Michel Bergère
email: michel.bergere@cea.fr.

Gaëtan Borot
email: gborot@mpim-bonn.mpg.de.

Bertrand Eynard
email: bertrand.eynard@cea.fr.

Rational differential systems, loop equations, and application to the q-th reductions of KP

published or not. The documents may come

Rational differential systems, loop equations, and

application to the q-th reductions of KP

Introduction

Let Lpxq be a d ˆd matrix with entries being rational functions of x, and P the set of poles of L. We consider matrix Ψpxq (whose columns form a basis of solutions) of the differential system:

Outline

The paper is organized in three parts.

Firstly, in Section 2, we associate to any dˆd invertible matrix Ψpxq solution of a linear differential system: ' a d ˆd matrix Kpx, yq, called kernel.

' an infinite family of functions W k p a1 x 1 , . . . , a k

x k q, indexed by integer a 1 , . . . , a k P v1, dw, called k-points correlators, or shortly, correlators. and we show that the k-point correlators satisfy a set of linear equations (Theorem 2.1) and a set of quadratic equations (Theorem 2.2). We use the name loop equations to refer collectively to those set of equations. We also introduce a notion of "insertion operator" (Definition 2.5) allowing the derivation of k-linear loop equations for k ď d (the size of the differential system) from the master ones. These results are of purely algebraic nature and hold for any system (1-1). When L depends on a set of parameters t preserving the monodromy of the solutions, we can also associate to Ψpx, tq a Tau function T p tq, defined up to a constant prefactor.

Secondly, in Section 3, we study the semiclassical expansion in powers of and describe in detail the monodromy of its coefficients (Section 3.2-3.4). We introduce in Definition 3.3 the notion of "expansion of topological type" -also referred to as the TT property -and show that the expansion can be computed by the topological recursion of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] when the TT property holds. In practice, the main consequence of our theory is Theorem 3.1, and in presence of isomonodromic times, this also allows the computation of the expansion of ln T p tq (Corollary 4.2).

Finally, in Section 5, we apply our theory to the linear system associated to the q-th reduction of KP, and illustrate it more specifically with examples of the pp, qq models (Section 6). As a motivation, those hierarchies are believed to describe the algebraic critical edge behavior that can be reached in the two hermitian matrix model, and universality classes of 2d quantum gravity coupled to conformal field theories [START_REF] Moore | Geometry of the string equations[END_REF][START_REF] Douglas | Strings in less than one dimension[END_REF][START_REF] Gross | A nonperturbative treatment of two-dimensional gravity[END_REF][START_REF] Di Francesco | 2d gravity and random matrices[END_REF]. In any q-th reduction of KP, we show ( § 5.6-5.8) that the TT property holds, and that our Theorem 3.1 can be applied.

Comments

The earlier work [START_REF] Bergère | Determinantal formulae and loop equations[END_REF] described the construction of Section 2 for general 2 ˆ2 rational systems, but implicitly assumed the TT property. It was illustrated for p2m `1, 2q systems in [START_REF]Universal scaling limits of matrix models and pp, qq Liouville gravity[END_REF], and entails a rigorous proof -modulo checking the TT property, which had not been performed so far -of an equivalence between the three usual approaches of quantum gravity, namely topological gravity (in relation with intersection theory on the moduli space of curves), random maps, and p2m `1, 2q models (see [START_REF] Di Francesco | 2d gravity and random matrices[END_REF] for a review on those equivalences in physics). Again taking the TT property as an assumption, [START_REF] Cafasso | Double scaling limits of random matrices and minimal p2m, 1q models: the merging of two cuts in a degenerate case[END_REF] treated the models p2m, 1q, in relation with the merging of two cuts in random matrix theory. The TT property was made explicit and checked by integrability arguments in [START_REF] Borot | Tracy-Widom GUE law and symplectic invariants[END_REF] for a 2 ˆ2 linear system associated to the Painlevé II equation [START_REF] Flaschka | Monodromy-and spectrum-preserving deformations, I[END_REF], justifying the computation of asymptotics of the GUE Tracy-Widom law by the topological recursion. The same approach -with a justification of the TT property -was applied more recently [START_REF] Bergère | The sine-law gap probability, Painlevé 5, and asymptotic expansion by the topological recursion[END_REF] to the 2 ˆ2 linear system of associated to Painlevé V [START_REF] Jimbo | Monodromy preserving deformation of linear ordinary differential equations with rational coefficients[END_REF], relevant to get the GUE sine kernel law. So far, this concerned only 2 ˆ2 systems.

The present work aims at presenting a complete theory for general d ˆd rational systems, and developing tools to study the TT property. Its application to the pp, qq models can then be used to establish rigorously the equivalence between the three quantum gravities for all pp, qq models. For clarity, this will appear in a separate work [BE].

In [START_REF]Geometry of spectral curves and all order dispersive integrable system[END_REF], the two last authors have made a conjecture to construct an integrable system out of the topological recursion of a given spectral curve. The present work aims at the converse: showing that the semiclassical expansion of linear differential systems satisfying the TT property are computed by the topological recursion of their semiclassical spectral curve.

The TT property is neither expected to hold in general -even among integrable systems -nor obvious to establish for a given system. Our proof that it holds for the q-th reduction of the KP hierarchy depends in an essential way on the integrability of the latter, i.e. on the existence of another system B t Ψpx, tq " Mpx, tqΨpx, tq with rational coefficients in x, which is compatible with (1-1), but also on the specific form of Mpx, tq. This is clear from the technical results of Section 5.7 and 5.8.

Within the TT property, the structure of the asymptotic expansion of correlators is identified in Theorem 3.1, but when the semiclassical spectral curve has genus g ą 0, it features an unknown "holomorphic part" H pgq n pz 1 , . . . , z n q, which are basically the moduli of the space of solutions of loop equations. A given solution Ψpxq knows which H pgq n pz 1 , . . . , z n q is chosen. It thus remains to investigate which conditions have to be added to the loop equations to determine completely the holomorphic part. They probably should take the form of period conditions. Actually, for many interesting solutions Ψpxq, we expect the TT property to breakdown if the semiclassical spectral curve has genus g ą 0.

We stress that, even when the TT property does not hold, the loop equations of Theorem 2.1 and 2.2 are still satisfied and provide some constraints on the asymptotic expansion of Ψpxq. In particular, the existence of a non-trivial moduli space of solutions of loop equations -which, in the context of expansion in powers of , can be parametrized by a "holomorphic part" at each order in -can be responsible for the breakdown of expansion in powers of , since the moduli may depend on in a complicated way. This mechanism explains for instance the oscillatory asymptotics in matrix models [START_REF] Eynard | Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence[END_REF][START_REF] Borot | Asymptotic expansion of beta matrix models in the multi-cut regime[END_REF]. It is also implicit in [START_REF]Geometry of spectral curves and all order dispersive integrable system[END_REF], where the candidate Tau function is constructed as a sum over a lattice of step in the moduli space of solutions of the loop equations: the interferences between the terms of the sum create in general an oscillatory Ñ 0, described by Theta functions evaluated at an argument proportional to 1{ . This suggest that in general when Ñ 0, the "fast variables" live in the moduli space of solutions of loop equations, whereas the dependence in the "adiabatic variables" is captured by the loop equations themselves.

An important, open problem, would be to show that the asymptotics of (bi)orthogonal polynomials are given by certain Baker-Akhiezer functions of an integrable system, which depend on the universality class. Around a point where the density of zeroes vanishes like a power p{q, the integrable system should be related to the pp, qq models. This remains beyond the scope of the present investigation.

2 Linear differential systems and loop equations 2.1 Kernel, determinantal formulae and correlators Definition 2.1 The kernel is a d ˆd matrix depending on two variables x 1 , x 2 P p CzP, defined by:

Kpx 1 , x 2 q " Ψ ´1px 1 qΨpx 2 q x 1 ´x2 , (2-1)
It is often referred to as the "parallel transport" operator, because it satisfies Ψpx 2 q " px 1 x2 q Ψpx 1 q Kpx 1 , x 2 q, i.e. it transports Ψpx 1 q to Ψpx 2 q.

It obviously satisfies a replication formula:

Kpx 1 , x 2 qKpx 2 , x 3 q " x 1 ´x3 px 1 ´x2 qpx 2 ´x3 q Kpx 1 , x 3 q, (2-2)
and it has a simple pole at coinciding points:

Kpx 1 , x 2 q " x1Ñx2 1 d x 1 ´x2 . (2-3) Definition 2.2
The n-point correlators are a family of symmetric functions in n variables, indexed by a 1 , . . . , a n P v1, dw, defined as follows:

W 1 p a xq " lim x 1 Ñx ˆKa,a px, x 1 q ´1 x ´x1 ˙, (2-4) @n ě 2, W n p a1 x 1 , . . . , an x n q " p´1q n`1 ÿ σ"n-cycles n ź i"1 K ai,a σpiq px i , x σpiq q, (2-5)
and the "non-connected" n-point correlators by:

W n p a1
x 1 , . . . , an

x n q " "det" K ai,aj px i , x j q, (2-6)

where "det" means that each occurrence of K ai,ai px i , x i q in the determinant should be replaced by

W 1 p ai x i q.
For instance, we have for any a, b P v1, dw, with a ‰ b:

W 1 p a xq " ´ ´1`Ψ ´1pxqLpxqΨpxq ˘a,a , (2-7) W 2 p a1 x 1 , a2
x 2 q " ´Ka1,a2 px 1 , x 2 qK a2,a1 px 2 , x 1 q, (2-8)

lim x1Ñx W 2 p a x 1 , b xq " ´ ´2`Ψ ´1pxqLpxqΨpxq ˘a,b `Ψ´1 pxqLpxqΨpxq ˘b,a . (2-9)
We may give another representation for the correlators, by: Definition 2.3 We define the projectors on state a:

Pp a xq " Ψpxq E a Ψ ´1pxq, (2-10)
where E a " diagp0, . . . , 0, a 1, 0, . . . , 0q denotes the diagonal matrix with a th -entry 1, and zero entries elsewhere. (2-12)

We observe that Pp

Proposition 2.1 The correlators can be written:

W 1 p a xq " ´ ´1Tr Pp a xqLpxq, (2-13) W 2 p a1 x 1 , a2 x 2 q " Tr Pp a1 x 1 qPp a2 x 2 q px 1 ´x2 q 2 " ´Tr pPp a1 x 1 q ´Pp a2 x 2 qq 2 2 px 1 ´x2 q 2 `1 px 1 ´x2 q 2 , (2-14)
and for n ě 3

W n p a1 x 1 , . . . , an x n q " p´1q n`1 ÿ σ"n-cycles Tr Pp a1 x 1 qPp a σp1q
x σp1q qPp a σ 2 p1q

x σ 2 p1q q ¨¨¨Pp a σ n´1 p1q

x σ n´1 p1q q ś n i"1 px i ´xσpiq q .

(2-15) l For instance, we can deduce if a 1 ‰ a 2 :

lim x1Ñx2 W 2 p a1 x 1 , a2 x 2 q " ´ ´2 Tr Pp a1 x 2 qLpx 2 qPp a2
x 2 qLpx 2 q, (2-16)

W 3 p a1 x 1 , a2 x 2 , a3 x 3 q " Tr Pp a1 x 1 qrPp a2 x 2 q, Pp a3 x 3 qs px 1 ´x2 qpx 2 ´x3 qpx 3 ´x1 q . (2-17)
Although it is not clear from the definition, the n-point correlators do not have poles at coinciding points when n ě 3. If I " v1, nw, px i q iPI and pa i q i P v1, dw I , we denote a I

x I the family p

ai x i q iPI .
Proposition 2.2 For any n ě 3, any a 1 , . . . , a n P v1, dw, and

1 ď i ‰ j ď n, W n p a I
x I q is regular when

x i Ñ x j .
Proof. By symmetry, it is enough to consider i " 1 and j " 2. The definition of W k px a I I q implies that it can have at most simple poles when x 1 Ñ x 2 . Let us compute its residue from (2-15):

Res x1Ñx2 W n p a I x I q " p´1q n`1 ! ÿ σ"n´cycle σp1q"2
Tr Pp a1

x 2 qPp a2

x 2 qPp a σp2q

x σp2q q ¨¨¨Pp a σ n´3 p2q

x σ n´3 p2q qPp a σ n´2 p2q

x σ n´2 p2q q px σ n´2 p2q ´x2 qpx 2 ´xσp2q q ¨¨¨px σ n´3 p2q ´xσ n´2 p2q q ´ÿ σ"n´cycle σ ´1p1q"2

Tr Pp a2

x 2 qPp a1

x 2 qPp a σ 2 p2q

x σ 2 p2q q ¨¨¨Pp a σ n´2 p2q

x σ n´2 p2q qPp a σ n´1 p2q

x σ n´1 p2q q px σ n´1 p2q ´x2 qpx 2 ´xσ 2 p2q q ¨¨¨px σ n´2 p2q ´xσ n´1 p2q q

) .

(2-18)

Using the relation Pp a1

x 2 qPp a2

x 2 q " δ a1,a2 Pp a2

x 2 q, we can rewrite:

Res x1Ñx2 W n p a I x I q " p´1q n`1 δ a1,a2 ! ÿ σ"n-cycle σp1q"2
Tr Pp a2

x 2 qPp a σp2q

x σp2q q ¨¨¨Pp a σ n´3 p2q

x σ n´3 p2q qPp a σ n´2 p2q

x σ n´2 p2q q px σ n´2 p2q ´x2 qpx 2 ´xσp2q q ¨¨¨px σ n´3 p2q ´xσ n´2 p2q q ´ÿ σ"n-cycles σ ´1p1q"2

Tr Pp a2

x 2 qPp a σ 2 p2q

x σ 2 p2q q ¨¨¨Pp a σ n´2 p2q

x σ n´2 p2q qPp a σ n´1 p2q

x σ n´1 p2q q px σ n´1 p2q ´x2 qpx 2 ´xσ 2 p2q q ¨¨¨px σ n´2 p2q ´xσ n´1 p2q q

) .

(2-19)

The two sums range over the set of pn ´1q-cycles, and are actually equal. We conclude that W k p a I

x I q is regular when x 1 Ñ x 2 . l

Loop equations

We first show that the correlators satisfy a set of linear equations.

d ÿ a"1 W n p a x, c I y I q " p´1q n`1 ÿ σ"n-cycle 1 px ´yσp1q qpy σ ´1p1q ´xq Tr Pp c σp1q y σp1q q ¨¨¨Pp c σ n´1 p1q y σ n´1 p1q q ś n´2 i"1 py σ i p1q ´yσ i`1 p1q q (2-23)
is a rational function of x, which vanishes in the limit x Ñ 8. Singularities can only arise as simple poles at x " y i for i P I, but their residue is 0 according to Proposition 2.2. Hence, the left hand side vanishes identically. l Theorem 2.2 (Quadratic loop equations) For any n ě 1, any c 2 , . . . , c n P v1, dw,

ÿ 1ďaăbďd ´Wn`1 p a x, b x, c I y I q `ÿ JĎI W |J|`1 p a x, c J y J qW n´|J| p b x, c IzJ y IzJ q ¯" P n px; c I y I q (2-24)
is a rational function of x, with possible poles at x " x i for i P I and poles of L.

As illustration, we give the formulas for P n up to n " 3:

P 1 pxq " 1 2 2 `
P 3 px; c1 y 1 , c2 y 2 q " ´1 Tr rPp c1 y 1 qPp c2 y 2 q `Pp c2 y 2 qPp c1 y 1 qsLpxq px ´y1 qpx ´y2 q `py 1 ´y2 q 2 W 2 p c1 y 1 , c2 y 2 q `1 px ´y1 q 2 px ´y2 q 2 . (2-27)
Proof. Notice that the left hand side makes sense even if n " 1, because the function W We claim:

Lemma 2.1 @n ě 1, @a P v1, dw, Ă W n`1 p a x, a x, c I y I q `ÿ JĎI W |J|`1 p a x, c J y J qW n´|J| p a x, c IzJ y IzJ q " 0.
(2-28)

The proof of the lemma will be given below. We deduce that:

P k px; c I y I q " 1 2 d ÿ a,b"1 Ă W n`1 p a x, b x, c I y I q `ÿ JĎI 1 2 ´d ÿ a"1 W |J|`1 p a x, c J y J q ¯´d ÿ b"1 W n´|J| p b x, c IzJ y IzJ q ¯.
(2-29)

The last term is given by the linear loop equations (Theorem 2.1): it vanishes when n ě 5, and is a rational function of x with poles at x " x i for some i P I, or at poles of L. We now focus on the first term, which is by definition:

Q k px; c I y I q " 1 2 d ÿ a,b"1 Ă W n`1 p a x, b x, c I y I q " p´1q n 2 d ÿ a,b"1 ! (2-30) ´pΨ ´1LΨq a,b pxq ÿ σ"pn`1q-cycle σp1q"2 K c σ ´1p1q ,a py σ ´1p1q , xqK b,c σp2q px, y σp2q q n´2 ź i"1 K c σ i p2q ,c σ i`1 p2q py σ i p2q , y σ i`1 p2q q ´pΨ ´1LΨq b,a pxq ÿ σ"pn`1q-cycle σp2q"1 K a,c σp1q px, y σp1q qK c σ ´1p2q ,b py σ ´1p2q , xq n´2 ź i"1 K c σ i p1q ,c σ i`1 p1q py σ i p1q , y σ i`1 p1q q `ÿ σ"pn`1q-cycle σp1q‰2, σp2q‰1 K a,c σp1q px, y σp1q q ¨¨¨K c σ ´1p2q ,b py σ ´1 p2q , xqK b,c σp2q px, y σp2q q ¨¨¨K c σ ´1p1q ,a py σ ´1 p1q , xq
) .

The two first lines are equal by symmetry. Performing the sum over a and b, and replacing the kernels involving the variable x by their definition, we find:

Q n px; c I y I q (2-31) " p´1q n`1 ! ÿ σ"pn`1q-cycle σp1q"2 ´rΨ ´1px σ ´1p1q qLpxqΨpx σp2q qs c σ ´1p1q ,c σp2q n´3 ź i"1 K c σ i p2q ,c σ i`1 p2q py σ i p2q , y σ i`1 p2q q `ÿ σ"pn`1q-cycle σp1q‰2, σp2q‰1 ź j"1,2
py σ ´1pjq ´yσpjq qK c σ ´1pjq ,c σpjq py σ ´1pjq , y σpjq q 2px ´yσ ´1pjq qpx ´yσpjq q

ˆn´2 ź i"1 σ i`1 p1q‰1,2 K c σ i p1q ,c σ i`1 p1q px σ i p1q , x σ i`1 p1q q ) .
This expression is a rational function of x which can have poles only at x i for i P I, and at poles of L. Therefore, we proved that P n px; c I y I q is a rational function of x which can have poles only at those very points.

Proof of Lemma 2.1. We have the analog of (2-30) for a " b:

Ă W n`1 p a x, a x, c I y I q " p´1q n ! ´2pΨ ´1LΨq a,a pxq (2-32) ˆÿ σ"pn`1q-cycles σp1q"2 K a,c σp2q px, y σp2q q " n´1 ź i"1 K c σ i p2q ,c σ i`1 p2q py σ i p2q , y σ i`1 p2q q ı K c σ n´1 p2q ,a py σ n´1 p2q , xq `ÿ 1ďj,kďn j`k"n ÿ σ"pn`1q-cycles σ j`1 p1q"2 K a,c σp1q px, y σp1q q " j´1 ź i"1 K c σ j p1q ,c σ i`1 p1q py σ i p1q , y σ i`1 p1q q ı K c σ j p1q ,a py σ j p1q , xq ˆKa,c σp2q px, y σp2q q " k´1 ź i"1 K c σ i p2q ,c σ i`1 p2q py σ i p2q , y σ i`1 p2q q ı K c σ k p2q ,a py σ k p2q , xq
) .

We recognize in the first line

´2W 1 p a xqW n p a x, c I y I q.
Besides, the two last lines amounts to a sum over two disjoint cycles of length pj `1q and pk `1q, and we recognize each term correlators up to a sign factor. Namely:

Ă W n`1 p a x, a x, a I x I q " ´2W 1 p a xqW n p a x, c I y I q ´ÿ ∅ĂJĂI W |J|`1 p a x, c J y J qW n´|J| p a x, c IzJ y IzJ q.
(2-33)

The first term completes the sum with the terms J " ∅ and J " I, hence the result. l Detailed example. Let us redo the computation in the case n " 1 to illustrate the method of the proof. We have:

P 1 pxq " 1 2 ÿ 1ďaăbďd ´`Ψ ´1LΨ ˘a,b pxq `Ψ´1 LΨ ˘b,a pxq ``Ψ ´1LΨ ˘a,a `Ψ´1 LΨ ˘b,b pxq. (2-34)
Notice that the summand vanish if a " b. We can thus write:

P 1 pxq " 1 2 2 d ÿ a,b"1 ´`Ψ ´1LΨ ˘a,b pxq `Ψ´1 LΨ ˘b,a pxq ``Ψ ´1LΨ ˘a,a pxq `Ψ´1 LΨ ˘b,b pxq " 1 2 2 `´Tr L 2 pxq `rTr Lpxqs 2 ˘. (2-35) l 2.3 Spectral curve Definition 2.4
The spectral curve is the plane curve S of equation detpy ´Lpxqq " 0.

The eigenvalues of Lpxq are algebraic functions.

Proposition 2.3 The spectral curve can be expressed in terms of correlators:

detpy ´Lpxqq " d ÿ k"0 y d´k ÿ 1ďa1ă...ăa k ďd W k p a1 x , . . . , a k x q.
(2-36)

Proof. We first write the coefficients of a characteristic polynomial as a sum over minors:

detpy ´Lpxqq " detpy ´Ψ´1 pxqLpxqΨpxqq " d ÿ k"0 y d´k ÿ 1ďa1ă...ăa k ďd det 1ďi,jďk r´Ψ ´1LΨs ai,aj pxq " d ÿ k"0 y d´k k ÿ 1ďa1ă...ăa k ďd det 1ďi,jďk r K ai,aj px, xq, (2-37)
where we have defined r

K a,b px, xq " ´ ´1pΨ ´1LΨq a,b pxq. Notice that r K a,b px, xq " K a,b px, xq when a ‰ b, whereas r K a,a px, xq " W 1 p a xq.
And, the specialization of the definition of non-connected correlators (2-6) to x i " x for i P v1, dw and a 1 ă . . . ă a k yields:

W k p a1 x , . . . , a k x q " det 1ďi,jďk r K ai,aj px, xq, (2-38)
whence the announced formula. l

We remark that the coefficients of y d´2 was already identified in Eqn. 2-25.

Gauge transformations

If Ψ is a solution of (1-1), and G is a matrix depending on x, r Ψ " GΨ will also be solution of similar equation, with:

r L " p B x GqG ´1 `GLG ´1.

Any two arbitrary d ˆd matrices Ψpxq and r

Ψpxq can be related by a gauge transformation Gpxq " r ΨpxqΨpxq ´1. Therefore, the concept of gauge transformations is only meaningful if we impose some restriction on the form of Gpxq. Here, the natural restriction to impose is that p B x GqG ´1 is rational, and its poles should occur at poles of L with a lower (or equal) degree than in L.

Gauge transformations in general completely change the kernel and the correlators. However, there are two special gauge transformations under which the correlators do not change. If G is independent of x:

r L " GLG ´1, r P " GPG ´1, r K " K, Ă W n " W n .
(2-39) (where, for bookkeeping, we included the gauge transformation of matrix Γ defined in section 3.6). If G depends on x but is scalar G " G1 d :

r L " L ` B x ln G, r P " P, r Kpx, yq " Gpyq Gpxq Kpx, yq, Ă W n " W n .
(2-40)

Insertion operator

Let pCpxq, B x q be the differential ring generated by rational functions. We consider a Picard-Vessiot ring B of the differential system B x Ψpxq " LpxqΨpxq [START_REF] Van Der Put | Galois theory of linear differential equations[END_REF]. It is is a simple extension of pCpxq, B x q by the matrix elements of Ψpxq and `det Ψpxq ˘´1 . Let B n the n-variable analog of B, i.e. the differential ring generated by rational functions in n variables x 1 , . . . , x n and by the matrix elements of Ψpx i q and `det Ψpx i q ˘´1 . We denote the projective limit B 8 " lim nÑ8 B n . By construction, the matrix elements of Pp a xq or of Lpxq are in B, those of Kpx 1 , x 2 q are in B 2 , and the n-point correlators W n p a1 x 1 , . . . , an x n q are in B n . Definition 2.5 An insertion operator is a collection of derivations pδ a y q 1ďaďd over B 8 , commuting with B xi , with the following properties:

' δ a y pB n q Ď B n`1 . ' δ a y pCpx i qq " 0.
' there exists matrices Up a y q with entries in B, so that:

δ a y Ψpxq " ´Pp a y q x ´y `Up a y q ¯Ψpxq, ( 2
δ a y W n p a1 x 1 , . . . , an x n q " W n`1 p a y , a1
x 1 , . . . , an

x n q.

(2-45)

Proof. Easy computations, done in appendix A. l

The fact that the insertion operator sends W n to W n`1 justifies the name "insertion operator". We remark that equations (2-43) and (2-45) are independent of U.

Remark. Because of relation (2-44), det Ψ is not constant regarding the action of the insertion operator. Notice that in general, up to a scalar gauge transformation, one can always chose det Ψpxq to be a constant. What this means here, is that the insertion operator δ a y doesn't commute with gauge transformations.

Let us define the semi-connected correlators:

W k;n p a1 x 1 , a2 x 2 , . . . , a k x k ; b1 y 1 , . . . , bn y n q " ÿ I$v1,kw ÿ J1 9 Y¨¨¨9 YJ pµq "v1,nw pIq ź j"1 W |Ij |`|Jj | p a I j
x Ij , b J j y Jj q.

(2-46)

Here, I is a partition of v1, kw, i.e. a set of pIq non-empty, pairwise disjoint subsets I i Ď v1, kw whose union is v1, kw, whereas the subsets J i Ă v1, nw could be empty.

y n q (2-47)
is a rational function of x, with poles at x " y j for some j and at poles of L.

Proof. The case n " 0 is Proposition 2.3. The cases n ě 1 are obtained by recursively applying δ bj yj , for any insertion operator δ. l

Asymptotics and topological expansion

Loop equations form an infinite system of equations, in general difficult to solve. In many applications, correlators have an asymptotic expansion (or are formal series) in powers of , and if this expansion is of "topological type" (Definition 3.3 below), loop equations can be solved recursively in powers of , by the topological recursion of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF]. This claim is justified in this section. We assume that Lpxq has an asymptotic expansion in powers of , of the form:

Lpxq " ÿ kě0 k L rks pxq, (3-1)
which is uniform for x in some domain of the complex plane, or alternatively, Lpxq P Crr ss is defined as a formal power series in . Let us denote

Λpxq " diagpλ 1 pxq, . . . , λ d pxqq (3-2) the diagonal matrix of eigenvalues of Lpxq counted with multiplicities and ordered arbitrarily. Λpxq also has an expansion in powers of :

Λpxq " ÿ kě0 k Λ rks pxq.
(3-3)

The semiclassical spectral curve

The semiclassical spectral curve is the locus of leading order eigenvalues:

Definition 3.1 The semiclassical spectral curve is defined as:

S r0s " px, yq P C 2 | detpy 1 d ´Lr0s pxqq " 0 ( . (3-4)
It can be seen as the immersion of a compact Riemann surface S r0s into C ˆC, through the maps x : S r0s Ñ C and y : S r0s Ñ C. If x is of degree d (the degree in y of the algebraic equation defining S r0s , i.e. the size of the matrix L r0s pxq), then the preimage of x 0 P C is denoted:

x ´1ptx 0 uq " tz 0 px 0 q, . . . , z d´1 px 0 qu Ď S r0s .

(3-5)

In other words, S r0s is realized as a branch covering of C of degree d by the projection x : S r0s Ñ C. The zeroes of dx in S r0s are the ramification points, and their x-coordinate are the branchpoints.

Branchpoints β i P C occur when z a pβ i q " z b pβ i q for at least two distinct indices a and b, and we then denote r i " z a pβ i q " z b pβ i q. Let us call r the set of ramification points. λ r0s a pxq are the eigenvalues of L r0s pxq, i.e. by definition they are the y coordinates of points of S r0s , i.e. they are the y image of some z a pxq: ypz a pxqq a P v1, dw ( " λ r0s a pxq a P v1, dw

( . (3-6)
Double points α i P C occur where two or more eigenvalues collide, i.e. ypz a pα i qq " λ r0s a pα i q " λ r0s b pα i q " ypz b pα i qq for at least two distinct indices a ‰ b, but dxpz a pα i qq ‰ 0 and dxpz b pα i qq ‰ 0 -a fortiori, z a pα i q and z b pα i q must be distinct points in S r0s .

The space H 1 pS r0s q of holomorphic 1-forms on S r0s is a complex vector space of dimension g, where g is the genus of S r0s . In particular, if g " 0, H 1 pS r0s q " t0u and a meromorphic form on C is completely determined by the singular behavior at its poles. Definition 3.2 Let BpS r0s q the set of fundamental bidifferentials of the second kind, i.e. Bpz 1 , z 2 q which are symmetric 2-form in pS r0s q 2 , with no residues, and a double pole at z 1 " z 2 with behavior in any local coordinate ξ: Bpz 1 , z 2 q " z1Ñz2 dξpz 1 q dξpz 2 q `ξpz 1 q ´ξpz 2 q ˘2 `Op1q.

(3-7)

Since one can add to B any symmetric bilinear combination of holomorphic forms, BpS r0s q is an affine space, whose underlying vector space is Sym 2 rH 1 pS r0s qs, so it has complex dimension gpg `1q{2.

Expansions in powers of

We now assume that S r0s is a regular plane curve, i.e. dx and dy do not have common zeroes. Therefore, L r0s pxq has simple eigenvalues for any x which is not a branchpoint or double point, hence is diagonalizable. So must be Lpxq at least when is small and x stays away from the branchpoints or double points. We can thus find a matrix of eigenvectors Vpxq:

Lpxq " VpxqΛpxqV ´1pxq, (3-8)
which admits an expansion in powers of :

Vpxq " ÿ kě0 k V rks pxq. (3-9)
Such a matrix is defined up to transformations Vpxq Ñ VpxqDpxqΣ, where Dpxq is a diagonal matrix and Σ a permutation matrix. We can use the first freedom to impose: @a P v1, dw, `V´1 pxq B x Vpxq ˘a,a " 0.

(3-10)

and we then say that Vpxq is a normalized matrix of eigenvectors. Any two such matrices are related by a transformation Vpxq Ñ VpxqDΣ, where D is a constant diagonal matrix and Σ a permutation matrix.

We would like to study solutions of (1-1) which have an expansion in powers of . For this purpose, we fix a base point o, an invertible matrix of constants C, and introduce a matrix p Ψpxq such that:

Ψpxq " Vpxq p Ψpxq exp ´1 ż x o Λpx 1 qdx 1 ¯C. (3-11)
Ψpxq is a solution of (1-1) if and only if:

B x p Ψpxq " ´ Tpxq p Ψpxq `rΛpxq, p Ψpxqs, (3-12)
where Tpxq " Vpxq ´1B x Vpxq also has an expansion in powers of derived from (3-9):

Tpxq " ÿ kě0 k T rks pxq.

(3-13) Proposition 3.1 Eqn. 3-12 has a unique solution which is a formal power series in of the form:

p Ψpxq " 1 d `ÿ kě1 k p Ψ rks pxq (3-14)
up to transformations p Ψ rks pxq Ñ p Ψ rks pxq `p C rks , where p C rks is a diagonal matrix of constants. A priori, the entries of p Ψ rks pxq are multivalued functions of x with monodromies around branchpoints, double points, and poles at the poles of pL rjs pxqq jě0 .

Proof. Inserting the ansatz (3-14) in (3-12) and collecting the terms of order k`1 yields, for any a, b P v1, dw:

B x p Ψ rks a,b " ´k ÿ j"0 pT rk´js p Ψ rjs q a,b `pλ r0s a ´λr0s b q p Ψ rk`1s a,b `k ÿ j"0 pλ rk´js a ´λrk´js b q p Ψ rjs a,b . (3-15)
Since we assume that S r0s is regular and x is away from a branchpoint or a double point, we have λ This equation determines the off-diagonal part of p Ψ rk`1s in terms of p Ψ rjs for j P v0, kw. For a " b in (3-15), we rather find:

B x p Ψ rk`1s a,a " ÿ c"1 c‰a T r0s a,c p Ψ rk`1s c,a `k ÿ j"0
pT rk`1´js p Ψ rjs q a,a .

(3-17)

We took into account the normalization 4 (3-10), so that the right hand side involves only off-diagonal entries of p Ψ rk`1s , or the entries of p Ψ rjs for j P v0, kw. We proceed by recursion starting from the initial condition p Ψ r0s " 1 d . Assuming that p Ψ rjs are completely known for j P v0, kw, we obtain the off-diagonal part of p Ψ rk`1s from (3-16), and solving the first order differential equation (3-17) we then obtain the diagonal part of p Ψ rk`1s up to a diagonal matrix of integration constants p C rk`1s . It is clear that the singularities of p Ψ rks can only occur at singularities of λ rjs a pxq and T rjs pxq, i.e. either at semiclassical branchpoints or poles of pL rjs pxqq jě0 , or at double points where λ Ψpxq, all have a power series expansion in , whose coefficients are such that their a th -column vector is the evaluation of meromorphic function on S r0s at z a pxq. In particular, there exists a vector ψrks pzq such that:

r Ψ i,a pxq " ´Vpxq p Ψpxq ¯i,a " ÿ kě0 k ψrks i pz a pxqq. (3-18)
Proof. For the diagonal matrix Λ, we have already seen in (3-6) that λ r0s a pxq " ypz a pxqq. Solving detpλ a pxq 1 d ´Lpxqq " 0 with Lpxq " ř kě0 k L rks pxq and λ a pxq " ř kě0 k λ rks a pxq, by recursion on k, shows easily that each λ rks a pxq is a meromorphic function λ rks pz a pxqq for all k. Similarly, Kramers formula for computing the eigenvectors of Lpxq, shows that up to a normalization factor, the eigenvector corresponding to the a th -eigenvalue λ a pxq, has also a power series expansion in whose coefficients are meromorphic functions of z a pxq at each order. In Then, notice that any symmetric meromorphic function of pz 1 pxq, . . . , z d pxqq is a meromorphic function of x, and thus a meromorphic function of any z a pxq. And, any symmetric meromorphic function of pz 1 pxq, . . . , z d pxqq â (i.e. all z j pxq's except z a pxq), is a meromorphic function of x and of z a pxq, and thus is a meromorphic function of z a pxq. In particular, this implies that the determinant of p Vpxq is a power series of whose coefficients are meromorphic function of z a pxq, and the inverse matrix p V ´1pxq takes the form: where each trks pzq is a meromorphic function on the semi-classical spectral curve S r0s .

4 Notice that we only need (3-10) at leading order here.

We chose to normalize our basis of eigenvectors Vpxq " p Vpxq Dpxq where Dpxq is some diagonal matrix, so that (3-10) is satisfied, i.e. we have to choose Dpxq satisfying: D ´1 a,a pxq B x D a,a pxq " ´´p V ´1pxq B x p Vpxq ¯a,a " ´ÿ kě0 k trks pz a pxqq.

(3-23)

This shows that D a,a pxq also has a power series expansion in whose coefficients are meromorphic functions of z a pxq. Finally, this shows that Vpxq has the form:

V i,a pxq " ÿ kě0 k v rks i pz a pxqq, (3-24)
where each v rks i pzq is a meromorphic function on the semi-classical spectral curve. If we choose C to be diagonal, we see that:

r Ψpxq " Vpxq p Ψpxq " Ψpxq C ´1 exp ´´1 ż x α Λpx 1 q dx 1 ¯(3-25)
obeys:

B x r
Ψpxq " Lpxq r Ψpxq ´r Ψpxq Λpxq.

(3-26)

The equation for the a th -column of r Ψpxq involves only Λ a,a pxq, and thus is order by order in analytical in z a pxq, and since we know that r Ψpxq has only meromorphic singularities, we see again that the column vectors of r

Ψpxq have an expansion such that the coefficients are meromorphic functions of z a pxq. l Corollary 3.1 The coefficients ψrks i pzq appearing in the expansion of r Ψ i,a pxq, are meromorphic functions of z P S r0s whose poles occur only at values of z such that Da ‰ b and x P C with z " z a pxq " z b pxq, or at poles of L rls pxq for l ď k. In other words, φ rks i pzq can be singular only at ramification points, at preimages in S r0s of double points, or at poles of L rls on the semi-classical spectral curve S r0s .

Proof. r

Ψpxq was constructed so that it has at most meromorphic singularities at poles of Lpxq. Then, one can see in (3-16) that singularities can occur only when λ r0s a pxq " λ r0s b pxq for some a ‰ b, i.e. at branchpoints or double points. l

Expansion of the correlators

In this section, we consider the projectors, the correlators, etc. (see Section 2.1) associated to the solution Ψpxq deduced from Proposition 3.1 via (3-11). and there exists a sequence of matrices p rks pzq of meromorphic functions in z P S r0s , with poles at ramification points, at preimages in S r0s of double points, and at poles of pL rjs pxqq jě0 , such that p rks pz a pxqq " P rks p a xq.

Proof. Since we assume C to be diagonal, the exponentials -which might have prevented the existence of an expansion in powers of -disappear:

Pp a xq " Vpxq p Ψpxq exp ´1 ż x o Λpx 1 qdx 1 ¯CE a C ´1 exp ´´1 ż x 0 Λpx 1 qdx 1 ¯p Ψ ´1pxqV ´1pxq " Vpxq p ΨpxqE a p Ψ ´1pxqV ´1pxq
" r Ψpxq E a r Ψ ´1pxq.

(3-28)

From Proposition 3.2, r Ψpxq has an expansion in , so Pp a xq has an expansion in . Moreover, r Ψpxq E a r Ψ ´1pxq involves only the a th column of r Ψpxq and the a th line of r Ψ ´1pxq, i.e. the coefficients of the expansion are meromorphic functions of z a pxq. From Corollary 3.1, those meromorphic functions can be singular only at ramification points, at preimages in S r0s of double points, or at poles of Lpxq in S r0s . l

Notice that to leading order, r Ψpxq " 1 d `Op q and:

P r0s p a xq " pV r0s pxqq ´1E a V r0s pxq (3-29)
is the projection on the a-th eigenspace of L r0s pxq. From the expression of the correlators in terms of the projectors, we deduce:

Corollary 3.2 For any a P v1, dw, W 1 p a xq has an expansion in powers of , of the form:

W 1 p a xq " ÿ kě´1 k W rks 1 p a xq, (3-30)
and there exist meromorphic functions w rks 1 pzq in z P S r0s , with poles at the ramification points, or at preimages in S r0s of double points, or at poles of pL rjs pxqq jě0 , so that w Corollary 3.3 For any n ě 2, any a 1 , . . . , a n P v1, dw, the correlators have an expansion in powers of :

W n p a1 x 1 , . . . , an x n q " ÿ kě0 k W rks n p a1 x 1 , . . . , an x n q (3-32)
and there exist symmetric meromorphic functions w rks n pz 1 , . . . , z n q in pz 1 , . . . , z n q P pS r0s q n , with poles when z i is at a ramification point or at a double pole or at a pole of pL rjs pxqq jě0 , and so that w rks n pz a1 px 1 q, . . . , z an px n qq " W rks n p a1

x 1 , . . . , an

x n q.

On top of that, w r0s 2 pz 1 , z 2 q has a double pole at z 1 " z 2 , and behaves as:

w r0s 2 pz 1 , z 2 q " z1Ñz2
x 1 pz 1 qx 1 pz 2 q `xpz 1 q ´xpz 2 q ˘2 `Op1q.

(3-33)

Expansion in with poles assumptions

Many interesting systems have the property that their leading asymptotic behavior at the poles of Lpxq is governed by the Ñ 0 limit, i.e. in some sense that L rjs pxq for j ą 0 is somewhat "smaller" than L r0s pxq. When this holds, only W where the coefficients, for j ą 0, are such that:

D r0s pxq Q rjs px, yq " ÿ pm,nqPinteriorpN q Qrjs m,n´1 x m y n´1 , (3-36)
where D r0s pxq is the common denominator of all coefficients of Q r0s px, yq, N is the envelope of the Newton's polytope of D r0s pxq Q r0s px, yq. Corollary 3.5 ω p0q 2 pz 1 , z 2 q " w r0s 2 pz 1 , z 2 qdxpz 1 qdxpz 2 q defines an element of BpS r0s q (see Definition 3.2).

For instance, we have from Proposition 2.1 and (3-29):

W r0s 2 p a1 x 1 , a2
x 2 q " rpV r0s q ´1px 1 qV r0s px 2 qs a1,a2 rpV r0s q ´1px 2 qV r0s px 1 qs a2,a1 px 1 ´x2 q 2 . (3-37)

Expansion of topological type and topological recursion

Definition 3.3 (TT property) We say that the correlators have an expansion of topological type (or have the TT property) when they have:

' the Ø ´ symmetry: pW n q ´ " p´1q n pW n q .

' the n´2 property: for any n ě 2, W n P Op n´2 q. When these two properties are satisfied, the expansion of the correlators looks like:

@n ě 1, W n " ÿ gě0 2g´2`n W pgq n .
(3-38)

• the pole property: when pg, nq ‰ p0, 1q, p0, 2q, the ω pgq n have poles only at the ramification points. In particular they must have no pole at the preimages in S r0s of double points, or at the poles of L rks pxq. And ω p0q 2 pz 1 , z 2 q has a double pole at z 1 " z 2 , and no other pole.

In the Section 4, we shall study some sufficient conditions (related to integrable systems) to have the TT property, and in Section 5, we shall show that q-th reductions of the KP hierarchy, have the TT property. We believe that the TT property is closely related to integrability, but we do not have a proof of such a statement. Let us just mention that the n´2 property is a highly non-trivial one. For example large random matrices, it is related to the "concentration" property [START_REF]Asymptotic expansion of β matrix models in the one-cut regime[END_REF].

When the TT property is satisfied, one can plug the expansion (3-38) into the loop equations to obtain a set of equations satisfied by W pgq n . The key point is that those equations can be solved recursively on 2g ´2`n. The prototype of such a result is known since [ACM92, ACKM93, ACKM95]. The solution is given by the topological recursion developed in [START_REF]Topological recursion in random matrices and enumerative geometry[END_REF]. The topological recursion associates to a plane curve pS r0s , x, yq (algebraic in our case) and ω p0q 2 P BpS r0s q, a sequence of symmetric meromorphic n-forms ω pgq n on pS r0s q n , defined by a recursion on 2g ´2 `n in terms of the geometry of the curve S r0s . It was first presented under the assumption that ramification points are simple [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF], and extended to arbitrary ramification points in [BHL `13]. Then, it was shown [START_REF] Bouchard | Think globally, compute locally[END_REF] that the general formula of [BHL `13] is a limiting case of the formula of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] for simple ramification points. For instance, the semiclassical spectral curve of r-KdV has one ramification point of order r. For readability, we present now the case of simple ramification points, and refer to [START_REF] Bouchard | Think globally, compute locally[END_REF] for the case of arbitrary ramifications.

Theorem 3.1 If the correlators have an expansion of topological type, and dx has only simple zeroes on the semiclassical spectral curve S r0s : detpy 1 d ´Lr0s pxqq " 0, then the coefficients of (3-38) are given by: W pgq n p a1

x 1 , . . . , an

x n qdx 1 ¨¨¨dx n " ω pgq n pz a1 px 1 q, . . . , z an px n qq (3-39)

and ω pgq n satisfy:

ω pgq n pz 1 , z 2 , . . . , z n q (3-40) " ÿ rPr Res zÑr K r pz 1 , zq " ω pg´1q n`1 pz, σ r pzq, z 2 , . . . , z n q `1 ÿ h`h 1 "g I 9 YI 1 "v2,nw ω phq 1`|I| pz, z I q ω ph 1 q 1`|I 1 | pσ r pzq, z I 1 q ı `Hpgq n pz 1 , . . . , z n q, (3-41)
where H g n pz 1 , . . . , z n q is some symmetric holomorphic n-form on pS r0s q n , ř 1 means that we exclude ph, Iq " p0, Hq and ph 1 , I 1 q " p0, Hq, r are the ramification points (i.e. the zeroes of dx), σ r is the local Galois involution near the ramification point r, i.e. the holomorphic map defined in the vicinity of r, such that x ˝σr " x and σ r ‰ id. And, the recursion kernel is:

K r pz 1 , zq " 1 2 ş z σrpzq ω p0q 2 pz 1 , ¨q ω p0q 1 pzq ´ωp0q 1 pσ r pzqq (3-42)
where ω p0q 1 " ´ydx on S r0s .

Corollary 3.6 If furthermore S r0s has genus 0, H pgq n " 0 (since there are no holomorphic 1-forms on S r0s ) and ω pgq n are exactly given by the topological recursion of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] applied to the initial data ω Proof. The proof is essentially done in [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF][START_REF] Borot | Abstract loop equations, topological recursion, and applications[END_REF]. To be self-contained, we redo it in Appendix B. l

Symmetry Ø ´

Here we give a sufficient condition for the existence of an Ø ´ symmetry. We do not know whether this criterion is also a necessary condition.

Proposition 3.3 Assume there exists an invertible matrix Γ, independent of x, such that:

ΓL T pxqΓ ´1 " L ´ pxq. (3-43)
Then, if Ψ `is a solution of (1-1), Ψ ´" ΓpΨ ´1 `qT is a solution of (1-1) with Ñ ´ . The projector associated to the two solutions are related by P `" ΓP T ´Γ´1 , and the correlators by pW n q `" p´1q n pW n q ´for any n ě 1.

Proof. The relation between the projectors is an easy computation, and given Proposition 2.1 for the n-point correlators, we deduce pW n q `" p´1q n pW n q ´for any n ě 2. For n " 1, we check it directly:

pW 1 q ´pa xq " rΨ ´1 ´pxqL ´ pxqΨ ´pxqs a,a " Tr Ψ ´1 ´pxqL ´ pxqΨ ´pxqE a " Tr P ´pa xqL ´ pxq " Tr ΓP T `pa xqΓ ´1L pxq " Tr P T `pa xqL T pxq " Tr Pp a xqL pxq " ´pW 1 q `pa xq. (3-44) l 4

Case of isomonodromic integrable systems

We believe that integrable systems is the good setting to have the TT property satisfied. We give some arguments here, and then show in section 5 that the special case of q-th reduction of KP fits in our framework.

Behavior at the poles and isomonodromic times

In this paragraph, we review classical results from the theory of linear differential systems. A d ˆd invertible matrix Ψpxq solution to B x Ψpxq " LpxqΨpxq can have singularities only at poles of Lpxq. For any p P P, it can be put locally around x " p in the form5 :

Ψpxq " r Ψ p pxq exp ´Bp lnpx ´pq `Ap pxq ¯Cp , A p pxq " mp ÿ k"1 A p;k px ´pq k , r Ψ p pxq " xÑp 1 d , (4-1)
where A p pxq and B p are Jordanized matrices. Such asymptotics can only be valid in an angular sector near x " p, and the constant matrix C p depends on the sector. B p describes the monodromy around p of the right hand side of (4-1).

Imagine that Lpxq depends smoothly on parameters t " pt α q α , generically called "times". One can always define a matrix M α pxq " B tα Ψpxq Ψpxq ´1, so that Ψpxq satisfy on top of (1-1) the compatible systems:

@α, B tα Ψpxq " M α pxqΨpxq. (4-2)
Requiring that M α pxq be rational is equivalent to requiring that the local monodromies do not depend on t. If B tα B p " 0 for any p P P, we say that t α is an isomonodromic time. Integrable systems in Lax form provide examples of such rational compatible differential systems. A second realization of this setting in the realm of formal series in t can be achieved by deformation of any given Lpxq and solution Ψpxq (independent of parameters) [BBT02, Chapter 5]. The latter might not be the restriction of an integrable system in Lax form (for Ψpx, tq might not be defined as a function of t). Our formalism applies equally well to the two cases.

Isomonodromic Tau function

In this section, we assume that Lpxq depends on a family of isomonodromic times t " pt α q α . If there is more than one time, we first need a remark. Let us define:

Υ α p
B tα ln T p tq " Υ α p tq. (4-5)
It is defined up to a constant independent of t.

Tau functions play an important role in the theory of integrable systems and its applications, and they have been extensively studied, we refer to [START_REF] Babelon | Introduction to classical integrable systems[END_REF] and references therein.

Case of an integrable system: expansion of the Tau function

If L depends on isomonodromic times t, an isomonodromic Tau function T p tq has been defined in Definition (4.1). It is a consequence of Corollary 3.2 and the formula (4-3) for the isomonodromic Tau function that: where: The existence of an insertion operator compatible with all times is not something obvious, but if it exists it is quite useful. For the q-th reduction of KP, we construct in § 5.7.3 a compatible differential operator, which enables to prove the Op n´2 q axiom of the TT property.

B tα F rks p

Application to finite reductions of KP

In this section, we show an important application of the former formalism, namely to the q-th reductions of the KP hierarchy. They are related to the Drinfeld-Sokolov hierarchies [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF], and they contain as a more special case the pp, qq models exemplified in Section 6. They appear in one of the formulation of 2d quantum gravity [START_REF] Douglas | Strings in less than one dimension[END_REF], and conjecturally describe the algebraic critical points which can arise in hermitian multi-matrix models. In physics, the pp, qq models are expected to describe thermodynamic observables in the coupling of Liouville theory to the pp, qq minimal models of conformal field theory [START_REF] Di Francesco | 2d gravity and random matrices[END_REF], the latter corresponding to the classification of finite representations of the conformal Virasoro symmetry of central charge c " 1 ´6pp ´qq 2 {pq [START_REF] Di Francesco | Conformal field theory[END_REF]. The q-th reduction of KP is related to perturbations of this coupled theory by primary operators.

5.1 Pseudo-differential approach to the q-th reduction of KP Let t be a 1-dimensional variable, and C 8 denote an algebra of smooth functions of t. Let D " C 8 r B t , ´1B ´1 t s be the graded algebra of pseudodifferential operators. Let D `" C 8 pRqr B t s its subalgebra of differential operators, graded by the degree. We say that D P D is monic of degree r ě 0 if

D " r B r t `r´1 ÿ k"´8 a k ptqp B t q k .
We then recall that there exists a unique pseudodifferential operator denoted D 1{r , which is monic of degree 1 and satisfies pD 1{r q r " D. We denote D `, the projection of any D P D to D `.

The string equation is a relationship rP, Qs "

(5-1) between differential operators P and Q. It can be written as the compatibility condition of two differential equation for a function ψpx, tq:

xψpx, tq " Qψpx, tq,

´ B x ψpx, tq " P ψpx, tq.

(5-2)

We call (5-2) the associated linear system.

Proof. By multilinearity, we can differentiate the minors ∆ i,j line by line:

B t ∆ i,j " det ¨ψm . . . p B t q i´2 ψ m p B t q i ψ m p B t q i`1 ψ m . . . p B t q q´1 ψ m ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' m‰j `det ¨ψm . . . p B t q i´1 ψ m p B t q i`1 ψ m . . . p B t q q´2 ψ m p B t q q ψ m ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' m‰j `q´2 ÿ k"0 k‰i det ¨ψm . . . p B t q k´2 ψ m p B t q k ψ m p B t q k ψ m . . . . . . p B t q q´1 ψ m ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' m‰j .
(5-70)

The non-zero contributions arise only from the terms where:

' the i-th line is differentiated. We then recognize the definition of D i´1,j px, tq.

' the pq ´1q-th line is differentiated. Since ψ 1 , . . . , ψ q are solutions of (5-2), we can replace p B t q q ψ m by a xψ m ´řq´2 l"0 u l ptq p B t q k . By subtraction of the other lines, we may keep in the latter only the term involving a derivative of order i-th, which was absent from the minor. We thus recreate a minor D q´1,j px, tq, with a prefactor pxδ i,0 ´ui ptqq, and up to a sign p´1q q´i taking into account the ordering of the lines.

We therefore arrive to (5-68), and (5-69) follows by recursion. In particular, we obtain at the last step of the recursion (k " q `1): 0 " ∆ ´1,j px, tq " ´q`1 ÿ l"1 p´1q l`1 p B t q k´l u q´l`1 ptq `p´1q q x ¯∆q´1,j px, tq " p´1q q px ´Q: q∆ q´1,j px, tq.

(5-71) Accordingly, r φ j px, tq " ∆ q´1,j px, tq provides a solution of (5-62) for any j P v1, qw. To show that p r φ j q j is a basis, we define the matrix Φpx, tq " rp Bq i´1 r φ j s 1ďi,jďq and compute its determinant. Thanks to (5-68), we may write:

det r Φ " det ¨∆q´1,m B t ∆ q´1,m . . . p B t q q´1 ∆ q´1,m ‹ ‹ ‹ ' 1ďmďj " det ¨∆q´1,m ∆ q´2,m `pu q´1 ptq ´xδ q,1 q∆ q´1,m . . . p B t q q´1 ∆ q´1,m ‹ ‹ ‹ ' , (5-72)
and upon subtracting the first line in the second line, we can replace the second line by r∆ q´2,m s 1ďmďq . We find recursively that the i-th line can be replaced by ∆ q´i,m , and thus: det r Φ " detr∆ q´k,j s 1ďj,kďq " pdet Ψq q det " p´1q j´1 ∆ k´1,j det Ψ ı 1ďj,kďq " pdet Ψq q´1 .

(5-73) So, p r φ j q j is a basis of solutions of (5-62) if and only if pψ j q j is a basis of solutions of (5-2) l

In order to obtain Theorem 5.2, we exploit the freedom to choose a normalization of φ j px, tq depending on x. As we shall see, an appropriate choice is: φ j px, tq " p´1q j r φ j px, tq det Ψpxq " p´1q j ∆ q´1,j px, tq ∆pxq " p´1q q´1 Ψ ´1 j,q´1 px, tq,

(5-74)

and we define the matrix:

Φpx, tq " ¨φ1 px, tq ¨¨¨φ q px, tq ´ B t φ 1 px, tq ¨¨¨´ B t φ q px, tq . . . . . . p´ B t q q φ 1 px, tq ¨¨¨p´ B t q q φ q px, tq ‹ ‹ ‹ '

.

(5-75)

It remains to show that:

C i,j px, tq " q ÿ k"1
rp B t q i´1 φ k px, tqs rp B t q j´1 ψ k px, tqs, i, j P v1, qw (5-76)

does not depend on x. For this purpose, we first observe:

@j P v1, qw, C 1,j " q ÿ k"1 p´1q q Ψ ´1 k,q´1 Ψ i´1,k " p´1q q´1 δ i,q .
(5-77)

Besides, from the very structure of (5-76), we observe: @i, j P v1, q ´1w, B t C i,j " C i,j`1 ´Ci`1,j , (5-78) and when j " q, we use the fact that ψ j is solution to the system (5-2) to write:

@i P v1, q ´1w, B t C i,q´1 " ´Ci`1,q´1 ´q´2 ÿ l"0 pu l ptq ´δl,0 xqC i,l`1 .
(5-79)

Considering (5-77) as an initial condition for (5-79), we obtain by recursion that C i,j " 0 whenever i `j ď q. Hence, ř q´2 l"0 δ l,0 C i,l`1 always vanish. This implies that the recursion relation (5-79) does not depend on x. Since C i,j is determined uniquely from (5-79) with the constant initial condition (5-77), we conclude that C does not depend on x, which completes the proof of Theorem 5.2.

The n´2 property

We are going to construct a suitable insertion operator allowing to prove the n´2 property.

A useful decomposition

The very special form (5-12) of the matrix Mpx, tq in pp, qq models allows a decomposition: and C depends on , is Op1q, and is expressible in terms of matrix elements of Ppx, tq and their time derivatives.

Proof. The projectors P satisfy the evolution equation:

B t Pp a
x, tq " rMpx, tq, Pp a x, tqs.

(5-83)

We have:

M l,m px, tq " δ m,l`1 `δl,q `x δ m,1 ´um´1 ptq x n q P Op n´2 q.

(5-108)

Proof. If δ a y commutes with B t , we also have for any k ě 0: (5-109) Since δ a y itself is expressible in terms of elements of the matrices P and their time derivatives, we can apply repeatedly (5-109) to show that each application of the insertion operator to Pp a xq increases at least by one the order in . Now, starting from the expression (2-14) of W 2 and by successive applications of the insertion operator to compute W n according to (2-45), we obtain that W n P Op n´2 q. where we used the convention u q ptq " ´1, define the unique insertion operator which commutes with B t .

Proof. (5-113)

6.1 pp, qq " p3, 2q: pure gravity

Here we chose q " 2 and p " 3 Q " p B t q 2 ´2u, P " p B t q 3 ´3u B t ´3 2 9 u `v.

(6-1)

The string equation rP, Qs " implies that 9 v " 0 and the Painlevé I equation for uptq:

´1 2 2 : u `3u 2 " t, v " t 1 .

(6-2)

It has the expansion:

u " c t 3 ´ 2 48 t ´2
´49 4 2 9 3 3{2 t ´9{2 ´52 7 2 6 2 11 3 2 t ´7 `Op 8 q.

(6-3)

The Lax pair is given by Mpx, tq " ˆ0 1 x `2u 0 ˙, (6-4) and Lpx, tq " ˆ1 2 9 uptq ´t1 x ´u px ´uqpx `2uq `1 2 2 : u ´1 2 9 u ´t1 ˙.

(6-5)

The spectral curve is: detpy 1 2 ´Lpx, tqq " py `t1 q 2 ´px `2uqpx ´uq 2 ´1 2 2 : u px ´uq ´1 4 2 9 u 2 . (6-6)

To leading order in , the eigenvalues of L r0s px, tq are thus:

y " ´t1 ˘px ´ur0s q a x `2u r0s , (6-7)

and they are parametrized by: " Xpzq " z 2 ´2u r0s Y pzq " z 3 ´3u r0s z ´t1 with u r0s " c t 3 . (6-8)

Notice that with ζ " pu r0s q ´1{2 z, we recover the Chebyshev polynomials:

" Xpzq " u r0s pζ 2 ´2q " u r0s T 2 pζq Y pzq " pu r0s q 3{2 pζ 3 ´3ζq ´t1 " pu r0s q 3{2 T 3 pζq ´t1 . (6-9)

Applying the topological recursion gives the coefficients of expansion of the correlators: ω p0q 1 pzq " ´Y pzqdXpzq " ´2 pz 4 ´3u r0s z 2 ´t1 zq dz, ω p0q 2 pz 1 , z 2 q " dz 1 dz 2 pz 1 ´z2 q 2 , ω p0q 3 pz 1 , z 2 , z 3 q " ´1 6u r0s Then, we compute δ a y Kpx 1 , x 2 q " 1 x 1 ´x2 δ b y rΨ ´1px 1 qΨpx 2 qs " 1 x 1 ´x2 ´Ψ´1 px 1 qPp a y qΨpx 2 q y ´x1 `Ψ´1 px 1 qPp a y qΨpx 2 q x 2 ´y `Ψ´1 px 1 qUp a y qΨpx 2 q ´Ψ´1 px 1 qUp a y qΨpx 2 q " ´Ψ´1 px 1 qΨpyq

x 1 ´y E a Ψ ´1pyqΨpx 2 q y ´x2 " ´Kpx 1 , yqE a Kpy, x 2 q. (A.3) notice that U disappears in this computation. Similarly,

a

  xq form a basis of rank one projectors:

byy

  xq is well-defined when a ‰ b, and given by (2-9). When a ‰ b, W n p I q can be computed from Definition 2.2, using K a,b px, xq " ´ ´1pΨ ´1LΨq a,b pxq. We introduce a new quantity Ă I q, as follows:' when a " b, it is computed from Definition 2.2 where each occurrence of K a,a px, xq is replaced by ´ ´1pΨ ´1LΨq a,b pxq (which is also equal to W 1 p a xq),' when a ‰ b, it is equal to W n p

r0s a pxq ‰ λ r0s b

 r0s pxq when a ‰ b, which allows to write:

  Proposition 3.2 (Analytic continuation) The matrices Vpxq, Λpxq and r Ψpxq " Vpxq p

  ´p V ´1pxq B x p Vpxq ¯a,a " ÿ kě0 k trks pz a pxqq, (3-22)

Lemma 3. 1

 1 Assume that the constant matrix C in (3-11) is diagonal. Then, the projectors have an expansion in powers of , of the form:

a

  xq can have poles at the poles of Lpxq, all other W rgs n have no poles at the poles of Lpxq. Let us make it precise. Assumption 3.1 Let us assume that Lpxq " ř jě0 j L rjs pxq has the property that for any j ą 0 the poles of L rjs pxq are a subset of the poles of L r0s pxq, and the expansion of its eigenvalues λ a pxq " is such that, for any j ą 0, λ rjs a pxq Ñ 0 when x approaches a pole of Lpxq. Equivalently, this means that the characteristic polynomial of Lpxq satisfies Qpx, yq " det `y 1 d ´Lpxq ˘" ÿ jě0 j Q rjs px, yq, (3-35)

Corollary 3. 4

 4 When assumption 3.1 is satisfied, only W r0s 1 p a xq can have poles at the poles of Lpxq, all other W rks n are regular at the poles of Lpxq.

Corollary 4. 1

 1 If A p " ´1 A r0s p is diagonal for any pole p, we have an expansion of the form: ln T p tq " ÿ kě´2 k F rks p tq, (4-6)

  If δ a y is a compatible insertion operator such that Up a y , tq " Bp a y , tq ` Vp a y q and V depends on , is of order 1 and is expressible in terms of matrix elements of Pp a xq and their time derivatives, then: δ a1 y1 ¨¨¨δ a k y k Pp a xq P Op k q, (5-107) and: W n p a1 x 1 , . . . , an

  l5.7.3 Existence of a compatible insertion operatorIt is possible to construct explicitly an insertion operator which commutes with B t : Theorem 5.3 The choices: B t q l P k´m´l,q p a x, tq " B k,m p a x, tq `Op q,(5-110) δ a y u k ptq " P 1,k p a y , tq ´δk,1 P q,q p a

  ΨpxqqE b Ψ ´1pxq `ΨpxqE b pδ a y Ψ ´1pxqq Lpxq " pδ a y ` B x Ψpxq Ψ ´1pxq " B x `δa y Ψpxq ˘Ψ´1 pxq ´ B x Ψpxq δ a y `Ψ´1 pxq " a y q ¸Ψpxq ¸Ψ´1 pxq ´Lpxq δ a y `Ψ´1 pxq

  Theorem 2.1 (Linear loop equation) For any n ě 1, any c 2 , . . . , c n P v1, dw, we have:

	Proof. We first address the cases n " 1, 2 by direct computation starting from (2-13)-(2-14), and use
	the properties (2-11) of the projectors:
	d ÿ	W 1 p	a xq " ´ ´1 Tr	´d ÿ	Pp	a xq ¯Lpxq " ´ ´1 Tr Lpxq,	(2-21)
	a"1									a"1
	d ÿ a"1	W 2 p	a x,	c y q "	Tr	`řd a"1 Pp px ´yq 2 a xq ˘Pp y q c	"	Tr Pp y q c px ´yq 2 "	1 px ´yq 2 .	(2-22)
	For n ě 3, combining the representation (2-15) and the fact that	ř d a"1 Pp xq " 1 d , we find that: a
		d ÿ a"1	W n p x, a	c2 y 2 , . . . ,	cn y n q " ´δn,1 ´1 Tr Lpxq	`δn,2 px ´y2 q 2 .	(2-20)

  If δ a y is an insertion operator, for any n ě 1, any a, b, a 1 , . . . , a n P v1, dw, δ a y Kpx 1 , x 2 q " ´Kpx 1 , yqE a Kpy, x 2 q,

	Lemma 2.2 (2-43)
	δ a y Pp xq " b	" Pp y q a x ´y `Up	a y q, Pp xq b	ı ,
	δ a y Lpxq "	" Pp y q a x ´y `Up	a y q, Lpxq	ı	´Pp px ´yq 2 , a y q
	δ a y Tr Lpxq "	´1 px ´yq 2 ,
	δ a y ln det Ψpxq "	x	1 ´y `Tr U p	a y q,			δ a y ln	ˆdet Ψpxq det Ψpzq	˙" 1 x ´y	´1 z ´y ,	(2-44)
										-41)
	and such that U satisfies
			δ a x Up y q ´δb b y Up xq " rUp a	a xq, Up	b y qs.	(2-42)

  other words, one can chose a matrix p

							Vpxq of
	eigenvectors of Lpxq satisfying
				Lpxq " p VpxqΛpxq p V ´1pxq		(3-19)
	of the form
	p Vpxq "	ÿ	k	p V rks pxq	, p V rks pxq i,a "	vrks i pz a pxqq.	(3-20)
		kě0

  Ψpxqq e ´Appxq pB tα e Appxq q ‰ Proof. The definition of Υ α and this result is due to Jimbo, Miwa and Ueno for integrable systems in Lax form and diagonal A p,k (see also[START_REF] Babelon | Introduction to classical integrable systems[END_REF]). It was generalized to non-diagonal A p,k in[START_REF] Bertola | The partition function of the two-matrix model as an isomonodromic Tau function[END_REF]. The proof is essentially the same. l Definition 4.1 We define the isomonodromic Tau function as a function T p tq (or as a power series in t), such that:

	tq " Ψ ´1pxqpB x " ´ÿ pPP Res xÑp dx Tr " ´ÿ pPP Res xÑp dx Tr " r Ψ ´1 p pxqpB x r Ψ p pxqq e ´Appxq pB tα e Appxq q ‰
	"	´ÿ pPP	Res xÑp	a"1 d ÿ	"	W 1 p	a xq `e´Appxq B tα e Appxq ˘a,a	‰	.	(4-3)
	Lemma 4.1
			@α, β,		B t β Υ α p tq " B tα Υ β p tq.			(4-4)

  Proof. Indeed, when there is an expansion of topological type, we have W Compatibility of the insertion operator with isomonodromic deformationsThe definition of Picard-Vessiot rings is easily generalized to a family of compatible differential systems B x Ψpxq " LpxqΨpxq and B tα Ψpxq " M α pxqΨpxq. We amend Definition 2.5 of insertion operators:Definition 4.2 We say that an insertion operator δ is compatible if it commutes with all B tα , i.e. if it satisfies:

										r2g´1s 1	p	a xq dx " ω	pgq 1 pz a pxqq.
	l
	4.4 B tα Up y q " δ a a y M α pxq `rM α pxq, Up y qs a	`" M α pxq ´Mα pyq x ´y	, Pp	 y q a	.	(4-11)
	tq "	´ÿ pPP	Res xÑp	a"1 d ÿ	"	dx W 1 rk`1s	p xq B tα pA r0s a p pxqq a,a	‰ .	(4-7)
										l
	Corollary 4.2 In particular, if the TT property holds, then only even powers of appear:
		ln T p tq "	ÿ	2g´2 F pgq p tq,	(4-8)
						gě0
	where
	B tα F pgq p tq "	´ÿ pPP		Res zÑp	"	ω 1 pzq f α pzq pgq	‰ ,	(4-9)
	with	df α pzq dxpzq	" B tα ypzq ˇˇxpzq .	(4-10)

  If δ ay is an insertion operator, we now prove the following formulae. For any n ě 1, any a, b, a 1 , . . . , a n P v1, dw, δ a y Kpx 1 , x 2 q " ´Kpx 1 , yqE a Kpy, x 2 q, Proof of Lemma 2.2. First we have by the Leibniz rule δ b y pΨ ´1pxqΨpxqq " 0, which leads to:

	A Proof of Lemma 2.2
		δ a y Pp	b xq "	" Pp y q a x ´y `Up y q, Pp a	ı xq b ,
		δ a y Lpxq "	" Pp y q a x ´y `Up y q, Lpxq a ı	´Pp y q a px ´yq 2 ,
		δ a y Tr Lpxq "	´1 px ´yq 2 ,
	δ a y ln det Ψpxq "	x	1 ´y `Tr U p y q, a
	δ a y ln	ˆdet Ψpxq det Ψpzq	˙"	x	1 ´y	´1 z ´y ,
	δ a y W n p
	δ a y Ψ ´1pxq " ´Ψ´1 pxqpδ a y ΨpxqqΨ ´1pxq "	Ψ ´1pxqPp y ´x	a y q	´Ψ´1 pxqUp
					dz 1 dz 2 dz 3 z 2 1 z 2 3 2 z 2	,
							˜1	¸,
	ω 4 pz 1 , . . . , z 4 q " p0q	1 36 pu r0s q 3	dz 1 dz 2 dz 3 dz 4 z 2 1 z 2 2 z 2 3 z 2 4	`4 ÿ i"1	3u r0s z 2 i

a1

x 1 , . . . ,

an x n q " W n`1 p a y , a1

x 1 , . . . , an x n q.

(A.1) a y q. (A.2)

When p " 8, the factors px ´pq should be replaced by 1{x.

The choice u q´1 " v p´1 " 0 can always be achieved by a redefinition of the variable t. And then u q´2 {q " v p´2 {p follows from the string equation, and we denote u " ´uq´2 {q " ´vp´2 {p.

This work benefited from the support of the Fonds Européen S16905 (UE7-CONFRA), the Swiss NSF, a Forschungsstipendium of the Max-Planck-Gesellschaft, and the Simons foundation. B.E. thanks the CRM, the FQRNT grant from the Québec government, Piotr Su lkowski and the ERC starting grant Fields-Knots.

Let pp, qq be a couple of positive integers distinct from p1, 1q. The pp, qq model is a hierarchy of 1-dimensional nonlinear differential equations for a sequence of functions uptq, u k ptq for k P v1, p ´3w, and v l ptq for l P v1, q ´3w, ensuing by looking 6 for a solution of a string equation of the form:

Q " q ÿ l"0 u l ptq p B t q l , u q " 1, u q´1 " 0, u q´2 " ´qu.

(5-4)

We thus have: " P " p B t q p ´puptq p B t q p´2 `řp´3 l"0 v l ptq p B t q l Q " p B t q q ´quptq p B t q q´2 `řq´3

k"0 u k ptq p B t q l .

(5-5)

When P and Q assume the form (5-3), it is well-known that:

Theorem 5.1 [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF][START_REF] Di Francesco | 2d gravity and random matrices[END_REF] The most general solution of (5-1) is of the form:

(5-6)

for some constants t l and r t k (with t p " 1 and r t q " 1).

For coprime pp, qq, the pp, qq model is defined by the choice P " pQ p{q q `. The string equation rP, Qs " usually implies a non-linear equation for uptq.

Example of PDEs for the pp, qq " p3, 2q model. Let us denote 9 uptq " B t uptq. We have:

P " p B t q 3 ´3u B t `v Q " p B t q 2 ´2u (5-7)

and the string equation implies v " ´3 2 9 u `t1

(5-8)

for some constant t 1 , and the Painlevé I equation for uptq:

´1 2 2 : u `3u 2 " t.

(5-9)

Constructing the Lax pair by "Folding"

In this paragraph we show that the associated linear system is an integrable system in Lax form, i.e. it can be written: B x Ψpx, tq " Lpx, tqΨpx, tq, B t Ψpx, tq " Mpx, tqΨpx, tq, (5-10) for a matrix Ψpx, tq " ¨ψ1 px, tq ¨¨¨ψ q px, tq p B t qψ 1 px, tq ¨¨¨p B t qψ q px, tq . . . . . . p B t q q´1 ψ 1 px, tq ¨¨¨p B t q q´1 ψ q px, tq ‹ ‹ ‹ ' .

(5-11)

where the ψ j pxq are independent solutions of the associated linear system (5-2).

It is easy to achieve the second equation with the companion matrix:

Mpx, tq " ¨1 1 . . . where we recall that u q´2 " ´qu, and up to a redefinition of time t we can chose u q´1 " 0. We now construct the matrix Lpx, tq to realize the first equation. Naively, B x B k t ψ can be expressed by the action of a differential operator of order pp `kq on ψ. But, if we want to write an equation like (5-10) with Lpx, tq having coefficients which are functions of x -and not differential operators -, only derivatives of order smaller than pq ´1q are allowed. To bypass this restriction, we can use the first member of (5-2) to express any q-th order derivative of ψ in terms of derivatives of lower order. This can be systematized with the notion of folding operators.

Definition 5.1 We define for any integer l the folding operators:

F l,j px, tq p B t q j P D `rxs,

(5-13) by the following recursion: F 0 px, tq " 1, F l`1 px, tq " p B t qF l px, tq `Fl,q´1 px, tqpx ´Qq.

(5-14)

They have the property that for every solution ψ l of (5-2) @ i P Z `, @ l " 1, . . . , q , p B t q i ψ l px, tq " q´1 ÿ j"0 F i,j px, tq p B t q j ψ l px, tq (5-15)

in other words they express any time derivative in terms of only up to order q ´1 derivatives.

Notice that F l px, tq " p B t q l for l P v0, q ´1w, but: F q px, tq " p B t q q `x ´Q " x ´quptq p B t q q´2 ´q´3 ÿ k"0 u k ptq p B t q k .

(5-16)

Lemma 5.1 For any integer l, F l,j px, tq " 0 whenever j ě q. Besides, for every solution ψ of (5-2)

´ B x ψpx, tq " P ψpx, tq " ´p ÿ l"0 v l ptqF l px, tq ¯ψpx, tq (5-17)

Proof. Since Q is monic of degree q, the last term in (5-17) prevents F l px, tq to have terms of degree higher than pq ´1q, as one can show by recursion. Then, recall that px ´Qqψpx, tq " 0, so these operators satisfy p B t q l ψpx, tq " F l px, tqψpx, tq, hence (5-17). l.

Definition 5.2 For any integer k, we define the operators:

L k,j px, tq p B t q j P D `rxs (5-18) by the following recursion:

v l ptqF l px, tq, L k`1 px, tq " p B t qL k px, tq `Lk,q´1 px, tqpx ´Qq.

(5-19)

We have similarly:

Lemma 5.2 For any integer k, L k,j px, tq " 0 whenever j ě q. l

We are now in position to conclude:

Proposition 5.1 The first equation of (5-10) is achieved with Lpx, tq " pL k,j px, tqq 0ďk,jďq´1 . l

In particular, the string equation is equivalent to the compatibility condition of this system: rMpx, tq, Lpx, tqs " B t Lpx, tq ´ B x Mpx, tq.

(5-20)

By a gauge transformation, one can chose u q´1 ptq " 0, i.e. Mpx, tq traceless and therefore det Ψpx, tq independent of t. If an initial condition Ψpx, t 0 q is invertible, Ψpx, tq remains invertible for all t.

Example of folding for the p3, 2q model. We have:

(5-21)

for which the string equation rP, Qs " implies the Painlevé I equation for uptq: ´1 2 2 : u `3u 2 " t. The first folding operators are

This gives

(5-23) and consequently

Lpx, tq "

and Mpx, tq "

(5-25)

Semiclassical spectral curve and formal expansion

We consider formal solutions of the string equation, i.e. u k and v l which have a formal series expansion in . Let us denote:

ptq.

(5-26)

(5-27)

(which are the Ñ 0 semiclassical limit of Q and P ). The leading order in of the string equation becomes a Poisson bracket:

which gives an algebraic constraint on u r0s k and v r0s l .

Proof. The leading order of rP, Qs is:

(5-29)

i.e. this means that Y 1 pzq 9 Xpzq ´X1 pzq 9 Y pzq " 1.

(5-30) l Lemma 5.4 A solution of (5-28) is obtained as follows

Xpzq " pu r0s q q{2 f `z pu r0s q ´1{2 ˘, Y pzq " pu r0s q p{2 g `z pu r0s q ´1{2 ˘,

(5-31)

where u r0s " pt{ρq 2 p`q´1 , and the functions f and g satisfy:

qf pζqg 1 pζq ´pgpζqf 1 pζq " pp `q ´1qρ, (5-32)

and ρ is chosen such that at large ζ the solution of (5-32) behaves as f pζq " ζ q p1 ´qu r0s ζ ´2 `Opζ ´3qq and gpζq " ζ p `1 ´pu r0s ζ ´2 `Opζ ´3q ˘. We call it the homogeneous solution.

Proof. The result is claimed in [START_REF] Di Francesco | 2d gravity and random matrices[END_REF]. Let us prove it directly. If we assume the form (5-31), and write ζ " z pu r0s q ´1{2 , then we have

B z Y " pu r0s q pp´1q{2 g 1 pζq, B t Y " 1 2 B t u r0s ´p pu r0s q pp´2q{2 gpζq ´pu r0s q pp´3q{2 g 1 pζq ¯.

(5-34)

It follows:

1

which is satisfied if u r0s " pt{ρq 2 p`q´1 and qf g 1 ´pgf 1 " pp `q ´1qρ. l Lemma 5.5 If p `q ě 4, this implies when ζ Ñ 8 that:

gpζq " f pζq p{q ´ρ q ζ 1´q ˆ1 `ur0s ˆq ´2 `2 p `q `1 ˙ζ´2 `Opζ ´3q ˙.

In particular: f " pg q{p q `, g " pf p{q q `.

(5-36)

Proof. Write f " g q{p h, the equation then gives:

(5-37) and upon integration:

(5-38)

Then, using p `q ě 4 to ensure that 2pp `q ´1q ą p `q `1, we can exponentiate:

(5-39)

We then multiply by g q{p " ζ q `1 ´qu r0s ζ ´2 `Opζ ´3q ˘and get f " g q{p `ρ p ζ 1´p ´1 `pp ´2 `2 p `q `1 q u r0s ζ ´2 `Opζ ´3q ¯.

(5-40)

We have the same proof for g. l

Special solutions

In the pp, qq model, we have P " pQ p{q q `and similarly Q " pP q{p q `. Therefore, at the semiclassical limit, we find Y pzq " pX p{q pzqq `and Xpzq " pY q{p pzqq `. The relation (5-28) can be solved explicitly in the case p " p2m `1qq ˘1 for some integer m [dFGZJ94]:

Γpp{q`1qΓpn´p{q`1q T p´2nq pζq gpζq " T q pζq , ρ " 2p.

(5-41)

where T l p2 cos θq " 2 cosplθq are the Chebyshev polynomials of the first kind. In particular, for the so-called "unitary" models p " q `1, we find: " f pζq " T q`1 pζq gpζq " T q pζq , ρ " 2pq `1q.

(5-42)

Semi-classical spectral curve

Proposition 5.2 In the semiclassical limit Ñ 0, the eigenvalues of Mpx, tq and Lpx, tq are given by the functions xpzq and ypzq defined in (5-27), by: z " eigenvalue of M r0s px, tq ðñ x " Xpzq "

(5-43)

y " eigenvalue of L r0s pXpzq, tq ðñ y " Y pzq "

(5-44)

The leading order spectral curve, i.e. the locus of eigenvalues of L r0s pxpzq, tq, is a genus 0 algebraic plane curve.

Proof. Since Mpx, tq is a companion matrix, its characteristic polynomial is 0 " det pz 1 q ´Mpx, tqq " x ´q ÿ k"0 u k ptq z k , (5-45) therefore in the limit Ñ 0, the eigenvalues of M r0s px, tq are the z's such that Xpzq " x:

where Xpzq is the function introduced in (5-27). It follows that in the limit Ñ 0, B t ψpx, tq " z ψpx, tq p1 `Op qq. The eigenvalues y of Lpx, tq, by definition are such that y ψpx, tq " ´ B x ψpx, tq " p ÿ l"0 v l ptq p B t q l ψpx, tq, (5-47)

and thus in the Ñ 0 limit, the eigenvalues of L r0s px, tq are such that y " Y pzq "

(5-48)

The spectral curve P px, yq " detpy 1 q ´Lr0s px, tqq is a polynomial of x and y, monic of degree q in y, which vanishes if and only if y is an eigenvalue of L r0s pxq, i.e. if and only if there exists some z such that x " Xpzq and y " Y pzq. Therefore P px, yq is proportional to the resultant of the polynomials Xpzq ´x and Y pzq ´y: . . . v

As mentioned above, it admits a parametric solution:

PpXpzq, Y pzqq " 0 (5-49) with x and y polynomials of z. This means that there is a holomorphic map z Þ Ñ pXpzq, Y pzqq from the Riemann sphere p C to the spectral curve (the locus of Ppx, yq " 0 in C ˆC). In particular this implies that the spectral curve is an algebraic plane curve of genus g " 0. l

Asymptotic expansion and TT property

As in Section 3.2, we look for asymptotics of the form:

Ψpx, tq " Vpx, tq p Ψpx, tq e 1 Spx,tq , (5-50)

where:

' Spx, tq " diagpSpz a qq 1ďaďq is such that B t S a pzq| Xpzq"x " z i are the eigenvalues of M r0s px, tq, where z " z a is related to x by

x " Xpzq " z q ´qu r0s ptq z q´2 `q´2 ÿ

(5-51)

Thanks to (5-28), it also satisfies:

where Y pz a q are the eigenvalues of L r0s px, tq.

' Vpx, tq is a matrix whose columns are eigenvectors of both M r0s px, tq and L r0s px, tq, normalized such that V ´1 B x Vpx, tq has a vanishing diagonal. Since M r0s px, tq is a companion matrix, Vpx, tq can be found rather explicitly, as a Vandermonde matrix, with columns normalized by a factor 1{ a X 1 pz a q:

V a,b px, tq " pz b pxqq a´1 a X 1 pz a q where x " Xpz b q "

(5-53)

Its inverse is pV ´1q a,b " pXpz a pxqqz a pxq ´bq àX 1 pz a pxqq "

9 X 1 pz a q X 1 pz a q " Opx ´2{q q.

(5-58)

' The matrix p Ψpx, tq " 1 q `Op q has a formal asymptotic series as Ñ 0. From B t Ψ ¨Ψ´1 " M " M r0s ´eq pu´u r0s q T , where e q " p0, 0, . . . , 0, 1q and u " pu 0 , . . . , u q´1 q, we get the equation for p Ψ involving the diagonal matrix Z " diagpz 1 , . . . , z q q of eigenvalues of M r0s :

i.e.

(5-60)

This equation uniquely determines p Ψ " 1 q `Op q as its asymptotic expansion in powers of . In fact it also uniquely determines p Ψ " 1 q `Opx ´1{q q as an asymptotic series at large x, in powers of x 1{q . From B x Ψ ¨Ψ´1 " L we also get an ODE for p Ψ:

(5-61)

We observe that the semiclassical spectral curve has genus 0. Therefore, we will be able to apply Theorem 3.1 if we can show:

' the existence of a Ø ´ symmetry. This is a technical but simple check done in § 5.6.

' that the n-point correlators W n px 1 , . . . , x n q are Op n´2 q after a suitable gauge transformation. This is a non-trivial property of pp, qq models, that we establish in § 5.7 by constructing an insertion operator δ a x which is compatible with B t .

' the pole property, i.e. that ω pgq n have poles only at ramification points, established in § 5.8.

The consequences of Theorem 3.1 for the pp, qq models are gathered in Section 5.10.

Ø ´ symmetry

The goal of this subsection is that the pp, qq models admit conjugated solutions in the terminology of § 3.6:

Theorem 5.2 For any invertible solution Ψpx, tq of (5-10) with coupling constant , there exists a solution Φpx, tq of (5-10) with coupling constant ´ , such that γpx, tq " Φpx, tqΨ T px, tq is independent of x.

This theorem is proved below, but in order to do so, we need some intermediate results and definitions:

We first introduce a conjugation operator:

Definition 5.3 There is a unique antilinear operator : : D Ñ D, such that:

' p B t q : " ´p B t q : . ' for any D 1 , D 2 P D, pD 1 D 2 q : " D : 2 D : 1 .

In particular, if P, Q P D `satisfy rP, Qs " , then rP : , Q : s " ´ . Moreover, if P and Q are differential operators of the form (5-5), so are P : and Q : . To summarize, : puts in correspondence the models with coupling constant and ´ .

The linear system associated to pP : , Q : q is: xφpx, tq " Q : φpx, tq, B x φpx, tq " P : φpx, tq.

(5-62)

If φ 1 px, tq, . . . , φ q px, tq denotes a family of solutions of (5-62), we can define a matrix:

Φpx, tq " ¨φ1 px, tq ¨¨¨φ q px, tq p B t qφ 1 px, tq ¨¨¨p B t qφ q px, tq . . . . . .

As before, we can represent (5-62) in Lax form, and we denote L ´ px, tq and M ´ px, tq the corresponding Lax matrices:

´ B x Φpx, tq " L ´ px, tqΦpx, tq, ´ B t Φpx, tq " M ´ px, tqΦpx, tq.

(5-64)

The following result gives a correspondence between solutions of the associated linear systems of pP, Qq and pP : , Q : q.

Proposition 5.3 Let ψ 1 , . . . , ψ q be a basis of solutions of (5-2), Ψpx, tq as defined in (5-11), and define:

∆pxq " det Ψpx, tq, (5-65) ∆ i0´1,j0 px, tq " det " p B t q i´1 ψ j px, tq ‰ i‰i0, j‰j0 1ďi,jďq , (5-66) r φ j px, tq " ∆ q´1,j px, tq.

(5-67) then p r φ j px, tqq 1ďjďq is a basis of solutions of (5-62).

The proof of this proposition relies on a technical result:

Lemma 5.6 Let j P v1, qw. With the convention ∆ ´1,j " 0, we have for any i P v0, q ´1w, B t ∆ i,j px, tq " ∆ i´1,j px, tq `p´1q q´j pu i ptq ´δi,0 xq∆ q´1,j px, tq, (5-68)

and for any k P v1, q `1w, B t ∆ q´k,j px, tq " ´k ÿ

l"1 p´1q l`1 p B t q k´l ru q´l`1 ptq∆ q´1,j px, tqs ¯`δ k,q`1 p´1q q x∆ q´1,j px, tq.

(5-69)

Omitting to precise the variables, (5-83) implies the relations:

u l´1 P l,n ´Pq,n´1 ´px δ n,1 ´un´1 qP q,q .

These relations give an expression of the elements P l,n in terms of the elements P k,q of the last column and their time derivatives. If we introduce:

(5-85)

we find for elements above and on the diagonal:

and for elements below the diagonal:

(5-87)

Consequently, we may write:

with:

(5-93)

We now prove the commutation relations. We claim that, for any θ P C generic, the matrix

(5-94) has a basis of eigenvectors which independent of x and a. This will imply:

from which the relations (5-80)-(5-82) can be deduced by identification of the coefficients of θ. Let pζ i q 1ďiďq be the roots of:

(5-96)

For generic θ, the roots are simple, so that the column vector v i pzq " pζ j i q 0ďjďq´1 form a basis of C q . Let us set:

(5-97)

Considering the second line:

(5-98) but since B 2,1 " Γ 1 , A 2,1 " ´u0 Γ 1 and A 1,d " Γ 1 , using the polynomial equation (5-96) for ζ i , it must vanish. If we proceed to the k-th line, we have:

Using:

(5-100)

we may collect the terms relative to a given Γ m and we obtain: With the expression (5-12) of Mpx, tq for pp, qq models, we compute:

Mpx, tq ´Mpy, tq x ´y " E q,1 (5-114)

δ a y M k,m px, tq " ´δk,q δ a y u m´1 ptq.

(5-115)

The equation (5-113) gives a strong constraints upon the matrix Up a y , tq. For instance, it cannot be zero since: rE q,1 , Pp a y , tqs k,m " δ k,q P 1,m p a y , tq ´δm,1 P k,q p a y , tq.

(5-116)

We compute: (5-117)

The condition (5-113) is an affine function of x. With the choice U k,q " U 1,m " 0 for any k, m P v1, q, the coefficient of x vanish. The remaining constraint reads:

´δk,q δ a y u m´1 ptq " U k,m´1 p a y , tq ´Uk`1,m p a y , tq ´δk,q

a y , tq ´δk,q P 1,m p a y , tq `δm,1 P k,q p a y , tq.

(5-118)

Omitting the dependence in y, a and t, we have for k ‰ q:

U k`1,m " U k,m´1 `δm,1 P k,q ´ B t U k,m .

(5-119)

The solution at leading order in is:

which coincides with the definition of the matrix B in (5-91). Eqn.

(5-119) can be solved recursively, and we find that its unique solution is given by (5-110). To define completely an insertion operator, it remains to specify how it acts on the functions u k ptq. The commutativity condition prescribes (5-111).

l Although we did not make use of this property, we show for completeness that insertion operators pairwise commute: Lemma 5.8 For any a, b P v1, qw, we have rδ a

x , δ b y s " 0.

Proof. This condition is equivalent to: x, tq `Op q and δ a x Up b y q P Op q owing to Lemma 5.2, the commutation relation

(5-81) implies (5-121) at leading order. l

The pole property

We need to prove that ω g n has poles only at ramification points, in particular, no pole at 8 or at double zeroes. For this purpose, we will use the observations of § 5.5.

Double points

Lemma 5.9 In the q-th reduction of KP, for any n, g, ω g n are regular at preimages in S r0s of double points.

Proof. We remind that this property is not obvious because equations (3-16) and (3-17), which allow the computation of the WKB expansion of Ψpx, tq " V p Ψ e S{ C, may have a pole 1{pλ r0s a px, tq λr0s b px, tqq, i.e. at the double points. However, this analysis was performed for the differential equation with respect to x. But now, we have a second differential equation B t Ψpx, tq " Mpx, tqΨpx, tq,

(5-122) from which we can perform a similar WKB analysis. One notices that solving (5-60) for p Ψpx, tq " 1 q `řkě1 k p Ψ rks px, tq recursively, the only denominators are of the form 1{pz a ´zb q, and thus the only poles that are produced are when x Ñ α such that z a pαq " z b pαq for a ‰ b, i.e. when z goes to a ramification point. The conclusion is that poles at double points in x (and thus at preimages of double points in z P S r0s ) do not occur. l

Behavior at z Ñ 8

Lemma 5.10 The q-th reduction of KP satisfies Assumption 3.1.

Proof. We now expand Ψ at large x as

where:

(5-124)

Moreover, as in Section 3.2, the equation B x Ψ " L Ψ implies that there is also a large x expansion of the form:

where B x r S " DiagpΛ i pxqq, and r V ´1B x r V " Opx ´1{q q and r Ψ " 1 q `Opx ´1{q q. This implies that:

Λ " Λ r0s `Opx ´1{q q, (5-126)

and thus the pole property of Assumption 3.1 is satisfied. This implies that, for any g, n ‰ p1, 0q, the ω pgq n pz 1 , . . . , z n q are regular when z i " 8.

Tau function

It is well known that for pp, qq model we have [dFGZJ94]:

Theorem 5.4 2 B 2 t ln T ptq " uptq.

(5-127)

Application of the topological recursion

Theorem 3.1, and in particular Corollary 3.6 (since our spectral curve has genus 0), implies that the correlators have the expansion:

x 1 , . . . , an

x n qdx 1 , . . . dx n " ÿ gě0 2g´2`n ω pgq n pz a1 px 1 q, . . . , z an px n qq, (5-128)

where the ω pgq n pz 1 , . . . , z n q are computed by the topological recursion. The initial data is:

(5-129)

To justify the second equation, we know from Corollary 3.5 that ω p0q 2 P BpS r0s q, and there is a unique such object on a genus 0 curve, which can be written as in the second equation in any uniformization variable z.

In particular, we can retrieve the expansion of the Tau function with Corollary 4.2.

ln T "

Y " 1, we find that B t Y | Xpzq " ´dz{dX, hence:

(5-131)

Remember that T is defined up to a multiplicative constant, so the constant of integration to get F pgq from (5-131) is irrelevant here. A direct integration can be done explicitly for F p0q [Dub96] and F p1q [EKK05], but the formulas are complicated to state. In simple examples, it is more efficient to rely on (5-131).

Case of the homogeneous solution

For the homogeneous solution, we have

Xpzq " pu r0s q q{2 f pζq, Y pzq " pu r0s q p{2 gpζq, ζ " z pu r0s q ´1{2 , (5-132) and where u r0s ptq " pt{ρq 2 p`q´1 . By homogeneity of the topological recursion (see [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF][START_REF]Topological recursion in random matrices and enumerative geometry[END_REF]) this implies:

n pz 1 , . . . , z n q " pu r0s q p2´2g´nqpp`qq{2 ωpgq n pζ 1 , . . . , ζ n q " pt{ρq p2´2g´nqpp`qq{pp`q´1q ωpgq n pζ 1 , . . . , ζ n q.

(5-133) where ωpgq n is computed as if u r0s was equal to 1. In particular for n " 0 @g ‰ 1, F pgq ptq " t p2´2gqpp`qq{pp`q´1q F pgq p1q.

(5-134)

For F p1q , we have: (5-136)

where the arbitrary integration constant was set to 0 for t " 1.

For the homogeneous solution, we observe that the Ñ 0 expansion coincides with a t Ñ 8 expansion:

T " exp ´ÿ gě0 2g´2 F pgq ptq ¯" t c exp ´ÿ gě0 p t ´pp`qq{pp`q´1q q 2g´2 F g p1q ¯.

(5-137)

We see that can be absorbed in a redefinition of the variable t. We also have:

uptq " 2 B 2 t ln T " t 2 p`q´1 ÿ gě0 p t ´pp`qq{pp`q´1q q 2g u tgu p1q,

(5-138) where u tgu p1q " pp `qqp2 ´2gq `pp `qqp2 ´2gq ´1p p `q ´1q 2 F g p1q.

(5-139)

In particular we see that u t0u p1q " ρ ´2{pp`q´1q , F p0q p1q " 1 2 pp `q ´1q 2 pp `qqpp `q `1q ρ ´2{pp`q´1q .

(5-140)

Examples

The q-th reductions of KP, and in particular the pp, qq models describe universal behavior -provably or conjecturally -in statistical physics, random matrix theory, and integrable systems. For those reasons, many of them have received names referring to the problems where they appear. The p1, 2q model is known to appear when studying the double scaling limit of random matrices at a generic edge of the spectral density, and is related to the Airy process [START_REF] Prähofer | Scale invariance of the PNG droplet and the Airy process[END_REF]. The p3, 2q model was shown, first in physics [Moo90, DS90], then rigorously [START_REF] Its | An isomonodromic approach in the theory of twodimensional quantum gravity[END_REF], to describe generating series of random maps with generic critical weights, and thus was called "pure gravity". The p4, 3q (resp. the p6, 5q model) is expected to describe the generating series of random maps carrying an Ising model (resp. 3-Potts model) with non-generic critical weights, and in fact, the theory we developed allows a proof of those conjectures [BE].

All the pp, qq models are conjectured to describe the double-scaling limit in random matrices around an edge a where the spectral density behaves like |x ´a| p{q . This is also relevant for systems of vicious walkers via Dyson Brownian motion [START_REF] Dyson | A Brownian-motion model for the eigenvalues of a random matrix[END_REF], and this is related to 2d quantum gravity for reasons dating back to [START_REF] Brézin | Planar diagrams[END_REF]. This has been proven so far in a handful of case (see e.g. [START_REF] Kuijlaars | Universality[END_REF] and references therein), but mainly for q " 2 cases -which correspond to the Gelfand-Dikii hierarchies [GD75]. This conjecture is based on an ansatz [START_REF] Moore | Geometry of the string equations[END_REF] for the convergence of operators P and Q -interpreted as differentiation and multiplication in the vector space generating by orthogonal polynomials -which has not been justified rigorously so far. Our methods do not provide a proof that double scaling limits exist. However, once this existence is granted and it is characterized in terms of a Lax pair, it can actually prove that the semiclassical expansion of the limit laws are computed by the topological recursion. Moreover, if the semiclassical spectral curve of the Lax pair can be identified with a blow-up of the large N spectral curve of the matrix model when parameters become critical, it shows -combining the results of [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] and [START_REF]Asymptotic expansion of β matrix models in the one-cut regime[END_REF] that the semiclassical expansion of the double-scaling limit does coincide with a limit of coefficients in an off-critical 1{N expansion when approaching criticality. This crossover is expected and we are able to justify it only relying on loop equations, i.e. by algebraic methods. We refer to [START_REF] Borot | Tracy-Widom GUE law and symplectic invariants[END_REF][START_REF] Bergère | The sine-law gap probability, Painlevé 5, and asymptotic expansion by the topological recursion[END_REF] for applications relying on those ideas.

In the remaining of the text, we illustrate some pp, qq models, by describing the non-linear PDEs they generate, the spectral curves and the first few coefficients in the Ñ 0 expansion of the correlators and of the Tau function.

1 pzq " ´7 2 10 3 5 pu r0s q 7 dz z 10 `4z 8 `12u r0s z 6 `36 pu r0s q 2 z 4 `87 pu r0s q 3 z 2 `135 pu r0s q 4 ˘, ω p3q 1 pzq " ´7 2 15 3 9 pu r0s q 12 dz z 16 ´1400z 14 `4200u r0s z 12 `12600pu r0s q 2 z 10 `34740pu r0s q 3 z 8 `85860pu r0s q 4 z 6 `181764pu r0s q 5 z 4 `297297pu r0s q 6 z 2 `289575pu r0s q 7

¯.

The expansion of the Tau function ln T " ř gě0 2g´2 F pgq is obtained from:

and the solution u " u r0s `řgě1 2g u tgu from u tgu " B 2 t F pgq . We emphasized that 1 " 6u r0s 9 u r0s to facilitate the integration. That gives:

2 15 3 9 pu r0s q 12 " 5 2 7 2 9 u r0s 2 11 3 8 pu r0s q 11 ñ F p3q " ´5¨7 2 2 12 3 8 pu r0s q 10 " ´5¨7 2 2 12 3 3 t 5 ñ u t3u " ´52 7 2 2 11 3 2 t 7 .

(6-11) These results agree with the direct expansion of the solution of the Painlevé I equation (6-3).

pp, qq " p2, 3q

Here, we consider pure gravity again, but exchange the role of P and Q, namely we chose p " 2 and q " 3. This gives the 3 ˆ3 Lax pair:

Lpx, tq "

(6-13)

The spectral curve is: det `y 1 3 ´Lpx, tq ˘" y 3 ´2u 2 y ´2u 3 `px ´t1 q 2 `1 2 2 py: u ´1 2 9 u 2 `u: uq.

(6-14)

To leading order the spectral curve is thus: y 3 ´2pu r0s q 2 y `px ´t1 q 2 ´2pu r0s q 3 " 0, (6-15) which admits the parametrization: " Xpzq " pu r0s q 3{2 T 3 pζq " z 3 ´3u r0s z Y pzq " ´ur0s T 2 pζq " 2u r0s ´z2 u r0s " pt{3q 1{2 .

(6-16)

The ramification points are at ζ " a ˘" ˘1, they correspond to Xpa ˘q " ¯2 pu r0s q 3{2 . The local Galois conjugate near a " ˘1 is:

The topological recursion gives (we denote ζ " pu r0s q ´1{2 z) for the expansion of the correlators: ω p0q 1 pzq " ´Y pzqdXpzq " 3 pu r0s q 5{2 pζ 2 ´2qpζ 2 ´1q dζ,

It is necessary to compute ω p1q 2 in order to obtain p2q 1 , but we omitted its expression for conciseness. The expansion of the Tau function ln T " ř gě0 2g´2 F pgq and the solution u " u r0s `řgě1 2g u tgu from u tgu " B 2 t F pgq . We may use 6u r0s 9 u r0s " 1 to perform the integration. That gives:

B t F p1q " 6 pu r0s q 3{2 9 u r0s 144 pu r0s q 5{2 " 9 u r0s 24 u r0s ñ F p1q " ln u r0s 24 " 1 48 lnpt{3q, ñ u t1u " ´1 48 t 2 . B t F p2q " 6 pu r0s q 3{2 9 u r0s 7168 2 18 3 5 pu r0s q 15{2 " 7 9 u r0s 2 7 3 4 pu r0s q 6 ñ F p2q " ´7 2 7 3 4 5 pu r0s q 5 " ´7

(6-18)

This again perfectly agrees with the direct expansion of the solution of the Painlevé I equation (6-3), and this agrees with the p3, 2q model, as an illustration of the pp, qq Ñ pq, pq duality.

pp, qq " p4, 3q: Ising model

The model is defined by:

where u, u 0 , v 1 , v 0 are functions of t. The string equation implies

where w is a function of t, and:

where w satisfies 12uw ´2 2 : w " t 3 , (6-22) and then uptq satisfies

where t 1 , t 2 , t 3 are integration constants. A particular choice is t 1 " t 2 " t 3 " 0 and w " 0, in which case we have 1 6

The first few orders of expansion are: (6-26)

The Lax pair is:

x ´t1 ´w ´u pt 1 ´x ´3wqu ´u2 `t2 x ´t1 ´w px ´t1 q 2 `2px ´t1 qw ´3w 2 ´2pt 1 ´x `3wqu ´u2 `t2

w ` `7 4 9 u 2 `5 2 u: u ˘` : w ´1 6 2 ˙˙˙u ´u 9 u ` : w ´1 6 2 ˙˙u ´9 w ´1 6 : u '.

In the particular case where t 1 " t 2 " t 3 " w " 0, we have:

(6-28)

The spectral curve is:

To leading order the spectral curve is thus: y 3 ´3pu r0s q 4 y " x 4 ´4pu r0s q 3 x 2 `2pu r0s q 6 , (6-30)

i.e. in terms of Chebyshev polynomials:

T 3 `y{pu r0s q 2 ˘" T 4 `x{pu r0s q 3{2 ˘, (6-31) which admits the parametrization: " Xpzq " pu r0s q 3{2 T 3 pζq " z 3 ´3u r0s z Y pzq " pu r0s q 2 T 4 pζq " z 4 ´4u r0s z 2 `2pu r0s q 2 u r0s " pt{4q 1{3 .

(6-32)

The ramification points are at ζ " a ˘" ˘1, they correspond to Xpa ˘q " ¯2. The local Galois conjugate near a " ˘1 is:

The topological recursion gives (we denote ζ " z{ ? u r0s ) for the expansion of the correlators: ω p0q 1 pzq " ´Y pzqdXpzq " ´3 pu r0s q 7{2 pζ 4 ´4ζ 2 `2qpζ 2 ´1q dζ,

dζ 1 dζ 2 pζ 1 ´ζ2 q 2 , ω p0q 3 pz 1 , z 2 , z 3 q " ´dζ 1 dζ 2 dζ 3 24 pu r0s q 7{2 ´1 pζ 1 ´1q 2 pζ 2 ´1q 2 pζ 3 ´1q 2 `1 pζ 1 `1q 2 pζ 2 `1q 2 pζ 3 `1q 2 ¯, ω p1q 1 pzq " ´dζ 576 pu r0s q 7{2 ´7 `7ζ `3ζ 2 pζ `1q 4 `7 ´7ζ `3ζ 2 pζ ´1q 4 ¯, ω p2q 1 pzq " ´5dζ 2 13 3 5 pu r0s q 21{2 1 pζ 2 ´1q 10 ´791 `10831ζ 2 `5642ζ 4 `8010ζ 6 ´5060ζ 8 `6556ζ 10 ´4098ζ 12 `1982ζ 14 ´539ζ 16 `77ζ 18 ¯, ω p3q 1 pzq " ´5dζ 2 19 3 9 pu r0s q 35{2 1 pζ 2 ´1q 16 ´1534020 `51852480ζ 2 `139051115ζ 4 `126732801ζ 6 `14026336ζ 8 `136206860ζ 10 ´165273597ζ 12 `227618305ζ 14 ´221591820ζ 16 `175823400ζ 18 ´107773575ζ 20 `51069755ζ 22 ´17959320ζ 24 `4465420ζ 26 ´701415ζ 28 `53955ζ 30 ¯. ), but since their expression is lengthy we do not copy them here. The expansion of the Tau function ln T " ř gě0 2g´2 F pgq and the solution u " u r0s `řgě1 2g u tgu from u tgu " B 2 t F pgq . We can use 12pu r0s q 2 9 u r0s " 1 to perform the integration. That gives:

The computation of ω

B t F p1q " 12 pu r0s q 5{2 9 u r0s 1 2 5 3 pu r0s q 7{2 " 9 u r0s 8 u r0s , ñ F p1q " ln u0 8 " ln pt{4q 24 . ñ u t1u " ´1 24t 2 , B t F p2q " 12 pu r0s q 5{2 9 u r0s 5¨7¨11 2 13 3 5 pu r0s q 21{2 " 5¨7¨11 2 11 3 4 pu r0s q 8 ñ F p2q " ´5¨11 2 11 3 4 pu r0s q 7 " ´55 1296 p2tq 7{3 . ñ u t2u " ´1925 1458 p2tq 13{3 , B t F p3q " 12 pu r0s q 5{2 9 u r0s 5 2 11¨109 2 19 3 7 pu r0s q 35{2 " 5 2 11¨109 9 u r0s 2 17 3 6 pu r0s q 15 ñ F p3q " ´52 11¨109 2 18 3 6 7 pu r0s q 14 " ´29975

This matches (6-25).

"

´ px ´yq 2 Pp a y q `« Pp a y q x ´y `Up a y q, Lpxq ff .

(A.5)

To compute the action of δ a y on the correlators, we consider n " 1 separately:

Then, for n ě 2, we can use formula (2-5):

K aj ,a px j , yqK a,a σpjq py, x σpjq q

We assume that all ramification points are simple (see [START_REF] Bouchard | Think globally, compute locally[END_REF] for the case or higher ramifications), the embedding of the curve S r0s Ñ C 2 by the functions px, yq is regular, and that TT is satisfied. We shall prove the topological recursion using the linear (Theorem 2.1) and quadratic (Theorem 2.2) loop equations only. This is already done in [EO07, BEO13], but we present here a self-contained proof. Contrarily to [START_REF] Borot | Abstract loop equations, topological recursion, and applications[END_REF] which is more general, we take advantage here that the semiclassical spectral curve S r0s is a compact Riemann surface of genus g, to identify more precisely the possible holomorphic term in (3-41).

From the TT hypothesis, we have that every ω pgq n with pg, nq ‰ p0, 1q, p0, 2q has poles only at the ramification points. We have called r " tr 1 , . . . , r m u the set of ramification points. Let r P r be a ramification point, by definition and assumption there are exactly two indices a ‰ b such that z a prq " z b prq, and we define the local Galois involution σ r in a vicinity of r, as the map z a pxq Þ Ñ z b pxq. Let J " t2, . . . , nu and z J " pz j q jPJ , and define:

where ř 1 means that we exclude the cases ph, Iq " p0, Hq and ph, Iq " pg, Jq, i.e.

1 Near a ramification point r, we have:

Proof of the lemma. To simplify notations, we can always label 1 and 2 the sheets meeting at the ramification point r. I.e. if z " z 1 , we have σ r pzq " z 2 . Let us decompose the sum over indices as:

The linear loop equation implies that:

and thus:

The last two lines have no poles at the ramification point, hence the announced result. l

Remark. Since the analytic term in r in (B.4) is a quadratic differential in z invariant under Galois involution, it must actually have a double zero at r. Qpgq n pz, σ r pzq; z J q `holomorphicpz 1 q. (B.8)

Proof. First, Lemma B.1 together with the quadratic loop equation imply that Q pgq n pz, σ r pzq; z J q has no pole at the ramification point r. This means that: Qpgq n pz, σ r pzq; z J q " ´ωp0q 1 pzq ω pgq n`1 pσ r pzq, z J q ´ωpgq n`1 pz, z J q ω p0q 1 pσ r pzqq `analytical at r. (B.9) Moreover, using again the linear loop equation we have that ω pgq n`1 pσ r pzq, z J q " ´ωpgq n`1 pz, z J q `analytical at r, (B.10) and thus Qpgq n pz, σ r pzq; z J q "

" ω p0q 1 pzq ´ωp0q 1 pσ r pzqq ‰ ω pgq n`1 pz, z J q `analytical at r. (B.11)

According to the previous remark, the remainder has actually a double zero at z " r. We remind that ω p0q 1 " ydx, and since we assume that the embedding of S r0s in C 2 by px, yq is regular, dyprq ‰ 0. Combined with the assumption that x has simple ramification points, this implies that " ω p0q 1 pzq ώp0q

1 pσ r pzqq ‰ has exactly a double zero at z " r. Therefore, we find: Now, observe that ω pgq n`1 pz, z J q has poles only at the ramification points, whereas ω p0q 2 pz, z 1 q has a pole only at z " z 1 (a double pole). We may move the integration contours from surrounding the poles of ω pgq n`1 pz, z J q to surrounding the poles of ω p0q 2 pz, z 1 q, i.e. using the Riemann bilinear identity: -

where the cycles A i , B j are chosen to form a basis of 2g non-contractible cycles on S r0s , with canonical intersections A i X B j " δ i,j . Observe that ´şz o ω p0q 2 pz 1 , ¨q¯h as a simple pole at z 1 " z with residue 1, so the first term is: Qpgq n pz, σ r pzq; z J q " ω pgq n`1 pz 1 , z J q `holomorphic pz 1 q. (B.17)

This finishes the proof of Theorem 3.1. l