve is hyperelliptic, and produces in general the so-called algebro-geometric solutions of KP. We slightly reformulate this construction by defining a "spinor kernel" ψ cl (z 1 , z 2 ), globally defined on the spectral curve (Section 3.1). Baker-Akhiezer functions and the integrable kernel can be recovered from ψ cl . We recall the construction of the Tau function T cl and its properties. This integrable system is called "classical", or "dispersionless".
• The goal of this article is to propose explicit formulas for another type of integrable system, called "dispersive" [Dub96]. The times are now (not necessarily independent) moduli of a spectral curve, which evolve at speed of order 1/N around a given algebraic spectral curve S. We define a function T which is an ad hoc solution to "loop equations" (Section 5), a spinor kernel ψ(z 1 , z 2 ) via a Satolike formula (Section 5.3), and correlators W n (z 1 , . . . , z n ) encoding variations of T with respect to all possible moduli (Section 6). We present the conjecture that ψ is self-replicating, and see its consequences: Hirota equations for T (Section 7), determinantal formulas for the correlators, Christoffel-Darboux formula for the integrable kernel, and a Lax pair formulation (Section 8).
• Eventually, in Section 9, we recall how the results known about asymptotics of matrix integrals are compatible with special cases of our construction.

Geometry of the spectral curve
See [FK07,Fay70] for an introduction to algebraic geometry.

Definition
For us, a spectral curve S is the data of a complex curve C and two meromorphic functions X, Y on C: S = (C, X, Y ) (2.1) An essential example is given by the zero locus of a polynomial: E(x, y) = 0 (2.2) If deg y E = d, then for every value x, there are d values (with multiplicities) y = y 1 (x), . . . , y d (x) solutions of that equation. Those d functions can also be viewed together, as d branches of one multivalued 4 function x → y(x). The equation E(x, y) = 0 can also be seen as the embedding of a Riemann surface C into C × C ∪ {∞}. In other words, we prefer to represent the spectral curve in a parametric form: This means that there are two analytical functions (meromorphic) X and Y on C, which generate S.
In this article, we only work with spectral curves for which C is compact. Then, there exists a polynomial E such that E(X, Y ) = 0. We also assume C connected without losing generality.

Some notations and properties 2.2.1 Topology and holomorphic 1-forms
The curve C is either simply connected, and then this is the Riemann sphere C = P 1 = C ∪ {∞}, or it has genus g > 0. Then, any maximal open contractible subset of C is called a fundamental domain. If it is of genus g > 0, there exist 2g independent non-contractible cycles (see Fig. 1), and we can choose them in such a way (but not unique) that: A basis satisfying these intersection relations is called "symplectic". From the topological point of view, a genus g > 0 compact Riemann surface with a basis (A i , B i ) 1≤i≤g is a 4g closed polygon Γ, with edges glued by pairs.Γ is a fundamental domain of C. It is a classical result, that on a curve of genus g, there exists g independent holomorphic 1-forms du i (holomorphic means in particular having no poles), and they can be normalized on the A-cycles: Figure 1: A symplectic basis of 2g non-contractible cycles on a Riemann surface of genus g.
Then, the g × g matrix τ i,j : is known to be symmetric τ i,j = τ j,i and its imaginary part is definite positive: τ is called the Riemann matrix of periods of C.

Riemann bilinear identity
It is well-known that, for any meromorphic 1-form ω defined on C: This can be generalized to some forms which are not globally defined on C. Let ω 1 , ω 2 be two meromorphic 1-form defined on C, such that Res ω 2 = 0 for every pole of ω 2 , and let φ 2 a primitive of ω 2 defined on a fundamental domain of C, i.e. dφ 2 = ω 2 . For any symplectic basis of homology (A, B), we have the Riemann bilinear identity: (2.10)

Theta functions
Given any symmetric matrix τ such that Im τ > 0, one can define the Riemann Theta function: θ(u|τ ) = n∈Z g e 2iπn·u e iπ n t ·τ ·n . (2.11) Since Im τ > 0, it is a well-defined convergent series for all u in C g . Most often we will not write the τ dependence of θ: The Theta function has nice quasi-periodicity properties. If n, m ∈ Z g we have: It also satisfies the heat equation: In this equation, τ i,j and τ j,i are considered independent.

Jacobian and Abel map
Let us choose a generic basepoint o ∈ C, which will in fact play no role. For any point z ∈ C, we define: where the integration path is chosen such that it does not intersect any A-cycle or B-cycle. Then we define the vector: The application z → u(z) mod (Z g + τ Z g ) is well-defined and analytical, it maps the spectral curve into the Jacobian J = C g /(Z g + τ Z g ). This defines the Abel map: The Jacobi inversion theorem states that every w ∈ J can be represented as w = g j=1 u(p j ) for some points p 1 , . . . , p g ∈ C.
The Theta function can be used with τ the Riemann matrix of periods of a Riemann surface C, and u the Abel map of a point on C. In this case, it enjoys other important properties. Its zero locus has the following description: there exists k ∈ C g , so that θ(w|τ ) = 0 iff there exists g − 1 points z 1 , . . . , z g−1 ∈ C satisfying w = g−1 j=1 u(z j ) + k. k is called a "Riemann vector of constants", and it depends on the basepoint o used to define the Abel map u.

Prime form
An odd characteristics c is a vector of the form: The Theta function vanishes at odd characteristics: θ(c) = 0, and the following holomorphic form: has only double zeroes on C, so that we can define its squareroot, and thus one can define the prime form: There exists choices of c such that E is not identically 0 (we say c is "non singular"), and E is in fact independent of such c. It is a (−1/2, −1/2)-form on C, and it vanishes only at z 1 = z 2 . In any local coordinate ξ(z) we have: Because of the Theta function, E(z 1 , z 2 ) is not a global (−1/2, −1/2) form on C. It transforms according to Eqn. 2.13.
The Theta function associated to a Riemann surface satisfies a non-linear relation called Fay identity [Fay70]: for any z 1 , z 2 , z 3 , z 4 ∈ C, any w ∈ C g ,

Bergman kernel
The Bergman kernel [BS53] is the "fundamental (1,1)-form of the second kind" [Fay70], defined as: It is independent of the choice of a non-singular, odd characteristics c. It is a globally defined, symmetric (1,1)-form, having a double pole at z 1 = z 2 with no residue, and no other pole. It is normalized so that: Near z 1 = z 2 , it behaves, in any local coordinate ξ(z), like: We also define the fundamental 1-form of the third kind: where the integration contour is chosen so that it does not intersect any A-cycle or B-cycle. It is a 1-form in the variable z, and a function of the variable z 1 , z 2 , and it satisfies: It has a simple pole at z = z 1 with residue +1, a simple pole at z = z 2 with residue −1, and no other pole. In other words, in any local coordinate ξ(z): Notice that in the variable z it is globally defined for z ∈ C (it has no monodromy if z goes around a non-contractible cycle), whereas in the variable z 1 (resp. z 2 ) it is defined only on the fundamental domain, it has monodromies when z 1 (resp. z 2 ) goes around a non-contractible cycle B j : X is then a meromorphic function on C of degree d. z i (x) is sometimes called the preimage of x in the i-th sheet.
Definition 2.1 We call "ramification points of order k", the zeroes of order k ≥ 1 of the meromorphic 1-form dX. If a ∈ C is a ramification point, the corresponding value X(a) is called a branchpoint. All the other points z ∈ C at which X(z) is analytical, are called "regular points".
Definition 2.2 We say that a branchpoint x a is simple if X −1 ({x a }) consists in d − 1 points, one of them being a ramification point of order 1, and all the remaining ones being regular points.

Definition of local ccordinates
Near a ramification point a of order k, ξ a (z) = (X − X(a)) 1/(k+1) defines a local coordinate on the curve. Simple branchpoints play a special role in Sections 3.7, 5.1 and 6.1. For a simple branchpoint we have Since X is a meromorphic function of degree d, it has d poles with multiplicities, i.e.: Near ∞ i , a good local variable is: Besides, we will need to consider also poles of a meromorphic form ω. If p is a pole of ω, but not a pole of X, neither a zero of dX, a good local variable is: In this case, the multiplicity of p is d p = −1. We shall now always use the local coordinates ξ(z) defined above. Notice that they depend only on the function X(z) and not on Y (z).
Definition 2.3 Given a meromorphic 1-form ω(z) which has no pole at ramification points, let us call: To any p ∈ P, we have associated a coordinate patch ξ p on C centered on p.

Poles and times, filling fractions
Definition 2.4 For any p ∈ P, we define the "times" near p as the coefficient of the negative part of the Laurent series expansion of ω near p: We also write collectively: Notice that the times t p,0 = Res p ω are not independent, because the sum of residues of ω must vanish: p∈P t p,0 = 0 (2.37) Definition 2.5 We define the "filling fractions" (also called "conserved quantities") by:

Form-cycle duality
To each time t k , we associate a differential meromorphic form ω k (z), as well as a dual cycle ω * k , and a dual orthogonal cycle ω * ⊥ k : In such a way that: The symbol | X(z) means that we differentiate keeping the local coordinates ξ fixed (their definition depend only on the function X). More explicitly we have: • Filling fractions ǫ i −→ ω j = 1st kind differential = 2iπdu j : , • Residues t p,0 −→ ω p,0 = third kind differential: where o is an arbitrary basepoint on C, and C p is a small circle surrounding p with index 1. As we mentioned, the t p,0 are not independent variables, and only (t p,0 −t p 0 ,0 ) p =p 0 for a fixed p 0 are independent. As a consequence, we see that only differences ω p,0 − ω p ′ ,0 and ω * p,0 − ω * p ′ ,0 are independent of a choice of basepoint o.
• Higher times t p,j with j ≥ 1 −→ ω p,j = second kind differential: The form ω is a linear combination of those meromorphic forms, and almost by definition we have: We also define its dual cycle t p,j ω * p,j .

The prepotential
The fact that ω * i ω * j B(z, z ′ ) is symmetric, implies that there exists a function F 0 ( t) called the "prepotential" such that: The problem (this is why we write quotation marks) is that those integrals are not well-defined for times associated to third kind differentials. Such a statement is correct after an appropriate regularization. When z is in the vicinity of a pole p, we define: It is such that ω − dV p has at most a simple pole at p. Given an arbitrary base point o ∈ C, the following integral is well-defined: µ p depends on the base point o, but only by an additive constant independent of p.
Since p t p,0 = 0, the sum p t p,0 µ p is thus independent of o. In some sense, µ p is a regularized version of ω * p,0 ω (which does not exists). Since for all the other cycles, ω * k ω is well-defined, we can now define the prepotential.
Definition 2.6 The prepotential is: The first derivatives of F 0 are given by, for j ≥ 1: The proof of this theorem has appeared in many works, in particular [Dub92, Ber03, EO07a]. The definition of F 0 is thus equivalent to write which means that F 0 is homogeneous of degree 2. Another classical result is: The second derivatives of F 0 are given by: except for the following cases: The second derivatives of F 0 do not depend on the form ω, and thus do not depend on the times. Thus we have:

Reconstruction formula
In this section, we review the reconstruction of a Lax matrix and an isospectral integrable system from the spectral curve. This so-called algebro-geometric reconstruction was first discovered by Krichever [Kri77], and we merely rephrase it. The only difference is that, instead of Baker-Akhiezer functions, we prefer to use a "Baker-Akhiezer spinor kernel", which turns out to be a more intrinsic object.

Baker-Akhiezer spinor kernel
Define the 1-form: which depends linearly on the times (and not on the filling fractions). By construction χ is normalized on A-cycles: Then we define the vector ζ( t ) = [ζ i ( t )] 1≤i≤g with coordinates: which we write for short as: It can be decomposed as: The vector ζ( t ) is a linear function of the times t k and is independent of the filling fractions ǫ i . In other words, it follows a linear motion with constant velocity v k in the Jacobian, as a function of any of the times. A well-known property of integrable systems is that, in appropriate variables, the motion (with any of the time t k ) is uniform and linear. The algebraic reconstruction takes the linear evolution in the Jacobian of C as starting point, and produces more intricated quantities whose evolution is described by a Lax system.
Definition 3.1 We now define the Baker-Akhiezer spinor kernel as the (1/2, 1/2) form: where c is a non-singular, odd characteristics.
We write a subscript cl to distinguish the algebraic construction (associated to a "classical" or dispersionless integrable system) from the one proposed in the second part of the article. This kernel is also known as the Szegő kernel [Sze21], twisted by the exponential term e z 1 z 2 χ(z; t ) .
• It has a simple pole at z 1 = z 2 : in any local coordinate ξ(z) • It has essential singularities when z 1 (resp. z 2 ) approaches a pole of ω.
Proof. The behavior at z 1 → z 2 is obvious, and the essential singularities at the poles of ω come from the exponential term. Next, we need to prove that ψ cl (z 1 , z 2 ; t) is unchanged when z 1 (resp. z 2 ) goes around a non-trivial cycle. When z 1 (resp. z 2 ) goes around an A-cycle, the vector u(z 1 ) (resp. u(z 2 )) is translated by an integer vector, θ is thus unchanged, and thanks to Eqn. 3.2, ψ cl is unchanged when z 1 (resp. z 2 ) goes around an A-cycle. When z 1 (resp. z 2 ) goes around a B-cycle, the vector u(z 1 ) (resp. u(z 2 )) is translated by a lattice vector of the form τ · n with n ∈ Z g , and θ gets multiplied by a phase according to Eqn. 2.13. Remember that the prime form E(z 1 , z 2 ) is also a θ function, and also gets a phase given by Eqn. 2.13. ψ cl is thus changed by: and because of Eqn. 3.3, i.e. ζ = 1 2iπ B χ, we see that ψ cl is unchanged when z 1 (resp. z 2 ) goes around a B-cycle.

Duality equation
Then we construct the following spinor matrix of size d × d: where we recall that z i (x) are the d preimages of x on the curve C, i.e. X(z i (x)) = x, and d = deg X. These preimages are distinct and this matrix is well-defined when x 1 (or x 2 ) is not at a branchpoint.
Theorem 3.2 We have the "duality" equation: is a meromorphic function of z. Indeed, the product of two (1/2)-forms is a 1-form, and when we divide by dX, we get a function. The essential singularities coming from the exponentials cancel in the product, so this function can only have poles, i.e. it is meromorphic. The only possible poles are at z = z i (x 1 ) or z = z j (x 3 ) or at the zeroes of dX(z). Then, summing over all sheets, we see that is a symmetric sum of a meromorphic function over all sheets of x 2 , therefore it is a meromorphic function of x 2 ∈ C, i.e. a rational function of the complex variable x 2 . It remains to find its poles. 1/dX(z k (x 2 ))) behaves like O(x 2 − X(a i )) −1/2 at ramification points, and since a rational function of x 2 cannot have a singularity of power −1/2, this means that this rational function has no pole at branchpoints. Its only poles can then be at x 2 = x 1 or x 2 = x 3 , and they are simple poles. The residues of the corresponding poles are easily computed and give the theorem.
Theorem 3.3 We have a refined version of the duality equation: Notice that Theorem 3.2 is a corollary of Theorem 3.3. Indeed the duality equation (Theorem 3.2) can be obtained by summing the equation above on all sheets z = z k (x), because k du i (z k (x)) = 0 and Proof. Notice that ψ cl (z 1 , z ; t)ψ cl (z, z 2 ; t) is a meromorphic 1-form in z, since it has no essential singularity. It has simple poles at z = z 1 and z = z 2 , with residues ∓ψ cl (z 1 , z 2 ; t), and it has no other pole. This means that ψ cl (z 1 , z ; t)ψ cl (z, z 2 ; t) + ψ cl (z 1 , z 2 ; t) dS z 1 ,z 2 (z) is a holomorphic 1-form, with no poles, therefore it must be a linear combination of the du i (z)'s, which we choose to write: (3.11) The left hand side is a well-defined spinor of z 1 and z 2 , whereas in the right hand side, dS z 1 ,z 2 (z) = z 1 z 2 B(z, ·) gets some shifts when z 1 or z 2 go around non-trivial cycles. This implies the following relations for the coefficients α j (z 1 , z 2 ): (3.13) Moreover, we must have α j (z 1 , z 1 ; t) = 0, and α j (z 1 , z 2 ; t) may have poles when ψ cl (z 1 , z 2 ; t) = 0. Apart from those poles, α j (z 1 , z 2 ; t) has no other singularities.
The following quantity has all the required properties: So, the difference of α j and that quantity should be a meromorphic function of z 1 and z 2 without poles, i.e. a constant, and its value is zero by looking at z 1 = z 2 .

Link with Baker-Akhiezer functions
Given g points p 1 , . . . , p g ∈ C in generic position, the Baker-Akhiezer function ψ BA (z) is usually defined as the unique (up to a multiplicative constant) function on C, with poles at the points p 1 , . . . , p g , and essential singularities at the poles q of a meromorphic 1-form σ on C, such that ln Ψ BA (z) = z σ+O(1) when z → q. It can be readily written as: We can obtain such a function by specializing the spinor kernel associated to the 1-form σ: Indeed, by Jacobi inversion theorem, we can always represent u(p 1 )−c−k as g j=1 u(q j ) for some points q 1 , . . . , q g ∈ C. Moreover, according to the description of the zero set of the Theta function, q 1 , . . . , q j are the g zeroes of θ(u(z) − u(z 1 ) + c), in particular p 1 ∈ {q 1 , . . . , q g }.

Eigenvectors and Baker-Akhiezer functions
The usual formulation of integrable systems is obtained by specializing one of the points to x = ∞. In some sense, we would like to consider: The problem is, that the expression in the right hand side is divergent, and thus we again need regularization. The definitions in this paragraph also apply to the spinor kernel constructed in Section 5.3, so we drop here the cl index. We define: There are d pairs I = (i, j) such that 0 ≤ j ≤ d ∞ i − 1, and therefore the vector: is a d-dimensional vector, and the matrix: Similarly, we define the dual Baker-Akhiezer functions: There are d pairs I = (i, j) such that 0 ≤ j ≤ d i − 1, and therefore the vector: is a d-dimensional vector, and the matrix:

Christoffel-Darboux relations
Theorem 3.4 The matrix is invertible, and independent of x. The matrix A cl is called the Christoffel-Darboux matrix. This can also be written: Proof. This is a mere application of Theorem 3.2, up to a conjugation. Indeed: If i = i ′ , the following quantity has a well-defined limit when z 1 → ∞ i ′ and z 2 → ∞ i , and the term and therefore we get 0, i.e.: If i = i ′ , we first take the limit z 1 → ∞ i , and again the term with 1 X(z)−X(z 1 ) vanishes. Then, remember that ψ cl (z 1 , z 2 ) has a simple pole at z 1 = z 2 , and thus the derivative with respect to z 1 , can generate a pole of degree k ′ at z 2 = ∞ i . Therefore, we are , the only non-vanishing contribution is: The matrix A −1 cl has thus typically the shape: It is made of (Inverted) triangular blocks. Since the diagonal of each triangle is nonzero, this proves that the matrix A −1 cl is invertible. Then, if i = i ′ and k + k ′ ≥ d ∞ i + 1, we write that: and we see that the leading term 1 X(z 1 ) is independent of X(z), and the part which A non vanishing contribution to the part which depends on X(z) could occur only if k + k ′ > 2d ∞ i , which can never happen since we assumed k, k ′ ≤ d ∞ i . This proves that A cl is independent of X(z).
As a consequence, ψ cl (z 1 , z 2 ) can be identified with an integrable kernel. This means that we have: Theorem 3.5 Christoffel-Darboux relation: Proof. This is an application of Theorem 3.4 and Theorem 3.2. Indeed, we have from the duality equation, for any x, and for any invertible matrix C: The very definition of the ψ cl|I 's, means exactly that there exists a matrix C cl (x) such that: and similarly, there exists a matrixC cl such that: and therefore: This means that: (3.39)

Lax matrix
As corollary of Theorem 3.2: Corollary 3.2 If x 1 and x 2 are not branchpoints, and if x 2 = x 1 , then the matrix Ψ cl (x 1 , x 2 ) is invertible, and: Proof. Take x 3 = x 1 in the duality relation.
For every x 1 , the matrix: is a rational function of x. Its characteristic polynomial is independent of the times t, and its zero locus defines the spectral curve: Changing x 1 just amounts to a conjugation: Theorem 3.6 The matrix L x 1 (x; t) obeys the Lax equation: with respect to any of the times t k , and where the matrix M t k ;x 1 is: Therefore, from any plane curve C : E(x, y) = 0, we have constructed a Lax system. Since the spectral curve here coincide with C and is independent of the times, the evolution is said "isospectral". This reconstruction of the Lax matrix from the spectral curve, is well-known in the theory of integrable systems [BBT02]. Usually, it is presented with special values of x 1 , for instance x 1 is most often chosen as a pole of X.

Differential systems
Baker-Akhiezer functions satisfy simultaneously several first order differential systems with respect to the spectral parameter and the times.
Theorem 3.7 The matrix Ψ cl (x 1 , x; t) is solution of a linear ODE with respect to the spectral parameter x: where M x;x 1 is a rational function of x having poles only on P, with the same order as those of ω(z)/dX(z).
Proof. Write x = X(z), we have: In the right hand side, the essential singularities cancel, and only meromorphic singularities remain. Since we perform the sum over all sheets, the result is necessarily a rational function of X(z). Poles could occur at singularities of ψ, or also at zeroes of dX(z), or at X(z) = x 1 . ψ cl has a simple pole at x = x 1 , i.e. at z k = z i (x 1 ) for some k. Taking the derivative yields a double pole, and multiplying by (X(z) − x 1 ) 2 cancels the double pole. If i = j, there is also a simple pole coming from ψ cl (z k , z j (x 1 )), and if i = j there is no pole. It is easy to check that the residue of the pole at x = x 1 is −δ i,j . This implies that M x;x 1 (x) has no pole at x = x 1 . From its definition, M x,x 1 (X(z)) must behave as o((X(z) − X(a)) −1/2 ) at a ramification point a, and since it is a rational fraction of X(z), this must actually be O(1), meaning that M x;x 1 (x) has no poles when x approaches a branchpoint.
Near z → p ∈P, the only singularity comes from the exponential term and we have This shows that dx M x;x 1 (x; t) has poles inP, of order at most that already present in ω/dX.
Theorem 3.8 We have with respect to the times t p,j (j ≥ 1): (ii) M t p,j ;x 1 is a rational function of x, with possible poles only at x = X(p).
One could write a similar theorem for ∂ t p,0 − ∂ t p ′ ,0 .
Proof. We have: and again the essential singularities cancel, and since we sum over all sheets, the result is a rational fraction of x. Since the pole of ψ cl at x = x 1 has a constant residue, the derivative with respect to t p,j has no pole at x = x 1 . Since we divide by dX(z), poles at branchpoints could occur, but dX(z) vanishes as X(z) − a, and a rational fraction of x cannot have half-integer singularities, so there is no pole at branchpoints. Therefore, poles may only occur through ∂ t p,j e z 1 z ω , and we find: which shows that M t p,j ;x 1 (x; t) has poles only at x = X(p), and of order at most j. If X(p) = ∞, then (x − x 1 ) 2 /dx has a pole of order d p − 1, and thus M t p,j ;x 1 (x; t) has a pole in z of order at most d p + j − 1, and thus in x it has a pole of order at most is a rational fraction of x, the order of its pole must be integer and is thus at most 1 + ⌊(j − 1)/d p ⌋. Again, in the usual presentation of integrable systems, the ODE's are presented after sending x 1 to ∞, i.e. in terms of Ψ(x; t) and Φ(x; t).

Isomonodromic problem
We see that the matrix Ψ cl (x 1 , x; t) is a simultaneous solution of several compatible linear differential systems with rational coefficients. This leads to some isomonodromic problem.
We have A general property of linear differential systems, is that the solution Ψ cl (x 1 , x; t) is analytical wherever M x;x 1 (x) is analytical, and may have essential singularities at singularities of M x;x 1 (x). The solution Ψ cl (x 1 , x; t) may have monodromies around the singularities. Let p be one of the poles, then the monodromy after going around p is: where x − and x + mean the same value of x before and after going around the singularity.
Since M x;x 1 (x; t) is a rational function of x, it has no monodromy and we have and it is easy to see that this implies that In other words, the monodromy is independent of x, this is why the system is called "isomonodromic". Similarly, consider any time t p,j , and write: Since M t p,j ;x 1 (x) is a rational function of x, it has no monodromy, and this implies that In other words, the monodromy is independent of the times t p,j . Here actually, the monodromy group has finite order since Ψ(x 1 , x; t) lives in a bundle over the fixed, compact algebraic curve C). This is a very special case, which is implied by the fact that the evolution is isospectral [BBT02].

Dispersionless Tau function
Definition 4.1 We shift the prepotential and define: Definition 4.2 We define: where ω = k t k ω k is the 1-form depending linearly on the times and the filling fractions, andF 0 is the shifted prepotential associated to ω. We shall also use as notation T cl ( t) depends as well on the data of a non-singular odd characteristics for the Theta function. It is the Tau function associated to the solution of the problem of isospectral evolution described in Section 3.6. We call here Tau function, any function satisfying Hirota bilinear equation, and linked to Baker-Akhiezer functions by Sato relation [JMU81a,JMU81b,JMU81c]. Sections 4.1 and 4.3 prove that T cl fits in this definition.
In this article, we also call T cl the "semiclassical" Tau function.

Sato relation
Sato relation [SS83] means that the Baker-Akhiezer kernel ψ cl (z 1 , z 2 ) is obtained from the Tau function by a Schlesinger transformation. This can be formulated as: Proof. We add two simple poles at z 1 and z 2 by considering: According to Theorem 2.2 we have and according to Theorem 2.3 we have Since F 0 is homogeneous of degree 2 we have It is also easy to see that Taking λ = 1 implies the theorem.

Expansion near poles
In order to recover the usual presentation of Sato relation, let p ∈ P, and U p ⊆ C an open neighborhood of p on which the local coordinate ξ p is well-defined. The Schlesinger transformation: can also be written by Taylor expanding dS z 1 ,z 2 for z 1 , z 2 ∈ U p : Thus, the times associated to p after Schlesinger transformation are: In the literature, a notation [z] p is introduced for this special collection of time: And since dS z 1 ,z 2 (z) has vanishing A-cycle integrals, the filling fractions are not changed. For example, Theorem 4.1, together with Eqn. 3.18 gives:

Hirota bilinear equation
For any point z ∈ C, we define the insertion operator δ z , which acts on functions of a meromorphic 1-form on C, by considering the one parameter deformation consisting in adding a Bergman kernel B(z, ·)/dX(z): Locally, if z is near a pole p, we have: δ z is obviously a derivation.
Theorem 4.2 For any two 1-forms ω andω defined on the same Riemann surface C, we have: Res Proof. ψ cl (z, z ′ ; ω)ψ cl (z ′ , z ;ω) has a double pole at z = z ′ , and thus evaluating the residue computes a derivative.
As a corollary, if we chooseω = ω + dS z 1 ,z 2 , the expression on which we take the residue is a meromorphic form in z (it has no essential singularity). Its only other pole is located at z = z 2 , it is simple, and by moving the integration contour, we find: Corollary 4.1 For any data (C, ω), the Baker-Akhiezer kernel is self-replicating: This self-replication property is the analog of the Ricatti equation in [BBT02]. Notice that it can also be obtained by a straightforward computation of δ z ln ψ cl (z 1 , z 2 ) from Definition 3.1) and comparison with the refined duality equation (Eqn. 3.3). Associated to any derivation ∂ t , we may define a Hirota operator D t [Hir71] acting on two functions f (t), g(t), such that: This allows us to reformulate the self-replication property as: Theorem 4.3 In terms of T cl , for any pole p, the self-replication property is equivalent to the Hirota bilinear difference equation: Actually, this property for T cl is equivalent to the Fay identity (see § 2.2.3) satisfied by the Theta function θ(·|τ ). The Hirota equation can also be written in a more symmetric way by setting: (4.14) Namely: The procedure to translate it into a set of differential equations (with respect to the times) is well-known, it is merely obtained by Taylor expansion in ξ p (z), ξ p (z 1 ) and ξ p (z 2 ) (see e.g. [JM83]). This gives an infinite set of partial differential equations involving derivatives of the Tau function with respect to times t p,j , which are equations of a KP hierarchy. These equations are equivalent to Hirota bilinear equation.

Dispersive deformation
So far, the spectral curve (C, X, Y ) was fixed once for all, and was equipped with a 1-form ω = k t k ω k depending linearly on times. Now, we shall let the spectral curve itself change around (C, X, Y ), and in particular we may vary the complex structure of the curve C. Y dX shall play the role of the 1-form.

More geometry: symplectic invariants
For any spectral curve S = (C, X, Y ), a sequence of symplectic invariants F g (S), and of symplectic covariant forms ω (g) n (S) was defined in [EO07a]. Let us recall their definition and main properties.

Topological recursion
Let a i be the ramification points of S, i.e. the zeroes of dX. We assume that the spectral curve S is regular, i.e. a i are simple zeroes of dX and dY (a i ) = 0. Then, Y behaves like a squareroot near ramification points: and there is a unique other point z = z such that X(z) = X(z), at least for z in a neighborhood U a i of a i . Then we define the recursion kernel [EO07a]: In the variable z 0 , it is a meromorphic 1-form defined globally for z 0 ∈ C, and in the variable z, it is the inverse of a 1-form, only defined in i U a i .
An important property proved in [EO07a], is that: n (z 1 , . . . , z n ) is symmetric in its n variables, and it is a meromorphic form in each variable, having poles only at ramification points, with vanishing residues. The poles are of order at most 6g + 2n − 4.
For instance, for (g, n) = (0, 3), this definition amounts to: Definition 5.2 The "symplectic invariants" F g are numbers associated to S, as follows.
• For g = 1, F 1 is defined in terms of the Bergman Tau function T B,X introduced by Kokotov and Korotkin [KK03]: . F 1 is related to the logarithm of the determinant of the Laplacian on C with metrics |ydx| 2 .
As a notation we write ω The name "symplectic invariants" came from the important property, proved in [EO07b], that F g and the cohomology class of ω where R is any rational function and λ ∈ C.

Special geometry
Let us consider an infinitesimal deformation of the spectral curve S. It implies an infinitesimal deformation of the complex structure of the Riemann surface C, and a deformation of the form Y dX: where Ω is a meromorphic form. We can decompose 5 as in Theorem 2.1 There is a dual cycle associated to Ω: such that: n+1 (·, z 1 , . . . , z n ) and the derivative is evaluated at X(z i ) fixed.
5 ω k here belongs to the basis of meromorphic forms defined in Section 2.3.4. It has nothing to do with the invariants ω (g) n .

Candidate for a dispersive Tau function
Let us consider: where F g (S, n ′ ) are the symplectic invariants associated to the curve S and the vector of filling fractions n ′ = (n ′ 1 , . . . , n ′ g ). ln Z[S, n] is a formal Laurent series in N (in particular we emphasize that it contains no oscillatory terms, by opposition with the definition to come). Then, we would like to define the sum over all integer filling fractions: The intuition behind this definition is Whitham averaging. The filling fractions are considered as fast evolving moduli in the system, so that we do not see a fixed spectral curve, but rather a statistical mixture of spectral curves with different filling fractions. When g > 0, ln T [µ,ν] (S) is no more a Laurent series in N.
To give a precise meaning to this expression, one has to collect the terms order by order in 1/N. Initially, such an object has been introduced with heuristic arguments to compute the large N asymptotic of hermitian matrix integrals, and it leads to theta functions to the leading order [BDE00], and a whole series of corrections involving derivatives of theta functions [Eyn09]. Taking this series as a definition of an object T [µ,ν] (S), some of its properties related to modularity and holomorphic anomaly equations were studied in [EM08]. In this article, we argue that T [µ,ν] (S) is a good candidate for a Tau function.

Preliminaries
We shall consider the dependence of the F g 's in the filling fractions 6 : F g ≡ F g (ǫ 1 , . . . , ǫ g ) and we shall write for short F ′ g for the vector of first derivatives of F g with respect to ǫ i 's: and F ′′ g , F ′′′ g , . . . , F (k) g , denote likewise the tensor of higher derivatives. We need Theta functions, and we shall introduce an appropriate notation for our purposes. We define the Θ [µ,ν] function as: Again, in this equation, τ i,j and τ j,i are considered independent. From now on, the dependence in the characteristics will be implicit. Given a spectral curve (C, X, Y ) of genus g, and the choice of a symplectic basis (A, B), we set: B the Riemann matrix of periods of C and we define to shorten formulas: as the tensor of k th derivatives of Θ. In the definition of Θ, notice that w 0 appears through the combination: which is modular of weight 1, and already appeared in Section 3.1.

Definition and comments
Definition 5.3 For any (µ, ν) ∈ C g and any spectral curve S = (C, X, Y ) of genus g, we define: This function is defined formally order by order in N (in the coefficient of 1/N k , each term Θ (j) is considered formally of order 1). It can be seen as a genuine asymptotic series when the reference filling fractions ǫ are chosen such that w 0 = NF ′ 0 is of order O(1).
When µ + τ ν is a non-singular half-integer odd characteristics, the leading term when N → ∞ of T (S) coincides with the classical Tau function of Section 4 computed for the differential form ω = N Y dX, up to an exponential factor: Notice that the Fay identity for θ [Fay70] presented in § 2.2.3 is also true if we multiply θ by an exponential factor of its argument. Therefore, Theorem 4.3 ensures that the large N limit of T (S) satisfies the Hirota bilinear equation (Theorem. 4.3), i.e. is a dispersionless Tau function to leading order. This is also true for arbitrary characteristics [µ, ν], although the times have to be shifted by a (maybe complex) constant in this case. We conjecture (see Section 7) that T [µ,ν] is actually a Tau function, satisfying Hirota equations to all orders.
We also mention that under a modular transformation (i.e. a change of choice for the A and B cycles), T [µ,ν] changes like the Theta function of characteristics [µ, ν] (see [EM08]):

the Tau function transforms as
where ζ [µ,ν] (A, B, C, D) is the phase factor, independent of the spectral curve, which already appears in the modular transformation of the Theta function.
ψ(z 1 , z 2 ) is again defined formally, order by order in 1/N, or as an asymptotic series for appropriate ǫ. The leading order coincides with ψ cl introduced in Definition 3.1. Let us give the first few orders: Lemma 5.1 ψ(z 1 , z 2 ; S) is a well-defined spinor in z 1 and z 2 . Furthermore: • ψ(z 1 , z 2 ) has a simple pole at z 1 = z 2 : • It has an essential singularity near any pole p of Y dX, of the form e • At all orders in 1/N (except at leading order), ψ(z 1 , z 2 ) has poles at the ramification points a i . Their order increase with the order of 1/N.
Proof. When z 1 goes around an A-cycle, dS z 1 ,z 2 is unchanged. When z 1 goes around a cycle B j , dS z 1 ,z 2 is shifted by a holomorphic form dS z 1 ,z 2 → dS z 1 ,z 2 + 2iπdu j , which is dual to a ∂/∂ǫ j , and since the Tau function is background independent (it was proved in [EM08] that ∂T /∂ǫ i = 0), then it is unchanged. This shows that ψ(z 1 , z 2 ; S) is a well-defined spinor. Then we compute each term of T (S + [z 1 ] − [z 2 ]) by writing the Taylor expansion: (5.26) which we need to evaluate at λ = 1. The n-th derivatives of F g at λ = 0 are computed by the special geometry relations theorem 5.2, using the dual cycle (dS z 1 ,z 2 ) * = [z 2 , z 1 ]: n with 2 − 2g − n < 0 are meromorphic, and have poles only at ramification points, without residues. This implies that their contribution to ψ(z 1 , z 2 ) provides only poles at ramification points. The only terms involving ω • ∂ F 0 ∂λ λ=0 = z 1 z 2 Y dX, which contributes to ψ as the essential singularity e = − ln (E(z 1 , z 2 ) 2 dX(z 1 ) dX(z 2 )) which contributes to ψ as 1/E(z 1 , z 2 ).
) which does not yield any singularity.
All the other terms have 2 − 2g − n < 0, and contribute order by order, only as combinations of meromorphic forms and derivatives of Theta functions, having poles at ramification points conveyed by the ω (g) n 's with 2 − 2g − n < 0 Notice that as a corollary of Proposition 5.1, ψ has nice modular properties:

Second kind deformations of S
Let us recall the definition of the insertion operator δ z , already encountered in Section 4.3 and adapted now for varying spectral curves.
Definition 6.1 We define the insertion operator δ z , acting on a functional f (S) of a spectral curve S, as follows: where S λ is the family of spectral curves obtained from S by: The dual cycle of B(z, ·)/dX(z) is the contour surrounding z with index 1: Then, the relations of special geometry (theorem 5.2) for ω (g) n implies, for any n, g ≥ 0: δ z ω (g) n (z 1 , . . . , z n ) = Res For instance: 1 (z ′ ) = B(z, z ′ ) (6.3)
Each coefficient, order by order in 1/N, is a meromorphic form with poles at ramification points

Examples
For instance the three first orders of W 1 are: where we recall that: For the 2-point correlator, the three first orders are: For n ≥ 3, the leading order of the n-point correlator is a O(1), and is obtained by successive applications of δ z to the (ln Θ) ′′ term.

Loop equations
Theorem 6.1 The dispersive Tau function obeys the loop equations. Namely, let us denote by Γ a contour separating the poles of Y dX, from the set of preimage of a point x ∈ C.
where Q is a rational function of x, whose only poles are those of Y dX, and with degree one less than that of Y dX.
Those loop equations can be written in terms of correlators by applying δ z 2 · · · δ zn : Theorem 6.2 Let J = {z 2 , . . . , z n }. The correlators satisfy (i) The linear loop equations. For all n ≥ 1: The quadratic loop equations. For all n ≥ 1: defines a quantity Q n (x; J) which is a rational function of x, whose only poles are located at those of Y dX.
The important information in loop equations, is that those particular combinations of W n 's have no monodromies in the variable x. Here in particular, they are regular at the ramification points a i . Since every W n has poles at ramification points to all orders in 1/N, this is a highly non-trivial property.
Proof. The formula:

Hirota equations
In the dispersionless case, we have seen that the self-replication property for ψ cl is equivalent to an infinitesimal Fay identity, which is known in turn to be equivalent to Hirota equations. In a similar way, we conjecture here: Conjecture 7.1 ψ is self-replicating: We have not been able to prove this conjecture. We prove in appendix A that it holds up to o(1/N). We argue in Section 9 that it is compatible with what is known for spectral curves coming from the one matrix model (hyperelliptic curves), or the two matrix model. Besides, these matrix models do not allow to reach all plane curves S. The difficulty in finding a proof of Conjecture 7.1 comes from the singularities at ramification points. For instance, one can always write: Since the integrand is a differential form on the Riemann surface Σ underlying S, we can move the contour to the poles at the ramification points, and the pole at z ′ = z 2 : Then, it remains to show that the sum of residues at ramification points vanishes. So, conjecture 7.1 is equivalent to: In Appendix A, we check that this residue at each a i is o(1/N). This involves already non-trivial identities between Theta functions associated to a Riemann surface, like Fay identity or its degenerations, and involves the precise value of ω (0) 3 . We have yet not been able to find a general way to show that this residue is 0 to all orders in 1/N.
In terms of the T function, Conjecture 7.1 can be rephrased: Conjecture 7.3 T satisfy a "differential Fay identity": There is also a global version of the former conjecture. First, notice from our definition in Section 5 that: so omitting the parentheses makes sense, but: This sign comes from the fact that the definition of T contains a regularization procedure (for , whose result depends on the way we form the pairs of simple poles to add to S.
Conjecture 7.4 T satisfies a "Fay identity": Provided our conjectures hold, T and ψ are actually the Tau function and the Baker-Akhiezer spinor kernel of a dispersive integrable system.

Proof of equivalence of Conjectures 7.3 and 7.4.
We can obtain Conjecture 7.3 from Conjecture 7.4 by letting z 1 and z 2 merge to a point z. In the other direction, we use condensed notations and we apply Conjecture 7.3 to the spectral curve S + [z 3 ] − [z 4 ]: Exchanging the roles of z 1 ↔ z 3 and z 2 ↔ z 4 also gives and comparing the two, we may get rid of the terms involving δ z : Let us define U 1234 = T T 1234 − T 12 T 34 + T 14 T 32 (7.7) which is the quantity which should vanish at the end of our computation. Notice that lim z 1 →z 2 U 1234 = 0 (7.8) Indeed: where E ij = E(z i , z j ), and this expression vanish by application of Conjecture 7.3 to the spectral curve S. Notice also that the remark about T made above Conjecture 7.4 implies U ijkl = U klij = −U ilkj . Let us rewrite Eqn. 7.6 in terms of U ijkl only: The left hand side may have simple poles when z i → z j due to T ij , but not higher degree poles since we know that U ijkl is actually regular when z i → z j or z l . For Eqn. 7.10 to hold, in particular, the coefficient of the simple pole when z 2 → z 3 must vanish: which we can also write after reindexing the points and using the symmetries of U: Similarly, the coefficient of the simple pole when z 1 → z 2 must vanish: Now, we may combine the latter with Eqn. 7.11, set z = z 2 for convenience, and isolate U 1234 : If U 1234 were not identically zero, we would have (by continuity of all the coefficients of the series) for any points z 1 , z 2 , z 3 , z 4 on the curve: U 1234 = T 32 T 14 T 31 T 24 (T 24 T 31 − T 21 T 34 ) + (T 14 T 32 − T 12 T 34 ) (7.14) But matching the coefficient of the simple pole when z 1 → z 4 in this equation yields: which is not true. Therefore, U 1234 ≡ 0.

Exponential formula
We start with a remark which does not rely of the conjectures of Section 7, but which is natural to present now. Recall that the kernel is defined by Sato's formula: In the last step, we have used the expression for the leading order of W n found in Section 6.3. Then, the second line can be resummated into ln Θ 12 , and the whole result exponentiated leads to the formula we announced.
Notice that, when expanding the exponential, to any given order O(N −k ), all the W n give a contribution involving derivatives of theta functions contracted with tensor products (u(z 1 ) − u(z 2 )) ⊗n . These contributions have to be resummated into a single theta function (or derivatives thereof) with argument shifted by u(z 1 ) − u(z 2 ), then producing an expression at the order sought involving only a finite number of terms.

Determinantal formulas
Conversely, the correlators W n (z 1 , . . . , z n ) can be recovered from the spinor kernel ψ(z 1 , z 2 ), they are the determinantal correlation functions built with ψ(z i , z j ). Let us consider first the case of W 1 , which does not rely on the conjectures of Section 7.
Proof. First, we notice that adding a double pole to Y dX can be realized by adding two simple poles and take the limit where the two simple poles collapse. More precisely, when z ′ → z, we have We can thus use the definition of δ z with λ = X(z ′ ) − X(z). For any regular functional f (Y dX) of the spectral curve, we thus have: In some sense we trade a variation of Y dX by a second kind differential with a variation with a third kind differential.
Proof. The formula for W 1 is proved in Lemma 8.1. Then, we get the formula for W n by recursively applying δ z i and using the self-replication of ψ.
It is thus clear that this determinantal structure relies on the (conjectured) existence of Hirota equations for T , i.e. on integrability.

Duality equation
Theorem 8.2 If Conjecture 7.1 holds, we have for any spectral curve S: Proof. For S = (C, x, y), we introduce an auxiliary spectral curve: To compute the matrix element, we use the self-replication of ψ: Now, we use the fact that δ z is a derivation, and by definition, The linear loop equation (Theorem 6.1) tells us the sum over sheets of W 1 [z m (x 2 ) ;Ŝ ij ]: The spinor kernel ψ cl (z 1 , z 2 ) of the dispersionless integrable system of Section 3 was regular at ramification points. So, we could find a formula for ψ cl (z 1 , z)ψ cl (z, z 2 ) even before the sum over the sheets where z is located (refined duality equation, Theorem 3.3). Here, the spinor kernel ψ(z 1 , z 2 ) does have, order by order in 1/N, poles at ramification points. So, we do not have a simple expression for ψ(z 1 , z)ψ(z, z 2 ). However, Theorem 8.2 shows all contributions from the ramification points cancel in the sum over sheets.

Christoffel-Darboux relations
The matrix Ψ(x 1 , x 2 ; S) is invertible, since it is defined as a series for which the leading term coincides with Ψ cl (x 1 , x 2 ; S) which is invertible (see Lemma 3.2). We also have a duality relation to express the inverse: Corollary 8.1 If Conjecture 7.1 holds, Theorem 8.3 If Conjecture 7.1 holds, the matrices Ψ(x ; S) and Φ(x ; S) are invertible, and we have the Christoffel-Darboux relation: where the matrix A is invertible, independent of x and given by: The proofs can be copied from the classical case (Section 3.5), because they are based only on the duality equation.

Differential systems
Ψ(x 1 , x 2 ; S) is the solution of a system of differential equations with respect to the positions of the poles X(p) (x = ∞ is a fixed pole) and the times t p,j .
Theorem 8.4 If Conjecture 7.1 holds, then for any deformation parameter λ = t p,j or λ = X(p) = ∞, there exists a d × d matrix M λ (x 1 , x 2 ; S), such that: (ii) M λ is a rational function of x 2 .
(iii) M λ has no pole at branchpoints.
These deformations are compatible (since Ψ is invertible) and isomonodromic.
Proof. By definition: When λ is a time t p,k (k ≥ 1), we have by self-replication: We compute: It is clear that M t p,k (x 1 , x 2 ) is a rational function of x 2 without poles at branchpoints. If we assume X(p) = ∞, it has a pole only at x 2 = X(p), and this pole is of order k + 1. If we assume X(p) = ∞, it has a pole only at x 2 = ∞, and this pole is of order 1 + ⌊k/d p ⌋ (we recall that d p is the multiplicity of order of the pole p of X). It is straightforward to adapt this proof for k = 0, namely ∂ λ = ∂ t p,0 − ∂ t p ′ ,0 . Now, we turn to λ = X(p) = ∞. We have assumed that dX(p) = 0, so the preimages p 1 , . . . , p d of X(p) are distinct, and (X(z) − X(p)) is a local coordinate near each p m . The Laurent expansion of Y dX(z) when z → p m is: where only a finite number of t p,k ′ = 0 are non zero. If we perform X(p) → X(p) + λ while the times are fixed, we change Y dX to (Y dX) λ , with Laurent expansion at p m : So, we can identify: which is a finite sum. Hence: This is a rational fraction of x 2 , with a pole at x 2 = X(p) of order 1 + ⌊(max m ord z m (p) Y dX)/d p ⌋.
One could also define a matrix L with the reconstruction formula presented in Section 3.6. But now, the function Y depends on the spectral curve, the evolution of L under the flows generated by the times is not isospectral. But the argument of Eqn. 3.8 is still valid, and the times in this construction are isomonodromic parameters. Such a reconstruction has also been performed at leading order by Bertola and Gekhtman [BG07], and our expressions up to o(1) match indeed their results.

Parallel with matrix models
Some unitarily invariant models of random matrices are special cases of this construction, that actually suggested to develop this formalism. We recall here the dictionary between the definitions in integrability (and in this article), and observables in matrix models. Let M be a random square matrix of size N × N, diagonalizable by a unitary conjugation, with eigenvalues restricted to some contour Γ in the complex plane. The probability measure dµ(M) is a data of the matrix model. Its

Examples of integrable matrix models
Important examples of integrable matrix models, described in details in [Meh04], include: The one hermitian matrix model M ∈ H N is a N × N hermitian matrix, with eigenvalues on the real axis is a rational function of its variable x, chosen such that R e −V (x) dx is absolutely convergent.
The one normal matrix model, with eigenvalues on a contour Γ We define and (x 1 , . . . , x N ) ∈ Γ N } (9.3) equipped with a "measure" (not necessarily real or normalized) 7 : where V ′ (x) ∈ C(x), i.e. V ′ (x) is a rational function of its variable x, chosen such that Γ e −V (x) dx is absolutely convergent. This means that for a given potential V (x), Γ must go to ∞ or to the poles of V ′ , only in sectors where Re V → +∞, i.e. in some "Stokes sectors". When Γ = R, this definition is the same as the Hermitian matrix model H N (R) = H N , with the usual Lebesgue measure on H N . When Γ is the unit circle S 1 , this definition is the same as the unitary matrix model H N (S 1 ) = U(N), with the usual Haar measure on U(N).
The two normal matrices model Given two paths Γ 1 , Γ 2 , we define a measure on where V 1 and V 2 are semiclassical potentials (V ′ 1 and V ′ 2 are rational functions), chosen such that integrals on Γ 1 × Γ 2 are absolutely convergent. Upon integration on M 2 , this measure induces a measure dµ(M) on M 1 that we rename M: The chain of matrices This is the natural generalization of the case of two matrices. Consider k paths Γ 1 , . . . , Γ k , and k semiclassical potentials V 1 , . . . , V k , and define the measure on All the listed matrix models above have the property to be integrable, in a sense explained below. There is a huge literature on the subject, let us mention among others the early works in physics [MMKZ92,Kos96], and in mathematics [HTW93, AM95,vM97]. For the one and two matrices models, it is established that the partition function Z N = dµ(M) is an isomonodromic Tau function [BEH03c, BEH06,BM09a] (such a result is not known at present for the chain of matrices). The so-called double scaling limit of matrix models has been also intensively from the point of view of isomonodromic deformations [IKF90, DS90, Moo90, FIKN06].

Correlators, spectral curves, loop equations
For all the matrix models above, the correlators are: Tr 1 Tr 1 where C means 'cumulant'. W n is often called n-point function, or connected n-point function, and W n is called disconnected n-point function.
The one point function W 1 (x) plays an important role, it is often called the resolvent. In all the matrix models above, under reasonable assumptions, it has a large N limit denoted: which is furthermore an algebraic function of x, i.e. there exists a polynomial E(x, y) such that E(x, Y(x)) = 0. (9.13) This algebraic equation defines an algebraic curve C, i.e. a compact Riemann surface C and two meromorphic functions X, Y : C → CP 1 , such that ∀ z ∈ C , Y(X(z)) = Y (z). (9.14) The "semiclassical spectral curve" S is then The correlators corresponding to Definition 6.2, are W n (z 1 , . . . , z n ) = N n W n (X(z 1 ), . . . , X(z n )) n i=1 dX(z i ) + δ n,2 dX(z 1 ) ⊗ dX(z 2 ) (X(z 1 ) − X(z 2 )) 2 (9.16) It is a classical result of random matrix theory (for instance it can be proved by integration by parts in the matrix integral) that, for all matrix models listed above, the correlators W n satisfy the loop equations of Theorem 6.1.

Large N asymptotic expansion
It is conjectured that the large N asymptotic expansion of the partition function Z N matches with that of the T function for the semiclassical spectral curve S introduced in Definition 5.3: i.e. has an asymptotic expansion The characteristics [µ, ν] of the Theta function Θ is determined by the choice of the integration contour Γ. This conjecture was derived heuristically in [Eyn09]. When the semiclassical spectral curve is of genus 0, there is no Theta function and the expansion involves only powers of 1/N (in fact, powers of 1/N 2 ). This happens for the so-called "one-cut regime", and for the one matrix model, the existence of such an expansion has been proved for the one matrix model for real-valued, analytic potential [APS01], and then the coefficients are necessarily given by the symplectic invariants of the semiclassical spectral curve [Eyn04]. Beyond the one-cut regime, the Riemann-Hilbert steepest descent analysis [IKF90,DZ95] has been applied to find explicitly the asymptotics up to o(1) in the one hermitian matrix model with real-valued, polynomial potential [BI99, DKM + 99]. It features in general a pseudo-periodic behavior with N, encoded in the Theta function, but the one-cut regime can also be recovered with this Riemann-Hilbert method [EM03]. These results have later been extended to the one normal matrix model with complex-valued polynomial potential [BM09b]. This method can be used in principle to find recursively the subleading orders, although it does not allow to write the answer a priori explicitly to all orders. Definition 5.3 is expected to give the correct answer to all orders.

Baker-Akhiezer spinor kernel
The spinor kernel is related to the expectation value of ratios of characteristic polynomials: ψ(z 1 , z 2 ) = det(X(z 1 ) − M) det(X(z 2 ) − M) dX(z 1 )dX(z 2 ) X(z 1 ) − X(z 2 ) (9.19) The dispersionless spinor kernel ψ cl (z 1 , z 2 ) is the large N limit of ψ(z 1 , z 2 ). When sending X(z 2 ) → ∞, one gets, after proper renormalization that the Baker-Akhiezer function is ψ(z 1 ) = "ψ(z 1 , ∞)" = det(X(z 1 ) − M) (9.20) which is clearly a polynomial in X(z 1 ) of degree N. It is a classical result of random matrix theory [Meh04] that this is the orthogonal polynomial of degree N, orthogonal with respect to the measure dµ for matrices of size 1. The dual Baker Akhiezer-function for matrix models is φ(z 1 ) = "ψ(∞, z 1 )" = 1 det(X(z 1 ) − M) (9.21) It can be derived from orthogonality relations of the orthogonal polynomials that those spinor kernels satisfy Hirota equations, also called "determinantal formula", for instance: In that case, N is an integer (the size of the matrices), and Hirota equation is an equality between sequences indexed by N, not only of formal asymptotic series. This relations have been proved in [FS03] for the one matrix model, and in [Ber,AP04] for the two matrices model, from which the case of chain of matrices can be deduced. It shows that, modulo the heuristic derivation of asymptotics of matrix integrals, Hirota equation as stated as in Conjecture 7.4) hold for the semiclassical spectral curves of matrix models (however, not all algebraic curves fall in this class).

Differential systems
The orthogonal polynomials, as well as their duals, in all cases above, do satisfy some ODE of order d (d depends on the degrees of potentials). As we have seen in Section 8, the proof is a consequence of Hirota equations. For instance for the one matrix model, this is a second order ODE (d = 2). For the two matrices model with potentials V 1 and V 2 , we always have d = 1 + deg V ′ 2 (in case V ′ 2 is a rational function, deg V ′ 2 is the sum of degrees of all poles). In all cases we have a d-dimensional vector: Ψ(z) = ψ(z 1 ) . . . (9.23) where the first entry is ψ(z 1 ). The other entries are obtained from ψ(z 1 ) with a procedure described in [BEH03a] and very similar to Section 3.4. This vector satisfies an ODE d d X(z) Ψ(z) = D(X(z)) Ψ(z) (9.24) where D(x) is a d × d matrix, whose entries are rational functions of x, that depend implicitly on N, on the coefficients of the potentials, and on the choice of integration contour Γ. The locus of eigenvalues of D(x), i.e. the polynomial equation E N (x, y) = det (y Id d×d − D(x)) = 0 (9.25) defines the finite N spectral curve. The semiclassical spectral curve corresponds to its large N limit: E(x, y) = lim There are also d ×d differential systems for derivatives with respect to all coefficients of the potentials, and there is also a linear recursion relation on N → N +1 (see [CI97] for the one matrix model). All these systems are compatible as shown in [BEH02] in full generality for the chain of matrices. Moreover, in all these matrix models, there is also a compatible Toda chain recursion equation relating Z N +1 , Z N and Z N −1 [GMM + 91]. We postpone to a future work the investigation of the presence of such relations in our construction.

Symplectic invariance
Notice that, in the 2-matrix model we have defined our semiclassical spectral curve S = (C, X, Y ) from the large N limit of the resolvent W 1 (x) = tr 1 x−M 1 associated to the matrix M 1 . Since M 1 and M 2 play a symmetric role, it is clear that we would have obtained the same partition function, starting from the semiclassical spectral curveS = (C,X,Ỹ ) associated to the resolvent of matrix M 2 , and thus we must have T (S) = T (S). (9.27) One can easily find that the two spectral curves S = (C, X, Y ) andS = (C,X,Ỹ ) are related by C =C andX = Y ,Ỹ = X, in other words they are symplectically equivalent, and the fact that T (S) = T (S) can be seen as a consequence of the symplectic invariance of the F g 's.
In fact, this is a manifestation at large N of an exact result for finite N. The orthogonal polynomials Ψ(x) associated to matrix M 1 satisfy an ODE of some order d (see Eqn. 9.23): d d X(z) Ψ(z) = D(X(z)) Ψ(z) (9.28) whereas the orthogonal polynomials Ψ (y) associated to matrix M 2 satisfy another ODE of some orderd (in generald = d): Ψ (z) =D(Y (z)) Ψ (z) (9.29) It was discovered in [BEH03b] that: det (y Id d×d − D(x)) = det (x Idd ×d −D(y)). (9.30) This also implies that the semiclassical spectral curve defined from M 1 or from M 2 are related as explained above.

Conclusion
For integrable systems with a small dispersive parameter 1/N, using the theory of symplectic invariants [EO07a], we have introduced a formal object T , which is conjectured to be a Tau function in the sense that it satisfies Hirota equations. It is challenging to find a full proof of Conjecture 7.4. This construction was strongly motivated by results known for matrix models, but developed independently. In some sense, it gives the answer to all order in 1/N of a Whitham averaging. All properties mentioned in this article, correspond to well-known or conjectured properties of matrix models. Our construction is thus an axiomatized extension of the properties of matrix models, to arbitrary algebraic spectral curves. One can wonder how to generalize the construction. First, to algebraic curves in C * × C * , having in view the mirror curve appearing in Gromov-Witten theory, which all are of the form Pol(e x , e y ) = 0. Second, to the bundles appearing in generalized matrix models like the O(n) model, which may be related to isomonodromic deformations on Riemann surfaces of positive genus and to Hitchin systems [Hit87]. And, even further to D-modules, for which an adapted topological recursion (the so-called β-deformation of the topological recursion) has been found with similiar properties [CEM09,CEM10].
Eventually, it remains to compare this construction with other approaches (Frobenius manifolds, Poisson bracket structures, Segal-Wilson formalism in the Grassmannian [SW85], etc.), study its consequences and better understand the underlying geometry. and B. Safnuk for useful and fruitful discussions on this subject. This work is partly supported by the ANR project Grandes Matrices Aléatoires ANR-08-BLAN-0311-01, by the European Science Foundation through the Misgam program, by the Quebec government with the FQRNT. B. E. thanks the CERN for its hospitality and G.B. thanks the SISSA for welcoming him while this work was pursued. where, to shorten notations, 1 means z 1 , 2 means z 2 , means the contour integral around B-cycles (indices are understood in tensor notations, i.e. contracted with the indices of derivatives of Θ), and Θ 12 means Θ 12 = Θ(w 0 + 2iπ(u(z 1 ) − u(z 2 )) | τ ), We remind that (Eqn. 5.3):