Puis Je Voudrais Remercier

Tamara Grava

Pol Vanhaecke D'

John Harnad

Marco Bertola

Mohamed Johan Le

Viet-Anh Celleux D'après : Alexis

Jérôme, Marine, Marius Axel David

Keywords: Lie algebras

Mes premiers remerciements, il va de soi, vont à Mattia d'avoir accepté de diriger ma thèse, ainsi qu'à Paolo et Volodya d'avoir accepté de me co-encadrer. Avoir trois encadrants de thèse m'a permis de ne jamais me sentir seulx, 1 d'avoir toujours eu quelqu'un vers qui me tourner mais aussi d'avoir à porter de main une variété de points de vue. Je les remercie tous les trois chaleureusement.

Organization of the thesis

Here we describe how the thesis is organized. Sections containing new results are preceded by a bold arrow →. The other sections contain material that can be found in the literature and are given here so as to describe the context of our results.

Chapter 1 Polynomial tau functions of the Korteweg-de Vries hierarchy Section 3.1: We detail the theory of pseudo-di erential operators. Section 3.2: We detail the theory of the Kadomtsev-Petviashvili hierarchy. Section 3.3: We detail the theory of the bilinear identity (and its equivalence with the KP hierarchy) and of the tau function of the KP hierarchy. Section 3.4: We detail the theory of the KdV hierarchy and its tau function.

→ Section 3.5: We exhibit the change of variables that transforms the tau functions of the KdV hierarchy into the Adler-Moser polynomials using mainly KP theory; a work published [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF].

Chapter 2 Polynomial tau functions of the Drinfeld-Sokolov hierarchies and Schur polynomials Section 4.1: We recall the theory of semisimple Lie algebras. Section 4.2: We recall the theory of untwisted a ne Kac-Moody algebras. Section 4.3: We detail the theory of the Drinfeld-Sokolov hierarchies and Wu's construction of a tau function [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF]. Section 4.4: We describe the construction of tau functions as Toeplitz determinants by Cafasso and Wu [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF][START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF].

→ Section 4.5: We expand the tau function of the DS hierarchies as a sum over partitions of integers; through this we de ne generalized Schur polynomials. As an application, we give a criterion for the polynomiality of tau functions; we compute examples of polynomial tau functions and give applications to detecting bilinear equations of the DS hierarchies; a work publihed in [START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF].

Chapter 1

Introduction en français

1.1 Aperçu historique de la théorie et de nos résultats

Mise à part quelques équations, la section 1.1 est plutôt historique et devrait être lisible par les non spécialistes. La présente thèse traite du calcul et des applications des fonctions tau des hiérarchies d'équations de Drinfeld-Sokolov. Les hiérarchies de Drinfeld-Sokolov sont des suites in nies d'équations aux dérivées partielles dont les ots commutent deux-à-deux. Pour cette raison, elles forment un système intégrable in ni-dimensionnel. Elles furent introduites par V. G. Drinfeld et V. V. Sokolov dans leur article fondateur [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF]. Dans ce dernier, ils montrèrent comment associer ces hiérarchies à n'importe quelle algèbre de Kac-Moody a ne [START_REF] Kac | Simple irreducible graded lie algebras of nite growth[END_REF][START_REF] Moody | A New Class of Lie Algebras[END_REF] g. Cette dernière, nous la réalisons comme l'extension centrale de l'algèbre de boucle d'une algèbre de Lie semisimple g, i.e.1 g = g ⊗ C[λ, λ -1 ] ⊕ Cc,

(1.1) avec le crochet de Lie approprié faisant de c un élément central ; voir (1.28). Donnons la forme générale de ces systèmes d'équations. Une solution d'une hiérarchie de Drinfeld-Sokolov est la donnée de fonctions u 1 , . . . , u N qui dépendent d'une in nité de variables {t i | i ≥ 1} avec x = t 1 ; le système d'équation de la hiérarchie a alors la forme

u α t i = A α i (u 1 , . . . , u N ) α ∈ {1, . . . , N }, i ≥ 1 , (1.2)
Introduction en français où u α t i = ∂ t i u α et où A α i est polynomial en les u β et leurs dérivées par rapport à x : u β x , u β x x , etc. Par exemple, la célèbre équation de Korteweg-de Vries [START_REF] Boussinesq | Essai sur la theorie des eaux courantes[END_REF][START_REF] Korteweg | On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves[END_REF] est de cette forme :

u t = 1 4 u x x x + 3 2 uu x .
(1.3)

Les hiérarchies de Drinfeld-Sokolov sont des suites d'équations ayant des propriétés similaires à celle de l'équation de Korteweg-de Vries et, comme nous le verrons, l'équation (1.3) joue un rôle crucial dans la théorie. Nous insistons sur le fait que bien que nous étudions des équations aux dérivées partielles, nous nous intéressons uniquement aux solutions algébriques de ces équations : polynômes, séries entières ou fractions rationnelles. Dans la prochaine section de cette introduction, nous développerons les principaux concepts sur lesquels cette thèse repose et expliquerons nos résultats : [Duc17, [START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF][START_REF] Du Crest De Villeneuve | Quantization of the D 4 Drinfeld-Sokolov hierarchy and Fan-Jarwis-Ruan-Witten theory[END_REF]. Mais avant cela, nous faisons une brève histoire des hiérarchies de Drinfeld-Sokolov et des fonctions tau et espérons ainsi incorporer nos résultats à ce dense domaine de la physique mathématique.

Tout d'abord, dans le cas où g est l'algèbre a ne associée à la l'algèbre de le Lie simple sl(n, C) (matrices complexes n × n de trace nulle), la hiérarchie de Drinfeld-Sokolov coïncide avec la n-ième hiérarchie de Gelfand-Dicket [START_REF] Gelfand | Fractional powers of operators and Hamiltonian systems[END_REF], ces dernières étant beaucoup plus simples ! La n-ième hiérarchie de Gelfand-Dickey admet la représentation suivante : considérons l'opérateur di érentiel L = ∂ n

x + u n-2 ∂ n-2

x + • • • + u 1 ∂ x + u 0 où les u i sont des fonctions lisses des paramètres {t i | i ≥ 1, i 0 mod n} avec t 1 = x. Alors la n-ième hiérarchie de Gelfand-Dickey consiste en la famille d'équations 2 ∂L ∂t i = (L i/n ) + , L i ≥ 1, i 0 mod n , (1.4) où L 1/n la racine n-ième de L par rapport à la composition des opérateurs di érentiels. 3 En particulier, pour sl(2, C) ( xant n = 2 dans (1.4)), on retrouve la hiérarchie de Kortewegde Vries [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF][START_REF] Miura | Korteweg-deVries Equation and Generalizations. I. Explicit Nonlinear Transformation[END_REF][START_REF] Gardner | Korteweg-de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion[END_REF] (dont le ot selon t 3 redonne l'équation de Korteweg-de Vries (1.3)). Les équations (1.4) sont appelées des paires de Lax scalaires. Pour la plupart, les hiérarchies de Drinfeld-Sokolov admettent une paire de Lax scalaire ainsi qu'une représentation bi-Hamiltonienne [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF], les rendant intégrables de nombreuses façons. Cependant, dans le cas général, elles sont construites génériquement en tant que hiérarchies d'équations de Lax matricielles (ou équations de courbure nulle) que nous introduisons dans la section suivante. Notons que la commutativité des ots des hiérarchies de Drinfeld-Sokolov résulte essentiellement de l'existence de sous-algèbres abéliennes in nie-dimensionnelles de l'algèbre de Kac-Moody g.

Le développement de la notion de fonction tau. La notion de fonction tau émergea au début des années 1980 avec les travaux de l'école de Kyoto menée par Sato ainsi que Date, Jimbo, Kashiwara, Miwa [JMS78, KM81, DKM81, DJKM81a] pour n'en citer que quelques uns. Originellement, la fonction tau est dé nie pour la hiérarchie de Kadomtsev-Petviashvili [START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF] qui contient toutes les hiérarchies de Gelfand-Dickey. 4 Une solution de la hiérarchie de Kadomtsev-Petviashvili est la donnée d'une in nité de fonctions ; la fonction tau est une seule fonction qui contient toute l'information de la solution. De plus, la fonction tau satisfait des équations di érentielles partielles de nature di érente de celles de la hiérarchie en soi. À savoir, les équations sur la fonction tau sont bilinéaires et intimement liées aux relations de Plücker. La hiérarchie de Kadomtsev-Petviashvili (KP) et sa fonction tau furent rapidement généralisées aux hiérarchies BKP et CKP, des hiérarchies de type KP liées aux algèbres de Lie de type orthogonale (B n , D n ) et symplectique (C n ) [START_REF] Date | Transformation groups for soliton equations VI: KP hierarchies of orthogonal and symplectic type[END_REF]. Pour les algèbres de Lie exceptionnelles, des hiérarchies de type KP furent dé nies par Kac et Wakimoto [START_REF] Kac | Exceptional hierarchies of soliton equations[END_REF]. Nous retiendrons que la fonction tau est équivalente à la solution de la hiérarchie, mais que les deux donnent deux approches très di érentes à la construction et la résolution de la hiérarchie. 5Dans le même temps, Edward Witten proposa sa célèbre conjecture de 1991 reliant une certaine fonction tau de la hiérarchie de Korteweg-de Vries à la fonction génératrice des nombres d'intersection sur l'espace de module des courbes stables M ,n [START_REF] Witten | Two-dimensional gravity and intersection theory on moduli space[END_REF]. La conjecture de Witten, prouvée par Kontsevich l'année suivante [START_REF] Kontsevich | Intersection theory on the moduli space of curves an the matrix Airy function[END_REF], initia une toute nouvelle façon d'approcher les hiérarchies intégrables que nous discutons ci-dessous.

Suite aux travaux de Drinfeld-Sokolov d'une part et de l'école de Kyoto et de Kac-Wakimoto d'autre part, plusieurs façons de dé nir les fonctions tau furent conçus pour les hiérarchies de Drinfeld-Sokolov. L'une d'elles, initiées par Kac [START_REF] Kac | In nite dimensional Lie algebras[END_REF] et Hollowood et Miramontes [START_REF] Hollowood | Tau functions and generalized integrable hierarchies[END_REF][START_REF] Miramontes | Tau-functions generating the conservation laws for generalized integrable hierarchies of KdV and a ne toda type[END_REF], fait usage des représentations de plus haut poids des algèbres de Kac-Moody a nes, dans cette approche, on retrouve les équations bilinéaires. Une autre approche est de construire une certaine famille de densités Hamiltoniennes dites tau symétriques et qui, essentiellement par le lemme de Poincaré, donnent une dé nition naturelle de la fonction tau [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF]. 6Dans la présente thèse, nous nous intéressons à deux classes de fonctions tau : les fonctions tau polynomiales et la fonction tau topologique. Les fonctions tau polynomiales forment une vaste classe de fonctions tau bien que les plus simples. Pour le cas particulier de la hiérarchie de Korteweg-de Vries, elles sont classi ées par les polynômes d'Adler-Moser [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF]. Les zéros de ces polynômes évoluent selon un système intégrable dit de Calogero-Moser [START_REF] Airault | Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem[END_REF] ; généralisé à certaines solutions de la hiérarchie de Kadomtsev-Petviashvili par Wilson [START_REF] Wilson | Collisions of Calogero-Moser particles and an adelic Grassmannian[END_REF]. Ainsi, les fonctions tau polynomiales des hiérarchies de Drinfeld-Sokolov devraient avoir des propriétés similaires qu'il nous faut encore découvrir. D'autre part, la fonction tau topologique d'une hiérarchie de Drinfeld-Sokolov donnée est l'objet principale sur lequel se construisent les ponts entre les hiérarchies intégrables et la géométrie algébrique. Nous y reviendrons cidessous ; discutons en premier lieu les fonctions tau polynomiales.

Fonctions tau polynomiales des hiérarchies de Drinfeld-Sokolov (traité au chapitre 4). En 2012 Chao-Zhong Wu, inspiré par [START_REF] Hollowood | Tau functions and generalized integrable hierarchies[END_REF], proposa une construction universelle de densités Hamiltoniennes tau symétriques pour n'importe quelle hiérarchie de Drinfeld-Sokolov associée à une algèbre a ne g [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF],7 donnant ainsi une fonction tau générique pour les hiérarchies de Drinfeld-Sokolov. Dans [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF][START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF], Cafasso et Wu exprimèrent la fonction tau de Wu comme la limite de déterminants de Toeplitz via une certaine représentation dèle de trace nulle de l'algèbre de Lie semisimple sous-jacente [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] (voir aussi l'appendice A). Cela généralisa en partie les premiers travaux de Cafasso [START_REF] Cafasso | Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies[END_REF] sur les hiérarchies de Gelfand-Dickey.

La construction de Cafasso-Wu associe une fonction tau à tout élément X ∈ g -de l'algèbre a ne g n'ayant que des puissances négatives du paramètre de boucle. Dans [START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF], à partir des résultats de Cafasso et Wu, nous avons montré comment développer la fonction tau de Wu comme une somme sur toutes les partitions d'entiers (théorème 4.5.2) ; la somme fait intervenir des polynômes qui généralisent les polynômes de Schur [START_REF] Schur | Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen[END_REF] (aussi [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF]). Conséquemment, nous avons obtenu un critère facile pour la polynomialité de la fonction tau : si X ∈ g -est tel que sa représentation matricielle est nilpotente (par rapport au produit usuel de matrices), alors la fonction tau associée est polynomiale. 8 Nous avons donnés quelques exemples non triviaux de fonctions tau polynomiales et décrit des applications aux équations bilinéaires des hiérarchies de Drinfeld-Sokolov. C'est l'objet du chapitre 4. Cela constituera un futur projet que d'étudier les propriétés de ces polynômes de Schur généralisés ainsi que de ces fonctions tau polynomiales.

Fonctions tau polynomiales de la hiérarchie KdV (traité au chapitre 3). Pendant que nous travaillons sur les fonctions tau polynomiales des hiérarchies de Drinfeld-Sokolov, nous nous intéressions au cas spéci que de la hiérarchie de Korteweg-de Vries (KdV). Cette dernière, comme nous l'avons décrit, est un cas particulier des hiérarchies de Drinfeld-Sokolov pour l'algèbre a ne associée à l'algèbre de Lie simple sl(2, C). De plus, toute solution de la hiérarchie de Korteweg-de Vries est aussi une solution de la hiérarchie de Kadomtsev-Petviashvili ; la réduction étant de mettre toutes les variables paires t 2i à zéro. Comme nous l'avons dit, les fonctions tau polynomiales de la hiérarchie de Korteweg-de Vries furent étudiées en profondeur et classi ées par Adler, McKean et Moser [START_REF] Airault | Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem[END_REF][START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF]. De plus, comme cela fut compris d'abord par Sato [START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF] (voir aussi [START_REF] Hirota | The Direct Method in Soliton Theory[END_REF]), chaque fonction tau polynomiale de la hiérarchie de Korteweg-de Vries peut être écrite en terme des polynômes de Schur.

Le principal résultat de [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF] est l'existence d'un unique changement des variables t i qui transforme les fonctions tau polynomiales de la hiérarchie de Korteweg-de Vries en une suite de polynômes qui satisfait une simple récurrence di érentielle, ce sont les polynômes d'Adler-Moser. Dans [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF], nous avons pu exhiber ce changement de variables (théorème 3.5.4 ; la preuve utilise essentiellement la théorie de la hiérarchie de Kadomtsev-Petviashvili. C'est l'objet du chapitre 3. Récemment, Kac et van de Leur contribuèrent de façon signi cative [START_REF] Kac | Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions[END_REF] en montrant que toutes les fonctions tau polynomiales de la hiérarchie de Kadomtsev-Petviashvili (incluant ainsi les hiérarchies de Gelfand-Dickey) peut être exprimée simplement comme un certain polynôme de Schur modulo un changement de variable. 9 Ils calculèrent aussi les fonctions tau polynomiales de la hiérarchie de Kadomtsev-Petviashvili modi ée, mais nous n'en parlerons pas ici.

La conjecture de Witten et la fonction tau topologique. Nous revenons à présent à la conjecture de Witten et ses conséquences. La conjecture de Witten stipule que la série génératrice des nombres d'intersection sur l'espace de module des courbes stable M ,n est en réalité une fonction tau de la hiérarchie de Korteweg-de Vries [START_REF] Witten | Two-dimensional gravity and intersection theory on moduli space[END_REF]. Cette conjecture fut prouvée par Maxim Kontsevich [START_REF] Kontsevich | Intersection theory on the moduli space of curves an the matrix Airy function[END_REF] et l'on appelle aujourd'hui cette fonction tau la fonction tau de Witten-Kontsevich. Depuis lors, les fonctions tau devinrent un objet essentiel connectant la géométrie complexe, la géométrie algébrique et les systèmes intégrables dans un domaine actif de la physique mathématique parfois appelé la symétrie miroir intégrable. Peu de temps après, la conjecture de Witten fut généralisée pour connecter les fonctions tau des hiérarchies de Gelfand-Dickey aux classes r -spin de Witten [START_REF] Witten | Algebraic geometry associated with matrix models of twodimensional gravity[END_REF]. En 2003 Dubrovin et Zhang montrèrent que la fonction génératrice des invariants de Gromov-Witten de la droite complexe projective P 1 est une fonction tau de la hiérarchie de Toda étendue [START_REF] Dubrovin | Virasoro Symmetries of the Extended Toda Hierarchy[END_REF] (qui n'est pas une hiérarchie de Drinfeld-Sokolov).

Les théorie cohomologiques des champs (CohFT pour cohomological eld theories), introduites par Kontsevich et Manin [START_REF] Manin | Gromov-Witten classes, quantum cohomology, and enumerative geometry[END_REF], sont des classes de cohomologie sur l'espace de module des courbes stables dé nies en vue d'uni er les classes r -spin de Witten ainsi que les théories de Gromov-Witten. Au début des années 2010, Fan, Jarvis et Ruan dé nirent les CohFT de Fan-Jarwis-Ruan-Witten en vue de généraliser les classes r -spin de Witten à n'importe quelle singularité complexe [START_REF] Fan | The Witten equation and its virtual fundamental cycle[END_REF][START_REF] Fan | The Witten equation, mirror symmetry, and quantum singularity theory[END_REF]. D'autre part, dans [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF], Dubrovin et Zhang montrèrent comment associer à n'importe quelle CohFT semisimple10 une hiérarchie qui coïncident, dans certains cas,11 à la hiérarchie de Drinfeld-Sokolov correspondante [START_REF] Givental | Simple singularities and integrable hierarchies[END_REF][START_REF] Frenkel | Soliton equations, vertex operators, and simple singularities[END_REF]. Dans la construction de Dubrovin-Zhang, la hiérarchie est dé nie de sorte que la série génératrice de la CohFT semisimple est une fonction tau. Cette fonction tau particulière est appelée la fonction topologique ; par exemple, la fonction tau de Witten-Kontsevich est la fonction tau topologique de la hiérarchie de Korteweg-de Vries. Depuis lors, la symétrie miroir intégrable devint une question de tisser des liens entre les CohFT et les hiérarchies intégrables. Bien que certains exemples marchent très bien, 12 (voir [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF] pour un clair exposé) il ne semble pas y avoir de schéma clair pour l'instant.

L'un des buts de la construction de Dubrovin-Zhang était de classi er les hiérarchies intégrables [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF][START_REF] Dubrovin | Frobenius manifolds and central invariants for the Drinfeld-Sokolov bihamiltonian structures[END_REF]. L'idée était d'utiliser la théorie de l'intersection sur l'espace de module des courbes stables 13 (et leurs généralisations : théorie de Gromov-Witten, théorie de Fan-Jarvis-Ruan-Witten) a n de dé nir des hiérarchies intégrables puis prouver que celles-ci sont équivalentes à des hiérarchies bien connues (telles que les hiérarchies de Drinfeld-Sokolov mais aussi d'autres). La relation d'équivalence entre les hiérarchies intégrables s'appelle une transformation de Miura : voir §1.4.2.

Les hiérarchies de double rami cation (traité au chapitre 5). Récemment, Alexander Buryak, inspiré par la théorie des champs symplectique [START_REF] Eliashberg | Introduction to symplectic eld theory[END_REF], a dé ni une nouvelle classe de hiérarchies appelées les hiérarchies de double rami cation [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF]. Elles sont construites à partir des cycles de double rami cation sur l'espace de module des courbes introduits par Graber et Vakil [START_REF] Graber | Relative virtual localization and vanishing of tautological classes on moduli spaces of curves[END_REF] (d'où le nom) et peuvent être dé nies pour n'importe quelle CohFT (pas seulement les CohFT semisimples, comme c'est le cas pour les hiérarchies de Dubrovin-Zhang). Dans ce même article, guidé par des exemples qu'il calcula, Buryak conjectura que dans le cas d'une CohFT semisimple, la hiérarchie de double rami cation est équivalente à la hiérarchie de Dubrovin-Zhang correspondante via une transformation de Miura. En particulier, pour la CohFT associée à la théorie FJRW d'une singularité complexe simple de type ADE, la hiérarchie de double rami cation, celle de Dubrovin-Zhang et celle de Drinfeld-Sokolov devraient coïncider selon la conjecture.

Les hiérarchies de double rami cation possèdent plusieurs propriétés qui leur donnent un intérêt particulier. (1) Elles sont dé nies pour n'importe quelle CohFT. (2) Elle sont construites en tant que hiérarchies Hamiltoniennes pour lesquelles le crochet de Poisson est très simple (appelée le crochet standard) ; en particulier, (3) une famille de densités Hamiltoniennes tau symétrique est aisément obtenue [START_REF] Buryak | Tau-structure for the Double Rami cation Hierarchies[END_REF]. (4) Leurs densités Hamiltoniennes satisfont des relations de récurrence fortes jusque lors inconnues [START_REF] Buryak | Recursion Relations for Double Rami cation Hierarchies[END_REF]. (5) Elles admettent une quanti cation relativement simple [START_REF] Buryak | Double Rami cation Cycles and Quantum Integrable Systems[END_REF][START_REF] Buryak | Integrable systems of double rami cation type[END_REF], mais nous n'en parlerons pas ici. De plus, l'équivalence conjecturale avec les hiérarchies de Dubrovin-Zhang a été augmentée en une transformation de Miura uniquement déterminée, d'où la conjecture DR/DZ forte 5.2.23 [START_REF] Buryak | Tau-structure for the Double Rami cation Hierarchies[END_REF]. Jusque là, l'équivalence DR/DZ forte a été con rmée dans plusieurs cas incluant les hiérarchies de Drinfeld-Sokolov de type A 1 à A 5 , la hiérarchie de Toda étendue et d'autres (voir 5.2.23).

Dans un papier accepté pour publication [START_REF] Du Crest De Villeneuve | Quantization of the D 4 Drinfeld-Sokolov hierarchy and Fan-Jarwis-Ruan-Witten theory[END_REF], nous avons con rmé l'équivalence DR/DZ forte pour la hiérarchie de Drinfeld-Sokolov associée à l'algèbre a ne de l'algèbre de Lie simple de type D 4 . La con rmation de la conjecture implique le calcul de la première structure Hamiltonienne de la hiérarchie de Drinfeld-Sokolov de type D 4 . Pour cela, nous avons utilisé la paire de Lax scalaire de la hiérarchie de type D n obtenue par Liu, Wu et Zhang [START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF]. Ces calculs, à notre connaissance, sont nouveaux vis-à-vis de a littérature.

La hiérarchie de Korteweg-de Vries : un exemple canonique

Dans cette section, nous présentons certains des aspect fondamentaux de l'équation et de la hiérarchie de Korteweg-de Vries (KdV). Cela servira d'illustration à de nombreuses propriétés des hiérarchies intégrables. Le chapitre 3 est dédié aux fonctions tau polynomiales de la hiérarchie KdV ; nous décrivons les résultats obtenus dans [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF] à la n de la section §1.2.

L'équation de Korteweg-de Vries

Mise à part ses applications en géométrie, l'équation KdV provient de mécanique des uides. Elle décrit l'évolution de la surface de l'eau dans un canal, ce dernier modélisant des eaux peu profondes s'écoulant essentiellement dans une seule direction. Si l'on ignore la direction transverse, on peut décrire la hauteur de la surface de l'eau par une fonction scalaire u(x, t) où t représente le temps et x représente la direction dans laquelle l'eau s'écoule. Alors u doit satisfaire l'équation KdV :

u t = 1 4 u x x x + 3 2 uu x , (1.5)
où u t = ∂ t u et u x = ∂ x u, etc. L'équation ci-dessus est en réalité une version normalisée de KdV où les constantes physiques ont été éliminées. L'équation KdV fut en premier théorisée par Boussinesq [START_REF] Boussinesq | Essai sur la theorie des eaux courantes[END_REF] puis proposée dans sa version actuelle par Korteweg et de Vries [START_REF] Korteweg | On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves[END_REF].

Pour la dérivation physique de l'équation (et celle de l'équation KP), nous référons à l'introduction de [START_REF] Ablowitz | Solitons, Nonlinear Evolutions Equations and Inverse Scattering[END_REF] par Ablowitz et Clarkson. Notons qu'en renormalisant u, x et t, on peut mettre des coe cients arbitraires devant chaque monôme u t , u x x x et uu x ; le choix 1, 1/4 et 3/2 deviendrait clair à la section §1.2.2. L'équation KdV peut être écrite de diverses façons, permettant ainsi des connections avec de nombreux domaines a priori déconnectés. Ici nous décrivons quatre représentations : (a) la paire de Lax scalaire, (b) la paire de Lax matricielle, (c) la représentation bi-Hamiltonienne et (d) la représentation bilinéaire.

(a) La paire de Lax scalaire. Considérons l'opérateur L = ∂ 2

x +u souvent appelé l'opérateur de Schrödinger. 14 Alors on peut réécrire l'équation (1.5) comme x := a t ∂ n x . En particulier, l'action de ∂ t sur les opérateurs satisfait la règle de Leibniz. Ainsi, ∂ t L = u t , mais c'est uniquement dû à la forme spéci que L. La paire (L, P) est ce qu'on appelle une paire de Lax scalaire pour l'équation KdV. Cette représentation est due à Peter D. Lax lui-même [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF].

∂L ∂t = [P, L] , P = ∂ 3 x + 3 2 u∂ x + 3 4 u x , (
Les équations (1.5) et (1.6) sont de nature di érente : l'équation (1.5) est une EDP non linéaire en u, tandis que l'équation (1.6) est une EDO linéaire en L et les deux côtés de l'équation sont des dérivations (des opérateurs de Leibniz) sur les opérateurs di érentiels. La formule pour l'opérateur P, à ce stade, semble plutôt ad hoc, mais elle entrera dans une élégant schéma avec la hiérarchie KdV. Un intérêt particulier de la paire de Lax scalaire est qu'elle permet de généraliser l'équation et la hiérarchie KdV à la hiérarchie de Kadomtsev-Petviashvili (voir §3.2).

(b) La paire de Lax matricielle. Considérons à présent l'opérateur di érentiel matricielle d'ordre 1 suivant :

L = ∂ x + 0 λ -u 1 0 . (1.8)
Ci-dessus, λ est une nouvelle indéterminée qu'on appelle le paramètre spectral. Toute identité doit est valable pour toute valeur complexe du paramètre λ. Considérons à présent la matrice suivante :

M = -1 4 u x λ 2 -1 2 uλ -1 2 (u) 2 -1 4 u x x λ + 1 2 u + 1 4 u x .
(1.9) (1.10) C'est équivalent à l'équation de courbure nulle

Alors ∂L ∂t -[L , M ] = 0 -u t + 1 4 u x x
∂ t N -∂ x M -[N , M ] = 0.
La paire (L , M ) est appelée une paire de Lax matricielle pour l'équation KdV. Cette représentation est due à Ablowitz, Kaup, Newell et Segur [START_REF] Ablowitz | The inverse scattering transform -Fourier analysis for nonlinear problems[END_REF]. Une fois de plus, les équations (1.6) sont (1.10) de nature di érente ; cette dernière n'implique que des opérateurs di érentiels d'ordre 1 et permet ainsi la représentation par courbure nulle. Un intérêt particulier est que les hiérarchies de Drinfeld-Sokolov sont construites génériquement comme des équations de Lax matricielles.

(c) La représentation bi-Hamiltonienne. On introduit l'anneau A = C[u, u x , u x x , . . .] dont nous appelons les éléments des polynômes di érentiels en u. L'opérateur ∂ x agissant sur A s'identi e alors avec l'opérateur ∂ x = k ≥0 u (k+1) ∂ u (k ) . Par ailleurs, on dé nit l'espace quotient suivant, 16 F := A/Im (∂ x ), dont nous appelons les éléments des fonctionnelles locales. Pour tout polynôme di érentiels f ∈ A, on note sa classe f ∈ F par f = ∫ f . 17 En tant que conséquence immédiate, on obtient une formule d'intégration par partie :

∫ f ∂ x = - ∫ ∂ x f , ou, plus généralement, pour k ≥ 1, ∫ f ∂ k x = (-1) k ∫ ∂ k x f . (1.11)
Puis l'on dé nit sur A l'opérateur δ /δu, appelé la dérivée variationnelle (aussi appelé l'opérateur d'Euler-Lagrange), par

δ f δu := k ≥0 (-1) k ∂ k x ∂ f ∂u (k) = ∂ f ∂u -∂ x ∂ f ∂u x + ∂ 2 x ∂ f ∂u x x -• • •
Alors pour tout f ne contenant pas de terme constant, δ f /δu = 0 si et seulement si f ∈ Im (∂ x ) (voir la proposition 5.1.1). En particulier, δ /δu est bien dé ni sur F . On dé nit ensuite deux applications, {-, -} 1 , {-, -} 2 : F ⊗ F → F by 18

f , 1 := ∫ δ f δu ∂ x δ δu , f , 2 := ∫ δ f δu 1 4 ∂ 3 x + u∂ x + 1 2 u x δ δu .
(1.12)

Ces applications sont antisymétriques et satisfont l'identité de Jacobi ; on appelle ce type d'applications des structures de Poisson locales. 19 De plus, toute combinaison linéaire λ{-, -} 1 + µ{-, -} 2 est encore une structure de Poisson locale ; on dit que les structures de Poisson sont compatibles. On dé nit aussi deux applications auxiliaires, notées pareillement {-, -} a : A ⊗ F → A, avec a = 1, 2, par

{ f , } 1 := k ≥0 ∂ f ∂u (k) ∂ k+1 x δ δu , { f , } 2 := k ≥0 ∂ f ∂u (k) ∂ k x 1 4 ∂ 3 x + u∂ x + 1 2 u x δ δu .
(1.13) Par construction, et grâce à l'équation (1.11), ces applications véri ent ∫ { f , } a = { f , } a , pour a = 1, 2. Ainsi, il est aisé de véri er que l'équation KdV (1.3) est équivalente aux deux équations suivantes :

u t = u, h 1 1 = ∂ x δh 1 δu , h 1 = ∫ 1 4 u 3 + 1 8 uu x x , (1.14)
u t = u, h 2 2 = 1 4 ∂ 3 x + u∂ x + 1 2 u x δh 2 δu , h 2 = ∫ 4 8 u 2 .
(1.15)

Les fonctionnelles locales h 1 et h 2 sont appelées des fonctionnelles Hamiltoniennes et les équations ci-dessus sont appelées des représentations Hamiltoniennes de l'équation KdV. Le fait que KdV admette deux représentations Hamiltoniennes avec deux structures de Poisson locales compatibles est ce qu'on appelle une représentation bi-Hamiltonienne. La représentation (1.14) fut trouvée simultanément en 1971 par Gardner [START_REF] Gardner | Korteweg-de Vries Equation and Generalizations. IV. The Korteweg-de Vries Equation as a Hamiltonian System[END_REF] et Fadeev et Zakharov [START_REF] Faddeev | Korteweg-de Vries Equation: A Completely Integrable Hamiltonian System[END_REF] (bien que Fadeev et Zahkarov l'attribuent à Gardner) ; on l'appelle la première représentation Hamiltonienne ou bien celle de Gardner-Zakharov-Fadeev. La représentation (1.15) est due à Magri [START_REF] Magri | A simple model of the integrable Hamiltonian equations[END_REF] et est appelée la seconde représentation ou bien celle de Magri. 20 Les représentations bi-Hamiltoniennes furent intensément étudiées par Dubrovin et Zhang (e.g. [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF][START_REF] Dubrovin | Frobenius manifolds and central invariants for the Drinfeld-Sokolov bihamiltonian structures[END_REF]), en particulier dans le but de classi er les hiérarchies intégrables. De plus, les hiérarchies de double rami cation, que nous étudions au chapitre 5, apparaissent naturellement dans une représentation Hamiltonienne.

(d) La représentation bilinéaire. Finalement, nous décrivons la représentation bilinéaire, ou de Hirota, de l'équation KdV. Contrairement aux représentations précédente, celle bilinéaire agit sur la fonction tau τ et non sur la solutions u. Comme mentionné précédemment, il y a diverses façons de dé nir la fonctions tau. Mais dans le cas de l'équation KdV, cela revient toujours à dé nir τ par la relation21 u = -2 ∂ 2 log τ ∂x 2 .

(1.16)

En particulier, à la donnée d'une solution u, la fonction tau τ n'est pas unique. On introduit à présent les dérivées bilinéaire, ou de Hirota, D t i avec i ≥ 1. On pose22 exp i ≥1

h i D t i (f , ) = f (t + h) • (t -h),
(1.17) où chaque côté de l'équation est développé en les puissances de h i et identi és en tant que séries en les h i . Par exemple, en posant x = t 1 et t = t 3 (les indices n'importent pas ici), on obtient

D x (f , ) = ∂ f ∂x -f ∂ ∂x , D x D t (f , ) = ∂ 2 f ∂x ∂t + f ∂ 2 ∂x ∂t - ∂ f ∂x ∂ ∂t - ∂ f ∂t ∂ ∂x .
Ainsi, c'est un peu long à véri er, mais l'équation KdV (1.5), en substituant u = -2∂ 2 x log τ , est équivalente à l'équation suivante, D 4

x -4D x D t (τ , τ ) = 0.

(1.18) L'équation ci-dessus est appelée une équation de Hirota, d'après Ryogo Hirota qui les introduisit dans [START_REF] Hirota | Direct method of nding exact solutions of nonlinear evolution equations[END_REF]. Cette équation est relativement compacte compte tenu du nombre de termes que les dérivées logarithmiques font apparaître ! D'aucuns se demanderaient quel est l'intérêt de ce type d'équations. C'est en réalité moins clair pour la hiérarchie KdV que pour d'autres hiérarchies plus compliquées. En particulier, lorsqu'une solution de la hiérarchie consiste en plusieurs fonctions u 1 , . . . , u N (et pas seulement u), il y a toujours une seule fonction τ associée à la solution. Ainsi, la représentation bilinéaire condense toute la hiérarchie sur une seule fonction. De plus, il se peut que la fonction tau soit plus simple à calculer que la solution elle-même. C'est par exemple l'approche des chapitres 4 et 4, bien qu'ils n'impliquent pas directement les équations bilinéaires.

La hiérarchie de Korteweg-de Vries

La hiérarchie KdV fut introduite en 1968 simultanément par Lax [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF] et Gardner-Kruskal-Miura-Zabusky [Miu68, GKM68, GKMZ70]. 23 Elle consiste en une suite d'équations commutant deux-à-deux (expliqué ci-dessous) et qui contient l'équation KdV (1.3), incluant ainsi l'équation KdV dans un système d'EDP in nie-dimensionnel. Dans ce qui suit, nous montrons comment construire la hiérarchie KdV en partant de (A) la paire de Lax scalaire, due Introduction en français à Lax [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF] ; puis nous discuterons (B) sa paire de Lax matricielle, (C) sa représentation bi-Hamiltonienne et (D) sa représentation bilinéaire. Sa paire de Lax matricielle est la plus technique ; nous la discuterons en réalité à la section 1.3 avec le cas général des hiérarchies de Drinfeld-Sokolov.

(A) Les paires de Lax scalaire. Pour les paires de Lax scalaires, il su t de considérer la racine carrée L 1/2 de l'opérateur de Schrödinger L = ∂ 2

x + u. Pour cela, nous commençons par étendre l'algèbre des opérateurs di érentiels en une algèbre

D = n i=-∞ a i ∂ i a i ∈ C[[x]], n ∈ Z , (1.19)
équipée du même produit dé ni à l'équation (1.7), mais où les coe cients binomiaux ont été généralisés à n k avec n < 0 en posant n k := n(n -1) • • • (n -k + 1)/k!. Nous appelons les éléments de D des opérateurs pseudo-di érentiels. Dans cette algèbre, il existe alors un opérateur

L 1/2 = ∂ x + k <0 p k ∂ k
x , avec des fonctions p k ∈ A, satisfaisant (L 1/2 ) 2 = L (voir théorème 3.1.8). Alors la hiérarchie KdV peut être formulée de la façon suivante,

∂L ∂t i = (L i/2 ) + , L i > 0 odd , (1.20)
où (-) + désigne la projection sur le sous-espace de D contenant seulement des puissances positives ou nulles de ∂ x (i.e. les opérateurs di érentiels). En particulier, (L 3/2 ) + = P = ∂ 3 x + 3 2 u∂ x + 3 4 u x , voir l'équation (1.6). Cette formulation de Lax scalaire de la hiérarchie KdV se généralise aisément à la n-ième hiérarchie de Gelfand-Dickey où l'opérateur L = ∂ 2

x + u est remplacé par l'opérateur L = ∂ n

x + u n-2 ∂ n-2

x + • • • + u 0 . Ainsi, les ots dé nis par le système (1.20) commutent deux-à-deux dans le sens où, pour tout entiers positifs i, j,

∂ ∂t i (L j/2 ) + , L = ∂ ∂t j (L i/2 ) + , L .
(1.21) C'est ce qu'on appelle la condition d'intégrabilité de la hiérarchie KdV. La preuve est classique et peut être trouvée par exemple dans [START_REF] Date | Solitons, Di erential equations, symmetries and in nite dimensional algebras[END_REF] ; nous donnons une preuve tout à fait similaire pour la hiérarchie KP au théorème 3.2.1. Gardant à l'esprit que L = ∂ 2 x + u, il s'en suit de l'équation (1.21) que ∂ t i ∂ t j u = ∂ t j ∂ t i u. C'est évidemment une condition nécessaire à l'existence de solutions au moins C 1 , sans parler de solutions lisses. Notons que la fonction u doit à présent dépendre d'une in nité de variables {t i | i > 0 odd}. Nous supposerons en général que

u ∈ C [[t 1 , t 3 , t 5 , . . .]] .
Comme nous l'avons dit, nous traiterons la paire de Lax matricielle à la section 1.3 avec les hiérarchies de Drinfeld-Sokolov. Traitons à présent la représentation bi-Hamiltonienne.

(C) La représentation bi-Hamiltonienne. Soit un opérateur pseudo-di érentiel A ∈ D, on note res (A) son coe cient en ∂ -1

x . Alors la représentation bi-Hamiltonienne est donnée par les deux mêmes structures de Poisson locales de l'équation (1.12), et par les densités Hamiltoniennes données par les résidus des puissances fractionnaires de L = ∂ 2

x + u, à savoir,

u t i = u, h 1,i 1 = ∂ x δh 1,i δu , h 1,i = 4 i + 2 ∫ res L i +2 2 , u t i = u, h 2,i 2 = 1 4 ∂ 3 x + u∂ x + 1 2 u x δh 2,i δu , h 2,i = 4 i ∫ res L i 2 ,
pour i > 0 impair. En particulier, en comparant avec les équations (1.14) et (1.15), h 1,3 = h 1 et h 2,3 = h 2 . De plus, les structures de Poisson locales peuvent être reformulées via l'opérateur L comme suit. Pour toute fonctionnelle locale f ∈ F , on note

δ f δ L = ∂ -1 x • δ f δ u ∈ A, 24 et soient X = δ f δ L et Y = δ δ L , alors { f , } 1 = ∫ res [X LY -LXY ] , { f , } 2 = ∫ res [(LX ) + LY -(X L) + Y L] .
(1.22) (D) La représentation bilinéaire. Finalement, pour la représentation bilinéaire, on doit introduire les polynômes de Schur élémentaires , notés p k (t) avec k ∈ Z >0 , dé nis par la relation

exp i ∈Z >0 t i z i = k ∈Z >0 p k (t)z k .
(1.23) par exemple, p 1 = t 1 , p 2 = 1 2 t 2 1 + t 2 , p 3 = 1 6 t 3 1 + t 2 t 1 + t 3 . Notons à présent pk le polynôme p k où toutes les variables t 2i ont été mises à 0. Alors la hiérarchie KdV (1.20), une fois e ectuée la substitution u = -2∂ 2

x log τ , est équivalente aux deux suites d'équations bilinéaires suivantes, 25

pk+1

([D]) -1 2 D x D t k (τ , τ ) = 0, D x D t k -D x pk ([D]) (τ , τ ) = 0, k > 0 odd , (1.24)
où [D] := (D x , 1 3 D t 3 , 1 5 D t 5 , . . .). Les équations ci-dessus peuvent être déduites des résultats de [START_REF] Date | Transformation groups for soliton equations[END_REF] (voir aussi [START_REF] Adler | Non-linear PDEs for gap probabilities in random matrices and KP theory[END_REF], lemme 4.1). En particulier, pour k = 3, pk+1

([D]) -1 2 D x D t k = 1 24 (D 4
x -4D x D t 3 ) et l'on retrouve la forme bilinéaire (1.18) de l'équation KdV.

Fonctions tau polynomiales de la hiérarchie KdV et les polynômes d'Adler-Moser

3, §3.5. Dans [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF], Adler et Moser introduisirent une suite de polynômes θ n , pour n ≥ 0, dépendant de paramètres r = {r i | i ∈ Z odd >0 }, r 1 = x, dé nis par la récurrence di érentielle suivante : θ 0 = 1; θ 1 = x; puis

∂θ n+1 ∂x θ n-1 -θ n+1 ∂θ n-1 ∂ x = (2n -1) (θ n ) 2 .
(1.25)

Lors du calcul de θ n , la constant d'intégration choisie est r 2n-1 . Par exemple, θ 2 = x 3 + r 3 et θ 3 = x 6 + 5r 3 x 3 + r 5 x -5r 2 3 . Ces polynômes sont désormais connus sous le nom de polynômes d'Adler-Moser. Dans ce même article, ils prouvèrent l'existence d'un unique changement de variables r → t transformant le polynôme d'Adler-Moser θ n (r) en une fonction tau τ n (t) de la hiérarchie KdV dans le sens de l'équation (1.16). De plus, d'après des résultats de [START_REF] Airault | Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem[END_REF], ils prouvèrent que via ce changement de variables, l'on retrouve toutes les fonctions tau polynomiales ainsi que toutes les solutions rationnelles de KdV, classi ant ainsi les solutions rationnelles. Cependant, cet unique changement de variables n'était pas connu. Dans [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF], nous avons prouvé le théorème suivant.

Théorème 1.2.1 ( [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF]). Le changement de variables suivant transforme les polynômes d'Adler-Moser en les fonctions tau polynomiales de la hiérarchie KdV :

r 1 = t 1 = x, et i ≥2 r 2i-1 α 2i-1 z 2i-1 = tanh i ≥2 t 2i-1 z 2i-1 . (1.26) où α 2i-1 = (-1) i-1 3 2 5 2 • • • (2i -3) 2 (2i -1).
Voir théorème 3.5.4 dans le document. On peut interpréter le théorème 1.2.1 comme une renormalisation des variables, plus le choix di érent, non trivial, d'une constante d'intégration pour la récurrence (1.25). Les premiers polynômes de la récurrence sont les suivants.

τ 0 = 1, τ 2 = x 3 -3t 3 , τ 1 = x, τ 3 = x 6 -15t 3 x 3 -45t 2 3 + 45t 5 x .
Ce sont des fonctions tau polynomiales bien connues, voir e.g. [START_REF] Hirota | The Direct Method in Soliton Theory[END_REF] où elles sont calculées par des méthodes similaires via les polynômes de Schur élémentaires (voir équation (1.23)). La preuve du théorème 1.2.1 est donnée à la section 3.5 ; elle repose essentiellement sur la théorie KP que nous développons à la section 3.2. Cela constituera un futur projet que de généraliser les polynômes d'Adler-Moser ainsi que les résultats de [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF] et le théorème cidessus aux hiérarchies de Drinfeld-Sokolov en général (i.e. pour une algèbre de Lie simple arbitraire).

Fonctions tau polynomiales des hiérarchies de Drinfeld-Sokolov

Ici nous décrivons les résultats obtenus dans [START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF]

g = n-⊕ h ⊕ n+ ,
où h est une sous-algèbre abélienne maximale dont les éléments H ∈ h sont tels que les applications adjointes ad H = [H , -] : g → g sont simultanément diagonalisables. La sous-algèbre h est appelée une sous-algèbre de Cartan de g et sa dimension r = dim h est appelée le rang de g ; le rang est unique. Un résultat classique est que g est complètement déterminée (à isomorphisme près) par une unique matrice (C i j ) ∈ Mat(r, Z), appelée sa matrice de Cartan, à laquelle on peut associer un ensemble de générateurs {H i , E i , F i | 1 ≤ i ≤ r } de g satisfaisant les relations suivantes :

[E i , F j ] = δ i j H i , [H i , E j ] = C i j E i , [H i , F j ] = -C i j F j , [H i , H j ] = 0,
Alors h est générée par les H i , tandis que n+ (resp. n-) est générée par les E i 's (resp. the F i 's). De tels générateurs s'appellent des générateurs de Weyl. On a deux éléments spéciaux E θ ∈ n + et E -θ ∈ n -qui satisfont les relations suivantes : pour tout i ∈ {1, . . . , r },

[E θ , E i ] = 0, [E -θ , F i ] = 0.
Par exemple, l'algèbre de Lie simple sl(2, C) (matrices complexes 2 × 2 de trace nulle) est de rang 1 et est générée par les matrices suivantes,

H = 1 0 0 -1 , E = 0 0 1 0 , F = 0 1 0 0 . Ces matrices satisfont les relations [H , E] = 2E, [H , F ] = -2F et [E, F ] = H de sorte que la matrice de Cartan de sl(2, C) est C = (2) ∈ Mat(1, Z). De plus, dans ce cas particulier, E θ = E et E -θ = F .
L'algèbre a ne associée. À l'algèbre nie-dimensionnelle semisimple g on associe une algèbre a ne (non twistée)

g := g ⊗ C[λ, λ -1 ] ⊕ Cc, (1.27)
qui est in nie-dimensionnelle. On équipe cette algèbre du crochet de Lie suivante : soient

X , Y ∈ g, p, q ∈ Z et a, b ∈ C, [X λ p + ac, Y λ q + bc] = [X , Y ]λ p+q + pδ p+q,0 (X , Y ) 0 • c, (1.28)
où (-, -) 0 est une certaine forme bilinéaire symétrique non dégénérée sur g. Clairement, c est un élément central de g (i.e. c commute avec tous les autres éléments de g), faisant de Cc un idéal non trivial de g. Si l'algèbre de Lie semisimple g est de type X n , on note X (1) n l'algèbre a ne associée g. Les algèbres a nes furent intensément étudiées par V. Kac, qui les introduisit parallèlement à R. Moody [START_REF] Kac | Simple irreducible graded lie algebras of nite growth[END_REF][START_REF] Moody | A New Class of Lie Algebras[END_REF]. Les détails sont donnés dans §4.2.

L'algèbre a ne g est générée par les éléments suivants :

e i = E i , f i = F i , αi = H i , e 0 = E -θ λ, f 0 = E θ λ -1 , α0 = H 0 + c, où H 0 = [E θ , E -θ ]
, avec i ∈ {1, . . . , r } (r = dim h étant le rang de g). Puis on dé ni deux gradations importantes : la gradation principale et la gradation homogène, comme suit, Principal:

deg e i = 1, deg f i = -1, deg αi = 0, g = k ∈Z g k , Homogeneous: deg e i = δ i,0 , deg f i = -δ i,0 , deg αi = 0, g = k ∈Z g k .
La gradation homogène, comme son nom l'indique, revient à poser deg λ = 1 et deg g = deg c = 0. On note g + (resp. g + ) le sous-espace de des éléments de degré principal (resp. homogène) ≥ 0 ; on pose g -= gg + et g -= gg + . On introduit à présent l'élément cyclique principal a ne

Λ = e 0 + e 1 + . . . + e r = E -θ λ + E 1 + . . . + E r .
(1.29)

En particulier, Λ ∈ g 1 (degré principal égal à 1). Un résultat fondamental (voir [START_REF] Kac | In nite dimensional Lie algebras[END_REF]) sur les algèbres a nes est la décomposition de g comme

g = s + Im (ad Λ ) , s ∩ Im (ad Λ ) = Cc.
où s = Ker (ad Λ ). On note s k = s ∩ g k le sous-espace de degré principal k. On appelle exposants a nes de g les entiers k ∈ Z tels que dim s k 0 et on note m k = dim s k la multiplicité de k.

On note

E = k ∈ Z dim s k 0 le multi-ensemble des exposants a nes de g, c'est-à-dire que l'élément k ∈ E a multiplicité m k . On note Λ k , Λ k , Λ k , .
. . (m k fois) les éléments engendrant s k linéairement. Ces générateurs Λ k (on omet les primes pour l'instant) peuvent être choisis de sorte qu'ils satisfassent les relations de Heisenberg

[Λ i , Λ j ] = iδ i+j,0 • c.
(1.30)

On appelle s la sous-algèbre de Heisenberg principale de g. On note ν ∈ C \ {0} le nombre complexe tel que Λ 1 = ν Λ. Par exemple, pour l'algèbre a ne g = sl(2, C) ⊗ C[λ, λ -1 ] ⊕ Cc l'élément cyclique principal a ne est

Λ = F λ + E = 0 λ 1 0 .
Les exposants a nes de g sont exactement les nombres impairs et ont tous multiplicité 1, i.e. E = Z odd . Les générateurs de la sous-algèbre de Heisenberg principale sont simplement les

Λ 2i+1 = Λ 2i+1 = λ i Λ pour i ∈ Z.
Les hiérarchies de Drinfeld-Sokolov. Nous avons à présent tout le matériel nécessaire à la description des hiérarchies de Drinfeld-Sokolov. Premièrement, on note E + = E ∩ Z >0 l'ensemble des exposants a nes positifs de g. On note alors t

= {t i | i ∈ E + }, avec t 1 = ν -1 x, et l'on considère l'algèbre suivante, g ⊗ C [[t]] = m ≥0 i 1 , ...,i m ∈E + k 1 , ...,k m ≥1 ξ k 1 , ...,k m i 1 , ...,i m • t k 1 i 1 • • • t k m i m ξ k 1 , ...,k m i 1 , ...,i m ∈ g .
L'algèbre ci-dessus (et ses sous-algèbres) jouera le rôle d'« espace de fonctions » de la théorie.

Le crochet de Lie est dé ni de façon similaire à celui sur g ⊗ C[λ, λ -1 ], i.e. on pose [ξt i , ηt j ] = [ξ , η]t i t j puis on étend par multiplicativité. On considère à présent l'opérateur matricielle d'ordre 1 suivant,

L = ∂ x + Λ + Q, Q ∈ b [[t]] ,
où b = h ⊕ n+ (typiquement, pour sl(n, C), la sous-algèbre b consiste en les matrices triangulaires inférieures de trace nulle). Nous appelons de telles opérateurs des connexions (bien qu'il n'y ait pas de variété ni de bré pour ainsi dire) et nous notons op g l'ensemble de ces connexions. 26 On considère ensuite le groupe multiplicatif suivant :

N := exp (ad S ) | S ∈ n [[t]]
et le faisons agir sur op g par

e ad S L = ∂ x + e ad S (Λ + Q) -∇ S S, ∇ S S := k ≥0 1 (k + 1)! (ad S ) k ∂ x (S).
Les raisons pour dé nir cette action et le fait qu'elle est bien dé nie sont expliqués à la section §4.3.1. Alors il n'est pas di cile de montrer, qu'en e et, pour tout L ∈ op g , on a e ad S L ∈ op g . Les hiérarchies de Drinfeld-Sokolov sont dé nies sur les classes d'équivalences de ces connexions L ∈ op g , plutôt que sur les connexions elles-mêmes. À présent nous devons étendre notre espace de fonctions de la façon suivante :

g := g + ⊕ g -, g + = g ≥0 = k ≥0 g k , g -= k <0 g k .
De la même façon, on dé nit naturellement g [[t]] et ses sous-algèbres. De telles dé nitions sont implicites pou g + et g -.

Un résultat fondamental qui fut prouvé pour la première fois par Drinfeld et Sokolov [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] (ici nous donnons une version un peu di érente due à Wu [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF]) est l'existence d'une unique fonction U ∈ g -[[t]] telle que la connexion L 0 := e -ad U L prenne la forme

L 0 = ∂ x + Λ + H , H ∈ s -[[t]] , (
∂L ∂t i = (e ad U Λ i ) + , L i ∈ E + . (1.32)
L = ∂ x + Λ + Q, Λ = 0 λ 1 0 , Q = 0 -u 0 0 , Λ 2i+1 = λ i Λ = 0 λ i+1 λ i 0 .

Fonctions tau comme déterminants de Toeplitz

Ω = i ∈E + Ω i i Λ -i dé nie par ∂ x Ω = -H , où H ∈ s -[[t]
] est dé nie à l'équation (1.31). Ainsi le fonctions Ω j ne sont pas des polynômes di érentiels en les composantes de Q, cependant les fonctions ∂ t i Ω j le sont et satisfont la relation de fermeture

∂ t i Ω j = ∂ t j Ω i . En posant H = i ∈E + -i -1 hi Λ -i , on obtient ∂ hi ∂t j = ∂ hj ∂t i . ( 1
T(γ ) = (γ I -) I, ≥0 = γ 0 γ -1 γ -2 γ 1 γ 0 γ -1 . . .
Ξ(t; λ) := j ∈E + t i Λ i ∈ s + [[t]] .
Alors (1) la fonction X (t) = e Ξ(t;λ) e X est une série formelle bien dé nie de gl

(n)[[λ, λ -1 , t]] 29 ; (2) la limite lim N →∞ det T N ( X (t)) est une série formelle bien dé nie de C [[t]] et (3) la fonction τ X (t) := lim N →∞ det T N ( X (t)) κ (1.36)
est une fonction tau de la hiérarchie g-DS au sens de l'équation (1.34). Ici κ est un nombre associé à la représentation π : g → gl(n) par (Y , Z ) 0 = κ • tr(π (Y )π (Z )) pour tous Y , Z ∈ g (avec (-, -) 0 la forme bilinéaire standard sur g). Typiquement, pour les algèbres de Lie de type

A n et C n , on a κ = 1 et pour les types B n et D n on a κ = 1/2.
Plus précisément, la fonction Θ de l'équation (1.34) à laquelle la fonction tau τ X (t) est associée est celle telle que Θ| t=0 = e X . Il est facile de montrer que pour tout X ∈ g -, il existe une telle fonction Θ ; voir le lemme 4.4.2.

Fonctions tau polynomiales et applications

Nous expliquons à présent les résultats de [START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF] sur les fonctions tau de Drinfeld-Sokolov et applications ; voir chapitre 4, §4.5. On choisit un élément X ∈ g -et la fonction tau associée τ X (t) de l'équation (1.36). Notre premier résultat est une expansion en série entière de τ X (t) sur les partitions d'entiers (ou, de façon équivalente, sur les diagrammes de Young) ; on note Y l'ensemble des partitions d'entiers. Pour ce faire, il nous faut introduire la matrice de Laurent L(γ ), qui est aussi constante le long des diagonales, mais de taille « Z × Z, » à savoir

L(γ ) = γ I -I, ∈Z = . . . . . . . . . . . . γ 0 γ -1 γ -2 . . . γ 1 γ 0 γ -1 . . . γ 2 γ 1 γ 0 . . . . . . . . . . . . .
On considère donc les matrices de Laurent r := L(e X ) et s(t) := L(e Ξ(t;λ) ) ainsi que les déterminants suivants,30

r µ := det r i-µ i -1; j-1 i, j ∈[1, (µ)] ∈ C, s µ (t) := det s i-1; j-µ j -1 i, j ∈[1, (µ)] ∈ C[t], (1.37)
où µ ∈ Y est n'importe quelle partition de longueur (µ). On appelle r µ les coordonnées de Plücker de e X et on appelle s µ (t) les polynômes de Schur généralisés de type (g, π ). Dans le cas où g = sl(n, C) et π est la représentation fondamentale, les polynômes s µ (t) coïncident avec les polynômes de Schur classiques [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF], une fois que l'on a posé t k = 0 pour tout k = 0 mod n. Nous avons le prouvé le théorème suivant. C'est une application de la formule de Cauchy-Binet et une généralisation des travaux fondateurs de Sato sur les fonctions tau de la hiérarchie KP [START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF].

Théorème 1.3.1 ([CDY18]
). Soit X ∈ g -, la fonction tau de l'équation (1.36) admet l'expansion suivante en terme de partitions d'entiers :

τ X (t) 1 κ = µ ∈Y s µ (t) • r µ = 1 + s r + s r + s r + s r + • • • (1.38)
Voir théorème 4.5.2 dans le document. À partir de l'équation (1.38) on trouve un critère facile pour la polynomialité de la fonction tau comme suit. Si la donnée initiale X ∈ g -est telle que π (X ) est une matrice nilpotente, alors les composantes de la matrice de Laurent r = L(e X ) s'annulent lorsque l'on s'éloigne su samment de la diagonale principale. Conséquemment, pour une partition µ su samment longue, les coordonnées de Plücker r µ de X nissent par s'annuler. Cela prouve l'assertion suivante, ∀ X ∈ g -, π (X ) est une matrice nilpotente =⇒ τ X (t) Cela constituera une futur projet que d'étudier les propriétés de ces polynômes. Notamment, en les voyant comme des polynômes en x avec paramètres t * = {t i | i 1}, l'on devrait étudier comment les zéros de ces polynômes évoluent par rapport à t * . Comme noté à la section §1.1, dans le cas A 1 , les zéros des polynômes d'Adler-Moser évoluent en formant un système de type Calogero-Moser [START_REF] Airault | Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem[END_REF][START_REF] Wilson | Collisions of Calogero-Moser particles and an adelic Grassmannian[END_REF]. C'est encore à comprendre pour les hiérarchies de Drinfeld-Sokolov en général.

Fonctions tau polynomiales et équations bilinéaire. Les fonctions polynomiales ont le mérite d'être des expressions nies et sont ainsi bien plus simples à manipuler dans les calculs pratiques. Comme application, nous pouvons détecter les équations bilinéaires (i.e. similaires à celles de l'équation (1.24)) que les fonctions tau de la hiérarchie doivent satisfaire. La forme bilinéaire des hiérarchies de Drinfeld-Sokolov est connue explicitement pour toutes les hiérarchies de Gelfand-Dickey (i.e. pour sl(n, C)), voir équation (4.98). Cependant, à notre connaissance, de telles équations bilinéaires explicites pour toute hiérarchie de Drinfeld-Sokolov ne sont pas données dans la littérature. Cependant, elles doivent être déductibles des résultats de la série de papier fondatrice « Transformation groups for soliton equations » [KM81]- [START_REF] Date | Transformation groups for soliton equations[END_REF].

Nous avons donc injecté les fonctions tau polynomiales que nous avions calculées dans des ansatz d'équations bilinéaires et avons requis que ces équations s'annulent. Pour des fonctions tau polynomiales de degré su samment haut, cela détermina les coe cients des dites équations bilinéaires. Bien entendu, cela ne signi e pas que toutes les fonctions tau de la hiérarchie doivent satisfaire ces équations. Cependant, nous pensons que cela donne de bonnes indications pour identi er ces équations.

Dans ce qui suit, HDS

[deg≤ ] X n désigne l'espace vectoriel des équations bilinéaires de degré ≤ que les fonctions de la hiérarchie DS de type X n (A n ou B n , etc.) satisfont. Le degré des dérivées de Hirota (introduites à l'équation (1.17)) est dé ni par deg D i = i où D i = D t i . Nous avons obtenu les propositions suivantes ; premièrement pour le cas B 2 .

Proposition 1.3.2. On a les estimations de dimension suivantes,

dim C HDS [deg≤4] B 2 = 0, dim C HDS [deg≤6] B 2 ≤ 1, dim C HDS [deg≤8] B 2 ≤ 2.

De plus, les éléments de HDS

[deg≤8] B 2 sont combinaisons linéaires des équations suivantes,

(D 6 1 -5D 3 1 D 3 -5D 2 3 + 9D 1 D 5 )(τ , τ ) = 0, (D 8 1 + 7D 5 1 D 3 -35D 2 1 D 2 3 -21D 3 1 D 5 -42D 3 D 5 + 90D 1 D 7 )(τ , τ ) = 0.
Les équations ci-dessus peuvent être trouvées dans [START_REF] Date | Transformation groups for soliton equations II: Vertex operators and tau functions[END_REF] comme étant les deux premières équations bilinéaires non triviales de la hiérarchie BKP. Cette dernière contient toutes les hiérarchies de type B n et D n (les types orthogonaux spéciaux). Dans la même façon, la hiérarchie KP contient toutes les hiérarchies de type A n .

Nous avons une proposition similaire pour la hiérarchie de type D 4 . Notons que pour le cas D 4 , on a deux variables associées à l'exposant 3 : t 3 et t 3 . C'est dû au fait que la sous-algèbre de Heisenberg s de type D 4 possède un sous-espace dimension 2 pour tout exposant dans 3Z.

Proposition 1.3.3. On a les estimations de dimension suivantes,

dim C HDS [deg≤4] D 4 = 0, dim C HDS [deg≤6] D 4 ≤ 3.

De plus, les éléments de HDS

[deg≤6] D 4 sont combinaisons linéaires des équations suivantes,

(2D 3 1 D 3 + 4D 3 D 3 -3D 2 3 )(τ , τ ) = 0, (D 3 1 D 3 -D 3 1 D 3 + D 3 D 3 -D 2 3 )(τ , τ ) = 0, (D 6 1 + 9D 1 D 5 -10D 3 1 D 3 + 5D 3 1 D 3 -5D 3 D 3 )(τ , τ ) = 0.
i : M ,n → M ,n+1 en posant s i (x) = xi ∈ π -1 (x)
, où xi est l'i-ème marque sur la courbe π -1 (x).

Puis on les brés suivants. Les brés en droite L i → M ,n , pour i ∈ {1, . . . , n}, sont construits de telle façon que les bres

L i | x en un point x ∈ M ,n satisfont L i | x T * xi π -1 (x), où xi = s i (x).
Ces brés sont appelés les brés tautologiques. Le bré H → M ,n est dé ni de sorte que H| x T * π -1 (x). Ce bré est appelé le bré de Hodge et est de rang . On peut considérer les classes de Chern de ces brés : (i) c ,n est équivariante par rapport à la permutation simultanée des copies de V dans V ⊗n et celle des marques sur les courbes ; (ii) c 0,3 (e 1 , e α , e β ) = η α β ; (iii) π * c ,n (e α 1 , . . . , e α n ) = c ,n+1 (e α 1 , . . . , e α n , e 1 ); (iv) σ * c ,n (e α 1 , . . . , e α n ) = c 1 ,n 1 +1 (e 1 , . . . , e n 1 , e µ )η µν c 2 ,n 2 +1 (e ν , e 1 , . . . , e n 2 ); (v) τ * c +1,n (e α 1 , . . . , e α n ) = η µν c ,n+2 (e µ , e ν , e α 1 , . . . , e α n ). In ne, on demande à ce que les classes c ,n « se comportent naturellement » par rapport aux applications π , σ et τ , considérant la métrique η α β .

ψ i := c 1 (L i ) ∈ H 2 (M ,n , Q), λ j := c j (H) ∈ H 2j (M ,n , Q).
Soit une CohFT {c ,n | , n ≥ 0}, on dé nit les nombres d'intersection (ou corrélateurs) associés à la CohFT par

τ d 1 (e α 1 ) • • • τ d n (e α n ) := ∫ M , n c ,n (e α 1 , . . . , e α n ) • ψ d 1 1 • • •ψ d n n ∈ C. avec α i ∈ {1, . . . , N } et d j ≥ 0.
Ces nombres s'annulent si le degré cohomologique de l'intégrande n'est pas égal à 3 -3 + n. On dé nit alors le potentiel F de la CohFT comme la série génératrice des nombres d'intersection, c'est-à-dire,

1 n! d 1 , ...,d n ≥0 τ d 1 (e α 1 ) • • • τ d n (e α n ) • t α 1 d 1 • • • t α n d n . (1.39) et F (t * * ; ε) = ≥0 ε 2 F (t * * ). Cela introduit des variables t * * := {t α i d j | α i ∈ {1, . . . , N }, d j ≥ 0} parfois appelées les variables topologiques.
Par exemple, la CohFT triviale a pour espace vectoriel V = C, métrique η = 1 et classes c ,n = 1. Elle joue un rôle important car elle est reliée à la hiérarchie KdV [START_REF] Witten | Two-dimensional gravity and intersection theory on moduli space[END_REF] comme nous le verrons ci-dessous. Dans la même veine, la CohFT de Hodge a les mêmes caractéristiques, sauf que les classes sont données par c ,n = Λ(s) := 1 + j=1 λ j s j . Cette dernière est reliée à la hiérarchie d'« onde longue intermédiaire » [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF].

Un autre exemple important et la théorie de Gromov-Witten d'une variété projective lisse X équipée d'une classe d'homologie β ∈ H 2 (X , Z). Elle est construite à partir de l'espace de module M ,n (X , β) des courbes de genre , avec n marques (C, (x i )), avec une application stable f : C → X telle que f * [C] = β. Nous rappelons la construction à l'exemple 5.2.3. L'espace vectoriel de la CohFT est donné par V = H * (X , C) et avec la métrique η( , w) = ∫ P( ) w, où P : H * (X , C) → H * (X , C) est la dualité de Poincaré. Les classes sont dé nies par c ,n (e α 1 , . . . , e α N ) = p * ev * (e α 1 , . . . , e α N ),

(1.40)

où p : M ,n (X , β) → M ,n envoie (C, (x i ), f ) sur (C, (x i )) (oublie l'application f ) et où ev : M ,n (X , β) → X n envoie (C, (x i ), f ) sur (f (x i )) (évalue l'application f ). La dé nition formelle est relativement complexe et implique la classe fondamentale virtuelle [M ,n (X , β)] vir intro- duite par Behrend et Fantechi [BF97].
Théorie de Fan-Jarvis-Ruan-Witten. Un dernier exemple important pour la suite est la théorie FJRW d'une paire (W , G). Ici W ∈ C[z 1 , . . . , z m ] est un polynôme quasi-homogène avec poids q 1 , . . . , q n possédant une singularité isolée à l'origine. Et G est un groupe de matrices diagonales γ = (γ 1 , . . . , γ m ) qui laissent le polynôme W invariant, et qui contient l'élément = (e 2iπ q 1 , . . . , e 2iπ q m ). L'espace vectoriel est dé ni par

V = γ ∈G (Q W γ ⊗ dz γ ) G , où W γ = W | Fix(γ ) et Q W γ et son anneau Jacobien et où dz γ = z i ∈Fix(γ ) dz i .
On donne une brève revue de la théorie FJRW associée et de la CohFT associée à la section §5.2.1, voir [START_REF] Fan | The Witten equation and its virtual fundamental cycle[END_REF][START_REF] Fan | The Witten equation, mirror symmetry, and quantum singularity theory[END_REF]. Puisque les polynômes quasi-homogènes possédant une singularité isolée sont classi ées par les diagrammes de Dynkin de type ADE, on a une correspondance entre les théories FJRW et les algèbres de Lie semisimples du type correspondant.

Les hiérarchies de Dubrovin-Zhang. Les hiérarchies de Dubrovin-Zhang (DZ) sont des hiérarchies d'EDP du même type que celui des hiérarchies de Drinfeld-Sokolov décrites à la section §1.3.1. Elles sont dé nie pour tout CohFT semisimple. Cette dernière condition signi e que le potentiel F de la CohFT de l'équation (1.39) doit être tel que l'algèbre complexe Ndimensionnelle dé nie par les constantes de structure

f α βγ = η α µ ∂ 3 F 0 ∂t µ 0 ∂t γ 0 ∂t β 0 t * p >0 =0
est semisimple. On note

Ω α,p; β,q (t * * ; ε) = ∂ 2 F ∂t α p ∂t β q = ≥0 Ω [2 ]
α,p; β,q ε 2 .

Puis l'on dé nit des densités Hamiltoniennes et une structure de Poisson locale par

h ,(0) α,p ( * ) := Ω [0] α,p+1; 1,0 (t * 0 = * , 0, 0, . . .), K α β ,(0) := η α β ∂ x .
Ces densités Hamiltoniennes commutent vis-à-vis du crochet ci-dessus [START_REF] Dubrovin | Virasoro Symmetries of the Extended Toda Hierarchy[END_REF]. Cela forme la hiérarchie DZ en genre 0, ou la hiérarchie principale de la CohFT. Pour construire la hiérarchie en genre supérieur, on utilise la solution topologique ; c'est la solution α de la hiérarchie DZ déterminée par la donnée initiale

α (x, t * * = 0) = δ α,1 x .
Alors il existe des fonctions rationnelles P α p telles que [Get02, BPS12]

F (t * 0 , t * 1 , t * 2 , . . .) = F (P * 0 ( * 0 , . . . , * 3 -2 ), . . . , P * 3 -2 ( * 0 , . . . , * 3 -2 ), 0, 0, . . .) | x =0 .
Considérons à présent le changement de coordonnées

φ( α ) = w α := α + ≥1 ε 2 ∂ 2 F ( * 0 , . . . , * 3 -2 ) ∂t α 0 ∂x . (1.41)
La hiérarchie DZ en genre supérieur est alors obtenue en appliquant la transformation inverse φ -1 à la hiérarchie en genre 0. C'est-à-dire,

h α,p (w * * ) := h ,(0) α,p • φ -1 (w * * ), K α β (w * * ) = ((φ -1 ) * K ,(0) ) α β (w * * ),
où (φ -1 ) * K ,(0) est l'opérateur di érentiel transformé ; la transformation est donnée à l'équation (5.18). Ces données forment un système Hamiltonien au sens de l'équation (1.14), i.e.

∂w α ∂t α p = {w α , h α,p } = N β =1 K α β δh α,p δu β . (1.42)
Les Hamiltoniens h α,p appartiennent à l'anneau

A := C u α , ε u α k >0
des polynômes di érentiels étendus. De façon similaire, on note F = A/Im(∂ x ). On dé nit une gradation sur cet anneau en posant deg u α k = k et deg ε = -1 de sorte que ε compte le nombre de dérivées dans une monôme de degré total 0. Le paramètre ε est souvent appelé le paramètre de dispersion et joue un rôle important dans ce qui suit. Il découle immédiatement de la dé nition que les Hamiltoniens h α,p sont tau symétriques dans le sens de [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF], i.e.

∂h α,p-1 ∂t β q = ∂h β,q-1 ∂t α p .
L'équation ci-dessus est à comparer avec l'équation (1.33). Par construction, le potentiel de la CohFT est une fonction tau de la hiérarchie DZ associée à la solution topologique. Il a été prouvé depuis que la hiérarchie DZ construite à partir de la théorie FJRW d'une singularité simple de type ADE coïncide (à certaines considérations près) avec la hiérarchie de Drinfeld-Sokolov associée à l'algèbre de Lie simple correspondante [START_REF] Givental | Simple singularities and integrable hierarchies[END_REF][START_REF] Frenkel | Soliton equations, vertex operators, and simple singularities[END_REF]. Voir [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF] pour un clair exposé sur le sujet. Cette correspondance32 entre CohFT et hiérarchies intégrables nous l'appellerons correspondance de Witten-Kontsevich généralisée.

Le premier exemple de telle correspondance (avant même la dé nition des CohFT et des hiérarchies DZ) est le célèbre théorème de Witten-Kontsevich [START_REF] Witten | Two-dimensional gravity and intersection theory on moduli space[END_REF][START_REF] Kontsevich | Intersection theory on the moduli space of curves an the matrix Airy function[END_REF] qui stipule que le potentiel de la CohFT triviale est la fonction tau topologique de la hiérarchie KdV (1.20) (qui coïncide avec la hiérarchie DZ associée à la CohFT triviale). Plus précisément, considérons le potentiel F (t * ) de la CohFT triviale :

F (t * ) = ≥0 n ≥0 2 -2+n >0 1 n! d 1 , ...,d n ≥0 t d 1 • • • t d n ∫ M , n ψ d 1 1 • • •ψ d n n .
Puis e ectuons les substitutions

F = -1 2 log(τ ) puis t i → (2i + 1)!!t 2i+1 où (2i + 1)!! = 1 • 3 • 5 • • • (2i + 1)
. Alors la fonction qui en résulte τ est un fonction tau de la hiérarchie KdV au sens où = -2∂ 2

x log(τ ) est solution de (1.20) (dans la substitution, t 0 → t 1 et on pose

t 1 = x). 33 De plus, satisfait (x, t i >1 = 0) = x.

Les hiérarchies de double rami cations

On explique à présent la construction des hiérarchies de double rami cation (DR) introduites par Buryak [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF]. Comme mentionné précédemment, il est conjecturé que ces dernières sont équivalentes aux hiérarchies DZ. Mais elles sont, dans une certaine mesure, plus facile à construire et satisfont de fortes relations de récurrence jusque lors inconnues. De plus, elles sont dé nie pour toute CohFT, pas seulement celle semisimples. Dans le cas de la théorie FJRW d'une singularité simple de type ADE, la conjecture DR/DZ forte stipule que les hiérarchies DR, DZ et DS coïncident. Ce fait a été véri é pour les hiérarchies de type A 1 à A 5 (et d'autre, voir ci-dessous) et nous l'avons con rmé dans le cas D 4 dans une prépublication à paraître décrite au chapitre 5.

Structure Hamiltonienne. Tout d'abord, la structure de Poisson locale est simplement donnée par l'opérateur η α β ∂ x , où η α β est la métrique de la CohFT, c'est-à-dire que pour deux fonc-

tionnelles locales f , ∈ F , { f , } = ∫ N α, β =1 δ f δu α η α β ∂ x δ f δu β .
Pour les densités Hamiltoniennes, on introduit les cycles de double rami cation. Soient des entiers a 1 , . . . , a n tels que a i = 0 et soit un entier ≥ 0. Le cycle de double rami cation DR (a 1 , . . . , a n ) est, heuristiquement, le Poincaré dual de la classe d'homologie du locus des courbes complexes marquées (C, (x i )) telles que a i x i soit le diviseur des zéros et pôles de quelque fonction méromorphe. On considère x i comme étant un pôle si a i > 0 et un zéro sinon. Une dé nition plus précise est donnée à la section §5.2.2.

Puis l'on dé nit les nombres d'intersection suivants : soient α ∈ {1, . . . , N } et , d ≥ 0,

P ;a α,d ;α := ∫ DR (-a i ,a) λ ψ d 1 c ,n+1 (e α , e α 1 , . . . , e α n ), où α = (α 1 , . . . , α n ) ⊂ {1, . . . , N } et a = (a 1 , . . . , a n ) ⊂ Z.
La classe λ est la classe de Chern c (H) de plus haut degré du bré de Hodge. Il peut être montré que ces nombres sont polynomiaux en les a i et homogènes de degré 2 de sorte que l'on peut les écrire comme

P ;a α,d ;α = b 1 , ...,b n ≥0 b j =2 P ;b α,d ;α a b 1 1 • • • a b n n ∈ C 2 [a 1 , . . . , a n ],
On peut à présent dé nir les densités Hamiltoniennes :

α,d := ,n ≥0 2 -2+n >0 ε 2 n! b 1 , ...,b n ≥0 b j =2 P ;b α,d ;α n j=1 u α j b j .
(1.43)

Comme précédemment, la donnée de la structure de Poisson locale η α β ∂ x et des densités Hamiltoniennes α,d forment un système Hamiltonien au sens de l'équation (1.42). Les densités α,d satisfont la relation de récurrence suivante : tout d'abord, on pose α,-1 := η α µ u µ , où η α µ est la matrice inverse de η α µ , puis l'on a [BR16b]

∂ x (D -1) α,p+1 = { α,p , 1,1 } = N β,γ =1 k ≥0 ∂ α,p ∂u β k η βγ ∂ k +1 x δ 1,1 δu γ , où D = ε∂ ε + s ≥0 u α s ∂ u α s .
Il peut être aisément montré que cela permet e ectivement de calculer les densités récursivement à partir de la seule donnée 1,1 . Nous insistons sur le fait que cette récurrence précisément est nouvelle vis-à-vis de la théorie des hiérarchies intégrables. D'autre part, ces densités Hamiltoniennes ne sont pas tau symétriques. Pour obtenir des densités tau symétriques, il su t de considérer les polynômes di érentiels

h α,p := δ α,p δu 1 .
Par rapport à la structure de Poisson η α β ∂ x , ces polynômes sont des densités Hamiltoniennes tau symétriques.

La conjecture d'équivalence DR/DZ forte. A n d'énoncer la conjecture, il nous faut dé nir la notion de transformation de Miura qui joue le rôle de « changement de coordonnées » dans ce contexte. Une transformation de Miura est toute application inversible φ : A → A,

α := φ(u α ) ∈ A [0] , telle que det ∂ α ∂u β ε =0 α, β ∈[1, N ] 0.
On choisit à présent des coordonnées particulières pour la hiérarchie que nous appelons normales suivant [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF]. À la donnée d'une structure tau h α,p , elles sont dé nies par ũα := η α β h α,-1 .

L'application φ : u α → ũα est une transformation de Miura. On note La paire de Lax matricielle de la hiérarchie DS de type D 4 . Ce que nous discutons ici se généralise aux algèbres a nes de type D (1) n . Rappelons que l'algèbre a ne g de type D (1) 4 est l'algèbre a ne associée à l'algèbre simple D 4 , cette dernière étant isomorphe à l'algèbre o(8) (matrices complexes 8 × 8 antisymétriques -non hermitiennes). I.e. D (1) 4 est isomorphe à

h α,p := h α,p • φ, K α β := φ * (η α β ∂ x ) (1.
∂ ũα ∂u 1 = δ α,1 , φ * (η α β ∂ x ) = K α β , 1,1 • φ = h 1,1 , (1
           ũ1 = u 1 + (-1 36 (u 3 1 ) 2 -1 36 u 3 u 3 2 )ε 2 -1 45 u 3 4 ε 4 ũ2 = u 2 ũ3 = u 3 ũ4 = u 4 (1.
g = o(8) ⊗ C[λ, λ -1 ] ⊕ Cc.
Cette algèbre a ne a pour ensembles d'exposants positifs E + = Z odd >0 ∪ (3Z odd >0 ) = {1, 3, 3 , 5, 7, 9, 9 , . . .}, ce qui veut dire que tout exposant de la forme 3k avec k impair, a multiplicité 2. On note Λ k (resp. Γ k ), avec k impair, le générateur de la sous-algèbre de Heisenberg associé à l'exposant 3k (resp. (3k) ). De façon similaire, on note t k (resp. tk ) la variable associée à Λ k (resp. Γ k ). Les Λ k et Γ k sont donnés aux équations (5.54) et (5.55). Cela donne les équations de Lax matricielles suivantes [DS84]

∂L ∂t k = (e ad U Λ k ) + , L , (1.47) ∂L ∂ tk = (e ad U Γ k ) + , L , (1.48) pour k ∈ Z odd >0 , où L = ∂ x + Λ + Q, avec Q ∈ b [[t]] et où Λ est l'élément cyclique principal donné à l'équation (1.29).
La paire de Lax scalaire la structure Hamiltonienne. Dans [DS84], Drinfeld et Sokolov décrivirent la structure Hamiltonienne par rapport aux variables t k , mais pas tk (il fallait une nouvelle algèbre d'opérateurs). La structure de Poisson locale, à la façon de l'équation (1.22), est dé nie comme suit [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF]. Premièrement, l'opérateur de Lax scalaire est donné par

L = ∂ 6 x + ∂ -1 x 3 µ=1 s µ ∂ 2µ-1 x + ∂ 2µ-1 x s µ + ∂ -1 x ϱ∂ -1 x ϱ, s 4 = (ϱ) 2 , (1.49)
Puis en notant L = ∂ x • L, la première structure de Poisson est donnée par

{ f , } 1 = ∫ res X [(∂ x Y + L ) --(L Y + ∂ x ) --(∂ x Y -L ) + + (L Y -∂ x ) + ] dx, (1.50) pour tout f , ∈ F , 34 où X = δ f δ L , Y = δ δ L sont
∂L ∂t k = (L k/6 ) + , L , ⇐⇒ ∂s α ∂t k = {s α , H k +6 } 1 , H k = 6 k ∫ res L k/
D = i ≥m j ≥m-i a i, j ∂ i x m ∈ Z, a i, j ∈ A j .
L'algèbre qui en résulte D, dont nous appelons les éléments des opérateurs pseudo-di érentiels du second type, contient des séries avec une in nité de termes de degrés positifs et négatifs. Le résultat est qu'il existe un unique opérateur L 1/2 ∈ D tel que (L 1/2 ) 2 = L satisfaisant certaines contraintes (avec L donné à l'équation (1.49)). Cela permet de dé nir la représentation de Lax scalaire de l'équation (1.48) [START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF] par par les variables topologiques t α,p , avec α ∈ {1, . . . , 4} et p ≥ 0. De façon similaire, nous devons dé nir des densités Hamiltoniennes tau symétriques h α,p au sens de l'équation (1.33). On a, pour µ ∈ {1, 2, 3} et p ≥ 0,

∂L ∂ tk = (L k /2 ) + , L . ⇐⇒ ∂s α ∂ tk = s α , Ĥk+2 1 , Ĥk = 2 k ∫ res L k/2 .
t µ,p = Γ µ,p • t 6p+2α -1 , h µ,p-1 = Γ -1 µ,p • res L (6p+2α -1)/6 , t 4,p = Γ 4,p • t2p+1 , h 4,p-1 = Γ -1 4,p • res L (2p+1)/2 .
Les facteurs Γ α,p sont donnés à l'équation (5.68). Il reste à substituer les fonctions s α par les coordonnées normales ũα = η α β h β,-1 de la hiérarchie DS de type D 4 (on note volontairement ces coordonnées comme celles de la hiérarchie DR). On note h α,p les densités Hamiltoniennes tau symétriques écrites dans les coordonnées normales ũα . De façon similaire, dans les coordonnées normales, la première structure de Poisson (1.50) prend la forme suivante

(K α β ) = 1 6 ũ3 2 ∂ x + 1 2 ũ3 1 ∂ 2 x + 1 3 ũ3 ∂ 3 x ε 2 + 4 15 ∂ 5 x ε 4 0 6∂ x 0 0 6∂ x 0 0 6∂ x 0 0 0 0 0 0 2∂ x , où (K α β ) est l'opérateur di érentiel matriciel tel que { f , } 1 = ∫ δ f δ ũα K α β δ δ ũ β .
En tenant compte du théorème de Buryak et Guéré (voir le début de la section §1.4.3), on a à présent tout le matériel nécessaire pour énoncer le théorème principal de cette section. Sa preuve (i.e. les calculs) est donnée à la section §5.3.2.

Chapter 2

Introduction in English

Historical overview of the theory and of our results

Apart from a few equations, section 2.1 is rather historical and shall be readable by non specialists.

The present thesis deals with the computation and applications of tau functions of the Drinfeld-Sokolov hierarchies of equations. The Drinfeld-Sokolov hierarchies are in nite sequences of partial di erential equations which ows commute pairwise. For that reason they form an in nite-dimensional integrable system. They were introduced in 1984 by V. G. Drinfeld and V. V. Sokolov in their seminal article [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF]. In the latter, they showed how to associate these hierarchies to any a ne Kac-Moody algebra g [Kac68, Moo68] g. The latter we realize as the central extension of the loop algebra of a semisimple Lie algebra g, i.e 1

g = g ⊗ C[λ, λ -1 ] ⊕ Cc, (2.1)
with the appropriate Lie bracket making c into a central element; see (2.28).

Let us give a simple picture of these systems of equations. A solution of a Drinfeld-Sokolov hierarchy is made of functions u 1 , . . . , u N which depend on in nitely many variables {t i | i ≥ 1} with x = t 1 ; the system of equations of the hierarchy then has the form

u α t i = A α i (u 1 , . . . , u N ) α ∈ {1, . . . , N }, i ≥ 1 , (2.2)
where u α t i = ∂ t i u α and where A α i is polynomial in the u β 's and their derivatives w.r.t. x: u

β x , u β x x ,
1 Notice that the a ne Kac-Moody algebra associated to the nite-dimensional semisimple Lie algebra g is actually a co-central extension of the algebra g in (2.1) (see [START_REF] Kac | In nite dimensional Lie algebras[END_REF] §7). However, the Drinfeld-Sokolov hierarchies do not involve the co-central extension; calling g an a ne algebra is somehow customary. etc. For example, the celebrated Korteweg-de Vries equation [START_REF] Boussinesq | Essai sur la theorie des eaux courantes[END_REF][START_REF] Korteweg | On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves[END_REF] is of this form:

u t = 1 4 u x x x + 3 2 uu x .
(2.3)

The Drinfeld-Sokolov hierarchies are sequences of equations that share many features with the Korteweg-de Vries equation and, as we will see, Equation (2.3) plays a pivotal role in the theory. We shall emphasize that although we study partial di erential equations, we are only looking for algebraic solutions of these equations: polynomials, power series or rational functions.

In the next sections of this introduction, we will develop the main concepts that this thesis is based on and explain our results: [Duc17, CDY18, DR19]. But before that, let us give a brief history of Drinfeld-Sokolov hierarchies and tau functions and how our works t into this dense area of mathematical physics.

First of all, in the case where g is the a ne algebra associated to the simple Lie algebra sl(n, C) (n × n complex traceless matrices), the Drinfeld-Sokolov hierarchy coincides with the n-th Gelfand-Dickey hierarchy [START_REF] Gelfand | Fractional powers of operators and Hamiltonian systems[END_REF], the latter being much simpler! The n-th Gelfand-Dickey hierarchy admits the following representation: rst consider the di erential operator

L = ∂ n x +u n-2 ∂ n-2 x + • • • +u 1 ∂ x +u 0
where the u i 's are smooth functions of parameters {t i | i ≥ 1, i 0 mod n} with t 1 = x. Then the n-th Gelfand-Dickey hierarchy consists of the family of equations2

∂L ∂t i = (L i/n ) + , L i ≥ 1, i 0 mod n , (2.4)
where L 1/n is the n-th root of L with respect to the composition of di erential operators. 3 In particular, for sl(2, C) (specializing n = 2 in (2.4)) it comes down to the Korteweg-de Vries hierarchy [Lax68, Miu68, GKM68] (whose ow along t 3 reads the Korteweg-de Vries equation (2.3)). Equations (2.4) are called scalar Lax pairs. For the most part, the Drinfeld-Sokolov hierarchies admit a scalar Lax pair along with a bi-Hamiltonian representation [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] making them integrable in many ways. However, in the general case, they are generically constructed as hierarchies of matrix Lax equations (or zero-curvature equations) which we introduce in the next section. Let us note that the commutation of the ows of the Drinfeld-Sokolov hierarchies essentially results from the existence of in nite-dimensional abelian subalgebras of the a ne Kac-Moody algebra g.

The development of the notion of tau function. The notion of tau function on the other hand emerged in the early 1980's with the works of the Kyoto school led by Sato along with Date, Jimbo, Kashiwara, Miwa [JMS78, KM81, DKM81, DJKM81a] to cite a few. Originally, the tau function is de ned for the Kadomtsev-Petviashvili hierarchy [START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF] which contains all the Gelfand-Dickey hierarchies. 4 A solution of the Kadomtsev-Petviashvili hierarchy is the data of in nitely many functions; the tau function is one single function that contains all the information of a solution. Moreover, the tau function satis es partial di erential equations that are di erent in nature to that of the hierarchy itself. Namely, the equations on the tau function are bilinear and deeply connected to Plücker relations. The Kadomtsev-Petviashvili (KP) hierarchy and its tau functions were soon generalized to the so-called BKP and CKP hierarchies which are connected to Lie algebras of orthogonal (B n , D n ) and symplectic (C n ) type [START_REF] Date | Transformation groups for soliton equations VI: KP hierarchies of orthogonal and symplectic type[END_REF]. For exceptional Lie algebras, KP-like hierarchies were de ned by Kac and Wakimoto [START_REF] Kac | Exceptional hierarchies of soliton equations[END_REF]. We shall retain that a tau function is equivalent to a solution of the hierarchy, yet that the two give very di erent approaches to constructing or solving the hierarchy. 5In the meantime, Edward Witten proposed his famous 1991 conjecture connecting a certain tau function of the Korteweg-de Vries hierarchy to the generating series of the intersection numbers on the moduli spaces of stable curves M ,n [START_REF] Witten | Two-dimensional gravity and intersection theory on moduli space[END_REF]. Witten's conjecture, proven by Maxim Kontsevich [START_REF] Kontsevich | Intersection theory on the moduli space of curves an the matrix Airy function[END_REF] the year after, initiated a brand new way of understanding integrable hierarchies which we discuss below.

Subsequently to the works of Drinfeld-Sokolov on the one hand and of the Kyoto school and Kac-Wakimoto on the other, several ways of de ning tau functions were designed for Drinfeld-Sokolov hierarchies. One of them, initiated by Kac [START_REF] Kac | In nite dimensional Lie algebras[END_REF] and Hollowood and Miramontes [START_REF] Hollowood | Tau functions and generalized integrable hierarchies[END_REF][START_REF] Miramontes | Tau-functions generating the conservation laws for generalized integrable hierarchies of KdV and a ne toda type[END_REF], makes use of highest weight representations of a ne Kac-Moody algebras, in this approach, it connects to the bilinear equations. A di erent approach is to construct a speci c family of Hamiltonian densities called tau symmetric and which, essentially via Poincaré lemma, give a natural de nition of the tau function [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF]. 6In the present thesis, we are interested in two classes of tau functions: the polynomial tau functions and the topological tau function. Polynomial tau functions of a given Drinfeld-Sokolov hierarchy form a vast class of tau functions yet the simplest ones. For the speci c case of the Korteweg-de Vries hierarchy, they are known to be classi ed by the Adler-Moser polynomials [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF]. The zeros of these polynomials evolve through an integrable system called of Calogero-Moser type [START_REF] Airault | Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem[END_REF]; generalized to certain solutions of the Kadomtsev-Petviashvili hierarchy by Wilson in [START_REF] Wilson | Collisions of Calogero-Moser particles and an adelic Grassmannian[END_REF]. Hence polynomial tau functions of the Drinfeld-Sokolov hierarchies are expected to present analogous features yet to be discovered. On the other hand, the topological tau function of a given Drinfeld-Sokolov hierarchy is the main object upon which bridges from integrable hierarchies and geometry are built. We will get to it below; let us rst discuss polynomial tau functions.

Polynomial tau functions of the Drinfeld-Sokolov hierarchies (treated in Chaper 4). In 2012 Chao-Zhong Wu, inspired by [START_REF] Hollowood | Tau functions and generalized integrable hierarchies[END_REF], proposed a universal construction of tau symmetric Hamiltonian densities for any of the Drinfeld-Sokolov hierarchies associated to an a ne Kac-Moody algebra g [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF]. 7 This gives a generic tau function for the Drinfeld-Sokolov hierarchies. In [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF][START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF], Cafasso and Wu expressed Wu's tau function as the limit of Toeplitz deteminants through a certain faithful traceless representation of the underlying sem-simple Lie algebra [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] (see also Appendix A). This generalized part of the early work of Cafasso [START_REF] Cafasso | Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies[END_REF] on the Gelfand-Dickey hierarchies.

Cafasso-Wu construction associates a tau function to any element X ∈ g -of the a ne algebra g that has only negative powers of the loop parameter. In [START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF], using the results of Cafasso and Wu, we showed how to expand Wu's tau function as a sum over all partitions of integers (Theorem 4.5.2); the summands involve polynomials that generalize Schur polynomials [START_REF] Schur | Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen[END_REF] (also [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF]). As a consequence, we obtained an easy criterion for polynomiality of the tau function: if X ∈ g -is such that its matrix representation is nilpotent (w.r.t. to the usual matrix power), then the associated tau function is polynomial. 8 We gave several non trivial examples of polynomial tau functions and described applications to the bilinear equations of the Drinfeld-Sokolov hierarchies. This is the object of Chapter 4. It will constitute a future project to study the properties of these generalized Schur polynomials and of these polynomial tau functions.

Polynomial tau functions of the Korteweg-de Vries hierarchy (treated in Chaper 3). While working on polynomial tau functions of the Drinfeld-Sokolov hierarchies, we became interested in the speci c case of the Korteweg-de Vries hierarchy. The latter, as we described, is a special case of the Drinfeld-Sokolov hierarchies for the a ne Lie algebra associated to the simple Lie algebra sl(2, C). Moreover, every solution of the Korteweg-de Vries hierarchy is also a solution of the Kadomtsev-Petviashvili hierarchy; the reduction being to set all evenly indexed variables t 2i to zero. As we have said, the polynomial tau functions of the Korteweg-de Vries hierarchy have been much studied and classi ed by Adler, McKean and Moser [START_REF] Airault | Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem[END_REF][START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF]. Moreover, as it was rst understood by Sato [START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF] (see also [START_REF] Hirota | The Direct Method in Soliton Theory[END_REF]), each polynomial tau function of the Korteweg-de Vries hierarchy can be written in terms of Schur polynomials.

The main result of [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF] is that there exists a change of the variables t i which transforms the polynomial tau functions of the Korteweg-de Vries hierarchy into a sequence of polynomials which satis es a much simple di erential recursion, the so-called Adler-Moser polynomials. In [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF] we were able to exhibit this change of variables (Theorem 3.5.4); the proof uses essentially the Kadomtsev-Petviashvili theory. This is the object of Chapter 3.

Recently, Kac and van de Leur made signi cant progress in the eld [START_REF] Kac | Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions[END_REF] by showing that all the polynomial tau functions of the Kadomtsev-Petviashvili hierarchy (thus including the Gelfand-Dickey hierarchies) can be expressed surprisingly simply as a Schur polynomial modulo a certain change of variables.9 They also worked out the polynomial tau functions of the modi ed Kadomtsev-Petviashvili hierarchy, but we will not discuss it here.

Witten's conjecture and the topological tau function. We now come back to Witten's conjecture and its consequences. Witten's conjecture states that the generating series of the intersection numbers on the moduli spaces of stable curves M ,n is actually a tau function of the Korteweg-de Vries hierarchy [START_REF] Witten | Two-dimensional gravity and intersection theory on moduli space[END_REF]. The conjecture was proven by Maxim Kontsevich [START_REF] Kontsevich | Intersection theory on the moduli space of curves an the matrix Airy function[END_REF] and this tau function is now called the Witten-Kontsevich tau function. Since then, tau functions became an essential object that connects complex geometry, algebraic geometry and integrable systems in an active area of mathematical physics sometimes referred to as integrable mirror symmetry. Soon after, Witten's conjecture was generalized to connect tau functions of Gelfand-Dickey hierarchies and Witten's r -spin classes [START_REF] Witten | Algebraic geometry associated with matrix models of twodimensional gravity[END_REF]. In 2003 Dubrovin and Zhang showed that the generating function of the Gromov-Witten invariants of the complex projective line P 1 is a tau function of the extended Toda hierarchy [START_REF] Dubrovin | Virasoro Symmetries of the Extended Toda Hierarchy[END_REF] (which is not a Drinfeld-Sokolov hierarchy).

Cohomological eld theories (CohFT), introduced by Kontsevich and Manin in [START_REF] Manin | Gromov-Witten classes, quantum cohomology, and enumerative geometry[END_REF], are cohomology classes on the moduli spaces of stable curves that are de ned in view of unifying Witten's r -spin classes as well as Gromov-Witten theories. In the early 2010's, Fan, Jarvis and Ruan de ned the so-called Fan-Jarvis-Ruan-Witten CohFTs in view of generalizing Witten's r -spin classes to arbitrary simple complex singularities [START_REF] Fan | The Witten equation and its virtual fundamental cycle[END_REF][START_REF] Fan | The Witten equation, mirror symmetry, and quantum singularity theory[END_REF].

On the other hand, in [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF], Dubrovin and Zhang showed how to associate to any semisimple 10 CohFT an integrable hierarchy that was shown to coincide, in some cases, 11 with the Drinfeld-Sokolov hierarchies [START_REF] Givental | Simple singularities and integrable hierarchies[END_REF][START_REF] Frenkel | Soliton equations, vertex operators, and simple singularities[END_REF]. In Dubrovin-Zhang construction, the hierarchy is designed such that the generating series of the semisimple CohFT it is based on is a tau function. This special tau function we call the topological tau function; for example, the Witten-Kontsevich tau function is the topological tau function of the Korteweg-de Vries hierarchy. Since then, integrable mirror symmetry became a matter of building bonds between CohFTs and integrable hierarchies. Although some examples work very well,12 (see [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF] for a clear expositions) there seem to be no universal pattern so far.

One of the goals of Dubrovin-Zhang construction was to classify integrable hierarchies [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF][START_REF] Dubrovin | Frobenius manifolds and central invariants for the Drinfeld-Sokolov bihamiltonian structures[END_REF]. The idea was to use the intersection theory of the moduli spaces of stable curves13 (and generalizations: Gromov-Witten theory, Fan-Jarvis-Ruan-Witten theory) to dene integrable hierarchies and then prove that they are equivalent to well-known hierarchies (such as the Drinfeld-Sokolov hierarchies but also others). The equivalence relation between integrable hierarchies is called a Miura transform [START_REF] Miura | Korteweg-deVries Equation and Generalizations. I. Explicit Nonlinear Transformation[END_REF]; see §2.4.2.

The double rami cation hierarchies (treated in Chapter 5). Recently, Alexander Buryak, inspired by ideas from symplectic eld theory [START_REF] Eliashberg | Introduction to symplectic eld theory[END_REF], de ned a new kind of integrable hierarchies he called the double rami cation hierarchies [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF]. They are constructed via the 10 The semisimplicity means that an associated algebra is semisimple, i.e. the direct sum of simple algebras. 11 These cases are those where the Drinfeld-Sokolov hierarchy is built from the a ne Lie algebra associated to a simple Lie algebra of type A n ≥1 , D n ≥4 or E 6 , E 7 , E 8 (the ADE classi cation). In the remaining cases, the Dubrovin-Zhang and Drinfeld-Sokolov hierarchies are still equivalent in the sense of quasi-Miura (i.e. rational) transformations.

double rami cation cycles on the moduli spaces of stable curves introduced by Graber and Vakil [START_REF] Graber | Relative virtual localization and vanishing of tautological classes on moduli spaces of curves[END_REF] (hence the name) and can be de ned for arbitrary CohFTs (not only semisimple ones, as is in the Dubrovin-Zhang hierarchies). In that same article, guided by examples he worked out, Buryak conjectured that, in the semisimple case, the double rami cation hierarchies are equivalent to the Dubrovin-Zhang hierarchies via a Miura transform. In particular, for the CohFT associated to the Fan-Jarvis-Ruan-Witten theory of a simple singularity of ADE type, the double rami cation, Dubrovin-Zhang and Drinfeld-Sokolov hierarchies are conjectured to coincide. Since then, the double rami cation hierarchies have been intensively studied by, for example but not only, A. Buryak, J. Guéré [START_REF] Buryak | Towards a description of the double rami cation hierarchy for Witten's r -spin class[END_REF], P. Rossi [BR16a, BR16b, BGR17] and B. Dubrovin [START_REF] Buryak | Integrable systems of double rami cation type[END_REF][START_REF] Buryak | Tau-structure for the Double Rami cation Hierarchies[END_REF].

The double rami cation hierarchies have several features that make them of particular interest. (1) They are de ned for arbitrary CohFTs. (2) They are constructed as Hamiltonian equations for which the Poisson bracket is very simple (called the standard bracket); in particular, (3) tau symmetric densities are easily obtained [START_REF] Buryak | Tau-structure for the Double Rami cation Hierarchies[END_REF]. (4) Their Hamiltonian densities satisfy strong recursion relations that are new to integrable hierarchies [START_REF] Buryak | Recursion Relations for Double Rami cation Hierarchies[END_REF]. (5) They admit a rather simple quantization [START_REF] Buryak | Double Rami cation Cycles and Quantum Integrable Systems[END_REF][START_REF] Buryak | Integrable systems of double rami cation type[END_REF], but we will not discuss it here. Moreover, the conjectural equivalence with the Dubrovin-Zhang hierarchies has been enhanced into a uniquely determined Miura transform, hence the strong DR/DZ conjecture 5.2.23 [START_REF] Buryak | Tau-structure for the Double Rami cation Hierarchies[END_REF]. So far, the strong DR/DZ equivalence has been con rmed in several cases including the Drinfeld-Sokolov hierarchies of type A 1 to A 5 , the extended Toda hierarchy and others (see 5.2.23).

In [START_REF] Du Crest De Villeneuve | Quantization of the D 4 Drinfeld-Sokolov hierarchy and Fan-Jarwis-Ruan-Witten theory[END_REF], we con rmed the strong DR/DZ equivalence for the Drinfeld-Sokolov hierarchy associated to the a ne algebra of the simple Lie algebra of type D 4 . This is the object of Chapter 5. The con rmation of the conjecture involves the computation of the rst Hamiltonian structure of the Drinfeld-Sokolov hierarchy of type D 4 . To do so, we used the scalar Lax pair representation of the hierarchies of type D n obtained by Liu, Wu and Zhang [START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF]. These computations, to the best of our knowledge, are new to the literature.

The Korteweg-de Vries hierarchy: a canonical example

In this section, we present some of the important features of the Korteweg-de Vries equation (KdV for short) and the KdV hierarchy. This will serve as an illustration of many of the properties of integrable hierarchies. Chapter 3 is dedicated to the polynomial tau functions of the KdV hierarchy; we describe the results we obtained in [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF] at the end of §2.2.

The Korteweg-de Vries equation

Aside from its application in geometry, the KdV equation arose from uid mechanics. It describes the evolution of the surface of the water in a canal, the latter modelling shallow water owing essentially in one direction. If one ignores the transverse direction, we can describe the height of the surface of the water by a scalar function u(t, x) where t represents time and x represents the one direction in which the water ows. Then u must satisfy the KdV equation:

u t = 1 4 u x x x + 3 2 uu x , (2.5)
where u t = ∂ t u and u x = ∂ x u, etc. The above equation is actually a normalized version of KdV where physical constants have been ruled out. The KdV equation was rst theorized by Boussinesq in 1877 [START_REF] Boussinesq | Essai sur la theorie des eaux courantes[END_REF] and then proposed in its current form by Korteweg and de Vries in 1895 [START_REF] Korteweg | On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves[END_REF]. For the physical derivation of the KdV equation (and that of the KP equation), we refer to the introduction of [START_REF] Ablowitz | Solitons, Nonlinear Evolutions Equations and Inverse Scattering[END_REF] by Ablowitz and Clarkson. Notice that by rescaling u, x and t, we can put arbitrary coe cients in front of each monomial u t , u x x x and uu x ; the choice 1, 1/4 and 3/2 will become clear in §2.2.2. The KdV equation can be written in many ways, thereby allowing connections to many a priori disconnected areas. Here we show four representations that we will use throughout the document: (a) the scalar Lax pair, (b) the matrix Lax pair, (c) the bi-Hamiltonian representation and (d) the bilinear representation.

(a) The scalar Lax pair. Consider the di erential operator L = ∂ 2

x + u often referred to as the Schrödinger operator. 14 Then we can rewrite Equation (2.5) as

∂L ∂t = [P, L] , P = ∂ 3 x + 3 2 u∂ x + 3 4 u x , (2.6)
where [L, P] = L • P -P • L is the commutator between di erential operators. The composition of operators is given by Leibniz's rule, de ned on monomials by15

a∂ n x • b∂ p x := n k=0 n k ab (k ) ∂ n+p-k x , (2.7)
where b (k ) := ∂ k x (b), and extended by linearity. We will omit the • sign when there is no ambiguity. The derivative ∂ t is de ned to act on operators by di erentiating each coe cient:

∂ t a∂ n x := a t ∂ n x .
In particular, the action of ∂ t on operators satis es Leibniz's rule. Therefore, ∂ t L = u t , but this is due to the speci c form L. The pair (L, P) is called a scalar Lax pair for the KdV equation. This representation is due to Peter D. Lax himself [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF].

Equations (2.5) and (2.6) are di erent in nature: Equation (2.5) is a PDE which is nonlinear in u, while Equation (2.6) is a linear ODE in L and both sides of the equation are derivations (Leibniz operators) on di erential operators. The formula for the operator P, at this stage, seems rather ad hoc, but it will t into a very clear and simple pattern with the KdV hierarchy. A particular interest of the scalar Lax pair is that it easily generalizes to the Kadomtsev-Petviashvili hierarchy (see Section 3.2).

(b) The matrix Lax pair. Now consider the following order 1 matrix-valued di erential operator:

L = ∂ x + 0 λ -u 1 0 .
(2.8)

In the above, λ is a new indeterminate which we call the spectral parameter. Any identity should hold for any complex value of λ. Now consider the following matrix:

M = -1 4 u x λ 2 -1 2 uλ -1 2 (u) 2 -1 4 u x x λ + 1 2 u + 1 4 u x .
(2.9)

Then ∂L ∂t -[L , M ] = 0 -u t + 1 4 u x x x + 3 2 uu x 0 0 .
In the above equation,

[L , M ] := ∂ x M + [N , M ] where N = L -∂ x and
∂L ∂t = [L , M ].
(2.10)

It is equivalent to the zero-curvature equation

∂ t N -∂ x M -[N , M ] = 0.
The pair (L , M ) is called a matrix Lax pair for the KdV equation. This representation is due to Ablowitz, Kaup, Newell and Segur [START_REF] Ablowitz | The inverse scattering transform -Fourier analysis for nonlinear problems[END_REF].

Once again, Equations (2.6) and (2.10) are di erent in nature; the latter only involves order 1 di erential operators and therefore allows this zero-curvature representation. A particular interest is that the Drinfeld-Sokolov hierarchies are generically constructed as matrix Lax pair equations.

(c) The bi-Hamiltonian representation. We now introduce the ring A = C[u, u x , u x x , . . .] whose elements we call di erential polynomials in u. The operator ∂ x on A identi es with the operator ∂ x = k ≥0 u (k+1) ∂ u (k ) . Besides, we de ne the following quotient space, 16

F := A/Im (∂ x ),
whose elements we call local functionals. Given a di erential polynomial f ∈ A we denote its class f ∈ F by f = ∫ f . 17 As an immediate consequence, we get an integration by parts 16 Notice that the quotient space F is only a vector space and not a ring since the operator ∂ x is linear yet not a homomorphism of algebras (it satis es Leibniz's rule). 17 We shall emphasize that the integral sign really is purely notational. Nonetheless, by construction ∫ ∂ x f = 0, and this can be realized by de ning solutions on the circle S 1 . Hence we sometimes talk about periodic solutions of the KdV equation (see e.g. [START_REF] Lax | Periodic Solutions of the KdV Equation[END_REF]). formula:

∫ f ∂ x = - ∫ ∂ x f , or, more generally for k ≥ 1, ∫ f ∂ k x = (-1) k ∫ ∂ k x f . (2.11)
From this we de ne on A the operator δ /δu, called the variational derivative (also called the Euler-Lagrange operator), by

δ f δu := k ≥0 (-1) k ∂ k x ∂ f ∂u (k) = ∂ f ∂u -∂ x ∂ f ∂u x + ∂ 2 x ∂ f ∂u x x -• • •
Then for any f that contains no constant term, δ f /δu = 0 if and only if f ∈ Im (∂ x ) (see Proposition 5.1.1). In particular, δ /δu is well de ned on F . We then de ne two maps, {-, -} 1 , {-, -} 2 :

F ⊗ F → F by 18 f , 1 := ∫ δ f δu ∂ x δ δu , f , 2 := ∫ δ f δu 1 4 ∂ 3 x + u∂ x + 1 2 u x δ δu .
(2.12)

These maps are skew symmetric and satisfy Jacobi's identity; we call such maps local Poisson structures. 19 Moreover, any linear combination λ{-, -} 1 + µ{-, -} 2 is still a local Poisson structure; we say that the Poisson structures are compatible. We also de ne two auxiliary maps, denoted similarly, {-, -} a : A ⊗ F → A, with a = 1, 2, by

{ f , } 1 := k ≥0 ∂ f ∂u (k) ∂ k+1 x δ δu , { f , } 2 := k ≥0 ∂ f ∂u (k) ∂ k x 1 4 ∂ 3 x + u∂ x + 1 2 u x δ δu .
(2.13) By construction, and thanks to Equation (2.11), these maps satisfy ∫ { f , } a = { f , } a , for a = 1, 2. Then it can be easily checked that the KdV equation (2.3) is equivalent to the following equations:

u t = u, h 1 1 = ∂ x δh 1 δu , h 1 = ∫ 1 4 u 3 + 1 8 uu x x , (2.14)
u t = u, h 2 2 = 1 4 ∂ 3 x + u∂ x + 1 2 u x δh 2 δu , h 2 = ∫ 4 8 u 2 . ( 2
h i D t i (f , ) = f (t + h) • (t -h), (2.17)
where both sides shall be expanded in powers of the h i 's and identi ed as series in he h i 's. For example, by setting x = t 1 and t = t 3 (the indices do not matter here), we get

D x (f , ) = ∂ f ∂x -f ∂ ∂x , D x D t (f , ) = ∂ 2 f ∂x ∂t + f ∂ 2 ∂x ∂t - ∂ f ∂x ∂ ∂t - ∂ f ∂t ∂ ∂x .
Then it is a bit lengthy to check, but the KdV equation (2.5), when substituting u = -2∂ 2 x log τ , is equivalent to the following equation,

D 4 x -4D x D t (τ , τ ) = 0.
(2.18) Equation (2.18) is called a Hirota equation, after Ryogo Hirota who introduced these derivatives [START_REF] Hirota | Direct method of nding exact solutions of nonlinear evolution equations[END_REF]. It is fairly compact considering the number of terms the logarithmic derivatives make appear.

One may naturally ask the interest of such equations. It is actually less clear for the KdV hierarchy than for other, more complicated hierarchies. In particular, when a solution of the 20 Magri actually brought much more to the eld as he rst discovered how to recover the entire hierarchy using the bi-Hamiltonian structure. This was an important step in understanding the richness of reconstruction techniques for integrable hierarchies. However, we will not use it in this document. 21 The (-2) factor is explained in Proposition 3.4.5; it cannot be ruled out since the KdV equation is nonlinear. 22 Notice importantly that when expanding identity (2.17) the symbol D t i 1 • • • D t in is an operator in itself and not a product of operators. hierarchy consists of several functions u 1 , . . . , u N (and not just u), there is still one single tau function τ . Therefore, the bilinear representation condenses the whole hierarchy on one single function. Moreover, it can be the case that the tau function is easier to compute than the solution itself. This is, for instance, the approach of Chapters 3 and 4, although they do not involve directly the bilinear equations.

The Korteweg-de Vries hierarchy

The KdV hierarchy was introduced in 1968 simultaneously by Lax [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF] and Gardner-Kruskal-Miura-Zabusky [Miu68, GKM68, GKMZ70]. 23 It consists of a sequence of equations that commute pairwise (explained below) and which contains the KdV equation (2.3), thus embedding the KdV equation whithin an in nite-dimensional system of PDEs. In what follows, we show how to construct the KdV hierarchy starting with (A) its scalar Lax pair, due to Lax [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF]; then we will discuss (B) its matrix Lax pair, (C) its bi-Hamiltonian representation and (D) its bilinear representation. The matrix Lax pair is the most technical one; we will actually discuss it in Section 2.3 along with the case of an arbitrary Drinfeld-Sokolov hierarchy.

(A) The scalar Lax pairs. For the scalar Lax pair, it su ces to consider the square root L 1/2 of the Schrödinger operator L = ∂ 2

x + u. For that, we rst extend the algebra of di erential operators into a larger algebra,

D = n i=-∞ a i ∂ i a i ∈ C[[x]], n ∈ Z , (2.19)
equipped with the same product de ned in Equation (2.7), but where the binomial coe cients have been generalized to n k with n < 0 by setting n k := n(n -1) • • • (n -k +1)/k!. The elements of D we call pseudo-di erential operators. In this larger algebra, there exists an operator

L 1/2 = ∂ x + k <0 p k ∂ k
x , with functions p k ∈ A, which satis es (L 1/2 ) 2 = L (see Theorem 3.1.8). Then the KdV hierarchy can be formulated as

∂L ∂t i = (L i/2 ) + , L i > 0 odd , (2.20)
where (-) + denotes the projection onto the subspace of D containing only nonnegative powers of ∂ x (i.e. di erential operators). In particular, (L 3/2 ) + = P = ∂ 3 x + 3 2 u∂ x + 3 4 u x , see Equation (2.6). This scalar Lax formulation of the KdV hierarchy easily generalizes to the n-th Gelfand-Dickey hierarchy where the operator L = ∂ 2

x +u is replaced by the operator

L = ∂ n x +u n-2 ∂ n-2 x +• • •+u 0 .
So, the ows de ned by the system (2.20) commute pairwise in the sense that for any odd positive integers i, j,

∂ ∂t i (L j/2 ) + , L = ∂ ∂t j (L i/2 ) + , L . (2.21)
This is the integrability condition of the KdV hierarchy. The proof is classic and can be found for example in [START_REF] Date | Solitons, Di erential equations, symmetries and in nite dimensional algebras[END_REF]; we give a very similar proof for the KP hierarchy in Theorem 3.2.1. Keeping in mind that

L = ∂ 2 x + u, it follows from Equation (2.21) that ∂ t i ∂ t j u = ∂ t j ∂ t i u.
This is of course a necessary condition for the existence of at least C 1 solutions, let alone smooth solutions. Notice that the function u must now depend on in nitely many variables {t i | i > 0 odd}. We will generally assume that

u ∈ C [[t 1 , t 3 , t 5 , . . .]] .
As we said, we will treat the matrix Lax pair in Section 2.3 along with the case of an arbitrary Drinfeld-Sokolov hierarchy. Let us now treat the bi-Hamiltonian representation.

(C) The bi-Hamiltonian representation. Given a pseudo-di erential operator A ∈ D, we denote by res (A) its coe cient in ∂ -1

x . Then the bi-Hamiltonian representation is given by the two same local Poisson structures of Equation (2.12), and by the Hamiltonian densities given by the residues of the fractional powers of L = ∂ 2

x + u, namely,

u t i = u, h 1,i 1 = ∂ x δh 1,i δu , h 1,i = 4 i + 2 ∫ res L i +2 2 , u t i = u, h 2,i 2 = 1 4 ∂ 3 x + u∂ x + 1 2 u x δh 2,i δu , h 2,i = 4 i ∫ res L i 2 ,
for i > 0 odd. In particular, comparing with Equations (2.14) and (2.15), h 1,3 = h 1 and h 2,3 = h 2 . Moreover, the local Poisson structures can be reformulated using the operator L as follows. For any local functional f ∈ F , denote

δ f δ L = ∂ -1 x • δ f δ u ∈ A, 24 and let X = δ f δ L and Y = δ δ L , then { f , } 1 = ∫ res [X LY -LXY ] , { f , } 2 = ∫ res [(LX ) + LY -(X L) + Y L] . (2.22) (D)
The bilinear representation. Finally, for the bilinear representation, we need to introduce the so-called elementary Schur polynomials, denoted p k (t) with k ∈ Z >0 , de ned by the relation

exp i ∈Z >0 t i z i = k ∈Z >0 p k (t)z k . (2.23) For instance, p 1 = t 1 , p 2 = 1 2 t 2 1 + t 2 , p 3 = 1 6 t 3 1 + t 2 t 1 + t 3
. Now denote pk the polynomial p k where all even variables t 2i are set to 0. Then the KdV hierarchy (2.20), once we substitute u = -2∂ 2

x log τ , is equivalent to the following two sequences of bilinear equations, 25

pk+1

([D]) -1 2 D x D t k (τ , τ ) = 0, D x D t k -D x pk ([D]) (τ , τ ) = 0, k > 0 odd , (2.24
) 24 The notation δ f δ L seems a bit arti cial here, but when one considers Drinfeld-Sokolov hierarchies for arbitrary simple Lie algebras (KdV being associated to sl(2, C)), it becomes clearer. 25 A much more compact form of Equations (2.24) is given by the celebrated bilinear identity (cf. §3.3), however it is less explicit than the one given here.

where [D] := (D x , 1 3 D t 3 , 1 5 D t 5 , . . .). The above can be deduced from results of [START_REF] Date | Transformation groups for soliton equations[END_REF] (see also [START_REF] Adler | Non-linear PDEs for gap probabilities in random matrices and KP theory[END_REF], Lemma 4.1). In particular, for k = 3, pk+1

([D]) -1 2 D x D t k = 1 24 (D 4 x -4D x D t 3
) so that we recover the bilinear form (2.18) of the KdV equation.

Polynomial tau functions of the KdV hierarchy and the Adler-Moser polynomials

Here we describe the result obtained in [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF] on the explicit transformation from the Adler-Moser polynomials to the polynomial tau functions of the KdV hierarchy; it is the object of Chapter 3, §3.5. In [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF], Adler and Moser introduced a sequence of polynomials θ n , for n ≥ 0, depending on parameters r = {r i | i ∈ Z odd >0 }, r 1 = x, de ned by the following di erential recursion:

θ 0 = 1; θ 1 = x; then ∂θ n+1 ∂x θ n-1 -θ n+1 ∂θ n-1 ∂ x = (2n -1) (θ n ) 2 .
(2.25)

When computing θ n , the integration constant is set to r 2n-1 . For example, θ 2 = x 3 + r 3 and θ 3 = x 6 +5r 3 x 3 +r 5 x -5r 2 3 . These polynomials are now known as the Adler-Moser polynomials. In that same article, they proved that there exists a unique change of variables r → t which transforms the Adler-Moser polynomial θ n (r) into a tau function τ n (t) of the KdV hierarchy in the sense of Equation (2.16). Moreover, from results of [START_REF] Airault | Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem[END_REF], they proved that through this change of variables, one recovers all the polynomial tau functions of KdV as well as all the rational solutions of KdV, thus classifying the rational solutions. However, this unique change of variables was not known. In [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF], we proved the following theorem.

Theorem 2.2.1 ( [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF]). The following change of variables transforms the Adler-Moser polynomials into the polynomial tau functions of the KdV hierarchy: r 1 = t 1 = x, and

i ≥2 r 2i-1 α 2i-1 z 2i-1 = tanh i ≥2 t 2i-1 z 2i-1 .
(2.26)

where α 2i-1 = (-1) i-1 3 2 5 2 • • • (2i -3) 2 (2i -1).
See Theorem 3.5.4 in the document. We can interpret Theorem 2.2.1 as rescaling the variables, plus choosing a di erent, nontrivial integration constant for recursion (2.25). The rst few tau functions read the following.

τ 0 = 1, τ 2 = x 3 -3t 3 , τ 1 = x, τ 3 = x 6 -15t 3 x 3 -45t 2 3 + 45t 5 x .
These are well-known tau functions of the KdV hierarchy, see e.g. [START_REF] Hirota | The Direct Method in Soliton Theory[END_REF] where they are computed using similar methods and the elementary Schur polynomials (see Equation (2.23)). The proof of Theorem 2.2.1 is given in Section 3.5; it relies essentially on KP theory which we develop in Section 3.2. It would constitute a further project to generalize the Adler-Moser polynomials as well as the result of [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF] and the above theorem to arbitrary Drinfeld-Sokolov hierarchies (i.e. for arbitrary simple Lie algebras).

Polynomial tau functions of the Drinfeld-Sokolov hierarchies

Here we describe the results obtained in [START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF] on polynomial tau functions of the Drinfeld-Sokolov hierarchies and applications; it is the object of Chapter 4, §4.5. We start by reviewing the construction of the Drinfeld-Sokolov hierarchies in the most straightforward, although technical way: the matrix Lax pair. In that way, they largely generalize representation (2.10) of the KdV equation. But rst we need to introduce some material on semisimple and a ne Lie algebras.

The Drinfeld-Sokolov hierarchies

Starting point: A semisimple Lie algebra. We start with a nite-dimensional semisimple Lie algebra g (see §4.1), i.e. a Lie algebra which is a direct sum of simple Lie algebras (i.e. algebras containing no nontrivial ideals). These algebras decompose nicely in the following way:

g = n-⊕ h ⊕ n+ ,
where h is a maximal abelian subalgebra whose elements H ∈ h are such that all the adjoint maps ad H = [H , -] : g → g are simultaneously diagonalizable. The subalgebra h is called a Cartan subalgebra of g and its dimension r = dim h is called the rank of g; the rank is unique. A classical result is that g is completely determined (up to isomorphism) by a unique matrix (C i j ) ∈ Mat(r, Z), called its Cartan matrix, to which we can associate a set of generators {H i , E i , F i | 1 ≤ i ≤ r } of g satisfying the following relations:

[E i , F j ] = δ i j H i , [H i , E j ] = C i j E i , [H i , F j ] = -C i j F j , [H i , H j ] = 0,
Then h is generated by the H i 's, while n+ (resp. n-) is generated by the E i 's (resp. the F i 's). Such generators are called Weyl generators. There are two special elements E θ ∈ n + and E -θ ∈ n - which satisfy the following relations: for all i ∈ {1, . . . , r },

[E θ , E i ] = 0, [E -θ , F i ] = 0.
For example, the simple Lie algebra sl(2, C) (2 × 2 complex traceless matrices) has rank 1 and is generated by the following matrices,

H = 1 0 0 -1 , E = 0 0 1 0 , F = 0 1 0 0 . They satisfy the relations [H , E] = 2E, [H , F ] = -2F and [E, F ] = H so that the Cartan matrix of sl(2, C) is C = (2) ∈ Mat(1, Z). Moreover, in this particular example, E θ = E and E -θ = F .
The associated a ne algebra. To the nite-dimensional semisimple Lie algebra g we associate the so-called (untwisted) a ne algebra

g := g ⊗ C[λ, λ -1 ] ⊕ Cc, (2.27)
which is in nite-dimensional. We equip this algebras with the following Lie bracket: given X , Y ∈ g, p, q ∈ Z and a, b ∈ C,

[X λ p + ac, Y λ q + bc] = [X , Y ]λ p+q + pδ p+q,0 (X , Y ) 0 • c, (2.28)
where (-, -) 0 denotes a certain nondegenerate symmetric bilinear form on g. Clearly, c belongs to the center of the algebra g (i.e. c commutes with all elements of g), making Cc into a nontrivial ideal of g. If the semisimple Lie lagebra g has type X n , we will denote by X (1) n the associated a ne algebra g. A ne algebras were much studied especially by V. Kac, who introduced them simultaneously with R. Moody [START_REF] Kac | Simple irreducible graded lie algebras of nite growth[END_REF][START_REF] Moody | A New Class of Lie Algebras[END_REF]. The details are explained in §4.2.

The a ne algebra g is generated by the following elements:

e i = E i , f i = F i , αi = H i , e 0 = E -θ λ, f 0 = E θ λ -1 , α0 = H 0 + c,
where

H 0 = [E θ , E -θ ]
, with i ∈ {1, . . . , r } (r = dim h being the rank of g). From these we de ne two important gradations on g, the principal gradation and the homogeneous gradation, as follows:

Principal:

deg e i = 1, deg f i = -1, deg αi = 0, g = k ∈Z g k , Homogeneous: deg e i = δ i,0 , deg f i = -δ i,0 , deg αi = 0, g = k ∈Z g k .
The homogeneous gradation, as the name suggests, amounts to setting deg λ = 1 and deg g = deg c = 0. We denote by g + (resp. g + ) the subspace of elements of principal (resp. homogeneous) degree ≥ 0; we set g -= gg + and g -= gg + .

We now introduce the so-called a ne principal cyclic element

Λ = e 0 + e 1 + . . . + e r = E -θ λ + E 1 + . . . + E r .
(2.29)

In particular, Λ ∈ g 1 (principal degree 1). A fundamental result (see [START_REF] Kac | In nite dimensional Lie algebras[END_REF]) on a ne algebras is that g decomposes as follows,

g = s + Im (ad Λ ) , s ∩ Im (ad Λ ) = Cc.
where s = Ker (ad Λ ). We denote by s k = s ∩ g k the subspace of principal degree k. We call the a ne exponents of g the integers k ∈ Z such that dim s k 0 and denote m k = dim s k called the multiplicity of exponent k. We denote by

E = k ∈ Z dim s k 0
the multiset of a ne exponents of g, meaning that elements k ∈ E have multiplicity m k .

We denote Λ k , Λ k , Λ k , . . . (as many as m k ) the spanning elements of s k . These generators Λ k (we omit the primes for now) can be chosen in such a way that they satisfy the Heisenberg relations

[Λ i , Λ j ] = iδ i+j,0 • c.
(2.30)

We call s the principal Heisenberg subalgebra of g. We denote ν ∈ C \ {0} the complex number such that Λ 1 = ν Λ.

For example, for the a ne algebra

g = sl(2, C) ⊗ C[λ, λ -1 ] ⊕ Cc the principal a ne cyclic element reads Λ = F λ + E = 0 λ 1 0 .
The a ne exponents of g are exactly the odd integers and all have multiplicity 1, i.e. E = Z odd . The generators of the principal Heisenberg subalgebra simply read

Λ 2i+1 = Λ 2i+1 = λ i Λ for i ∈ Z.
The Drinfeld-Sokolov hierarchies. We now have all the material we need to describe the Drinfeld-Sokolov hierarchies. First, we denote by

E + = E ∩ Z >0 the set of positive exponents of g. Then we denote t := {t i | i ∈ E + }, with t 1 = ν -1
x, and we consider the following algebra,

g ⊗ C [[t]] = m ≥0 i 1 , ...,i m ∈E + k 1 , ...,k m ≥1 ξ k 1 , ...,k m i 1 , ...,i m • t k 1 i 1 • • • t k m i m ξ k 1 , ...,k m i 1 , ...,i m ∈ g .
The above algebra (and its subalgebras) will serve as the "function space" of the theory. The Lie bracket on it is de ned similarly to that on g ⊗ C[λ, λ -1 ], i.e. we set [ξt i , ηt j ] = [ξ , η]t i t j and then extend by multiplicativity. Now consider the following degree 1 matrix-valued operator:

L = ∂ x + Λ + Q, Q ∈ b [[t]] ,
where b = h ⊕ n+ (typically, for sl(n, C), the subalgebra b consists of lower triangular traceless matrices). Such operators we call connections (even though there is no manifold or bre bundle per say) and denote the set of such connections L ∈ op g . 26 We then consider the following multiplicative group:

N := exp (ad S ) | S ∈ n [[t]]
and make it act on op g via

e ad S L = ∂ x + e ad S (Λ + Q) -∇ S S, ∇ S S := k ≥0 1 (k + 1)! (ad S ) k ∂ x (S).
The reasons for de ning this speci c action and the well-de nedness of the latter are explained in §4.3.1. Then it is not hard to show that, indeed, for any L ∈ op g , we have e ad S L ∈ op g . The Drinfeld-Sokolov hierarchies are de ned on equivalences classes of connections L ∈ op g , rather than on the connection itself. Now we need to extend our function space in the following way:

g := g + ⊕ g -, g + = g ≥0 = k ≥0 g k , g -= k <0 g k .
Through this we naturally consider g [[t]] and its subalgebras. We also imply similar de nitions for g + and g -.

A fundamental result that was proven by Drinfeld and Sokolov [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] (here we give a slightly di erent version due to Wu [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF]) is that there exists a unique function U ∈ g -[[t]] such that the connection L 0 := e -ad U L takes the form

L 0 = ∂ x + Λ + H , H ∈ s -[[t]] , (2.31)
and such that (e ad U ξ ) c = 0 for all ξ ∈ s + , where (-) c denotes the component along Cc in the decomposition (2.27) of g. The components of the functions U and H are computed recursively. The key point here is that, thanks to the Heisenberg relations (2.30), the algebra s -[[t]] is abelian! We can now formulate the matrix Lax representation of the Drinfeld-Sokolov hierarchies. Given a connection L ∈ op g , the Drinfeld-Sokolov hierarchy of type g (or the g-DS hierarchy) consists of the following system of equations:

∂L ∂t i = (e ad U Λ i ) + , L i ∈ E + .
(2.32)

As suggested above, if L is a solution of the hierarchy, then so does e ad S L for any S ∈ n [[t]]. If the multiplicity of exponent i ∈ E + is greater than 1, then we denote t i , t i , t i , . . . the associated variables. It is essentially the abelianity of s -[[t]] that makes the Drinfeld-Sokolov equations commute pairwise and thus integrable. The right-hand side of these equations are matrices whose components are di erential polynomials in the components of the matrix Q. Hence these equations eventually take the form of Equation (2.2). Finally, we easily show that the components of H are conserved densities in the sense that for any j ∈ E + ,

∫ ∂H ∂t j = 0.
For example, the KdV hierarchy (2.20) is equivalent to the system (2.32), with the following data: E + = Z odd >0 and

L = ∂ x + Λ + Q, Λ = 0 λ 1 0 , Q = 0 -u 0 0 , Λ 2i+1 = λ i Λ = 0 λ i+1 λ i 0 .

Tau functions as Toeplitz determinants

Here we explain the construction of a generic tau function for arbitrary Drinfeld-Sokolov hierarchies by C.-Z. Wu [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF]. Then we explain how to realize these tau functions as the limit of large size Toeplitz determinants by Cafasso and Wu [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF][START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF]. Finally, we describe our results of [START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF] on the computation and applications of polynomial tau functions based on the works of Cafasso and Wu.

Tau functions of the Drinfeld-Sokolov hierarchies. In [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF], C.-Z. Wu de ned a generic tau function for an arbitrary g-DS hierarchy as follows. First we introduce a function

Ω = i ∈E + Ω i i Λ -i de ned by ∂ x Ω = -H , where H ∈ s -[[t]
] is de ned in Equation (2.31). Then the functions Ω j are not di erential polynomials in the components of Q, however the functions ∂ t i Ω j are and they satisfy the closure condition

∂ t i Ω j = ∂ t j Ω i . By setting H = i ∈E + -i -1 hi Λ -i , we get ∂ hi ∂t j = ∂ hj ∂t i . (2.33)
The above equation is called the tau symmetry condition on the conserved densities hi . From Poincaré lemma, it follows that we can de ne a function

τ ∈ C [[t]] such that 27 ∂ log τ ∂x ∂t j = hj .
(2.34)

The function τ we call the tau function. Wu's tau function is generic in the sense that its construction works for arbitrary a ne algebras (previous de nitions where somehow caseby-case). Now by considering the formal power series Θ = e U e Ω (we show that is it well de ned), we can rede ne the tau function by the relation

∂ log τ ∂t j = (ΘΛ j Θ -1 ) c (2.35)
Tau functions as Toeplitz determinants. In [CW15a, CW15b], Cafasso and Wu showed how to express the tau function τ associated to the function Θ as the limit of large size Toeplitz determinants. First of all, we x a faithful traceless representation 28 π : g → gl(n)

27 Equation (2.34) is equivalent to setting ∂ t j log τ = -Ω j . We shall emphasize that since the Ω j 's are not di erential polynomials, it does not mean that the hj 's are exact derivatives of di erential polynomials. In particular, Equation (2.34) does not imply that ∫ hj = 0. 28 By gl(n) we mean Mat(n, C) (n × n complex matrices) equipped with Lie bracket de ned by the commutator. as in Appendix A. Toeplitz matrices are in nite matrices of size "Z ≥0 × Z ≥0 " that are constant along diagonals, i.e. of the form

T(γ ) = (γ I -) I, ≥0 = γ 0 γ -1 γ -2 γ 1 γ 0 γ -1 . . . γ 2 γ 1 γ 0 . . . . . . . . . . . . ,
where the γ I 's are certain coe cients that, in our case, we set to be elements of gl(n)

[[t]].
We gather these coe cients in a power series

γ = I ∈Z γ I λ I ∈ gl(n)[[λ, λ -1 , t]] (same λ as before).
Of course it is not clear that the determinants of T(γ ) can be de ned and it is a technical aspect of this approach. To do so, we consider the truncated Toeplitz matrices T N (γ ) = γ I -with I , ∈ {1, . . . , N }.

In §4.4 we explain the following result of [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF][START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF]. Consider an arbitrary element X ∈ g -and the following function that we call the matrix Xi function,

Ξ(t; λ) := j ∈E + t i Λ i ∈ s + [[t]] .
Then (1) the function X (t) = e Ξ(t;λ) e X is a well de ned power series in gl

(n)[[λ, λ -1 , t]]; 29 (2) the limit lim N →∞ det T N ( X (t)) is a well de ned power series in C [[t]] and (3) the function τ X (t) := lim N →∞ det T N ( X (t)) κ (2.36)
is a tau function of the g-DS hierarchy in the sense of Equation (2.34). Here κ is a number associated to the representation π : g → gl(n) by (Y , Z ) 0 = κ • tr(π (Y )π (Z )) for all Y , Z ∈ g (with (-, -) 0 the standard bilinear form on g). Typically, for simple Lie algebras of type A n and C n , we have κ = 1 and for types B n and D n we have κ = 1/2. More precisely, the function Θ of Equation (2.34) to which the tau function τ X (t) is associated is the one such that Θ| t=0 = e X . It can be easily shown that for any X ∈ g -, there exists such a function Θ; see Lemma 4.4.2.

Polynomial tau functions and applications

We now explain the results of [START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF] on polynomial tau functions of the Drinfeld-Sokolov hierarchies and applications; see Chapter 4, §4.5. We choose an element X ∈ g -and the associated tau function τ X (t) of Equation (2.36). Our rst point is a series expansion of τ X (t) over integer partitions (or equivalently, over Young diagrams); we denote by Y the set of integer partitions. To do so, we need to introduce the Laurent matrix L(γ ), which is also a diagonally constant matrix, yet of size "Z × Z, " namely,

L(γ ) = γ I -I, ∈Z = . . . . . . . . . . . . γ 0 γ -1 γ -2 . . . γ 1 γ 0 γ -1 . . . γ 2 γ 1 γ 0 . . . . . . . . . . . .
. So, consider the Laurent matrices r := L(e X ) and s(t) := L(e Ξ(t;λ) ) along with the following determinants30

r µ := det r i-µ i -1; j-1 i, j ∈[1, (µ)] ∈ C, s µ (t) := det s i-1; j-µ j -1 i, j ∈[1, (µ)] ∈ C[t], (2.37)
where µ ∈ Y is an integer partition of length (µ). We call r µ the Plücker coordinates of e X and we call s µ (t) the generalized Schur polynomials of type (g, π ). In the case where g = sl(n, C) and π is the fundamental representation, the polynomials s µ (t) coincide with the traditional Schur polynomials [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF], up to setting t k = 0 whenever k = 0 mod n. We proved the following theorem. It shall be seen as an application of the well-known Cauchy-Binet formula as well as a generalization of Sato's seminal result on the tau functions of the KP hierarchy [START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF].

Theorem 2.3.1 ([CDY18]
). For any X ∈ g <0 , the tau function of Equation (2.36) admits the following expansion in terms of integer partitions:

τ X (t) 1 κ = µ ∈Y s µ (t) • r µ = 1 + s r + s r + s r + s r + • • • (2.38)
See Theorem 4.5.2 in the document. From Equation (2.38) we nd an easy criterion for polynomiality of the tau function as follows. If the initial data X ∈ g -is such that π (X ) is a nilpotent matrix, then the matrix components of r = L(e X ) vanish as we move further away from the main diagonal. Consequently, for partitions µ of a higher length, the Plücker coordinates r µ of X eventually vanish. This proves that

∀ X ∈ g -, π (X ) is a nilpotent matrix =⇒ τ X (t) 1 κ is polynomial.
In §4.5.2 we give several examples of polynomial tau functions for the Drinfeld-Sokolov hierarchies associated to the simple Lie algebras of type A 1 , A 2 , B 2 and D 4 . For the A 1 casei.e. sl(2, C), i.e. the KdV hierarchy-all the tau functions we computed were, up to change of variables, the Adler-Moser polynomials which we discussed in the section 2.2.3 of this introduction. Although it has not been proven yet, we believe that our algorithm recovers all the Adler-Moser polynomials (up to the right change of variables). This shall be showed using the degree of these polynomials. For the remaining cases A 2 , B 2 , D 4 , the tau functions become rapidly fairly nontrivial.

It will constitute a future project to study the properties of these polynomials. In particular, seeing these polynomials as polynomials in x with parameters t * = {t i | i 1}, it should be investigated how the zeros of these polynomials evolve w.r.t. t * . As noted in §2.1, in the A 1 case, the zeros of the Adler-Moser polynomials evolve through time so as to form a system of Calogero-Moser type [START_REF] Airault | Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem[END_REF][START_REF] Wilson | Collisions of Calogero-Moser particles and an adelic Grassmannian[END_REF]. This is yet to be understood for arbitrary Drinfeld-Sokolov hierarchies.

Polynomial tau functions and bilinear equations. Besides being the simplest tau functions, the polynomial tau functions have the merit of being a nite expression and are therefore much easier to use in practical computations. As an application, we can detect possible bilinear equation (i.e. similar to Equation (2.24)) that the polynomial tau functions of the hierarchies must satisfy. The bilinear form of the Drinfeld-Sokolov hierarchies are known explicitly for all the Gelfand-Dickey hierarchies (i.e. for sl(n, C)), see Equation (4.98). However, to the best of our knowledge, such explicit bilinear equations for arbitrary Drinfeld-Sokolov hierarchies are not pointed in the literature. Although they must be deducible from results of the seminal series of paper "Transformation groups for soliton equations" [KM81]- [START_REF] Date | Transformation groups for soliton equations[END_REF].

So, we plugged the polynomial tau functions we computed in ansatz of bilinear equations and ask for the equations to return 0. With polynomial tau functions of su ciently high degree, this determined the coe cients of said bilinear equations. Of course, it does not mean that all tau functions must satisfy the bilinear equations we found. However, we believe it gives a good hint on who these equations must be.

In what follows, HDS

[deg≤ ] X n denotes the vector space of bilinear equations of degree ≤ that the tau functions of the DS hierarchy of type X n (A n or B n , etc.) satisfy. The degree of Hirota derivatives (introduced in Equation (2.17)) is simply set as deg D i = i where D i = D t i . We proved the following propositions; rst for the B 2 case.

Proposition 2.3.2. The following dimension estimates hold true

dim C HDS [deg≤4] B 2 = 0, dim C HDS [deg≤6] B 2 ≤ 1, dim C HDS [deg≤8] B 2 ≤ 2.

Moreover, the only possible elements in HDS

[deg≤8] B 2
are linear combinations of the following,

(D 6 1 -5D 3 1 D 3 -5D 2 3 + 9D 1 D 5 )(τ , τ ) = 0, (D 8 1 + 7D 5 1 D 3 -35D 2 1 D 2 3 -21D 3 1 D 5 -42D 3 D 5 + 90D 1 D 7 )(τ , τ ) = 0.
The above equations can be found in [START_REF] Date | Transformation groups for soliton equations II: Vertex operators and tau functions[END_REF] as the two rst nontrivial equations of the bilinear formulation of the BKP hierarchy. The latter contains all the hierarchies of type B n and D n (the special orthogonal types). In a similar way the KP hierarchy contains all the hierarchies of type A n . We also have the following proposition for the DS hierarchy of type D 4 . Notice that for the D 4 case, we have two variables associated to the exponent 3: t 3 and t 3 . This is due to the fact that the Heisenberg subalgebra s of type D n has subspaces of multiplicity 2 for all exponents in 3Z odd .

Proposition 2.3.3. The following dimension estimates hold true

dim C HDS [deg≤4] D 4 = 0, dim C HDS [deg≤6] D 4 ≤ 3.
Moreover, the only possible elements in HDS

(2D 3 1 D 3 + 4D 3 D 3 -3D 2 3 )(τ , τ ) = 0, (D 3 1 D 3 -D 3 1 D 3 + D 3 D 3 -D 2 3 )(τ , τ ) = 0, (D 6 1 + 9D 1 D 5 -10D 3 1 D 3 + 5D 3 1 D 3 -5D 3 D 3 )(τ , τ ) = 0.

Hierarchy of type D 4 and the strong DR/DZ equivalence

Finally, we describe our results on the con rmation of the strong Double Rami cation / Dubrovin-Zhang equivalence (DR/DZ); a work accepted for publication [START_REF] Du Crest De Villeneuve | Quantization of the D 4 Drinfeld-Sokolov hierarchy and Fan-Jarwis-Ruan-Witten theory[END_REF]. It is the object of Chapter 5. First we will brie y explain the concept of cohomological eld theories (CohFT), the example of Fan-Jarvis-Ruan-Witten theory (FJRW) and the construction of the Dubrovin-Zhang hierarchies.

Cohomological eld theories and the Dubrovin-Zhang hierarchies

Moduli spaces of stables curves. Essentially speaking, the moduli space M ,n is the space that parameterizes isomorphism classes of compact, possibly nodal, complex curves C with genus together with n points on it x 1 , . . . , x n ∈ C which we call markings. Markings cannot be nodes though and must be distinct from each other. That allows to de ne two such genus marked curves (C, (x 1 , . . . , x n )) and (C , (x 1 , . . . , x n )) to be isomorphic if there exists a biholomorphism ϕ : C → C such that ϕ(x i ) = x i . We say that such a curve is stable if its group of automorphism is nite. The stability assumption is necessary in order to have a well de ned moduli space; it translates into the following condition,

2 -3 + n ≥ 0.
We de ne the moduli space of stable genus curves with n marked points as the following quotient space,31

M ,n := { (C, (x 1 , . . . , x n ))}/ ∼
It is a compact complex orbifold, i.e., roughly speaking, a topological space that is locally homeomorphic an a ne Euclidean vector space quotiented by the action of a nite group. As an orbifold, it has a well de ned complex dimension

dim C M ,n = 3 -3 + n.
This orbifold comes with several important maps and bundles. The details can be found in §5.2.1; we refer to [START_REF] Zvonkine | An introduction to moduli spaces of curves and its intersection theory[END_REF]. First of all, the symmetric group S n act on M ,n by permuting the marked points. Then we have three natural maps:

The forgetful map, π : M ,n+1 → M ,n , which forgets the last marked point;

The separating attaching map, σ :

M 1 ,n 1 +1 × M 2 ,n 2 +1 → M 1 + 2
,n 1 +n 2 , which glues two curves on their last marked point;

The non-separating attaching map, τ : M ,n+2 → M +1,n , which glues a curve on itself by identifying its two last marked points. Let a point x ∈ M ,n , i.e., the isomorphism class of a curve, say C. Then it can be shown that π -1 (x) ∈ M ,n+1 is actually isomorphic to the curve C itself (together with the action of its own automorphism group). We can then de ne sections s i : M ,n → M ,n+1 by setting

s i (x) = xi ∈ π -1 (x),
where xi is the i-th marked point on the curve π -1 (x).

Then we have important bre bundles. The line bundles L i → M ,n , for i ∈ {1, . . . , n}, are constructed in such a way that the bres

L i | x at a point x ∈ M ,n satisfy L i | x T * xi π -1 (x)
, where xi = s i (x). These bundles L 1 , . . . , L n are called the tautological bundles. The bundle H → M ,n is de ned in such a way that H| x T * π -1 (x). This bundle H is called the Hodge bundle and is of rank . We can then consider the Chern classes of these bundles:

ψ i := c 1 (L i ) ∈ H 2 (M ,n , Q), λ j := c j (H) ∈ H 2j (M ,n , Q).
The classesψ 1 , . . . ,ψ n we call the psi classes and the classes λ 1 , . . . , λ we call the lambda classes.

Cohomological eld theory. Using the material above, we can de ne cohomological eld theories (CohFTs) which were introduced by Kontsevich and Manin [START_REF] Manin | Gromov-Witten classes, quantum cohomology, and enumerative geometry[END_REF]. A CohFT consists of the data of a nite-dimensional complex vector space V , with a special vector e 1 , and equipped with a metric η = (η α β ), together with maps

c ,n : V ⊗n → H * (M ,n , C)
which satisfy the following properties (we choose a base (e 1 , e 2 , . . . , e N ) and we use Einstein notation): (i) c ,n is equivariant w.r.t. to simultaneous permutation of copies of V in V ⊗n and of the marked points on the curves;

(ii) c 0,3 (e 1 , e α , e β ) = η α β ; (iii) π * c ,n (e α 1 , . . . , e α n ) = c ,n+1 (e α 1 , . . . , e α n , e 1 ); (iv) σ * c ,n (e α 1 , . . . , e α n ) = c 1 ,n 1 +1 (e 1 , . . . , e n 1 , e µ )η µν c 2 ,n 2 +1 (e ν , e 1 , . . . , e n 2 ); (v) τ * c +1,n (e α 1 , . . . , e α n ) = η µν c ,n+2 (e µ , e ν , e α 1 , . . . , e α n ).
All in all, we ask for the classes c ,n to "behave naturally" with respect to the maps π , σ and τ , considering the metric η α β .

Given a CohFT {c ,n | , n ≥ 0}, we de ne the intersection numbers (or correlators) associated to the CohFT by

τ d 1 (e α 1 ) • • • τ d n (e α n ) := ∫ M , n c ,n (e α 1 , . . . , e α n ) • ψ d 1 1 • • •ψ d n n ∈ C.
with α i ∈ {1, . . . , N } and d j ≥ 0. These numbers vanish if the cohomological degree of the integrand di ers from 3 -3 + n. We then de ne the potential F of a CohFT as the generating series of the intersection numbers, namely,

F (t * * ) := n ≥0 2 -2+n >0 1 n! d 1 , ...,d n ≥0 τ d 1 (e α 1 ) • • • τ d n (e α n ) • t α 1 d 1 • • • t α n d n .
(2.39)

and

F (t * * ; ε) = ≥0 ε 2 F (t * * ). That introduces variables t * * := {t α i d j | α i ∈ {1, .
. . , N }, d j ≥ 0} which we sometimes call the topological variables.

For example, the trivial CohFT has vector space V = C, metric η = 1 and classes c ,n = 1. It plays an important role for it is connected to the KdV hierarchy [START_REF] Witten | Two-dimensional gravity and intersection theory on moduli space[END_REF] as we will see below. In the same fashion, the Hodge CohFT has the same data except that the classes are c ,n = Λ(s) := 1 + j=1 λ j s j . The latter is connected to the "intermediate long wave" hierarchy (ILW) [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF] Another important example is the Gromov-Witten theory of a smooth projective variety X with a given homology class β ∈ H 2 (X , Z). They are constructed via the moduli spaces M ,n (X , β) of genus , n marked curves (C, (x i )) together with a stable map f : C → X such that f * [C] = β. We recall its construction in Example 5.2.3. The vector space of the CohFT is V = H * (X , C) and the metric is given by η( , w) = ∫ P( ) w, where P :

H * (X , C) → H * (X , C) is the Poincaré duality. The classes are de ned by c ,n (e α 1 , . . . , e α N ) = p * ev * (e α 1 , . . . , e α N ), (2.40)
where p : M ,n (X , β) → M ,n maps (C, (x i ), f ) to (C, (x i )) (forgets the map f ) and where ev :

M ,n (X , β) → X n maps (C, (x i ), f ) to (f (x i ))
(evaluates the map). The proper de nition is complex and involves the virtual fundamental class [M ,n (X , β)] vir introduced by Behrend and Fantechi [START_REF] Behrend | The intrinsic normal cone[END_REF].

Fan-Jarvis-Ruan-Witten theory. One last example which is of interest here is the FJRW theory of a pair (W , G). Here W ∈ C[z 1 , . . . , z m ] is a quasi-homogeneous polynomial with weights q 1 , . . . , q m which has an isolated singularity at the origin. And G is group of diagonal matrices γ = (γ 1 , . . . , γ m ) which leave the polynomial W invariant, and which contains the element = (e 2iπ q 1 , . . . , e 2iπ q m ).

The vector space is de ned by

V = γ ∈G (Q W γ ⊗ dz γ ) G , where W γ = W | Fix(γ )
and Q W γ is its Jacobian ring and where dz γ = z i ∈Fix(γ ) dz i . We give a brief review of FJRW theory and the associated CohFT in §5.2.1, see [START_REF] Fan | The Witten equation and its virtual fundamental cycle[END_REF][START_REF] Fan | The Witten equation, mirror symmetry, and quantum singularity theory[END_REF]. Since quasi-homogeneous polynomials with isolated singularities are classi ed by the Dynkin diagrams of type ADE, there is a correspondence between FJRW theories and semisimple Lie algebras of the corresponding type.

The Dubrovin-Zhang hierarchies. The Dubrovin-Zhang hierarchies (DZ for short) are hierarchies of PDEs of the same type as those of Drinfeld and Sokolov described in §2.3.1. They are de ned for any semisimple CohFT. The latter means that the potential F of Equation (2.39) must be such that the N -dimensional complex algebra de ned by the structure constants

f α βγ = η α µ ∂ 3 F 0 ∂t µ 0 ∂t γ 0 ∂t β 0 t * p >0 =0 is semisimple. Let us denote Ω α,p; β,q (t * * ; ε) = ∂ 2 F ∂t α p ∂t β q = ≥0 Ω [2 ]
α,p; β,q ε 2 .

Then we de ne Hamiltonian densities and local Poisson structure:

h ,(0) α,p ( * ) := Ω [0] α,p+1; 1,0 (t * 0 = * , 0, 0, . . .), K α β ,(0) := η α β ∂ x .
These Hamiltonian densities commute w.r.t. to the above bracket [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF]. They form the socalled genus 0 Dubrovin-Zhang hierarchy, or the principal hierarchy of the CohFT. To construct the full genus DZ hierarchy, we make use of the so-called topological solution; it is the one solution α of the DZ hierarchy that is determined by the initial data

α (x, t * * = 0) = δ α,1 x .
Then there exist rational functions P α p such that [Get02, BPS12]

F (t * 0 , t * 1 , t * 2 , . . .) = F (P * 0 ( * 0 , . . . , * 3 -2 ), . . . , P * 3 -2 ( * 0 , . . . , * 3 -2 ), 0, 0, . . .) | x =0 .
Now consider the change of coordinates

φ( α ) = w α := α + ≥1 ε 2 ∂ 2 F ( * 0 , . . . , * 3 -2 ) ∂t α 0 ∂x . (2.41)
The full genera DZ hierarchy is then obtained by applying the inverse transform φ -1 to the genus 0 hierarchy. Namely,

h α,p (w * * ) := h ,(0) α,p • φ -1 (w * * ), K α β (w * * ) = ((φ -1 ) * K ,(0) ) α β (w * * ),
where (φ -1 ) * K ,(0) is the transformed di erential operator; the transformation is given in Equation (5.18). These data form a Hamiltonian system in the sense of Equation (2.14), i.e.

∂w α ∂t α p = {w α , h α,p } = N β =1 K α β δh α,p δu β .
(2.42)

The Hamiltonians h α,p lie in the ring

A := C u α , ε u α k >0
of extended di erential polynomials. Similarly, F = A/Im(∂ x ). We de ne a gradation on it by setting deg u α k = k and deg ε = -1 so that ε counts the number of derivatives in a monomial of total degree 0. Parameter ε is often referred to as the dispersion parameter and plays an important role in what follows. It is immediate from the de nition that the Hamiltonians h α,p are tau symmetric in the sense of [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF], i.e.

∂h α,p-1 ∂t β q = ∂h β,q-1 ∂t α p .
The above equation is to be compared with (2.33). By construction, the potential of the CohFT is a tau function of the DZ hierarchy and is associated to the topological solution. It has been proven over the years that the DZ hierarchy constructed from the FJRW theory of a simple singularity of type ADE coincides (up to certain non trivial subtleties) with the Drinfeld-Sokolov hierarchy associated to the corresponding simple Lie algebra of the same type [START_REF] Givental | Simple singularities and integrable hierarchies[END_REF][START_REF] Frenkel | Soliton equations, vertex operators, and simple singularities[END_REF]. See [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF] for a clear exposé on that matter. This correspondence32 between CohFTs and integrable hierarchies we call genralized Witten-Kontsevich correspondence.

The very rst example of such correspondence (prior to the de nition of CohFTs and of the DZ hierarchy) is the celebrated Witten-Kontsevich theorem [START_REF] Witten | Two-dimensional gravity and intersection theory on moduli space[END_REF][START_REF] Kontsevich | Intersection theory on the moduli space of curves an the matrix Airy function[END_REF] which states (in our framework) that the potential of the trivial CohFT is the topological tau function of the KdV hierarchy (2.20) (which coincides with the DZ hierarchy associated the trivial CohFT). More precisely, consider the potential F (t * ) of the trivial CohFT:

F (t * ) = ≥0 n ≥0 2 -2+n >0 1 n! d 1 , ...,d n ≥0 t d 1 • • • t d n ∫ M , n ψ d 1 1 • • •ψ d n n .
Now substitute F = -1 2 log(τ ) and then t i → (2i+1)!!t 2i+1 where (2i+1

)!! = 1•3•5 • • • (2i+1).
Then the resulting function τ is a tau function of the KdV hierarchy in the sense that = -2∂ 2

x log(τ ) solves (2.20) (in the substitution, t 0 → t 1 and we set t 1 = x). 33 Moreover, satis es (x, t i >1 = 0) = x.

The double rami cation hierarchies

We now explain how to construct Buryak's double rami cation hierarchies (DR) introduced in [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF]. The latter, as mentioned previously, are conjectured to be equivalent to the DZ hierarchies. Yet they are, in some extent, easier to construct, and they satisfy strong recursion relations that are new to the theory of integrable hierarchies. Moreover, they can be de ned for arbitrary CohFTs, not only semisimple ones. In the case of the FJRW theory of a simple singularity of type ADE, it is conjectured that the DR, DZ and DS hierarchies coincide. A fact veri ed for hierarchies of type A 1 , . . . , A 5 (and others, see below) and which we con rmed for the case of D 4 in a forthcoming preprint and described in Chapter 5.

Hamiltonian structure. First of all, the local Poisson structure is simply given by the operator η α β ∂ x , where η α β is the metric of the CohFT, meaning that for two local functionals f , ∈ F ,

{ f , } = ∫ N α, β =1 δ f δu α η α β ∂ x δ f δu β .
For the Hamiltonian densities we introduce the double rami cation cycles. Let integers a 1 , . . . , a n be such that a i = 0. Given ≥ 0, the double rami cation cycle DR (a 1 , . . . , a n ) ∈ H 2 (M ,n , Q) is, heuristically, the Poincaré dual of the homology class of the locus of marked complex curves (C, (x i )) such that a i x i is the divisor of the zeros and poles of some meromorphic function. We consider x i a pole if a i > 0 and a zero otherwise. A more precise de nition is given in §5.2.2. Form this, we de ne the following intersection numbers: let α ∈ {1, . . . , N } and let , d ≥ 0,

P ;a α,d ;α := ∫ DR (-a i ,a) λ ψ d 1 c ,n+1 (e α , e α 1 , . . . , e α n ),
where α = (α 1 , . . . , α n ) ⊂ {1, . . . , N } and a = (a 1 , . . . , a n ) ⊂ Z. The class λ is the Chern class c (H) of highest degree of the Hodge bundle. It can be shown that these numbers are polynomial in the a i 's and homogeneous of degree 2 so that they can be written as

P ;a α,d ;α = b 1 , ...,b n ≥0 b j =2 P ;b α,d ;α a b 1 1 • • • a b n n ∈ C 2 [a 1 , . . . , a n ],
We can now de ne the Hamiltonian densities:

α,d := ,n ≥0 2 -2+n >0 ε 2 n! b 1 , ...,b n ≥0 b j =2 P ;b α,d;α n j=1 u α j b j .
(2.43)

Again, the data of the local Poisson structure η α β ∂ x and of the Hamiltonian densities α,d form a Hamiltonian system in the sense of Equation (2.42). The densities α,d satisfy the following recursion relation: First we set α,-1 := η α µ u µ , where η α µ is the inverse matrix of η α µ , then we have [BR16b]

∂ x (D -1) α,p+1 = { α,p , 1,1 } = N β,γ =1 k ≥0 ∂ α,p ∂u β k η βγ ∂ k +1 x δ 1,1 δu γ , where D = ε∂ ε + s ≥0 u α s ∂ u α s .
It can be easily shown that it indeed allows to compute the densities recursively from the data of 1,1 . It shall be emphasized that this speci c form of recursion is new to the theory on integrable hierarchies.

On the other hand, these Hamiltonian densities are not tau symmetric. To obtain tau symmetric densities, it su ces to consider di erential polynomials

h α,p := δ α,p δu 1 .
With respect to the Poisson structure η α β ∂ x , they are tau symmetric Hamiltonian densities.

The strong DR/DZ equivalence conjecture. In order to state the DR/DZ conjecture, we must de ne Miura transforms which will play the role of "change of coordinates" in this context. A Miura transform is any invertible map φ : A → A, α := φ(u α ) ∈ A [0] , which is such that

det ∂ α ∂u β ε =0 α, β ∈[1, N ] 0.
We now choose special coordinates for the hierarchy that we call normal following [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF].

Given the tau structure h α,p , they are de ned as

ũα := η α β h α,-1 .
The map φ : u α → ũα is a Miura transform. We denote ). For any semisimple CohFT there exists a unique di erential polynomial f ∈ A [-2] such that the DR and DZ hierarchies coincide up to the normal Miura transform de ned by

h α,p := h α,p • φ, K α β := φ * (η α β ∂ x ) (2.
ũα = w α + η α β ∂ x { f , h β,0 } ,
In particular, for the CohFT consisting of the FJRW theory of a simple singularity of type ADE, the conjecture states that the DR, DZ and DS hierarchies coincide. The strong DR/DZ equivalence has been con rmed in a number of cases [Bur15a, BG16, BDGR18]: The trivial CohFT; the full Hodge class CohFT; the hierarchies of type A 3 , A 4 , A 5 ; the Gromov-Witten theory of P 1 ; up to genus 5 for any CohFT of rank 1; up to genus 2 for any semisimple CohFT.

Hierarchy of type D 4 and the strong DR/DZ equivalence

Finally, we explain how we con rmed the strong DR/DZ equivalence in the case of the hierarchy of type D 4 . It amounts to proving that the DR and DS hierarchies of type D 4 coincide. We will compare them in normal coordinates. To do so, we used the following useful theorem of Buryak and Guéré [START_REF] Buryak | Towards a description of the double rami cation hierarchy for Witten's r -spin class[END_REF]: Given a semisimple CohFT, suppose there exists a Miura transform φ :

u α → ũα (u * * ; ε), such that ∂ ũα ∂u 1 = δ α,1 , φ * (η α β ∂ x ) = K α β , 1,1 • φ = h 1,1 , (2.45)
where φ * (η α β ∂ x ) is given in (5.18). Then the DR hierarchy, under the Miura transform φ, coincides with the DZ hierarchy. When computing the DR hierarchy associated to the FJRW CohFT of type D 4 , we nd that the Miura transform from the coordinates u α to the normal coordinates ũα reads

           ũ1 = u 1 + (-1 36 (u 3 1 ) 2 -1 36 u 3 u 3 2 )ε 2 -1 45 u 3 4 ε 4 ũ2 = u 2 ũ3 = u 3 ũ4 = u 4 (2.46)
This agrees with the rst condition of Equation (2.45). The Hamiltonian densities α,0 and 1,1 are given in §5.2.4.

So, it remains to explain how to compute the Drinfeld-Sokolov hierarchy of type D 4 . To do so, we will use the scalar Lax pair, in the sense of Equation (3.34), and the Hamiltonian structure found by Liu, Wu and Zhang [START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF]. Indeed, the DS hierarchies are constructed generically as hierarchies of matrix Lax pair, but speci cally for the case of the hierarchies of type D n , Drinfeld and Sokolov did not give a full picture of the (bi-)Hamiltonian structure. However they constructed the local Poisson structure. In order to state the problem, it is necessary to discuss the a ne algebra of type D (1) 4 .

The matrix Lax pair of the DS hierarchy of type D 4 . What we discuss here adapts to the a ne algebra of type D (1) n in general. Recall that the a ne algebra g of type D (1) 4 is the a ne algebra associated to the simple algebra D 4 , the latter being isomorphic to the algebra o(8) (8×8 antisymmetric-not antihermitian-complex matrices). That is, D (1) 4 is isomorphic to

g = o(8) ⊗ C[λ, λ -1 ] ⊕ Cc.
This a ne algebra has set of positive exponents E + = Z odd >0 ∪ (3Z odd >0 ) = {1, 3, 3 , 5, 7, 9, 9 , . . .}, meaning that all exponent of the form 3k, with k odd, has multiplicity 2. We denote Λ k (resp. Γ k ), with k odd, the generator of the Heisenberg subalgebra associated to the exponent 3k (resp. (3k) ). Similarly, we denote t k (resp. tk ) the variable associated to Λ k (resp. to Γ k ). The Λ k 's and Γ k 's are given in Equation (5.54) and (5.55). This gives the matrix Lax equations [DS84]

∂L ∂t k = (e ad U Λ k ) + , L , (2.47) ∂L ∂ tk = (e ad U Γ k ) + , L , (2.48) for k ∈ Z odd >0 , where L = ∂ x + Λ + Q, with Q ∈ b [[t]
] and where Λ is the principal cyclic elements given in Equation (2.29).

The scalar Lax pair and Hamiltonian structure. In their original paper [DS84], Drinfeld and Sokolov described the Hamiltonian structure of the ows w.r.t. the variables t k , but not tk (it needed a new algebra of operator). The local Poisson structure, in the fashion of Equation (2.22), is given as follows [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF]. First, the scalar Lax operator reads

L = ∂ 6 x + ∂ -1 x 3 µ=1 s µ ∂ 2µ-1 x + ∂ 2µ-1 x s µ + ∂ -1 x ϱ∂ -1 x ϱ, s 4 = (ϱ) 2 , (2.49)
Then by denoting

L = ∂ x • L, the rst Poisson structure reads { f , } 1 = ∫ res X [(∂ x Y + L ) --(L Y + ∂ x ) --(∂ x Y -L ) + + (L Y -∂ x ) + ] dx, ( 2
∂L ∂t k = (L k/6 ) + , L , ⇐⇒ ∂s α ∂t k = {s α , H k +6 } 1 , H k = 6 k ∫ res L k/6 .
The

D = i ≥m j ≥m-i a i, j ∂ i x m ∈ Z, a i, j ∈ A j .
The resulting algebra D, whose elements are called pseudo-di erential operators (PDOs) of the second type, allows for in nitely many terms of positive and negative degrees. The result is that there exists a unique operator L 1/2 ∈ D such that (L 1/2 ) 2 = L satsifying some constraints (with L given in (2.49)). It allows to de ne a scalar Lax pair representation of Equation (2.48) [START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF] as

∂L ∂ tk = (L k/2 ) + , L . ⇐⇒ ∂s α ∂ tk = s α , Ĥk+2 1 , Ĥk = 2 k ∫ res L k/2 .
Explicit computation of the Hamiltonian structure and comparison with the DR hierarchy. Finally, we give the explicit computations performed in §5.3.2 on the DS hierarchy of type D 4 and the result that they do coincide with the DR hierarchy of type D 4 , thus con rming the strong DR/DZ equivalence. To do so, we need to substitute the variables t k and tk , with k ∈ Z odd >0 , for the topological variables t α,p , with α ∈ {1, . . . , 4} and p ≥ 0. Similarly, we need to de ne tau symmetric Hamiltonian densities h α,p in the sense of Equation (2.33). We have, for µ ∈ {1, 2, 3} and p ≥ 0,

t µ,p = Γ µ,p • t 6p+2α -1 , h µ,p-1 = Γ -1 µ,p • res L (6p+2α -1)/6 , t 4,p = Γ 4,p • t2p+1 , h 4,p-1 = Γ -1 4,p • res L (2p+1)/2 .
The factors Γ α,p are given in (5.68). It remains to substitute the functions s α for the normal coordinates ũα = η α β h β,-1 of the DS hierarchy of type D 4 (we purposefully denote them like the normal coordinates of the DR hierarchy). We denote by h α,p the tau symmetric Hamiltonian densities h α,p written in normal coordinates ũα . Similarly, in normal coordinates, the rst Poisson structure (2.50) takes the following form,

(K α β ) = 1 6 ũ3 2 ∂ x + 1 2 ũ3 1 ∂ 2 x + 1 3 ũ3 ∂ 3 x ε 2 + 4 15 ∂ 5 x ε 4 0 6∂ x 0 0 6∂ x 0 0 6∂ x 0 0 0 0 0 0 2∂ x , where (K α β ) is the matrix-valued di erential operator such that { f , } 1 = ∫ δ f δ ũα K α β δ δ ũ β .
Considering Buryak and Guéré's theorem (see the beginning of §2.4.3), we now have all the material needed to state the main theorem of this section. Its proof (that is, the computations) is given in §5.3.2. (ii) The Hamiltonian densities h 1,1 and h 1,1 di er by a total x-derivative, i.e. h 1,1 = h 1,1 . Moreover, the level 0 Hamiltonian densities coincide: h α,0 = h α,0 for α ∈ {1, . . . , 4}.

Chapter 3

Polynomial tau functions of the Korteweg-de Vries hierarchies

Pseudo-di erential operators

In this section we de ne the ring of algebraic pseudo-di erential operators and give some of their essential properties that we will use through out the document. We refer to the classic books [START_REF] Date | Solitons, Di erential equations, symmetries and in nite dimensional algebras[END_REF], [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF], [START_REF] Dickey | Soliton Equations and Hamiltonian Systems[END_REF] for more details on pseudo-di erential operators and integrable hierarchies.

De nition and basic properties

De nition 3.1.1 (Pseudo-di erential operators). The algebra D of pseudo-di erential operators is the following set of formal power series in an indeterminate ∂ over the ring of complexvalued formal power series,

D = n i=-∞ a i ∂ i a i ∈ C[[x]], n ∈ Z ,
equipped with the non commutative product When there is no ambiguity, we will simply call them operators and omit the • notation for the product.1

a∂ n • b∂ m = k ≥0 n k ab (k) ∂ n+m-k , n k := (n -k + 1) • • • (n -1) • n k! , ( 3
For our purpose, the algebra D is de ned over C [[x]]. However, we can de ne pseudo-di erential operators over any commutative integral domain R equipped with a derivation δ that we denote DO (R, δ ). For instance, it is often de ned over C ∞ (R, C) with the derivation being the usual derivation w.r.t. the variable along R. Our approach is purely algebraic and involves only formal power series so that, in our case,

D = DO (C[[x]], ∂ = ∂ x ) .
The algebra of D is sometimes denoted

D = R((∂ -1 )) (where R = C[[x]]
). This notation is intended as a non-commutative version of the ring R((T -1 )) of formal Laurent series in T -1 , i.e. formal power series in T and T -1 with only nitely many nonzero coe cients of positive degree.

Equation (3.1) naturally generalizes the binomial coe cients n k for negative values of n. And if n < 0, then the sequence n k diverges. They still satisfy the recursion relation

n k = n -1 k -1 + n -1 k , n ∈ Z, k ≥ 0,
which extends Pascal's triangle as shown in Figure 3.1. The formula for binomial coe cients of Equation (3.1) readily implies the relation

n k = (-1) k k -n -1 k , n ≥ 0, k ∈ {0, . . . , n}, (3.2)
(so that kn -1 < 0). This relation can be easily seen in Figure 3.1 as the identi cation of coe cients (up to a sign) within the k-th column modulo a shift n → kn -1.

n \ k 0 1 2 3 4 -4 1 -4 10 -20 35 -3 1 -3 6 -10 15 -2 1 -2 3 -4 5 -1 1 -1 1 -1 1 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1
A ≥m = n k=m a k ∂ k .
In the same fashion, we call respectively the positive part and negative part of A the operators

A + = A ≥0 , A -= A -A + .
Finally, we call the residue of A its coe cient of degree -1 in ∂, i.e.,

res ∂ A = a -1
(we denote simply res A when there is no ambiguity).

The positive part A + of A is also called its di erential part while its negative part A -is also called its integral part (sometimes its purely pseudo-di erential part). We call respectively di erential operators and integral operators the elements of the following sets:

D + = n k =0 a k ∂ k , n ≥ 0 , D -= n k =-∞ a k ∂ k , n < 0 .
The product of pseudo-di erential operators stabilizes these sets, that is,

D + × D + • -→ D + and D -× D - • -→ D -. That makes D + and D -into subalgebras of D.

Order and valuation

Similarly to the case of usual (commutative) formal Laurent series, we can de ne the order and the valuation of a pseudo-di erential operator as follows.

De nition 3.1.3 (Order, valuation). Let A = a k ∂ k be a nonzero pseudo-di erential operator .

(i) We call order (or degree) of A the highest integer n such that a n 0, and denote it n = deg(A). The order of the 0 operator is set as deg(0) = -∞. (ii) We call valuation of A the least integer such that a 0, and denote it = val(A). If there is no such integer, then we set val(A) = -∞. The valuation of the 0 operator is set as val(0) = +∞; (iii) We say that A is monic if a n = 1. We will often use the "big O" notation as follows: For any integer m ∈ Z, we say that A is a "big O" of ∂ m , and we write

A = O(∂ m ) ⇔ deg A ≤ m.
In order to state the next proposition, it is necessary to set the following arithmetic rules for in nity. As usual, +∞ (respectively -∞) is set to be greater (respectively less) than any real number. Besides, we set +∞+∞ = +∞ and -∞-∞ = -∞. With these rules, and by integrality of the ring C[[x]], we get the following proposition. Proposition 3.1.4. For any pseudo-di erential operators A, B,

deg(A + B) ≤ max(deg(A), deg(B)), val(A + B) ≥ min(val(A), val(B)), (3.3) deg(A • B) = deg(A) + deg(B). (3.4)
Remark 3.1.5. We cannot get a formula of the type val(A • B) = val(A) + val(B) (which holds true for nonzero elements in the ring R((T -1 )), where R is a commutative integral domain). For example, ∂

• x = x ∂ + 1. So val(∂ • x) = 0, while val(∂) + val(x) = 1 + 0 = 1.
When we do explicit computations of products of pseudo-di erential operators, we often have to limit ourselves to computing some truncated part of said product. It is especially the case when we use computer programs. On the other hand, if one wants to compute one speci c coe cient of a product, say res AB, then the computation must involve only nitely many coe cients of A and B. Therefore, it is helpful to know precisely "how much" we need to truncate the input operators A and B to compute res AB. The following proposition gives the optimal truncation needed to compute a product of operators down to some valuation.

Proposition 3.1.6 (Optimal valuations). Let A, B be two pseudo-di erential operators:

A = p ≤n a a p ∂ p , B = p ≤n b b p ∂ p ,
i.e., A has order n a and B of order n b . Then for any integer ≤ n a + n b , the following truncated operators coincide:

(A • B) ≥ = A ≥ -n b • B ≥ -n a ≥ .
Moreover, these valuations are optimal in the sense that for any w a ≤ -n b and w b ≤ -n a , we get the same result:

A ≥w a • B ≥w b ≥ = A ≥ -n b • B ≥ -n a ≥ .
The way to interpret this proposition is that the coe cients of the operator A•B down to some degree depend only on a n a , . . . , a -n b and b n b , . . . , b -n a .

Proof. The proof is straightforward. The product of A and B reads

A • B = p ≤n a q ≤n b k ≥0 p k a p b q (k) ∂ p+q-k .
So the coe cient of degree of A • B reads

p ≤n a q ≤n b k ≥0 p k a p b q (k) δ p+q-k, .
We want to identify, in the above sum, the contribution in a p and b p with the least value of p.

Clearly, this contribution only happens when k = 0, that is, in the sum

p ≤n a q ≤n b a p b q δ p+q, = a n a b -n a + a n a -1 b -n a +1 + • • • a -n b b n b .
Therefore, the above sum only involves the terms a n a , . . . , a -n b and b n b , . . . , b -n a .

As a direct consequence of the above proposition, we get the following corollary for raising an operator to some power, down to some valuation, with optimal valuations of the factors. This formula can be helpful when programming with pseudo-di erential operators, since it gives an optimal knowledge of what is needed to compute.

Corollary 3.1.7. Let A be a pseudo-di erential operator of order m ∈ Z.

(i) For any integers q ≥ 1 and ≤ qm, the following formula uses optimal valuations:

(A q ) ≥ = A ≥w • • • • • (A ≥w • A ≥w ) ≥w +m • • • ≥ -m ≥ , (3.5
)

where w = -(q -1)m = m + -qm. (ii) As a consequence, if we denote A = k ≤m a k ∂ k , A q = k ≤qm a q,k ∂ k ,
then the coe cient a q, is a di erential polynomial in the coe cients a m , . . . , a m+ -qm .

In the above proposition, we called a di erential polynomial in a m , . . . , a m+ -qm a polynomial in these functions and their derivatives. More generally, given functions u 1 , . . . , u n we call di erential polynomials in u 1 , . . . , u n the elements of the polynomial algebra

A u 1 , . . . , u n := C u α (k ) 1 ≤ α ≤ n, k ≥ 0 .
The n-th root of an operator

In the study of integrable hierarchies, a key features of pseudo-di erential operators is the existence of a n-th root for any monic operator of order n ≥ 1. This will allow us to express the hierarchy in terms of the fractional powers of some pseudo-di erential operator.

Theorem 3.1.8 (n-th Root). Let A be a monic pseudo-di erential operator of order n ≥ 1, that is, of the form

A = ∂ n + k <n a k ∂ k . (i)
There exists a unique monic operator of order 1,

P = ∂ + k <1 p k ∂ k , such that P n = A, called the n-th root of A. (ii)
The computation of coe cients p 0 , . . . , p k , k ≤ 0 amounts to equating the operators

A ≥k+n-1 = P ≥k • • • • • (P ≥k • P ≥k ) ≥k+1 • • • ≥k+n-2 ≥k+n-1 , (3.6)
and p k is a di erential polynomial in a n-1 , . . . , a k +n-1 of the form

p k = 1 n a k +n-1 + R(a n-1 , . . . , a k+n-2 ).
Remark 3.1.9. If we relax the condition that P must be monic, then we have n distinct roots of the form P = e 2ik π /n ∂ + O(1), with k ∈ {0, . . . , n}. If moreover the operator A is not monic, i.e. A = a n ∂ n + O(∂ n-1 ), with a n {0, 1}, then there are n distinct roots of the form P = (a n ) 1/n e 2ik π /n ∂ + O(1).

Proof. We prove (ii). Denote = k + n -1 and assume that the operator P such that P n = A exists. Thanks to Corollary 3.1.7, we know that a k is a di erential polynomial in p 0 , . . . , p 1+k -n (substituting m = 1 and q = n in p m+ -qm in 3.1.7(ii)), which implies that a is a di erential polynomial in p 0 , . . . , p k . So if we want to compute a in terms of p 0 , . . . , p k , we only need to equate the two truncated operators

((P ≥k ) n ) ≥ = A ≥ . (3.7)
Then applying Equation (3.5), one nds Equation (3.6). On the other hand, developing Equation 3.7, we get

(P ≥k ) n = m 1 , ...,m n ∈ {k, ...,1} p m 1 ∂ m 1 • . . . • p m n-1 ∂ m n-1 • p m n ∂ m n = m 1 , ...,m n ∈ {k, ...,1} k 1 , ...,k n-1 ≥0 m 1 k 1 • • • m n-1 k n-1 p m 1 . . . p m n-1 p m n (k n-1 ) (k n-2 )
. . .

(k 1 ) ∂ m i -k i
Now if we select the term of order , that is, for which = m i -k i , we obtain that

a = m 1 , ...,m n ∈ {k, ...,1} k 1 , ...,k n-1 ≥0 = m i -k i m 1 k 1 • • • m n-1 k n-1 p m 1 . . . p m n-1 p m n (k n-1 ) (k n-2 )
. . .

(k 1 )
.

(3.8)

In the above equation, for the condition = m i -k i under the sum to be ful lled, and since the k i 's are nonnegative, it is necessary that

- n i=1 m i ≤ 0. (3.9)
Now we want to extract the coe cient p k from the big sum in Equation (3.8). To do so, let i 0 ∈ {1, . . . , n} be such that m i 0 = k. Then all the other m i 's must equal 1 since then, by de nition of ,

- n i=1 m i = -(n -1 + k) = 0,
which saturates Equation (3.9). As a saturation, it xes all the k i 's to 0, and we have n possible choices for m i 0 = k. Therefore,

p k = 1 n a - 1 n m 1 , ...,m n ∈ {k +1, ...,1} k 1 , ...,k n-1 ≥0 = m i -k i m 1 k 1 • • • m n-1 k n-1 p m 1 . . . p m n-1 p m n (k n-1 ) (k n-2 )
. . .

(k 1 )
.

(In the rst sum, the interval {k, . . . , 1} has been changed into {k + 1, . . . , 1}.) Consequently, by recursion on k ≤ 0, the coe cient p k is a di erential polynomial in a n-1 , . . . , a k+n-1 of the form p k = 1 n a k +n-1 + R(a n-1 , . . . , a k+n-2 ).

Example 3.1.10. The square root of the so-called Schrödinger operator L = ∂ 2 + u reads

∂ 2 + u 1/2 = ∂ + 1 2 u∂ -1 - 1 4 u (1) ∂ -2 + 1 8 u (2) -(u) 2 ∂ -3 + • • • (3.10)

Multiplicative inverse

Another important property of pseudo-di erential operators that is essential in the study of integrable hierarchies is the existence of a unique multiplicative inverse, as long as the leading coe cient of said operator is invertible within the ring

C[[x]].
The following theorem makes a more precise statement.

Theorem 3.1.11 (Multiplicative inverse). Let A be a monic operator of order n ∈ Z, that is, of the form

A = ∂ n + k <n a k ∂ k . (i)
There exists a unique monic operator of order -n that we denote

A -1 = ∂ -n + k <-n ãk ∂ k , such that AA -1 = A -1 A = 1. (ii)
The computation of coe cients ã-n-1 , . . . , ãk , with k < -n, amounts to equating the operators

A ≥2n+k • (A -1 ) ≥k ≥k +n = 1,
and ãk is a di erential polynomial in a n-1 , . . . , a 2n+k of the form ãk = -a 2n+k + (a n , . . . , a 2n+k-1 ).

The rst few terms of A -1 read

A -1 = 1 -a n-1 ∂ -1 -a n-2 -(a n-1 ) 2 -na n-1 (1) ∂ -2 -• • • Remark 3.1.12. If the operator A is not monic, i.e. if A = a n ∂ n + O(∂ n-1
), with a n {0, 1}, then by denoting A = a n B (with B monic), we easily see that A's inverse is

A -1 = (a n ) -1 B -1 .
Proof. We prove point (ii). Let a n = ã-n = 1 and let k < -n (so that 2n + k < n). Let us assume that there exists such an operator A -1 . Then by Proposition 3.1.6, we know that

A • A -1 ≥k +n = A ≥2n+k • (A -1 ) ≥k ≥k +n .
Consequently, in order to compute coe cient ãk , it is su cient to equate the truncated operators above. We get

A ≥2n+k • (A -1 ) ≥k = n m=2n+k -n m =k a m ∂ m • ãm ∂ m = n m=2n+k -n m =k ≥0 m a m ãm ( ) ∂ m+m -.
Clearly, the equation for k = -n simply reads 1 = 1. So set k < -n. If we select the term of degree n + k, we obtain

0 = n m=2n+k -n m =k ≥0 m a m ãm ( ) δ m+m -,n+k = n m=2n+k -n m =k m m + m -n -k a m ãm (m+m -n-k) 1 m+m -n-k ≥0 ,
where the symbol 1 m+m -n-k ≥0 means that we select only the terms for which m+m -n-k ≥ 0. If one saturates this inequality, i.e. imposes m + m = n + k, then the least admissible value of m is m = k, which imposes m = n. Consequently, the above expression contains the term a n ãk = ãk , and that is the only term involving ãk . In other words,

ãk = - n m=2n+k -n m =k +1 m m + m -n -k a m ãm (m+m -n-k) 1 m+m -n-k ≥0 .
By recursion on k < -n, we get that ãk is a di erential polynomial in a n-1 , . . . , a 2n+k . Finally, in the above equation, if m = 2n +k, then the inequality m +m -n -k ≥ 0 saturates and imposes m = -n. Consequently, ãk = -a 2n+k + (a n , . . . , a 2n+k -1 ), where is a di erential polynomial that does not depend on a 2n+k . One can check that A and A -1 commute.

Formal adjoint

Proposition 3.1.13 (Formal adjoint). The algebra D has a natural involution (-) * : D → D, called the formal adjoint, de ned on monomials by Notice that since (A * ) -1 A * = 1 and (A -1 ) * A * = (AA -1 ) * = 1, then

(a∂ k ) * = (-1) k ∂ k • a, ( 3
(A * ) -1 = (A -1 ) * .
Proof. We prove the anti-homomorphism property on monomials. We have,

(a∂ n • b∂ p ) * = k ≥0 n k (ab k ∂ n+p-k ) * = (-1) n+p k ≥0 (-1) k n k ∂ n+p-k • (ab k ).
On the other hand,

(b∂ p ) * • (a∂ n ) * = (-1) n+p ∂ p • b∂ n • a,
so that it amounts to proving that

∂ p • b∂ n • a = k ≥0 (-1) k n k ∂ n+p-k • (ab k ).
(3.12)

Now, using the relation

∂ n • b = k ≥0 n k b k ∂ n-k , we can write ∂ p • b∂ n • a = ∂ n+p • (ab) - k ≥1 n k ∂ p • b k ∂ n-k • a.
Using the same argument, we get that

∂ p • b∂ n • a = ∂ n+p • (ab) - n 1 ∂ n+p-1 • (ab 1 ) + k ≥2 n k ∂ p • b k ∂ n-k • a.
Using the same argument recursively, we prove Equation (3.12).

Lie algebra structure

Here we discuss the structure of Lie algebra with which D is naturally equipped. For more details on Lie algebras in general, see Section 4.1. As an associative algebra, D is naturally equipped with a structure of Lie algebra by de ning the bracket, for any operators A, B ∈ D,

[A, B] = A • B -B • A.
Because this bracket is the commutator, it immediately satis es the skew-symmetry property, [A, B] + [B, A] = 0, along with Jacobi's identity:

[A, [B, C]] + [B, [C, A]] + [C, [B, A]] = 0. (3.13)
It is worth noticing, by a direct check, that the commutator [A, B] contains no term of degree deg

A + deg B, so that deg [A, B] ≤ deg A + deg B -1. (3.14)
As in the case of linear Lie algebras, the following proposition holds.

Proposition 3.1.14. For any operator A ∈ D, the adjoint mapping ad A : B → [A, B] is a derivation of the algebra D for the product •, i.e for any B, C ∈ D,

[A, BC] = B[A, C] + [A, B]C.
As a direct consequence of Proposition 3.1.14, and similarly to the case of linear Lie algebras, we get the following corollary by recursion on q ≥ 1.

Corollary 3.1.15. For any operators A, B ∈ D and any integer q ≥ 1, the following equation holds,

[A q , B] = A q-1 [A, B] + A q-2 [A, B]A + • • • + [A, B]A q-1 .
In turns, Corollary 3.1.15 implies the following proposition which is necessary in order to inject the Gelfand-Dickey hierarchies, a fortiori the KdV hierarchy, into the KP hierarchy.

Corollary 3.1.16. Let A, B ∈ D be two operators. There exists an integer q ≥ 1 such that

[A q , B] = 0 if and only if [A, B] = 0.
Proof. Let us assume that [A, B] 0 and prove that it implies that [A q , B] 0 for any q ≥ 1.

We denote

A = k ≤n a k ∂ k and [A, B] = k ≤m c k ∂ k
, where a n 0 and c m 0. Then

[A q , B] = A q-1 [A, B] + A q-2 [A, B]A + • • • + [A, B]A q-1 , has leading term q(a n ) q-1 c m , which cannot vanish. Conversely, if [A, B] = 0, then for any q ≥ 1, [A q , B] = 0.

Local functionals

Recall that we set our functions to be formal power series in some indeterminate x. The derivation

∂ = ∂ x on C[[x]] de nes a quotient space C[[x]] := C[[x]]/(Im(∂) ⊕ C1),
whose elements are called local functionals. We denote by

∫ : C[[x]] → C[[x]] the canonical projection onto local functionals; the class of an element f ∈ C[[x]] is denotes alternatively by f = ∫ f = ∫ f (x)dx ∈ C[[x]]. Notice that C[[x]
] is not a ring for these equivalence classes do not respect the product in

C[[x]].
As a direct consequence, we get a "integration by parts" formula:

∫ f (1) = - ∫ f (1) , or more generally, ∫ f (k ) = (-1) k ∫ f (k) . (3.15)
The space of local functionals plays a pivotal role in the Hamiltonian structure of integrable hierarchies. A rst step towards it is the Adler trace, introduced by Mark Adler in [START_REF] Adler | On a Trace Functional for Formal Pseudo-Di erential Operators and the Symplectic Structure of the Korteweg-de Vries Type Equations[END_REF].

Proposition 3.1.17 (Adler trace). The Lie algebra D is naturally equipped with a linear form

-: D → C[[x]]
, called the Adler trace, de ned by, for any A ∈ D,

A = ∫ res A ∈ C[[x]],
which satis es the fundamental trace property [A, B] = 0. The Adler trace produces a nondegenerate symmetric bilinear form (a scalar product):

(A, B) = AB , that is ad-invariant, i.e. (A, [B, C]) = ([A, B], C
). This scalar product de nes an isomorphism of vector spaces (D + ) * D -.

Proof. We start by proving the trace property, which we can prove on monomials by linearity. Clearly, if both A, B ∈ D + or both A, B ∈ D -, then AB = 0. Let us assume that A = a∂ n , n ≥ 0 and B = b∂ p , p ≤ -1 and n + p ≥ -1, then

AB = ∫ res a∂ n b∂ p = ∫ res k ≥0 n k ab (k) ∂ n+p-k = ∫ n n + p + 1 ab (n+p+1) .
On the other hand,

BA = ∫ res b∂ p a∂ n = ∫ p n + p + 1 ba (n+p+1) = ∫ (-1) n+p+1 p n + p + 1 ab (n+p+1) ,
where we have used the integration part property (5.2). Now thanks to the symmetry property of binomial coe cients (3.2), the two expressions under the integral sign coincide. The ad-invariance property follows from the trace property. The non-degeneracy follows from the fact that a nonzero monomial a∂ n is non-degenerately paired with any nonzero monomial ∂ -n-1 b (as long as ab Im∂):

(a∂ n ) • (∂ -n-1 b) = ∫ res a∂ -1 b = ∫ ab
The above proves the duality (D + ) * D -.

The Kadomtsev-Petviashvili hierarchy

In this section we de ne the celebrated Kadomtsev-Petviashvili (KP) hierarchy, which rst appeared in [START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF]. Even though the main goal of this chapter is to discuss the polynomial tau functions of KdV (rather then KP), we will see that KP will be our main tool in proving our statements. Moreover, once the main theorems on KP are proved, then most of the job is done.

The KdV hierarchy then appears as a particular case of the KP hierarchy.

Let us rst have a preliminary discussion. As we will see, a solution to the equations of the KP hierarchy is expressed as a single pseudo-di erential operator Q ∈ D of the form

Q = ∂ + k <0 q k ∂ k = ∂ + q -1 ∂ -1 + q -2 ∂ -2 + • • • (3.16)
In order to be able to express the KP hierarchy, we ask the functions q i to depend on in nitely many variables t := (x; t 1 , t 2 , t 3 , . . .).

To avoid problems of convergence, we restrict our space of functions to setting

q i ∈ C[[t]] := C [[x, t 1 , t 2 , t 3 , . . .]] .
Thereby, we've extended our space of functions; in other words, from now on,

D = DO (C[[t]], ∂ = ∂ x ) .
This change does not a ect any of the results given above. We naturally de ne the action of ∂ t i on pseudo-di erential operators by setting, for any operator

A = k ≤n a k ∂ k ∈ D, ∂A ∂t i := k ≤n ∂a k ∂t i ∂ k .
This action automatically satis es Leibniz's rule

∂ ∂t i (A • B) = ∂A ∂t i • B + A • ∂B ∂t i .
In particular, the action of ∂ t i commutes with the composition by ∂:

∂ ∂t i (∂ • A) = ∂ • ∂A ∂t i .

Lax pairs of the KP hierarchy

Our rst theorem gives a de nition of the KP hierarchy in its pseudo-di erential operators formulation, one of many ways to de ne the hierarchy.

Theorem 3.2.1 (KP hierarchy [START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF]). Let Q ∈ D be an operator of the form

Q = ∂ + q -1 ∂ -1 + q -2 ∂ -2 + • • • . (3.17)
Then the following ows commute pairwise, producing the so-called KP hierarchy:

∂Q ∂t i = (Q i ) + , Q , i ∈ Z ≥1 . (3.18)
The operator Q is often called the Lax operator of the KP hierarchy. Equations (3.18) are said to be equations of Lax type and (Q,

(Q i ) + ) is called a Lax pair.
Proof. (of Theorem 3.2.1). We give the same proof as in [START_REF] Date | Solitons, Di erential equations, symmetries and in nite dimensional algebras[END_REF], pp. 14-15 (for the KdV case). First of all since [Q i , Q] = 0, we have

(Q i ) + , Q = -(Q i ) -, Q .
We easily check that deg

(Q i ) -, Q = deg ∂ t i Q = -1
, so that the equations of the hierarchy are well de ned. The ows of Equation (3.18) commute if for any i, j ≥ 1,

∂ t i ∂ t j Q = ∂ t j ∂ t i Q. Let us write ∂ t i ∂ t j Q = ∂ t i (Q j ) + , Q = ∂ t i (Q j ) + , Q + (Q j ) + , ∂ t i Q .
First, it is easy to see that

∂ t i (Q j ) + = (∂ t i Q j ) + = (Q i ) + , Q i + .
Indeed, prior to projecting onto the subspace D + , we have

∂ t i Q j = (∂ t i Q)Q j-1 + Q(∂ t i Q)Q j-2 + • • • + Q j-1 (∂ t i Q) = (Q i ) + , Q j .
Therefore,

∂ t i ∂ t j Q = (Q i ) + , Q j + , Q + (Q j ) + , (Q i ) + , Q . (3.19)
On the other hand, using the fact that for any A, B ∈ D, [A + , B + ] + = [A + , B + ] and [A -, B -] + = 0, we see that

(Q i ) + , Q j + = (Q i ) + , (Q j ) + + + (Q i ) + , (Q j ) -+ = (Q i ) + , (Q j ) + + Q i , (Q j ) -+ = (Q i ) + , (Q j ) + -Q i , (Q j ) + + .
Therefore,

∂ t i ∂ t j Q = (Q i ) + , (Q j ) + , Q -Q i , (Q j ) + + , Q + (Q j ) + , (Q i ) + , Q = (Q j ) + , Q i + , Q + Q, (Q j ) + , (Q i ) + + (Q j ) + , (Q i ) + , Q .
Now using Jacobi's identity (3.13) on the two rightmost terms of the above equation, we get

∂ t i ∂ t j Q = (Q j ) + , Q i + , Q + (Q i ) + , (Q j ) + , Q .
According to Equation (3.19), we just proved that

∂ t i ∂ t j Q = ∂ t j ∂ t i Q.
The ow along t 1 straightforwardly reads

∂ t 1 Q = [∂, Q] = ∂(Q), so that ∂ t 1 = ∂.
As a convention, we set t 1 := x .

The rst two ows along t 2 gives the following equations

∂ t 2 q -1 = q -1 (2) + 2q -2 (1) , ∂ t 2 q -2 = q -2 (2) + 2q -3 (1) + 2q -1 q -1 (1) .
Similarly, the ows along t 3 gives

∂ t 3 q -1 = q -1 (3) + 3q -2 (2) + 3q -3 (1) + 6q -1 q -1 (1) .
Eliminating q -2 and q -3 plus renaming = -2q -1 , t 2 = and t 3 = t, we nd

∂ 2 = 4 3 ∂ x ∂ t + 9 4 ∂ x - 3 4 ∂ 3 x .
The above equation is called the KP equation (if we also rename t 2 = and t 3 = t), introduced by Kadomtsev and Petviashvili in [START_REF] Kadomtsev | On the stability of solitary waves in weakly dispersive media[END_REF]; hence the KP hierarchy. In the next proposition, we give in nitely many conserved quantities expressed via the Adler trace.

Proposition 3.2.2. The local functionals H j = Q j are conserved quantities of the KP hierarchy in the sense that for any i ≥ 1,

∂H j ∂t i = ∫ res ∂Q j ∂t i = 0.
Proof. Since ∂ = ∂ x = ∂ t 1 commute with the other ∂ t i 's, we can permute ∂ and the integral sign so that, by linearity of the residue map,

∂ t i H j = ∫ res ∂ t i Q j .
Then by Equation (3.18),

∂H j ∂t i = ∫ res (Q i ) + , Q j .
Yet by Proposition 3.1.17, the above vanishes.

Equivalent forms of the KP hierarchy

The KP hierarchy admits multiple formulations: the isospectral deformation system, Sato's equation, the bilinear identity. But all these rely on the existence of the so-called "dressing operator. " The idea is to "dress up" the operator ∂ with the operator Q via an operator M: Q = M ∂M -1 . Then nding the operator M amounts to nding Q.

Proposition 3.2.3 (Dressing operator). Let Q be an operator of the form of the KP Lax operator, that is,

Q = ∂ + k <0 q k ∂ k .
There exists an operator M, called the dressing operator, of the form

M = 1 + k <0 m k ∂ k , such that Q = M ∂M -1 . (3.20)
Moreover, the operator M is determined by Q up to a transformation of the form

M → M •C, with C = 1 + k <0 c k ∂ k ,
where the functions c k do not depend on x (i.e. ∂(c k ) = 0).

Proof. First, since deg M = 0, then deg M -1 = 0 too. Therefore, deg M ∂M -1 = 1 = deg Q, so that the equation Q = M ∂M -1 is meaningful. Second, if N = MC, then N ∂N -1 = MC∂C -1 M -1 = MC ∂(C -1 ) + C -1 ∂M -1 = M ∂M -1 ,
where ∂(C -1 ) stands for the operator which coe cient of degree i is the derivative of the coe cient of degree i of C -1 (which are constant too). Consequently, transformation M → MC leaves Q invariant. To prove that Q determines M (up to M → MC), it su ces to equate M ∂ = QM. Then by methods similar to the proofs of Theorems 3.1.8 and 3.1.11, we nd that the coe cients m k are determined recursively.

Now that we have the existence of the dressing operator, we can translate the ows of the KP hierarchy on Q into the ows on M. The latter is called Sato's equation.

Theorem 3.2.4 (Sato's equation). Let Q be an operator of the form of Equation 3.16 and let M be an operator such that Q = M ∂M -1 . Then the operator Q satis es the KP hierarchy (3.18) if and only if the operator M satis es the following hierarchy of equations, called Sato's equations

∂M ∂t i = -(M ∂ i M -1 ) -M, i ≥ 1. (3.21)
Proof. Let us start from Equation (3.21) on M and prove that, equivalently, Q = M ∂M -1 satis es Equation (3.18). First, from Equation (3.21) it is not hard to see that

∂ t i M -1 = M -1 M ∂ i M -1
-. Therefore, we can write the ows of M and M -1 as

∂M ∂t i = -(Q i ) -M, ∂M -1 ∂t i = M -1 (Q i ) -.
Then, by de nition of M,

∂Q ∂t i = ∂M ∂t i ∂M -1 + M ∂ ∂M -1 ∂t i = -(Q i ) -M ∂M -1 + M ∂M -1 (Q i ) - = Q, (Q i ) -.
Clearly, these equations follow from one another by equivalences.

Our next important theorem is the equivalence between the KP hierarchy and what is called the linear system of isospectral deformation of KP (or simply its linear system). But before stating the theorem, we need to de ne an action of pseudo-di erential on some space of functions as follows. First we introduce the so-called xi function:

ξ (t; λ) = i ≥1 t i λ i (3.22)
Then, for any integer k ∈ Z, we set ∂ k e ξ (t;λ) := λ k e ξ (t;λ) , which naturally extends the usual de nition for k ≥ 0. We then extend this de nition on D and get that, for

A = k ≤n a k ∂ k ∈ D, A e ξ (t;λ) = k ≤n a k λ k e ξ (t;λ) .
Given this de nition, we can de ne the following "function space":

W = Ae ξ (t;λ) A ∈ D ⊂ C[[t, λ, λ -1 ]].
This space is, of course, completely ad hoc. It is also clearly in one-to-one correspondence with D. We can de ne an action of D on W by

A Be ξ (t;λ) = A • B e ξ (t;λ) .
Remark 3.2.5. Notice that still, the meaning of Ae ξ (t;λ) = ( k ≤n a k (t)λ k )( j ≥0 p j (t)λ j ) 2 is somehow unclear since if one preforms the natural Cauchy product of the two series in λ, then the series may not converge in the sense of the topology induced by the gradation in λ (i.e. each coe cient in λ may involve an in nite sum of elements of C [[t]]). Namely,

k ≤n a k λ k e ξ (t;λ) = i ∈Z λ i j ≥max{0,i-n } p j (t)a i-j (t).
To remedy this problem, we may prove that, thanks to the increasing degrees of the polynomials p j , each coe cient in λ i actually involves a nite sum of complex numbers for each coe cient of each monomial in the t i 's. Another way to remedy this is simply to state that the meaning we give to an expression of the form Ae ξ (t;λ) is that of a formal symbol whose derivatives (which is essentially what matters here) satisfy

∂ t i Ae ξ (t;λ) := (λ i A + ∂ t i A)e ξ (t;λ) .
We can now state the following theorem.

2 The polynomials p j (t) that appear here are called the elementary Schur polynomials; they are precisely de ned by e ξ (t;λ) = j ≥0 p j (t)λ j . They ready p 0 = 1, p 1 = t 1 , p 2 = 1 2 t 2 1 + t 2 , etc. They will be discussed in Section 3.5 and will be generalized in Section 4.5.

Theorem 3.2.6 (KP linear system). The KP hierarchy (3.18) on the operator Q holds if and only if there exists a function w ∈ W such that the following hierarchy of equations is satis ed,

Qw = λw; ∂ t i w = (Q i ) + w; i ≥ 1, λ ∈ C. (3.23)
This system is called the linear system of KP; the complex number λ is called the spectral parameter; the function w is called the wave function associated to λ.

Proof. Suppose there exists a function w ∈ W such that (3.23) holds. Then

∂Q ∂t i w + Q ∂w ∂t i = λ ∂w ∂t i ∂Q ∂t i w + Q(Q i ) + = (Q i ) + Qw ∂Q ∂t i + Q, (Q i ) + w = 0.
The above equation holds for arbitrary values of λ ∈ C, which means that the the operator

∂ t i Q + [Q, (Q i ) + ]
has in nitely many independent solutions (one w(λ) for each value of λ), while being independent of λ. Therefore, it vanishes, i.e

∂Q ∂t i + Q, (Q i ) + = 0.
We now prove the converse. Let an operator

M = 1 + k <0 m k ∂ k be such that Q = M ∂M -1 .
We will prove that the function w = M e ξ (t;λ) is a wave function of the linear system of Kp. Notice that by construction of the xi function, ∂ t i e ξ (t;λ) = ∂ i e ξ (t;λ) = λ i e ξ (t;λ) .

Now we easily see that

Qw = M ∂e ξ (t;λ) = λw, so that the function w is a eigenvector with eigenvalue λ. Now considering Sato's equation (3.21) on the dressing operator M, we get that

∂w ∂t i = ∂M ∂t i e ξ (t;λ) + M ∂e ξ (t;λ) ∂t i = -(Q i ) -M -e ξ (t;λ) + M ∂ i e ξ (t;λ) = -(Q i ) -w + M ∂ i M -1 w = -(Q i ) -w + Q i w = (Q i ) + w.
Therefore, there exist an eigenfunction w such that the linear system of KP holds.

The bilinear identity and tau functions

The bilinear identity

In this section, we follow the approach of [START_REF] Babelon | Introduction to Classical Integrable Systems[END_REF]. In the proof of Theorem 3.2.6, we saw an important fact: Let Q satisfy the KP hierarchy, then any dressing operator M = 1 + k <0 m k ∂ k , i.e. such that Q = M ∂M -1 , provides a wave function of the linear system of KP by setting

w(t; λ) := Me ξ (t;λ) = 1 + m -1 λ + m -2 λ 2 + • • • e ξ (t;λ) , (3.24)
On the other hand, since the KP hierarchy is equivalent to the existence of a wave function, we can chose the wave function to be of the form of Equation (3.51) without loss of generality. We also de ne the adjoint wave function by w * (t; λ) := (M * ) -1 e -ξ (t;λ) .

We get the following lemma.

Lemma 3.3.1. The adjoint wave function obeys the following hierarchy of equation:

Q * w = λw * ; ∂ t i w * = -(Q i ) * + w * ; i ∈ Z. (3.25)
Proof. By properties of the formal adjoint, we have that

Q * = -(M -1 ) * ∂M * (since ∂ * = -∂), so that
Q * w * = -(M -1 ) * ∂e -ξ (t;λ) = λw * .
On the other hand, since (M -1 ) * = (M * ) -1 , we have

∂ t i w * = ∂ t i M -1 * e -ξ (t;λ) + (M * ) -1 ∂ t i e -ξ (t;λ) = M -1 (Q i ) - * e -ξ (t;λ) + (M * ) -1 ∂ i e -ξ (t;λ) = (Q i ) * -(M * ) -1 e -ξ (t;λ) -(Q i ) * (M * ) -1 e -ξ (t;λ) = -(Q i ) * + w * .
We can now state the celebrated bilinear identity, rst introduced by R. Hirota's in his lecture notes [START_REF] Hirota | Direct method of nding exact solutions of nonlinear evolution equations[END_REF]. This theorem expresses an equivalence between the KP hierarchy and a bilinear formulation of the latter.

Theorem 3.3.2 (Bilinear identity). Let Q be an operator of the form (3.16) and let M be a dressing operator for Q. Let w be the function

w(t; λ) = Me ξ (t;λ) ∈ W.
Then w is a wave function of the linear system of KP (3.23) (i.e., Q satis es the KP hierarchy (3.18)) if and only if w satis es the following hierarchy of bilinear equations:

∮ dλ 2iπ ∂ j 1 t 1 • • • ∂ j m t m w(t; λ) • w * (t; λ) = 0, (3.26)
for any m ≥ 0 and j 1 , . . . , j m ≥ 0. Moreover, it is equivalent to the identity

∮ dλ 2iπ w(t; λ) • w * (t ; λ) = 0, ∀t, t .
In the above expressions, the contour integral denotes the residue around λ = ∞, i.e. the coe cient in λ -1 .

Remark 3.3.3. Everything we state is purely formal. In this fashion, it does not really make sense to say "∀ t, t . " What we actually mean by that is that the second form of the bilinear identity is an equality holding within the ring

C[[t, t , λ, λ -1 ]].
We proceed by a series of lemmas. Proof. Since w satis es (3.23), we have

∂ t i w = (Q i ) + w. And since (Q i ) + is a polynomial in ∂, any term of the form ∂ j 1 t 1 • • • ∂ j m t m w eventually reads a polynomial in the ∂ k (w)'s.
Consequently, it is su cient to prove the bilinear identity for integrand ∂ k (w) • w * , with k ≥ 0. Yet because w = Me ξ (t;λ) and because M only acts on t 1 = x, we can write

∮ dλ 2iπ ∂ k (w) • w * = ∮ dλ 2iπ ∂ k Me ξ (t;λ) • (M * ) -1 e -ξ (t;λ) = ∮ dλ 2iπ (∂ k M)e λx • (M * ) -1 e -λx = res ∂ (∂ k MM -1 ) = res ∂ ∂ k = 0,
where we have used Lemma 3.3.4. The compact form of the bilinear identity (the one with ∀t, t ) is obtained by a Taylor expansion around t = t , i.e by replacing t = t + Π ∨ and expanding the integrand.

For the nal step, we prove a statement that is a bit stronger then the implication that is converse to Lemma 3.3.5.

Proposition 3.3.6. Let w, w ∈ W be two formal power series of the form

w(t; λ) = Me ξ (t;λ) , w(t; λ) = Me -ξ (t;λ) ,
where the operators M, M ∈ D have the form

M = 1 + k <0 m k ∂ k , M = 1 + k <0 Mk ∂ k .
If the functions w and w satisfy the equation

∮ dλ 2iπ w(t; λ) • w(t ; λ) = 0, ∀t, t , (3.27)
then M = (M * ) -1 (so w = w * ) and w is a wave function of the linear system of KP (3.23) (equivalently, the operator Q = M ∂M -1 satsi es the KP hierarchy (3.18)).

Clearly, the above proposition, along with Lemma 3.3.5, implies Theorem 3.3.2.

Proof. Let us rst prove that if w and w satisfy Equation (3.27), then M = (M * ) -1 . By Lemma 3.3.4, we see that for any k ≥ 0,

res ∂ ∂ k MN * = ∮ dλ 2iπ ∂ k Me ξ (t;λ) • Me -ξ (t;λ) = ∮ dλ 2iπ ∂ k (w) • w = 0,
where we have used Equation (3.27) in the last step. On the other hand, M = 1 + O(∂ -1 ) and this readily implies that M * = 1 + O(∂ -1 ) as well. Consequently,

M M * = 1 + A, A ∈ D -,
so that res ∂ ∂ k A = 0 for any k ≥ 0. Therefore, A = 0, which means that M = (M * ) -1 . We now prove that M satis es Sato's equation (which is one equivalent form of the KP hierarchy, cf. Theorem 3.2.4), i.e. that

∂M ∂t i = -(Q i ) -M,
where Q = M ∂M -1 . First, we observe that the ows of w and e ξ are related by

∂M ∂t i + (Q i ) -M e ξ (t;λ) = ∂ ∂t i -(Q i ) + w. Indeed, using ∂ t i e ξ = ∂ i e ξ and M ∂ i = Q i M, we get ∂M ∂t i + (Q i ) -M e ξ (t;λ) = ∂w ∂t i -M ∂e ξ (t;λ) ∂t i + (Q i ) -w = ∂w ∂t i -M ∂ i e ξ (t;λ) + (Q i ) -w = ∂w ∂t i -Q i w + (Q i ) -w = ∂ ∂t i -(Q i ) + w.
Since (Q i ) + is a di erential operator in ∂ = ∂ t 1 and using Equation (3.27), we have, for k ≥ 0,

0 = ∮ dλ 2iπ ∂ k ∂ t i -(Q i ) + w • w = ∮ dλ 2iπ ∂ k ∂ t i M + (Q i ) -M e ξ (t;λ) • Me -ξ (t;λ) .
By Lemma 3.3.4, this is equivalent to

0 = res ∂ ∂ k ∂ t i M + (Q i ) -M • M -1 ,
where we have used that M * = M -1 . The above holding for any k ≥ 0, we get that

∂ t i M + (Q i ) -M = 0,
which is what was to be shown.

The tau functions of the KP hierarchy

Now that we have the bilinear identity, we can prove the essential property of the existence of a tau function for any solution of the KP hierarchy. This statement is formulated in the following theorem. We still follow the approach of [BBT03] (pp. 352-354).

w(t; λ) = τ (t -[λ -1 ]) τ (t) e ξ (t;λ) , w * (t; λ) = τ (t + [λ -1 ]) τ (t) e -ξ (t;λ) ,
where [λ -1 ] := ( 1 λ , 1 2λ 2 , 1 3λ 3 , . . .) and ξ (t; λ) = i ≥1 t i λ i . Remark 3.3.8. Notice that since τ (t) does not depend on λ, it does not contribute to the residue. Therefore, the bilinear identity can be reformulated as

∮ dλ 2iπ τ (t -[λ -1 ]) • τ (t + [λ -1 ]) exp i ≥1 (t i -t i )λ i = 0, ∀t, t .
To prove Theorem 3.3.7, we will proceed with two lemmas. For these lemmas, it will be convenient to introduce a function m ∈ W de ned by

w(t; λ) = Me ξ (t;λ) =: m(t; λ) × e ξ (t;λ) By construction, if M = 1 + k <0 m k ∂ k , then m(t; λ) = 1 + k <0 m k λ k .
Accordingly, we de ne a function m * by w * (t; λ) = (M * ) -1 e ξ (t;λ) =: m * (t; λ) × e ξ (t;λ)

Lemma 3.3.9. If w, w * satisfy the bilinear identity, then the function m satis es the functional equation

m(t -[µ -1 ]; ν ) m(t; µ) = m(t -[ν -1 ]; µ) m(t; ν ) (3.28)
Proof. We start by making two simple observations. First, for any function f (λ) = 1+ k <0 f k λ k , using the geometric series, we have

∮ dλ 2iπ f (λ) 1 -λ/µ = µ (f (µ) -1) . (3.29)
Second, using the power series expansion of the logarithm, i.e. log(

1 +x) = k ≥1 (-1) k+1 x k k -1 , we see that e -ξ (t-[µ -1 ];λ) = e -ξ (t;λ) 1 -λ/µ . (3.30)
Now, applying the bilinear identity to t and t = t -[µ -1 ] and using Equations (3.29) and (3.30), we get

0 = ∮ dλ 2iπ w(t; λ) • w * (t -[µ -1 ]; λ) = ∮ dλ 2iπ m(t; λ)e ξ (t;λ) • m * (t -[µ -1 ]; λ)e -ξ (t-[µ -1 ];λ) = ∮ dλ 2iπ m(t; λ) • m * (t -[µ -1 ]; λ) 1 -λ/µ = µ m(t; µ) • m * (t -[µ -1 ]; µ) -1 .
In other words,

m * (t -[µ -1 ]; µ) = 1 m(t; µ) . (3.31)
Similarly, we want to apply the bilinear identity to t and

t = t -[µ -1 ] -[ν -1 ]. Let us denote (t; λ) := m(t; λ) • m * (t -[µ -1 ] -[ν -1 ]; λ), then we get 0 = ∮ dλ 2iπ w(t; λ) • w * (t -[µ -1 ] -[ν -1 ]; λ) = ∮ dλ 2iπ (t; λ) (1 -λ/µ)(1 -λ/ν ) = ∮ dλ 2iπ ν ν -µ (t; λ) 1 -λ/µ - µ ν -µ (t; λ) 1 -λ/ν = µν ν -µ (t; µ) -(t; ν ) . Consequently, (t; µ) = (t; ν ), i.e. m(t; µ) • m(t -[µ -1 ] -[ν -1 ]; µ) = m(t; ν ) • m(t -[µ -1 ] -[ν -1 ]; ν ).
Using Equation (3.31), we get

m(t -[µ -1 ]; ν ) m(t; µ) = m(t -[ν -1 ]; µ) m(t; ν ) Lemma 3.3.
m(t; λ) = τ (t -[λ -1 ]) τ (t)
Proof. First of all, it is easy to check that m of this form satis es Equation (3.28). For the converse statement, let us take the logarithm of Equation (3.28):

h(t -[µ -1 ]; ν ) -h(t; ν ) = h(t -[ν -1 ]; µ) -h(t; µ), (3.32)
where h = log m. The function h is well de ned since m = 1 + O(λ -1 ). We introduce the generating function of time derivatives:

∇ λ := i ≥1 λ -i-1 ∂ t i .
Notice that for any function ϕ, we automatically get

(∂ λ -∇ λ )ϕ(t -[λ -1 ]) = 0. Applying (∂ ν -∇ ν ) to Equation (3.32) we get (∂ ν -∇ ν )h(t -[µ -1 ]; ν) -(∂ ν -∇ ν )h(t; ν ) = -∇ ν h(t; µ). Let us denote (∂ ν -∇ ν )h(t; ν ) =: i ≥1 γ i (t)ν -i-1 .
Then expanding in powers of ν , the above equation reads

γ i (t -[µ -1 ]) -γ i (t) = ∂ t i h(t; µ). (3.33)
We denote Γ i j = ∂ t i γ j -∂ t j γ i . Then by symmetry of the second partial derivatives, we can eliminate the right-hand side in Equation (3.33) and get

Γ i j (t -[µ -1 ]) -Γ i j (t) = 0.
Now, expanding in powers of µ, we get i ≥1 µ -i-1 ∂ t i Γ i j = Γ i j , so that Γ i j is actually independent of all the variables t i , i ≥ 1.

On the other hand, Γ i j is by construction a polynomial in the ∂ t k m 's. Using the KP hierarchy, we can express ∂ t k m as a polynomial in the ∂ p m 's so that Γ i j is a polynomial in the independent variables m p = ∂ p m . Because these variables are independent and because Γ i j is constant, it reduces to its constant term. Now notice that the function m ≡ 1 solves Equation (3.28). For this particular solution, h = log m = 0, so that Γ i j = 0 as well. Therefore, Γ i j = 0 for any solution (since it is constant). By Poincaré lemma, it means that there exists a function

F ∈ C[[t]] such that γ i = ∂ t i F ; we denote τ = exp F . Inserting this in Equation (3.33) we get that log τ (t -[µ -1 ]) -log τ (t) = log m(t; µ) + C, where C ∈ C is constant, i.e. m(t; µ) = C • τ (t -[µ -1 ]) τ (t) ,
where C = e -C . The constant C can be absorbed into τ (which is de ned, in particular, up to multiplication by a constant) to get the result. The equation on w * follows directly from that on w.

The two lemmas above prove Theorem 3.3.7.

3.4

The Korteweg-de Vries hierarchy

Lax pairs and other forms of the KdV hierarchy

As we already said, the KdV hierarchy can be seen a reduction of the KP hierarchy by imposing

L = Q 2 ∈ D + , i.e. Q 2 -= 0.
Let us rst state the KdV hierarchy. Theorem 3.4.1 (KdV hierarchy [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF]). Let L = ∂ 2 + u ∈ D + be a di erential operator. Then the following ows commute pairwise, producing the so-called KdV hierarchy:

∂L ∂t i = (L i/2 ) + , L , i ∈ Z odd ≥1 . (3.34)
The rst ows of the KdV hierarchy read u t 1 = u 1 (where

u t i = ∂ t i u and u 1 = u x , u 2 = u x x , etc.
), then

u t 3 = 1 4 u 3 + 3 2 uu 1 , u t 5 = 1 16 u 5 + 5 8 uu 3 + 5 4 u 1 u 2 + 15 8 u 2 u 1 , u t 7 = 1 64 u 7 + 7 32 uu 5 + 21 32 u 1 u 4 + 35 32 u 2 u 3 + 35 32 (u) 2 u 3 + 35 8 uu 1 u 2 + 35 32 (u 1 ) 3 + 35 16 (u) 3 u 1 .
The ow with respect to t 3 reads the so-called KdV equation, rst theorized by J. Boussinesq in 1877 [START_REF] Boussinesq | Essai sur la theorie des eaux courantes[END_REF] and then proposed in its current acceptation by D. J. Koretweg and G. de Vries in 1895 [START_REF] Korteweg | On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves[END_REF]. First of all, Theorem 3.1.8 ensures the existence of the square root of L. Second, as in the KP hierarchy and because [L i/2 , L j/2 ] = 0 for any i, j ≥ 1, we can rewrite the equations as

∂L ∂t i = L, (L i/2 ) -, i ∈ Z odd ≥1 .
Then one can easily show that these equations are well de ned. Just like for KP, there is a straightforward proof of Theorem 3.4.1 (see proof of Theorem 3.2.1). However, we will take a di erent approach to the problem and show that we can deduce the commutativity of the KdV ows from that of the KP ows. To do so, we start with an important lemma.

Lemma 3.4.2. Let P be a pseudo-di erential operator of the form P = ∂ + k <0 p k ∂ k . Then for any integer i ∈ Z odd ≥1 , the following equivalence holds:

∂P 2 ∂t i + (P i ) -, P 2 = 0 if and ony if ∂P ∂t i + (P i ) -, P = 0.
Proof. The proof is entirely similar to that of Corollary 3.1.16. Let us denote

A i = (P i ) -, P = k <0 a i,k ∂ k .
Suppose that ∂ t i P + A i 0 and denote m i its order, i.e. ∂ t i p m i + a i,m i 0. Then by Leibniz's rule, we can write

∂P 2 ∂t i + (P i ) -, P 2 = P ∂P ∂t i + ∂P ∂t i P + PA i + A i P = 2 ∂ t i p m i + a i,m i ∂ m i +1 + O(∂ m i ).
Therefore, ∂ t i P 2 + (P i ) -, P 2 0 as well. The converse is obvious.

Remark 3.4.3. With exactly the same arguments, we could have shown the following statement (that we use for the Geldfand-Dickey hierarchies): For any n ≥ 1,

∂P n ∂t i + (P i ) -, P n = 0 if and ony if ∂P ∂t i + (P i ) -, P = 0.
Proof. (of Theorem 3.4.1). Thanks to Lemma 3.4.2, the Lax equations of the KdV hierarchy can be interpreted as the Lax equations of the KP hierarchy on the operator P = L 1/2 . Then Theorem 3.2.1 ensures that the ows commute.

In the proof above, we implicitly proved that any operator P = ∂ + k <0 p k ∂ k such that (P 2 ) -= 0 and which satis es the KP hierarchy, provides a solution to the KdV hierarchy. In the same fashion, we can de ne the linear system of KdV as follows.

Theorem 3.4.4 (linear system of KdV). The KdV hierarchy (3.34) on the operator L = ∂ 2 + u holds if and only if there exists a function w ∈ W such that the following hierarchy of equations is satis ed,

Lw = λ 2 w; ∂ t i w = (L i/2 ) + w; i ∈ Z odd ≥1 , λ ∈ C. (3.35)
As in the KP case, one can explicitly construct a wave function w that satis es the linear system of KdV as follows. Let M = 1+ k <0 m k ∂ k be an operator such that P = M ∂M -1 where P = L 1/2 . Then

w(t; λ) = Me ξ (t;λ)
provides a wave function that solves the system (3.35). Here ξ (t; λ) = i ≥1 t i λ i as well, but we could have gotten rid of the even variables and still get a valid wave function.

As w is also a wave function for the linear system of KP, we still have the existence of a tau

function τ ∈ C[[t]] such that w(t; λ) = τ (t -[λ -1 ]) τ (t) e ξ (tλ) .
That de nes the tau functions of the KdV hierarchy. In the next section, we show that the constraint (P 2 ) -= 0 of the KdV hierarchy implies much simpler formulae for the tau function.

The tau functions of the KdV hierarchy

Proposition 3.4.5 (Tau function (KdV)). Let L = ∂ 2 + u be a solution of the KdV hierarchy. Let τ be the tau function associated to the square root operator P = L 1/2 (seen as a solution to the KP hierarchy). Then τ is determined by the constraint

u = -2 ∂ 2 ∂x 2 log τ = 2 (τ (1) ) 2 -ττ (2) (τ ) 2 . (3.36)
Proof. Let us recall how the tau function of KP is de ned starting from the operator P. First of all, P goes like this:

P = ∂ - 1 2 u∂ -1 + O(∂ -2 ) (Equation (3.10)). Since P = ∂ + O(∂ -1
), there exists an operator M = 1 + O(∂ -1 ) (the dressing operator) such that P = M ∂M -1 (Proposition 3.2.3). It is easy to show that

M = 1 -m -1 ∂ -1 -m -2 + m -1 2 ∂ -2 + O(∂ -3 ), such that ∂m -1 ∂x = - 1 2 u,
From the operator M, we de ne the function

w(t; λ) = Me ξ (t;λ) = 1 + m -1 λ -1 + O(λ -2 ) e ξ (t;λ) ∈ W,
where ξ (t; λ) = i ≥1 t i λ i , which produces a solution to the linear system of KP (3.23), the latter being equivalent to the KP hierarchy (Theorem 3.2.6). Then the fact that the wave function satis es Equation (3.23) implies the existence of the tau function

τ ∈ C[[t]] such that w(t; λ) × e -ξ (t;λ) = τ (t -[λ -1 ]) τ (t) = 1 + 1 τ ∂τ ∂x λ -1 + O(λ -2 ).
Comparing the three equations above, we get that

u = -2 ∂m -1 ∂x = -2 ∂ ∂x 1 τ ∂τ ∂x = -2 ∂ 2 ∂x 2 log τ .
As we saw, given an operator Q satisfying the KP hierarchy, one only has to impose the constraint (Q 2 ) -= 0 to produce a solution to the KdV hierarchy. It remains to know how that works at the level of tau functions. The next proposition is essential.

Proposition 3.4.6. Let τ be a tau function of the KP hierarchy. If ∂ t 2 τ = 0, then the function u = -2∂ 2 x log τ is a solution of the KdV hierarchy.

Remark 3.4.7. The converse is wrong: A tau function τ of KP can be a tau function of KdV and depend on t 2 (or other even variables). Indeed, since the tau function is de ned-in the KdV case-by u = -2∂ 2 x log τ , then any transformation of the form

τ (t) → τ (t) × exp [A(t)x + B(t)] , ∂A ∂x = ∂B ∂x = 0,
de nes the same function u, and therefore produces the same operator L = ∂ 2 +u that satis es the KdV hierarchy. Notice that, in practice, we often compute tau functions of KdV such that they do not depend on even variables. In particular, they do not depend t 2 . Then it su ces to prove that they satisfy the KP hierarchy, since Proposition 3.4.6 ensures that they will automatically be tau function of KdV.

As we will see in Section 3.5, it can actually be easier to prove that a tau function satis es the KP hierarchy, rather then the KdV hierarchy.

Proof of Proposition 3.4.6. We use the isospectral formulation of KP and consider the wave function w associated to τ via the relation

w(t; λ) = τ (t -[λ -1 ]) τ (t) e ξ (t;λ) .
Since ∂ t 2 τ = 0, we have ∂ t 2 w = λ 2 w. By Theorem 3.2.6, w satis es the system of equations

Pw = λw; ∂ t i w = (P i ) + w; i ∈ Z ≥1 .
In particular, (P 2 ) -w = 0; indeed,

0 = ∂ t 2 w -(P 2 ) + w = λ 2 w -(P 2 ) + w = (P 2 ) -w.
Since P is independent of λ, yet (P 2 ) -w = 0 for arbitrary values of λ ∈ C, it follows that (P 2 ) -= 0. Equivalently (P 2 ) + = P 2 ∈ D + and the operator L = P 2 =: ∂ 2 + u satis es the linear system system Lw = λ 2 w;

∂ t i w = (L i/2 ) + w; i ∈ Z odd ≥1 , (3.37)
which is equivalent to the KdV hierarchy.

The polynomial tau functions of KdV and the Adler-Moser polynomials

As described in the former section, to any solution u of the KdV hierarchy, we associate a tau function de ned by

u = -2 ∂ 2 ∂x 2 log τ = 2 (τ (1) ) 2 -ττ (2) (τ ) 2
If τ is polynomial then u is clearly rational. On the other hand, it was proven by Airault, McKean and Moser [START_REF] Airault | Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem[END_REF] that there are denumerably many rational solutions of KdV, each one being the orbit of the function u n = n(n + 1)/x 2 under the ows of the hierarchy.

In [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF], the authors constructed the Adler-Moser polynomials θ n (x = r 1 , r 3 . . . , r 2n-1 ) for n ≥ 0, de ned by the recursion

∂θ n+1 ∂x θ n-1 -θ n+1 ∂θ n-1 ∂x = (2n -1)(θ n ) 2 .
An important result of [AM78] (cf. Theorem 3.5.1) is that there exists a unique change of variables that transforms the Adler-Moser polynomials into polynomial tau functions of KdV and that we recover all rational solutions of KdV. But we did not know what this change of variables was.

In this section, we show that the following change of variables transforms the Adler-Moser polynomials into the polynomial tau functions of KdV: r 1 = t 1 = x, and

i ≥2 r 2i-1 α 2i-1 z 2i-1 = tanh i ≥2 t 2i-1 z 2i-1 .
where

α 2i-1 = (-1) i-1 3 2 5 2 • • • (2i -3) 2 (2i -1).
To do so, we apply this change of variables to the Adler-Moser polynomials to get some polynomials τ n (t 1 , t 3 , t 5 , . . .). Then by seeing them as functions τ n (t 1 , t 2 , t 3 , . . .) of even and odd times we show that they are tau functions of the Kadomstev-Petviasvhili hierarchy (KP). Then by Proposition 3.4.6, since the τ n 's actually depend only on odd times, they are indeed tau functions of KdV.

It is well known how to compute the polynomial tau functions of KdV without using the Adler-Moser polynomials. For instance, in [Hir04] R. Hirota constructs a family of tau functions of KP in terms of Wronskians of the elementary Schur polynomials, which can be reduced to recover the polynomial tau functions of KdV. But the Adler-Moser polynomials reveal a recursive structure in the space of rational solutions of KdV. It would be interesting to investigate how to generalize this to the Drinfeld-Sokolov hierarchies.

The Adler-Moser Polynomials

Let r = (x = r 1 , r 3 , r 5 , . . .) be a set of oddly-indexed variables. 3 We save the usual variables t = (x = t 1 , t 3 , t 5 , . . .) for the tau functions of KdV. The Adler-Moser polynomials form a sequence θ n (r 1 , r 3 . . . , r 2n-1 ), n ≥ 0, de ned by the following recursion [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF]:

θ 0 = 1, θ 1 = x and for n ≥ 1, ∂θ n+1 ∂x θ n-1 -θ n+1 ∂θ n-1 ∂x = (2n + 1)(θ n ) 2 . (3.38)
This recursion leaves an integration constant that is chosen to be r 2n-1 when computing θ n . We can check that θ n has degree d n = 1 2 n(n + 1) in x and r 2n-1 is actually the coe cient of x d n-2 in this polynomial. The rst few polynomials read

θ 0 = 1; θ 1 = x; θ 2 = x 3 + r 3 ;
θ 3 = x 6 + 5r 3 x 3 + r 5 x -5r 2 3 ; θ 4 = x 10 + 15r 3 x 7 + 7r 5 x 5 -35r 3 r 5 x 2 + 175r 3 3 x -7 3 r 2 5 + r 7 x 3 + r 3 r 7 ; θ 5 = x 15 + 35r 3 x 12 + 28r 5 x 10 + 175r 2 3 x 9 + 9r 7 x 8 -105r 3 r 5 x 7 + r 9 x 6 -147r 2 5 x 5 -63r 3 r 7 x 5 + 3675r 2 3 r 5 x 4 -18375r 4 3 x 3 -21r 5 r 7 x 3 + 5r 3 r 9 x 3 + 735r 3 r 2 5 x 2 -315r 2 3 r 7 x 2 -9 5 r 2 7 x + r 5 r 9 x -49r 3 5 + 42r 3 r 5 r 7 -5r 2 3 r 9 .

In that same article ([AM78], pp. 17-18), the authors state the following theorem.

Theorem 3.5.1 (Adler-Moser, 1978). There exists a unique change of variables r → t that transforms the Adler-Moser polynomials θ n (r) into the polynomial tau functions τ n (t) of KdV.

That is, the rational functions u n = -2∂ 2 log τ n de ne operators L n = ∂ 2 +u n that satisfy the Lax system of KdV:

∂L n ∂t 2i-1 = L 2i -1 2 n + , L n .
We aim to prove that the desired change of variables is given by: r 1 = t 1 = x, and

i ≥2 r 2i-1 α 2i-1 z 2i-1 = tanh i ≥2 t 2i-1 z 2i-1 .
where α 2i-1 = (-1) i-1 3 2 5 2 . . . (2i -3) 2 (2i -1). The latter coe cients α 2i-1 where already given in Adler and Moser's article. Notice that this change of variables amounts to simply changing the choice of the integration constant in the di erential recursion (3.38). To prove the statement, let us rst recall and prove some results stated in [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF]. These lemmas relate the Adler-Moser polynomials to a Wronskian representation through a multiplicative factor and a simple rescaling of the variables. Let s = (x = s 1 , s 3 , s 5 , . . .) be another set of variables and let ψ j (s), j ≥ 0 be functions de ned by:

j ≥1 ψ j z 2j-1 = sinh(xz) + cosh(xz) i ≥2 s 2i-1 z 2i-1 , (3.39)
and ψ 0 = 0. It readily implies the recursion ψ j

(2) = ψ j-1 , where as before ψ j (i) = ∂ i x ψ j . The rst few terms read

ψ 0 = 0; ψ 1 = 1; ψ 2 = 1 6 x 3 + s 3 3 ; ψ 3 = 1 120 x 5 + 1 2
x 2 s 3 3 + s 5 5 ; ψ 4 = 1 5040 x 7 + 1 24 x 4 s 3 3 + 1 2 x 2 s 5 5 + s 7 7 ; ψ 5 = 1 362880 x 9 + 1 720 x 6 s 3 3 + 1 24 x 4 s 5 5 + 1 2 x 2 s 7 7 + s 9 9 .

We now de ne the Wronskians of these functions:

W n := Wr(ψ 1 , . . . ,ψ n ) = det ψ j (i-1) i, j=1, ...,n .
Lemma 3.5.2. The Wronskians W n of the functions ψ j , satisfy the recursion

∂W n+1 ∂x W n-1 -W n+1 ∂W n-1 ∂x = (W n ) 2 . (3.40)
Proof. For any smooth function χ (s) with respect to x, denote W n (χ ) := Wr ψ 1 , . . . ,ψ n , χ .

Then one has the identity:

∂W n (χ ) ∂x W n+1 -W n (χ ) ∂W n+1 ∂x = W n+1 (χ )W n . (3.41)
This can be proven by noticing that the left-hand side is a linear di erential operator of order n + 1 which vanishes for χ = ψ 1 , . . . ,ψ n , as well as for ψ n+1 . Yet the functions ψ j being linearly independent, the left-hand side must be proportional to W n+1 (χ ). Comparing the highest coe cient, one obtain Equation (3.41). Now thanks to the relation ψ j

(2) = ψ j-1 and ψ 1 = x, we can compute that for χ = 1,

W n (1) = (-1) n W n-1 .
(3.42)

Indeed, rst we have W n (1) = Wr(ψ , . . . ,ψ n , 1) = (-1) n Wr(1,ψ 1 , . . . ,ψ n ). Then writing down the determinants one obtains that

Wr(1,ψ 1 , . . . ,ψ n ) = 1 0 0 • • • 0 x 1 0 • • • 0 ψ 2 ψ 2 (1) ψ 2 (2) • • • ψ 2 (n)
. . . . . .

ψ n ψ n (1) ψ n (2) • • • ψ n (n) = ψ 2 (2) • • • ψ 2 (n) . . . . . . ψ n (2) • • • ψ n (n) = ψ 1 • • • ψ 1 (n-2)
. . . . . . Now since W 0 = θ 0 = 1 and W 1 = θ 1 = x, the two sequences of polynomials di er only by a multiplicative factor that can be computed:

ψ n-1 • • • ψ n-1 (n-2) = W n , hence
θ n (r) = µ n W n (s), µ n = k j=1 (2k -2j + 1) j . (3.43)
Lemma 3.5.3. The parameters of the Adler-Moser polynomials θ n (r) and those of the Wronskians W n (s) are related via a rescaling:

s 2i-1 = r 2i-1 α 2i-1 , α 2i-1 = (-1) i-1 3 2 5 2 . . . (2i -3) 2 (2i -1) (3.44)
Proof. It all has to do with the choice of the normalization in the recursions (3.38) and (3.40). For θ n , the choice is such that r 2n-1 is the coe cient of x d n-2 . Moreover if θ n is a solution of (3.38), so is θ n + cθ n-2 so that the normalization can be expressed as

θ n = θn + r 2n-1 θ n-2 ,
where θn := θ n | r 2n-1 = 0. Similarly, de ne Wn := W n | s 2n-1 = 0. Since ψ n = ψ n + s 2n-1 , then by Equation (3.42),

W n = Wn + s 2n-1 Wr(ψ 1 , . . . ,ψ n , 1) = Wn + (-1) k-1 s 2n-1 W n-2 .
Comparing the last two equations and using Equation (3.43) one obtains the result.

A change of variables to the polynomial tau functions KdV

The aim of this section is to prove the following theorem.

Theorem 3.5.4 ( [START_REF] Du Crest De Villeneuve | From the Adler-Moser polynomials to the polynomial tau functions of KdV[END_REF]). The following change of variables transforms the Adler-Moser polynomials into the polynomial tau functions of KdV: r 1 = t 1 = x, and

i ≥2 r 2i-1 α 2i-1 z 2i-1 = tanh i ≥2 t 2i-1 z 2i-1 . (3.45)
where

α 2i-1 = (-1) i-1 3 2 5 2 • • • (2i -3) 2 (2i -1).
As a matter of fact, this change of variables does not a ect the rst few variables except for a rescaling. Here are the rst variables r 2i-1 in terms of the t 2i-1 's: r 1 = t 1 = x, and then

r 3 = -3 • t 3 , r 7 = -1575 • t 7 , r 11 = -9823275 • t 11 -t 2 3 t 5 , r 5 = 45 • t 5 , r 9 = 99255 • t 9 -1 3 t 3 3 , r 13 = 1404728325 • t 13 -t 2 3 t 7 -t 3 t 2 5 .
We voluntarily factorized by the constants α 2i-1 ; in turns, the variable r 2i-1 in terms of the s 2i-1 's is the expression within the parentheses, i.e. r 1 = s 1 = x and then

r 3 = s 3 , r 7 = s 7 , r 11 = s 11 -s 2 3 s 5 , r 5 = s 5 , r 9 = s 9 -1 3 s 3 3 , r 13 = s 13 -s 2 3 s 7 -s 3 s 2 5 .
Notice that the change of variables is homogeneous for the grading deg

r i = deg t i = deg s i = i,
as proven by Adler and Moser [START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF]. Under this change of variables, we obtain the following polynomial tau functions of KdV:

τ 0 = 1, τ 1 = x, τ 2 = x 3 -3t 3 , τ 3 = x 6 -15t 3 x 3 -45t 2 3 + 45t 5 x, τ 4 = x 10 -45t 3 x 7 + 315t 5 x 5 + 4725t 3 t 5 x 2 -4725t 3 3 x -4725t 2 5 -1475t 7 x 3 + 4725t 3 t 7 .
Then the polynomials τ n /µ n (cf. Equation (3.43)) correspond to the tau functions computed via the Wronskians of the elementary Schur polynomials following Hirota [START_REF] Hirota | The Direct Method in Soliton Theory[END_REF].

To prove Theorem 3.5.4, we introduce another sequence of functions de ned by the generating series.

j ≥1 ϕ j z 2j-1 = sinh(xz) + cosh(xz) tanh i ≥2 t 2i-1 z 2i-1 .
(3.46)

It amounts to applying the change of variables of Equation (3.45) to the functionsψ j of Equation (3.39). The rst few terms read ϕ 0 = 0; With these functions, we de ne another sequence of Wronskians:

ϕ 1 = 1; ϕ 2 = 1 6 x 3 + s 3 3 ; ϕ 3 = 1 120 x 5 + 1 2 x 2 s 3 3 + s 5 5 ; ϕ 4 = 1 5040 x 7 + 1 24 x 4 s 3 3 + 1 2 x 2 s 5 5 + s 7 7 ; ϕ 5 = 1 362880 x 9 + 1 720 t 3 3 x 6 + 1 24 t 5 5 x 4 + 1 2 t 7 7 x 2 -
τ n := Wr(ϕ 1 , . . . , ϕ n ), (3.47)
In what follows, we prove that these Wronskians are tau functions of the KP hierarchy. Yet because they depend only on odd times, they are tau functions of KdV (cf. Proposition 3.4.6), which proves Theorem 3.5.4. We use the same approach as in [START_REF] Itzykson | Combinatorics of the modular group. II. The Kontsevich integrals[END_REF] (also as in [START_REF] Cafasso | Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies[END_REF]). First we need the following lemma.

Lemma 3.5.5. The functions ϕ j satisfy the following relation for any integers i, j ≥ 1:

ϕ j (2i-1) - ∂ϕ j ∂t 2i-1 = j-i-1 k=1 ϕ k a j-i-k +1 , (3.48)
where ϕ j (2i-1) = ∂ 2i-1 x ϕ j . Here the a j 's are functions that do not depend on x and are de ned by

j ≥2 a j z 2j-1 = tanh i ≥2 t 2i-1 z 2i-1 .
Proof. It is a direct calculation: di erentiating Equation (3.46) and using a Cauchy product, one obtains

j ≥1 ϕ j (2i-1) - ∂ϕ j ∂t 2i-1 z 2j-1 = j ≥1 z 2j-1 j-i k=0 ϕ k a j-i-k +1 .
Then, noticing that ϕ 0 = 0 and a 0 = a 1 = 0, one gets the correct boundaries in the last sum.

Remark 3.5.6. This relation is to be compared with the one satis ed by the elementary Schur polynomials de ned by

exp i ≥1 t i z i = j ≥1
p j z j .

(3.49)

These polynomials satisfy the relation p j (i) = ∂ t i p j . Now de ne p j to be p 2j-1 where all even times are set to 0, i.e. p j = p 2j-1 |{t 2i = 0, ∀ i ≥ 1}. They satisfy the relation

p j (2i-1) = ∂ p j ∂t 2i-1
.

The last relation allows to prove that their Wronskians Wr( p1 , . . . , pn ) are the tau functions of KdV the same way we prove that the Wronskians of the ϕ j 's are (see for instance [START_REF] Itzykson | Combinatorics of the modular group. II. The Kontsevich integrals[END_REF]). In particular the Wronskians of the p j 's and those of the ϕ j 's coincide. Now let us introduce all the variables (x = t 1 , t 2 , t 3 , . . .) of the KP hierarchy (that is, odd and even). To prove that the τ n 's are tau functions of KP we use Sato's equation, which is equivalent to the KP hierarchy (see Theorem 3.2.4). We state Sato's equation in a equivalent form in the following proposition.

Proposition 3.5.7. Let ∆ n be the following di erential operator

∆ n (χ ) := Wr χ, ϕ 1 , . . . , ϕ n Wr (ϕ 1 , . . . , ϕ n ) = 1 τ n Wr χ, ϕ 1 , . . . , ϕ n ,
for any di erentiable function χ with respect to x. The following equation holds for any i ≥ 1:

∂∆ n ∂t i = ∆ n ∂ i ∆ -1 n + ∆ n -∆ n ∂ i . (3.50)
Then from Equation (3.50) we readily get Sato's equation as follows. Set M n = ∆ n ∂ -n , which clearly is a monic operator of order 1 (i.e.

M n = ∂ + O(1)), then ∂M n ∂t i = -(M n ∂ i M -1 n ) -M n .
Proof. It is su cient to prove the equality when acting on ϕ 1 , . . . , ϕ n which are n linearly independent functions. Yet these functions are solutions of the equation ∆ n (ϕ j ) = 0, so it amounts to proving that

∂∆ n ∂t i (ϕ j ) + ∆ n (ϕ j i ) = 0.
If i = 2 is even, then we only have to prove that ∆ n (ϕ j 2 ) = 0. Yet ϕ j 2 = ϕ j-1 so that ϕ j 2 = ϕ j-, or 0 if j ≤ . Eventually, it amounts to ∆ n (ϕ j-) = 0 which holds true. If i = 2 -1 is odd, by Lemma 3.5.5, it amounts to proving that

∂∆ n ∂t 2 -1 (ϕ j ) + ∆ n ∂ ∂t 2 -1 (ϕ j ) + j--1 k =1 ∆ n (ϕ k a j--k +1 ) = 0.
Yet the functions a j do not depend on x, so the last sum vanishes. And for the other terms it amounts to

∂∆ n ∂t 2 -1 (ϕ j ) + ∆ n ∂ ∂t 2 -1 (ϕ j ) = ∂ ∂t 2 -1 ∆ n (ϕ j ) = 0.
Finally, the following proposition states that the dressing operator M n is indeed related to the Wronskians τ n via the wave function, i.e. that τ n is a tau function of KP.

Proposition 3.5.8. The dressing operator M n and the Wronskian τ n satisfy the wave function relation

M n e ξ (t;λ) = τ n t -[λ -1 ] τ n (t) e ξ (t;λ) , (3.51)
where ξ (t; λ) = i ≥1 t i λ i and [λ -1 ] := ( 1 λ , 1 3λ 3 , 1 5λ 5 , . . .). To prove that, we need the following lemma.

Lemma 3.5.9. The shift of the functions ϕ j reads the following triangular relation for any j ≥ 1,

ϕ j (t -[λ -1 ]) = ϕ j (t) -λ -1 ϕ j 1 (t) + j-1 i=1 ϕ i (t) -λ -1 ϕ i 1 (t) b j-i (t), (3.52)
where the b j 's are functions that do not depend on x and are de ned by

j ≥0 b j z 2j = sech zλ -1 1 -zλ -1 tanh zλ -1 -zλ -1 tanh (η) + tanh zλ -1 tanh (η) -1 , (3.53) with b 0 = 1, and 4 η(t; z) := i ≥2 t 2i-1 z 2i-1 .
Here sech = 1/cosh and the exponent -1 stands for the multiplicative inverse of formal power series.

Proof. Using the fact that tanh -1 (zλ -1 ) = zλ -1 + i ≥2 z 2i-1 (2i -1)λ 2i-1 for zλ -1 ∈ (-1, 1), and applying the sum formulae of hyperbolic functions, one obtains that j ≥1

ϕ j t -[λ -1 ] z 2j-1 = sech zλ -1 j ≥1 ϕ j -λ -1 ϕ j 1 z 2j-1 1 -zλ -1 tanh (zλ -1 ) -zλ -1 tanh (η) + tanh (zλ -1 ) tanh (η)
.

Moreover, the above denominator has its constant term equal to 1 and only even powers, so is its multiplicative inverse. Therefore, the series j ≥0 b j z 2j in Equation (3.53) is well de ned and has indeed b 0 = 1, hence the triangular relation (3.52).

Remark 3.5.10. As in Lemma (3.5.5), this relation is to be compared with the one satis ed by the elementary Schur polynomials, namely,

p j (t -[λ -1 ]) = p j (t) -λ -1 p j-1 (t).
We can now prove Proposition 3.5.8 which concludes the proof of Theorem 3.5.4.

Proof of Proposition 3.5.8. We prove an equivalent equation which only uses di erential operator:

∆ n e ξ (t;λ) = λ n e ξ (t;λ) τ n t -[λ -1 ] τ n (t) .

Thanks to Lemma 3.5.9, we can rewrite the right-hand side as

λ n e ξ (t;λ) τ n ϕ 1 -λ -1 ϕ 1 1 • • • ϕ 1 n-1 -λ -1 ϕ 1 n . . . . . . ϕ n -λ -1 ϕ n 1 + j-1 i=1 ϕ i -λ -1 ϕ i 1 b j-i • • • ϕ n n-1 -λ -1 ϕ n n + j-1 i=1 ϕ i n-1 -λ -1 ϕ i n b j-i .
On the other hand, the left-hand side reads

∆ n e ξ (t;λ) = 1 τ n e ξ (t;λ) λe ξ (t;λ) • • • λ n e ξ (t;λ) ϕ 1 ϕ 1 1 • • • ϕ 1 n . . . . . . ϕ n ϕ n 1 • • • ϕ n n .
Using operations on rows and columns, we can easily check that these two expressions are equal.

Chapter 4

Polynomial tau functions of the Drinfeld-Sokolov hierarchies and generalized Schur polynomials

Semisimple Lie algebras: a brief review

In this section we brie y review the theory of semisimple Lie algebra. In the sequel, our starting point will often be the data of a semisimple Lie algebra together with a choice of a set simple roots and Weyl generators. We review these concepts. As stated above, for proofs of the statements below, we refer to the books [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF], [START_REF] Carter | Cambridge studies in advanced mathematics 96[END_REF], [START_REF] Erdmann | Lie Algerbas od Finite and A ne Type[END_REF] and to the unpublished quality notes [START_REF] Igusa | Lie Algebras[END_REF].

Abelian and semisimple Lie algebras

Let g be a complex Lie algebra,1 that is, an algebra over the eld of complex numbers C whose product, denoted (X , Y ) → [X , Y ] satis es the following properties: for any X , Y , Z ∈ g, Skew symmetry:

[X , Y ] + [Y , X ] = 0; Jacobi's identity: [X , [Y , Z ]] + [Y , [Z , X ]] + [Z , [X , Y ]] = 0.
Notice that in what follows, all Lie algebras are implicitly considered over complex numbers.

We say that g is abelian if its product is trivial, i.e. if [X , Y ] = 0 for any X , Y ∈ g. We say that g is simple if it is nite dimensional,2 non abelian and if it has no ideal other then {0} and g itself. We say that g is semisimple if it is a direct sum of simple Lie algebra; in particular, g is nite-dimensional. If g is a semisimple Lie algebra which decomposes into g = g1 ⊕ • • • ⊕ gm with the gi 's simple, then each gi is a simple ideal of g.

We will sometimes use the term nite Lie algebra to mean complex semisimple Lie algebra; this terminology will become clear later with the introduction of a ne Lie algebras.

Any algebra A can be naturally equipped with a structure of Lie algebra by de ning the bracket [X , Y ] := XY -Y X (the commutator) for any X , Y ∈ A. In particular, we denote by gl(n) the algebra Mat(n) of complex square matrices of size n equipped with the bracket de ned by the commutator. A nite-dimensional Lie algebra is called linear if it is a Lie subalgebra of gl(n) for some n.

Example 4.1.1 (Algebra sl(2)). The simplest example of a simple Lie algebra is the algebra sl(2) made of three generators {H , E, F } which satisfy the following relation,

[E, F ] = H , [H, E] = 2E, [H , F ] = -2F . (4.1)
This algebra can be realized with complex square matrices of size 2 as follows:

H = 1 0 0 -1 , E = 0 0 1 0 , F = 0 1 0 0 .

Automorphisms and the adjoint representation

A morphism from a Lie algebra g1 to another Lie algebra g2 is a map ϕ : g1

→ g2 such that ϕ([X , Y ]) = [ϕ(X ), ϕ(Y )]
for any X , Y ∈ g1 . An automorphism of a Lie algebra g is a morphism ϕ : g → g that is bijective (then its inverse is automatically a morphism too). The group of automorphisms (under composition) of a Lie algebra g is denoted Aut(g).

To any element X ∈ g we associate a linear endomorphism ad X : g → g de ned by ad X (Y ) = [X , Y ] and called the adjoint endomorphism of X . The adjoint map is not a morphism but a derivation, i.e. for any X , Y , Z ∈ g,

ad X [Y , Z ] = [ad X (Y ), Z ] + [Y , ad X (Z )],
the above being equivalent to Jacobi's identity. The above equation can also be rewritten equivalently as

ad [X,Y ] (Z ) = [ad X , ad Y ](Z ),
where [ad X , ad Y ] = ad X • ad Yad Y • ad X (the commutator within the algebra End(g) of endomorphisms of g). This allows to de ne the so-called adjoint representation ad : g -→ End(g) X -→ ad X .

To sum up, Jacobi's identity is equivalent to the fact that the adjoint endomorphism ad X : g → g, associated to any element X ∈ g, is a derivation; it is also equivalent to the fact that the adjoint map ad : g → End(g) is a morphism of algebras (i.e. de nes a representation of g on g itself as a vector space).

Killing form

The adjoint representation ad : g → End(g) de nes a symmetric bilinear form κ : g ⊗ g → C, called the Killing form

κ(X , Y ) = (X , Y ) := tr(ad X • ad Y ). (4.2)
Since the trace depends only on the endomorphism, the Killing form is well de ned. Cartan's second criterion ([EW06], p. 82) states that a complex Lie algebra is semisimple if and only if its Killing form is nondegenerate. Moreover, the Killing form is said to be invariant (ad-invariant or associative) in the sense that for any X , Y , Z ∈ g,

([X , Y ], Z ) = (X , [Y , Z ]).
Using Schur's lemma ([Hum72], p. 26), one shows that any invariant scalar product (i.e. a nondegenerate symmetric bilinear form) on g is necessarily a scalar multiple of the Killing form ([Hum72], p. 118).

Inner automorphisms

We say that an element X ∈ g is ad-semisimple if its adjoint endomorphism ad X : g → g is semisimple (by which we mean that it is diagonalizable). Accordingly, we say that X ∈ g is ad-nilpotent if ad X is nilpotent. For any ad-nilpotent element X ∈ g we can de ne a linear endomorphism of g exp(ad

X ) := k ≥0 1 k! (ad X ) k ,
often denoted e ad X . By expanding the exponential, it can be easily seen that exp(ad X ) : g → g is actually an automorphism of algebras with inverse exp(-ad X ) ([Car05], p. 26). Such an automorphism is called inner. The set of all inner automorphisms then forms a subgroup of Aut(g) that we denote Inn(g) = exp(ad X ) | X ∈ g ad-nilpotent .

The composition of two inner automorphisms is given by the celebrated Baker-Campbell-

Hausor (BCH) formula [Cam97, Bak02, Hau06]: for X , Y ∈ g ad-nilpotent, exp(ad X ) exp(ad Y ) = exp(ad X •Y ),
where

X • Y = n ≥1 (-1) n-1 n n i=1 r i +s i >0 [X r 1 Y s 1 • • • X r n Y s n ] n j=1 r j + s j • n k =1 r k !s k ! , (4.3)
where

[X r 1 Y s 1 • • • X r n Y s n ] = [X , • • • , [X r 1 , [Y , • • • , [Y s 1 , • • • [X , • • • , [X r n , [Y , • • • Y ] s n • • • ].
Equation (4.3) is due to Dynkin [START_REF] Dynkin | Calculation of the coe cients in the Campbell-Hausdorf formula[END_REF] (see [START_REF] Serre | Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University[END_REF] §7 as well) and clearly shows that the map (X , Y ) → X • Y is a homomorphism. The rst few terms read

X • Y = X + Y + 1 2 [X , Y ] + 1 12 ([X , [X , Y ]] + [Y , [Y , X ]]) + • • •

Cartan subalgebras and root system

A subalgebra h of a semisimple Lie algebra g is called a Cartan subalgebra if h Is abelian; Contains only semisimple elements; Is maximal with respect to the dimension. Any semisimple Lie algebra g contains a non-unique Cartan subalgebra. Moreover, for any two Cartan subalgebras h1 and h2 , there exists an ad-nilpotent element X ∈ g such that h2 = e ad X ( h1 ) (e.g. [START_REF] Carter | Cambridge studies in advanced mathematics 96[END_REF], p. 34); we say that h1 and h2 are conjugate to one another. In particular, all Cartan subalgebras have the same dimension r ; we call r the rank of g and denote rk g = dim h = r .

Let h be a Cartan subalgebra of a semisimple Lie algebra g. We call a root of g any nonzero linear functional α ∈ h * such that there exists a nonzero X ∈ g satisfying the eigenvalue equation

[H , X ] = α(H )X , ∀ H ∈ h (i.e.
we co-diagonalize the elements of h). We denote by ∆ the set of all roots and call it the root system. Since h is abelian and all its elements are ad-semisimple, then the ad H 's, for H ∈ h, are simultaneously diagonalizable. This provides the so-called root space decomposition of g:

g = h ⊕ α ∈∆ gα , gα = X ∈ g ∀ H ∈ h, [H , X ] = α(H )X .
We also denote g0 = h. Thanks to Jacobi's identity, for any α, β ∈ ∆ and X ∈ gα , Y ∈ gβ , we have [X , Y ] ∈ gα+β ; in other words, the root space decomposition induces a gradation:

gα , gβ ⊂ gα+β . (4.4)
As a consequence, any X ∈ gα is ad-nilpotent. Each eigenspace gα is a one-dimensional complex vector space ([EW06], p. 100). The Killing form κ satis es (g α , gβ ) = 0 for any α, β ∈ ∆ such that α + β 0. In particular, κ pairs gα and g-α nondegenerately.

Similarly, κ is also nondegenerate on h = g0 . Hence, we are provided with a natural isomorphism h * → h : ϕ → T ϕ such that ϕ(H ) = (T ϕ , H ) for any H ∈ h. The latter engenders a scalar product on h * de ned by, for any ϕ,ψ ∈ h * , (ϕ,ψ ) := (T ϕ ,T ψ ).

(4.5)

Using the the scalar product (4.5), we de ne a non symmetric pairing as follows, for any roots α, β ∈ ∆,

β, α := 2(β, α) (α, α) .
It automatically satis es α, α = 2. Among other properties, the roots of a semisimple Lie algebra satisfy the following. (i) The set ∆ spans h * ;

(ii) If α ∈ ∆, then -α ∈ ∆ and no other multiple of α does;

(iii) For any α, β ∈ ∆, the quantity β, α is an integer; (iv) The set ∆ is stable by the following re ections:

σ α (β) = β -β, α α ∈ ∆.
If one replaces h * by a generic vector space E, then properties (i) to (iv) de ne what is called an abstract root system on E. As a matter of fact, the classi cation of root systems classi es simple Lie algebras [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF][START_REF] Carter | Cambridge studies in advanced mathematics 96[END_REF].

Simple roots, Cartan matrix

From the root system ∆ we can always extract a (non unique) set of simple roots (or a basis) denoted Π = {α 1 , . . . , α r } ⊂ ∆, satisfying the following properties:

The set Π spans h * ; For any α ∈ ∆, there exist k 1 , . . . , k r ∈ Z such that

α = k 1 α + • • • + k r α r ,
such that the k i 's all have the same sign (or are null). (Note that if we relax the assumption that Π has r elements, the two properties above allow to prove it.) From the set of simple roots Π we naturally de ne a partition ∆ = ∆ + ∆ -, where ∆ + (resp. ∆ -) is made of all roots which decompose in Π with only nonnegative coe cients (resp. nonpositive); obviously Π ⊂ ∆ + .

If α, β are two distinct simple roots, then their pairing β, α is heavily constrained. Indeed, β, α ∈ {0, -1, -2, -3}; more precisely, up to exchanging α and β, Either β, α = 0 and then α, β = 0 as well; Or β, α = -1 and then α, β ∈ {-1, -2, -3}. By numbering the simple roots Π = {α 1 , . . . , α r } we can de ne, for any semisimple Lie algebra, a matrix (C i j ) ∈ Mat(r, Z), called the Cartan matrix, by

C i j = α j , α i .
(Notice that the indices are purposefully put in the "reverse" order.) The constraints on the pairings β, α induce similar constraints on the Cartan matrix as follows. First, C ii = 2, then if i j, then C i j ∈ {0, -1, -2, -3}; up to exchanging i and j, Either C i j = 0 and then C ji = 0 as well; Or C i j = -1 and then C ji ∈ {-1, -2, -3}. The Cartan matrix determines the semisimple Lie algebra g up to isomorphism.

Weyl generators, Chevalley involution

Let g be a rank-r semisimple Lie algebra with Cartan matrix (C i j ) and a set of simple roots

Π = {α 1 , . . . , α r }. A set of Weyl generators is a subset {E i , F i , H i | 1 ≤ i ≤ r } of
g which generates the algebra g and whose elements satisfy the following relations: for any 1 ≤ i, j ≤ r ,

[E i , F j ] = δ i j H i , [H i , E j ] = C i j E i , [H i , F j ] = -C i j F j , [H i , H j ] = 0, (4.6)
where δ i j is the Kronecker symbol. In particular the H i 's generate a Cartan subalgebra that we still denote h. Any semisimple Lie algebra admits a set of Weyl generators. For any two sets of Weyl generators {E i , F i , H i } and { Ẽi , Fi , Hi }, there exist an inner automorphism e ad X and a permutation σ of {1, . . . , r } such that for any 1

≤ i ≤ r , Ẽi = e ad X E σ (i) , Fi = e ad X F σ (i) , Hi = e ad X H σ (i) .
Besides the relations of Equation (4.6), the Weyl generators satisfy the so-called Chevalley-Serre relations:

ad E i 1-C i j E j = 0, ad F i 1-C i j F j = 0. (4.7)
Relations of Equations (4.6) and (4.7) constitute a complete set of relations in g. With any set of Weyl generators we associate the so-called Chevalley involution Ω : g → g (i.e. such that Ω 2 = id) de ned by

Ω(H i ) = -H i , Ω(E i ) = -F i , Ω(F i ) = -E i .
From Equation 4.6, we easily see that the derived algebra, namely g = [g, g] of g, is g itself:

g := [g, g] = g. (4.8)
This is speci c to semisimple algebra and does not necessarily happen in other types of Lie algebras.

Cartan, Borel and nilpotent subalgebras

As we have seen, for any root α ∈ ∆ (so α 0), each element of gα is ad-nilpotent. For that reason, we call the principal nilpotent subalgebra the subalgebra n+ (resp. n-) generated by the E i 's (resp. the F i 's) for 1 ≤ i ≤ r . To that we add the so-called Borel subalgebra b+ (resp. b-) generated by the H i 's and the E i 's (resp. the H i 's and the F i 's), along with the Cartan subalgebra h generated by the H i 's. To sum up, given a set of Weyl generators

{E i , F i , H i | 1 ≤ i ≤ r } come ve important subalgebras: n+ = E 1 , . . . , E r , b+ = E 1 , . . . , E r , H 1 , . . . , H r , h = H 1 , . . . , H r , n-= F 1 , . . . , F r , b-= F 1 , . . . , F r , H 1 , . . . , H r .
Considering the partition ∆ = ∆ + ∆ -between positive and negative roots (w.r.t. the set of simple roots Π),

n+ = α ∈∆ + gα , n-= α ∈∆ - gα ,
along with b+ = h ⊕ n+ and b-= h ⊕ n-. In the sequel, we will often denote n := nand b := bwhen there is no ambiguity. In the end, the semisimple Lie algebra g decomposes as

g = n-⊕ h ⊕ n+ .

Highest root, normalized invariant bilinear form

Let α ∈ ∆ be a root which we decompose as

α = k 1 α 1 + • • • +k r α r in the basis Π = {α 1 , . . . , α r }.
We call the height of α the integer ht(α

) = k 1 + • • • + k r .
There exists a unique root ([Car05], p. 251),

θ = a 1 α 1 + • • • + a r α r ∈ ∆ + (4.9)
called the highest root, such that for any root

α = k 1 α 1 + • • • + k r α r , we have k i ≤ a i . The integer h = 1 + a 1 + • • • + a r (4.10)
is called the Coxeter number of g (the +1 becomes clear later).

As stated at the end of the section on the Killing form, any invariant scalar product is a scalar multiple of the Killing form κ : g ⊗ g → C. So let

b 0 : g ⊗ g -→ C X ⊗ Y -→ (X , Y ) 0
be an invariant nondegenrate symmetric bilinear form on g and let µ ∈ C be such that b 0 = µ •κ; in particular, the restriction of b 0 on the Cartan subalgebra h is also nondegenerate. The form b 0 then induces an isomorphism h * → h : ϕ → T 0,ϕ de ned by (T 0,ϕ , H ) 0 = ϕ(H ), for all ϕ ∈ h * and H ∈ h. As usual, this naturally de nes a nondegenerate symmetric bilinear form on h * that we still denote b 0 : h * ⊗ h * → C, de ned by (ϕ,ψ ) 0 := (T 0,ϕ ,T 0,ψ ) 0 for all ϕ,ψ ∈ h * . We x the constant µ ∈ C by imposing the condition

(θ, θ ) 0 = 2.
The invariant nondegenrate symmetric bilinear form b 0 : h * ⊗ h * → C is called the normalized bilinear form on h * (see for instance [START_REF] Kac | In nite dimensional Lie algebras[END_REF], p. 85 §6.4).

Root space Z-gradation

Using the Coxeter number, we can re ne the root space decomposition into a Z-gradation as follows. Given the Weyl generators, we set

deg E i = 1, deg F i = -1, deg H i = 0.
Since the highest root is a a sum h-1 simple roots (where h is the Coxeter number, see Equation 4.10) we get the following Z-gradation:

g = k ∈Z gk , gk = {0} if k {1 -h, . . . , h -1}. (4.11)
Note that, again, g0 = h. This gradation amounts to setting g1 =

α i ∈Π gα i = Span{E 1 , . . . , E r }, g-1 = α i ∈Π g-α i = Span{F 1 , . . . , F r }.
In particular, dim h = dim g1 = dim g-1 = rk g = r . It easy to see from Equation (4.6) that the remaining subspaces have the following form: for any k ∈ {1, . . . , h -1},

gk = Span [E i 1 , [E i 2 , . . . , [E i k -1 , E i k ] . . .]] 1 ≤ i 1 , . . . , i k ≤ r . g-k = Span [F i 1 , [F i 2 , . . . , [F i k -1 , F i k ] . . .]] 1 ≤ i 1 , . . . , i k ≤ r .

Simple co-roots and associated Weyl generators

Starting from the set of simple roots Π, there is a constructive way of building a set of Weyl generators as follows. For any simple root α i ∈ Π, we have de ned the element T 0,α i ∈ h which is such that α i (H ) = (T 0,α i , H ) 0 for any H ∈ h. We then de ne the so-called simple coroots by

H i := 2T 0,α i (α i , α i ) 0 . (4.12)
Since b 0 is proportional to κ, we automatically get α j (H i ) = 2(α j , α i )/(α i , α i ) = C i j . Now let elements X j ∈ gα j and X -j ∈ g-α j , where j ∈ {1, . . . , r }, then we also get

[H i , X j ] = C i j X and [H i , X -j ] = -C i j X -j . It remains to choose E i ∈ gα i and F j ∈ g-α j such that [E i , F j ] = δ i j H i . First, for any H ∈ h H , [X i , X -j ] 0 = [H , X i ], X -j 0 = α j (H ) X i , X -j 0 = H ,T 0,α i 0 X i , X -j 0 .
Since b 0 is nondegenerate on h, it follows that [X i , X -j ] = (X i , X -j ) 0 T 0,α i . Moreover, (X i , X -j ) 0 = 0 unless i = j, and if so, b 0 pairs nondegenerately gα i and g-α i so that we can chose X -i such that (X i , X -i ) 0 = 1. Therefore, we obtain [X i , X -i ] = T 0,α i 0 and [X i , X -j ] = 0 if i j. Then it su ces to set E i = X i and

F j = 2X -j (α j , α j ) 0 to get the relation [E i , F j ] = δ i j H i . The family {E i , F i , H i | 1 ≤ i ≤ r } is then a set of Weyl generators. Notice that since C ii = α i , α i = 2, each triplet {E i , F i , H i } forms a subalgebra of g with relations [E i , F i ] = H i , [H i , E i ] = 2E i and [H i , F i ] = -2F i , i.
e. isomorphic to sl(2) (see Example 4.1.1). This exact construction also applies to any root α ∈ ∆ (not necessarily a simple root). Remark 4.1.2. Since the simple roots {α 1 , . . . , α r } form a basis of the vector space h * , then the simple co-roots {H 1 , . . . , H r } form a basis of h. On the other hand, by de nition of the roots and of the Cartan matrix, C i j E j = [H i , E j ] = α j (H i )E j . In other words, the vector (C 1j , . . . , C r j ) ∈ Z r reads the coordinates of the root α j in the basis Π ∨ = {H 1 , . . . , H r }; we rewrite this statement as follows:

C i j i, j ∈[1,r ] = Mat Π ∨ (α 1 , . . . , α r ).
In particular, the matrix (C i j ) is invertible.

Dynkin diagrams

Dynkin diagrams are a way to encode the entries of the Cartan matrix in a graph with simple, double or triple edges, and possibly a direction on the edges. Since the Cartan matrix determines the semisimple Lie algebra g up to isomorphism, so does the Dynkin diagram.

We start by numbering the simple roots Π = {α 1 , . . . , α r }, which de nes the Cartan matrix (C i j ) ∈ Mat(r , Z). The Dynkin diagram of g has r vertices, each one labelled by a simple root. Earlier we described all the di erent cases that can arise for the computation of C i j = α j , α i . According to these cases, we draw an edge (or not) between vertices α i and α j and add a direction on the edge as follows:

α i α j if C i j = 0 (which implies that C ji = 0 as well); α i α j if C i j = -1 and C i j = -1 as well; α i α j if C i j = -1 and C ji = -2; α i α j if C i j = -1 and C ji = -3. The Dynkin diagram of a simple Lie algebra is a connected graph. The Dynkin diagram of a semisimple Lie algebra is a graph which connected components are the Dynkin diagrams of its simple ideals. A case by case study of simple Lie algebra shows that all Dynkin diagrams (of simple Lie algebras) are exhausted in Table 4.1.

Untwisted a ne Kac-Moody algebras

In this section we review some of the properties of loop algebras and a ne algebras that we will use in the sequel. We refer to the books [START_REF] Kac | In nite dimensional Lie algebras[END_REF] and [START_REF] Carter | Cambridge studies in advanced mathematics 96[END_REF] for more details and proofs.

De nition and basic properties

The nite-dimensional data

In accordance with the previous section, we need the following material. Let g be a semisimple Lie algebra of rank r with Cartan matrix C ∈ Mat(r, Z). Let ∆ be the root system and Π =

i , F i , H i | 1 ≤ i ≤ r
} with H i is the simple co-root associated to α i (4.12) and E i , F i chosen accordingly. Let Ω : g → g be the Chevalley involution associated to our Weyl generators. As usual, we denote by h, b = band n = nthe Cartan, negative Borel and negative principal nilpotent subalgebras respectively. On top of that, we need two speci c vectors: E θ ∈ gθ and E -θ ∈ g-θ . Since both gθ and g-θ are one-dimensional, we need two constraints to x E θ and E -θ . We impose

Ω(E θ ) = -E -θ , (E θ , E -θ ) 0 = 1.
Notice that since θ is the highest root, we automatically get

[E θ , n+ ] = 0, [E -θ , n] = 0. (4.13)
Finally, we set α 0 := -θ (not a simple root) and de ne H 0 := T 0,α 0 , 3 the co-root of α 0 w.r.t. b 0 . Then using the invariance property of b 0 and the normalization (E θ , E -θ ) 0 = 1, we easily show that

H 0 = [E -θ , E θ ].
We decompose H 0 in the basis Π ∨ of h and α 0 in the basis Π of h * (similarly to Equation (4.9)) to get the following relations,

α 0 + a 1 α 1 + • • • + a r α r = 0, H 0 + a ∨ 1 H 1 + • • • + a ∨ r H r = 0.
(Notice that a i θ (H i ) and a ∨ i α i (T 0,θ ) since neither the basis Π nor Π ∨ is orthogonal.) The integer

h ∨ = 1 + a ∨ 1 + • • • + a ∨ r 3 Notice that H 0 = 2T 0,-θ /(θ |θ ) 0 since (θ |θ ) 0 = 2
we call the dual Coxeter number of g. Notice that since H 0 = T 0,α 0 and α 0 = -θ , we have

α 0 (H 0 ) = -θ (T 0,-θ ) = (T 0,-θ ,T 0,-θ ) 0 = (θ |θ ) 0 = 2, (4.14)
which also implies θ (T 0,θ ) = 2. The integers (a 0 , a 1 , . . . , a r ), with a 0 = 1 (resp. (a ∨ 0 , a ∨ 1 , . . . , a ∨ r ), with a ∨ 0 = 1), are called the Kac labels (resp. the dual Kac labels).

Structure of loop and a ne algebras

Given all these data, we call the loop algebra of g the vector space

g = g[λ, λ -1 ] := g ⊗ C[λ, λ -1 ] = n k =m X k λ k m, n ∈ Z, X k ∈ g ,
i.e. Laurent polynomials over g. This vector space is naturally equipped with a Lie bracket de ned by

[X λ p , Y λ q ] = [X , Y ]λ p+q ,
for any X , Y ∈ g and p, q ∈ Z. We extend the invariant scalar product b 0 : g → g on g by setting

(X λ p , Y λ q ) 0 := δ p+q,0 (X , Y ) 0 ,
which produces an invariant nondegenerate symmetric bilinear form on g. It is clearly isotropic since (X λ p , X λ p ) 0 = 0. We then proceed to a central extension of the loop algebra and call it the a ne algebra associated to g,4 namely,

g = g ⊕ Cc,
with the following Lie bracket: for any X , Y ∈ g, p, q ∈ Z and a, b ∈ C,

[X λ p + ac, Y λ q + bc] = [X , Y ]λ p+q + pδ p+q,0 (X , Y ) 0 c. (4.15)
Clearly, c belongs to the center of the algebra g (i.e. c commutes with all elements of g), making Cc into an ideal of g. Therefore, we have a natural surjective homomorphism π : g → g/Cc g, hence the central extension. We again extend b 0 on g to produce an invariant nondegenerate bilinear form on g de ned by

(X λ p + aK, Y λ q + bK) 0 := δ p+q,0 (X , Y ) 0 .
Remark 4.2.1. The a ne algebras that arise through the construction we described above are often called untwisted a ne algebras. But since we will not consider the twisted cases, we simply call them a ne algebras if there is no ambiguity.

Weyl generators and a ne Cartan matrix

Starting from the Weyl generators {E i , F i , H i | 1 ≤ i ≤ r }, we de ne the following elements of the a ne algebra g: for all 1 ≤ i ≤ r ,

e i = E i , f i = F i , αi = H i , (4.16) e 0 = E -θ λ, f 0 = E θ λ -1 , α0 = H 0 + c, (4.17)
where we recall that

H 0 = [E -θ , E θ ].
Then there exists a matrix (A i j ) ∈ Mat(r + 1, Z), called the a ne Cartan matrix5 of g, such that for any 0 ≤ i ≤ r ,

[e i , f i ] = δ i j αi , [ αi , e i ] = A i j e j , [ αi , f i ] = -A i j f j , [ αi , αj ] = 0, (4.18)
Notice that by Equation 4.15, [e 0 ,

f 0 ] = [E -θ , E θ ] + (E -θ , E θ ) 0 K = α0 , as stated above. Moreover, since [E θ , n+ ] = [E -θ , n] = 0, we have that [f 0 , n+ ] = 0, [e 0 , n] = 0, (4.19) (recall that n = n-).
It is well known [START_REF] Kac | In nite dimensional Lie algebras[END_REF] that the family {e i , f i , αi | 0 ≤ i ≤ r } generates the a ne algebra g; the same family-expect for α0 that we change for H 0 -generates the loop algebra g. These vectors also satisfy the so-called a ne Chevalley-Serre relations: for any 0 ≤ i ≤ 0,

ad e i 1-A i j e j = 0, ad f i 1-A i j f j = 0. (4.20)
Relations of Equations (4.18) and (4.20) form a complete set of relations in g. Finally, we can extend the the Chevalley involution Ω : g → g by

Ω( αi ) = -αi , Ω(e i ) = -f i , Ω(f i ) = -e i .

A ne Cartan subalgebra and Kac labels

By Equation 4.18, the vectors α0 , . . . , αr are all ad-semisimple (i.e. each ad αi is diagonalizable) and commute pairwise. We denote

h = Span{ α0 , . . . , αr } = h ⊕ Cc,
and call it the a ne Cartan subalgebra of the a ne algebra g. Since α0 = H 0 + c and since c is linearly independent from the loop algebra, then the family { α0 , . . . , αr } also forms a basis of h.

In Remark 4.1.2, we explained that the vector (C 1j , . . . , C r j ), of the Cartan matrix C of the semisimple algebra g, actually reads the components of the simple root α j in the basis Π ∨ = {H 1 , . . . , H r } of the Cartan subalgebra h. In particular, we saw that C is invertible. Now since for 1 ≤ i, j ≤ r , we have [ αi , e j ] = [H i , E j ] = C i j e j , it follows that A i j = C i j for all 1 ≤ i, j ≤ r . In particular, the matrix C is an invertible r × r submatrix of the matrix A. Second, A 00 = 2, as shown below,

[ α0 , e 0 ] = [H 0 , E 0 ]λ = α 0 (H 0 )E 0 λ = 2e 0 ,
where we have used the fact that c commutes with all elements, along with α 0 (H 0 ) = 2 (Equation 4.14). Finally, since α 0 = -θ , then for any 1

≤ i ≤ r , [ αi , e 0 ] = [H i , E 0 ] = -θ (H i )E 0 = - r j=1 a j α j (H i )e 0 ,
so that A i0 = -r j=1 a j C i j for all 1 ≤ i ≤ r . Similarly, we prove that A 0i = -r j=1 a ∨ i C i j . To sum up, the untwisted a ne Cartan matrix A has the form

A = 2 A 01 • • • A 0r A 10 C 11 • • • C 1r . . . . . . . . . A r 0 C r 1 • • • C r r , A i0 = - r j=1 a j C i j , A 0i = - r j=1 a ∨ i C i j .
where, again, C i j = α j (H i ) and C ii = 2 in particular. From this we show that dim KerA = 1; more precisely we show that A a 0 . . .

a r = 0 . . . 0 , a ∨ 0 • • • a ∨ r A = (0 • • • 0). (4.21)
We prove the equation on the left. It amounts to proving that for any 0 ≤ i ≤ r , we have r j=0 A i j a j = 0. If i 0, the identity follows directly from Equation (4.21). If i = 0, then it amounts to

r j=0 A i j a j = a 0 A 00 + r j=1 a j A 0j = 2 - r j,c=1 a j C k j a ∨ k = 2 - r j,c=1 a j α j (H k )a ∨ k = 2 -α 0 (H 0 ) = 0.
where we have used Equation (4.14).

Remark 4.2.2. Now we have seen to di erent types of Cartan matrix: the matrix C which is associated to the semisimple Lie algebra g and the matrix A, associated to the a ne algebra g. Cartan matrices can actually be generalized to any matrix with integer entries such that the diagonal entries equal 2, the o -diagonal entries are nonpositive, and such that if an odiagonal entry is 0, then the transposed entry is 0 too. These generalized Cartan matrices can be classi ed such that they correspond either to a semisimple Lie algebra (in which case they are called nite), an a ne algebra (in which case they are called a ne) or to none of those (in which case they are called inde nite). That is the reason why we call semisimple complex Lie algebras nite algebras.

Untwisted a ne Cartan matrices and Dynkin diagrams

If the underlying semisimple Lie algebra g is actually simple and has Dynkin diagram of type X n (X = A, B, C, . . .), then the Dynkin diagram of the untwisted a ne algebra g is denoted X (1) n . The untwisted a ne Dynkin diagram determines the Cartan matrix as in the case of semisimple Lie algebras. The Cartan matrices of these algebras are exhausted in Table 4.2.1. The white dot denotes the vertex labelled by the root α 0 = -θ . Type A (1) 1 is treated separately.

A (1) 1 sl(2) E (1) 6 A (1) n , n ≥ 2 sl(n + 1) Ẽ(1) 7 B (1) n , n ≥ 3 o(2n + 1) E (1) 8 C (1) n , n ≥ 2 sp(2n) F (1) 4 D (1) n , n ≥ 4 o(2n) G (1) 2

Gradations and principal Heisenberg subalgebra

Principal and homogeneous gradations For any vector s = (s 0 , s 1 , . . . , s r ) ∈ Z r +1 ≥0 we can associate a gradation on the a ne algebra g as follows. We set deg s e i = s i , deg s f i = -s i and deg s αi = 0 for all 0 ≤ i ≤ r . We denote by g k,s the space of elements of g of degree k ∈ Z and set g j,s , g k,s ⊂ g j+k,s .

Then we have the decomposition

g = k ∈Z g c,s .
We also de ne, for any m ∈ Z,

g ≥m,s = k ≥m g k,s , g <m,s = k <m g k <m,s .
Two gradations will be of particular interests: the principal gradation, which correspond to the vector s 1 = (1, 1, . . . , 1) and the homogeneous gradation, which correspond to the vector s 0 = (1, 0, . . . , 0). When there is no ambiguity, we will denote g k = g k,s 1 (resp. g k = g k,s 0 ) the degree-c subspace under the principal (resp. homogeneous) gradation. For the sake of clarity, these two gradations are de ned as follows, Principal:

deg e i = 1, deg f i = -1, deg αi = 0, g = k ∈Z g k ,

Homogeneous

:

deg e i = δ i,0 , deg f i = -δ i,0 , deg αi = 0, g = k ∈Z g k .
It follows that

g 0 = h = h ⊕ Cc, g 0 = g ⊕ Cc.
Clearly, both the principal and homogeneous gradations transfer to the loop algebra via the quotient by the ideal Cc. We denote g = k ∈Z g k (for the principal gradation) and g = k ∈Z g k (for the homogeneous gradation). Notice that that Z-gradation of g induced by the root space decomposition, i.e deg E i = 1, deg F i = -1 and deg H i = 0, is preserved by the inclusion map g → g. Similarly to the Zgradation of g induced by the root space decomposition, the principal gradation of g amounts to setting

g 1 = Span{e 0 , e 1 , . . . , e r }, g -1 = Span{ f 0 , f 1 , . . . , f r }.
In particular, dim h = dim g 1 = dim g -1 = r + 1. It easy to see from Equation (4.18) that the remaining subspaces have the following form: for any k > 1,

g k = Span [e i 1 , [e i 2 , . . . , [e i k -1 , e i k ] . . .]] 0 ≤ i 1 , . . . , i k ≤ r . (4.22) g -k = Span [f i 1 , [f i 2 , . . . , [f i k -1 , f i k ] . . .]] 0 ≤ i 1 , . . . , i k ≤ r . (4.23)
On the other hand, the homogeneous gradation amounts to setting g 1 = Ce 0 and g -1 = Cf 0 . The homogeneous subspace of degree k > 1 is generated by arbitrary long commutators

[e i 1 , [e i 2 , . . . , [e i k -1 , e i M ] . . .]],
with M ≥ k, such that the index 0 occurs exactly k times among the i 1 , . . . , i M . A similar assertion holds for g -k . Consequently, g k ⊂ g ⊗ Cλ k , and since the homogeneous gradation produces a decomposition of g, it follows that for any k ∈ Z,

g k = g ⊗ Cλ k ,
hence the name homogeneous gradation. In particular,

g >0 = λg[λ], g <0 = λ -1 g[λ -1 ].
Moreover, it is clear from the discussion above that for any k ≥ 1,

g k ⊂ j ≥1 g j , g -k ⊂ j ≥1 g -j .
As a consequence, g >0 ⊂ g >0 and g <0 ⊂ g >0 .

Finite exponents and principal Cartan subalgebra

The exponents can be de ned both in the case of the nite algebras (i.e. semisimple Lie algebra) and a ne algebras. The rst are called nite exponents and the latter a ne exponents. As we will see, the a ne exponents are related to the nite exponents via an arithmetical modular relation. We start with the nite exponents. We still use the same set of Weyl generators {E i , F i , H i | 1 ≤ i ≤ r } where the H i 's are the simple co-roots and the E i 's and F i 's are chosen accordingly. We de ne the so-called nite principal cyclic element:

= E 0 + E 1 + • • • + E r .
We call the principal Cartan subalgebra of g the subalgebra Ker(ad ). We then decompose Ker(ad ) through the gradation of Equation (4.11) that is induced by the simple roots:

Ker ad = h-1 c=1-h Ker(ad ) k , Ker(ad ) k = Ker(ad ) ∩ gk .
It is a classical result that [START_REF] Kostant | The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group[END_REF] that Ker(ad ) is a Cartan subalgebra of g, hence the name. In particular, dim Ker(ad ) = rk g = r .

Then an integer k ∈ {0, . . . , h -1} is said to be a nite exponent of g if Ker(ad

) k [h-1] {0}; if so, we say that k has multiplicity m k = dim Ker(ad ) k [h-1] . (Here, k [h -1] means k mod h -1.)
We gather the nite exponents into an ordered multi-set that we denote

F(g) = k m 1 1 , . . . , k m h-1 h-1 = k m k 1 ≤ k ≤ h -1, m k = dim Ker(ad ) k [h-1] .
A multi-set means that each element has a multiplicity; a null multiplicity means that the element does not belong to the set. The fact that F(g) is ordered simply means that k i < k j if i < j. Since dim Ker(ad ) = r , we must have m 1 + • • • + m h-1 = r . On the other hand, one can show that m 1 = m h-1 = 1. When there is no ambiguity, we will simply denote F = F(g). Table 4.2.2 table gives the nite exponents F(g) of all simple Lie algebras g. Notice if h is the Coxeter number of g, then h = max(F(g)) + 1.

Example 4.2.3 (Computation of the nite exponents of sl(n)). We start with the algebra g = sl(3). We can present sl(3) via the following system of generators:

E 1 = 0 0 0 1 0 0 0 0 0 , F 1 = 0 1 0 0 0 0 0 0 0 , H 1 = 1 0 0 0 -1 0 0 0 0 , E 2 = 0 0 0 0 0 0 0 1 0 , F 2 = 0 0 0 0 0 1 0 0 0 , H 2 = 0 0 0 0 1 0 0 0 -1 . ( 4
E 0 = [F 1 , F 2 ], F 0 = [E 1 , E 2 ] and H 0 = [E 0 , F 0 ],
which reads

E 0 = 0 0 1 0 0 0 0 0 0 , F 0 = 0 0 0 0 0 0 1 0 0 , H 0 = 1 0 0 0 0 0 0 0 -1 .
Consequently, the nite principal cyclic element, which is given by = E 0 + E 1 + E 2 , is the matrix

= 0 0 1 1 0 0 0 1 0 .
Then a direct calculation shows that the kernel of ad is 3] so that the nite exponents of sl(3) are (where all multiplicity are equal to 1)

Ker ad = C(E 0 + E 1 + E 2 ) ⊕ C(F 0 + F 1 + F 2 ) Yet E 0 + E 1 + E 2 = ∈ sl(3) 1 [3] and F 0 + F 1 + F 2 ∈ sl(3) 2 [
F(sl(3)) = {1, 2}.
Similarly, we could have found that the nite exponents of sl(2) read

F(sl(2)) = {1}.
One can easily generalize these results to nd that the nite principal cyclic element of sl(n) is given by

= E 1,n + n-1 i=1 E i+1,i , that is, = 0 • • • 0 1 1 0 0 1 . . . . . . . . . 0 • • • 0 1 0 ,
and that the set of nite exponents (where all multiplicities are equal to 1) is

F(sl(n)) = {1, 2, . . . , n -1}. (4.25)
A ne exponents and principal Heisenberg subalgebra

The a ne exponents are constructed in a similar way. First, we de ne the a ne principal cyclic element of the loop algebra

Λ = e 0 + e 1 + • • • + e r ∈ g.
We denote s its kernel, that is,

s := Ker ad Λ : g → g .
It is proven in [START_REF] Kac | In nite dimensional Lie algebras[END_REF] that s is an abelian algebra. We decompose s = k ∈Z s k through the principal gradation. Then we say that an integer j ∈ Z is an a ne exponent of g if s {0}, and if so, with multiplicity m j = dim s j . We gather the a ne exponents into a multi-set that we denote

E(g) = j m j j ∈ Z, m j = dim s j .
We also denote E + (g) = E(g) ∩ Z >0 the set of positive exponents (0 is never an exponent). The a ne exponents are related to the nite exponents via the following relation ([Kac90], p. 298 Proposition 14.3): E(g) = j + ph j ∈ F(g), p ∈ Z , the latter we often write in the condensed form E(g) = F(g) + hZ. We will denote E = E(g) when there is no ambiguity.

Following [START_REF] Kac | In nite dimensional Lie algebras[END_REF], we can lift the abelian algebra s ⊂ g in g as follows. There exists a subalgebra s ⊂ g such that, through the principal gradation, dim s k = dim s k for any exponent k ∈ E(g), and which admits a set of generators Λ j , j ∈ E(g), satisfying the Heisenberg commutation relations:

Λ i , Λ j = iδ i+j,0 • c, ∀ i, j ∈ E(g). (4.26)
Moreover, s/Cc = s. We call s the principal Heisenberg subalgebra of g. Notice that, in particular,

Λ i , Λ j = 0, ∀ i, j ∈ E + (g), (4.27)
which hold for i, j ∈ E(g) ∩ Z <0 as well, making s >0 and s <0 into abelian subalgebras of s. The condition (4.26) on the generators Λ j leaves yet one degree of freedom; we normalize them via the condition

Λ i , Λ j 0 = hδ i+j,0 , (4.28)
where h is the Coxeter number of g. We denote ν ∈ C the complex number such that Λ 1 = ν Λ.

Still in [START_REF] Kac | In nite dimensional Lie algebras[END_REF], it is proven that Λ ∈ g is ad-semisimple, i.e.

g = s ⊕ Im ad Λ . (4.29)
Now, on the level of g, since c is central, then c ∈ Ker(ad Λ ) = s. On the other hand, from Equation (4.26) we see that c = ν -1 ad Λ (Λ -1 ), where Λ 1 = ν Λ, so that c ∈ Im(ad Λ ). Since g = Ker(ad Λ ) ⊕ Im(ad Λ ) and g = g ⊕ Cc, it follows that

g = s + Im(ad Λ ), s ∩ Im(ad Λ ) = Cc. ( 4
e 0 = E 0 λ, e 1 = E 1 , e 2 = E 2 .
Then using the same Weyl generators of sl(3) as in Equation (4.24), the (projection of the) a ne principal cyclic element Λ = e 0 + e 1 + e 2 reads Λ = 0 0 λ 1 0 0 0 1 0 .

In order to compute s = Ker(ad Λ ), it is convenient to rst compute the powers of Λ:

Λ 2 = 0 λ 0 0 0 λ 1 0 0 , Λ 3 = λ 0 0 0 λ 0 0 0 λ
, from which we get that Λ 3k +1 = λ k Λ and Λ 3k +2 = λ k Λ 2 for any k ∈ Z. Knowing that, a direct computation shows that the a ne principal Heisenberg subalgebra decomposes as

s = k ∈Z CΛ 3k +1 ⊕ CΛ 3k+2 =        0 λc 2 λc 1 c 1 0 λc 2 c 2 c 1 0 c 1 , c 2 ∈ C[λ, λ -1 ]        .
Yet w.r.t. the principal gradation, Λ = e 0 + e 1 + e 2 ∈ sl(3) 1 , so that the set of exponents reads

E (sl(3)) = k ∈Z {3k + 1, 3k + 2} = Z -3Z.
With similar computations we can show that

E (sl(2)) = k ∈Z {2k + 1} = Z -2Z = Z odd .
One can easily generalize these results to nd that the a ne principal cyclic element of sl(n) is given by

Λ = 0 • • • 0 λ 1 0 0 1 . . . . . . . . . 0 • • • 0 1 0 ,
and that the set of a ne exponents (where all multiplicity are equal to 1) is

E (sl(n)) = k ∈Z {nk + 1, nk + 2, . . . , nk + n -1} = Z -nZ. (4.31)

The Drinfeld-Sokolov hierarchies

In their seminal paper [DS84], Drinfeld-Sokolov built the so-called Drinfeld-Sokolov hierarchies based on the loop algebra g = g[λ, λ -1 ]. Hollowood and Miramontes [START_REF] Hollowood | Tau functions and generalized integrable hierarchies[END_REF] showed that one can build the exact same hierarchies of equations based on the a ne algebra g = g ⊕ Cc and that this approach allows to de ne a notion of dressing operator. In the present section, we follow the approach of C.-Z. Wu [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF] who, inspired by [START_REF] Hollowood | Tau functions and generalized integrable hierarchies[END_REF], produced a systematic de nition of tau symmetric Hamiltonian densities (in the sense of [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF])-and therefore of a tau function-for the Drinfeld-Sokolov hierarchy associated to any a ne algebra g. We shal notice that the original construction of Drinfeld and Sokolov makes use the principal and homogeneous gradations, respectively de ned by the vectors s 1 = (1, . . . , 1) and s 0 = (1, 0 . . . , 0) = (δ 0, j ) j ∈[1,r ] (see Section 4.2.2), but also applies if the homogeneous gradation is replaced by any vector of the form (δ i, j ) j ∈[1,r ] . In this thesis, we are only intrested in those de ned by the homogeneous gradation.

One the other hand, de Groot, Hollowood and Miramontes genralized the DS hierarchies as follows. Take two vectors s = (s 0 , . . . , s r ) and s = (s 0 , . . . , s r ) of relatively coprime nonnegative integers. We say that s ≥ s if s i = 0 implies s i = 0 (e.g. s 1 ≥ s 0 ). Then de Groot-Hollowood-Miramonts showed, in particular, how to adapt the Drinfeld-Sokolov procedure when one replaces s 1 by s and s 0 by s [START_REF] De Groot | Generalized Drinfel'd-Sokolov hierarchies[END_REF][START_REF] Hollowood | Tau functions and generalized integrable hierarchies[END_REF]. They also generalized the construction to the case when the principal cyclic element Λ = e 0 +• • • e r is replaced by any constant element of g which is semisimple; and further generalizations [START_REF] De Groot | Generalized Drinfel'd-Sokolov hierarchies[END_REF]. This gives rise to the generalized Drinfeld-Sokolov hierarchies. Notice that the work of de Groot-Hollowood-Miramontes applies to untwisted a ne algebras.

Finally, Kac and Wakimoto also generalized the DS hierarchies in the following sense [START_REF] Kac | Exceptional hierarchies of soliton equations[END_REF]. For each a ne Lie algebra g together with an integrable highest weight representation V of g and a vertex operator construction R of V , Kac and Wakimoto formulated a hierarchy of soliton equations. These equations can be written down in terms of Hirota bilinear equations and their super analogue [START_REF] Kac | Exceptional hierarchies of soliton equations[END_REF]. When g is the untwisted a nization of a simply laced nite Lie algebra, the Kac-Wakimoto hierarchy coincides with the corresponding generalized Drinfeld-Sokolov hierarchy de ned by Groot, Hollowood and Miramontes [START_REF] De Groot | Generalized Drinfel'd-Sokolov hierarchies[END_REF][START_REF] Hollowood | Tau functions and generalized integrable hierarchies[END_REF].

In particular, if the highest weight representation is the basic one, and the vertex operator realization is constructed from the principal Heisenberg subalgebra s of g, then the Kac-Wakimoto hierarchy is equivalent to the Drinfeld-Sokolov hierarchy associated to g for the homogeneous gradation.

Matrix Lax equations of the DS hierarchies

Lie algebra of g-valued formal power series

Let g be a semisimple Lie algebra, g = g[λ, λ -1 ] be the associated loop algebra and g = g ⊕ Cc be the a ne algebra with the material described in the previous section. We consider the vector space

g [[x]] := g ⊗ C [[x]] = k ≥0 ξ [k] x k ξ [k ] ∈ g
of g-valued formal power series in x. The elements of this vector space will be what we call our functions. We equip this space with the Z-gradation in powers of x:

g [[x]] = k ∈Z g ⊗ Cx k ;
subspaces of degree k < 0 are null. This vector space admits a natural bracket

k ≥0 ξ [j] x j , k ≥0 η [k ] x k = j,k ≥0 ξ [j] , η [k ] x j+k . (4.32)
where the bracket ξ [j] , η [k] in g is given by Equation (4.15) (Equation (4.32) is actually the same kind of bracket as the one on the loop algebra). The bracket (4.32) is well de ned for each term of degree c in g ⊗ Cx k is clearly a nite sum.

Next we consider the multiplicative group

G = {exp (ad T ) | T ∈ g [[x]] ad-nilpotent } .

This group acts in on g [[x]

] in a similar way the group of inner automorphisms of g acts on g:

for T ∈ G and ξ ∈ g [[x]], e ad T ξ = k,n ≥0 1 n! j ≥0 ad T [j] x j n ξ [k] x k ,
where as before, ad T [j] x j ξ [k ] x k = T [j] , ξ [k ] x j+k . This action is well de ned since, again, each term of a given degree involves only nitely many terms.

Gauge equivalence classes of connections

The initial data of the Drinfeld-Sokolov hierarchy of g-type (or simply the g-DS hierarchy) consists of a connection on the algebra g [[x]] of the speci c form

L = ∂ + Λ + Q, Q ∈ b [[x]] , (4.33)
where ∂ = ∂ x and Λ = e 0 + e 1 + • • • + e r ∈ g is the a ne principal cyclic element. This type of connection rst appeared in [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] (p. 2010). The set of such connections we denote by

op g = ∂ + Λ + Q Q ∈ b [[x]] .
L , ξ ⊗ x k = ξ ⊗ kx k -1 + [Λ + Q, ξ ] ⊗ x k , (4.34)
for any ξ ∈ g and k ≥ 0, where [Λ +Q, ξ ] is given by Equation (4.32). We call them connections in reference to connections on a vector bundle on a manifold. Here we use the terminology and some notations of [START_REF] Lacroix | A ne Gaudin models and hypergeometric functions on a ne opers[END_REF].

For L = ∂ + Λ + Q ∈ op g and T ∈ g [[x]] ad-nilpotent, we de ne e ad T L := ∂ + e ad T (Λ + Q) -∇ T T , (4.35)
where

∇ T T = e ad T -1 ad T ∂T := k ≥0 1 (k + 1)! (ad T ) k ∂(T ), (4.36)
and where ∂(T ) is the derivative of T w.r.t. x component-wise (the rst equality is simply notational). For the moment, e ad T L is considered simply as a derivation on g [[x]] and may not belong to op g anymore.

Remark 4.3.2. Equations (4.35) and (4.36) become clear when we choose a matrix realization of g. In that case, e T is well de ned as a matrix exponential, and, by Leibniz's rule,

e ad T ∂ = e T ∂ • e -T = ∂ + e T ∂(e -T ).
The derivative of the matrix exponential is given by the formula

∂e A = e A 1 -e -ad A ad A ∂A,
for A any square matrix (see for instance [START_REF] Rossmann | Lie Groups -An Introduction Through Linear Groups[END_REF], Theorem 5 §1.2). Replacing A by -T , one nds that

e ad T ∂ = ∂ + 1 -e ad T ad T ∂T = ∂ -∇ T T , (4.37)
hence Equation (4.36).

Since any element of n is ad-nilpotent and since n is a subalgebra of g, we can de ne the following subgroup N of G:

N = exp (ad S ) | S ∈ n [[x]] ⊂ G.
The following proposition holds. Proof. We need to prove that for any S ∈ n [[x]] and any

L = ∂ + Λ + Q ∈ op g , there exists Q ∈ b [[x]] such that e ad S L = ∂ + Λ + Q. From Equation (4.35), it amounts to nding Q ∈ b [[x]] such that Λ + Q = e ad S (Λ + Q) -∇ S S.
Developing the exponentials, we get

Q = k ≥1 1 k! (ad S ) k Λ + k ≥0 1 k! (ad S ) k Q - k ≥0 1 (k + 1)! (ad S ) k ∂S.
First, using [e 0 , n] = 0 (4.19), we see that

k ≥1 1 k! (ad S ) k Λ = k ≥1 1 k! (ad S ) k (e 1 + • • • + e r ).
Yet using the root space decomposition of g = k gk , we have e 1 , . . . , e r ∈ g1 and S ∈ g<0

[[x]], so that ad S (e 1 + • • • + e r ) ∈ b [[x]] ,
where we used b = g≤0 . Similarly, since

Q ∈ b [[x]], it follows that ad S Q ∈ b [[x]] as well. Finally, ∂S ∈ n [[x]] by linearity of ∂, which implies that ad S ∂S ∈ n [[x]] ⊂ b [[x]]. We conclude that Q ∈ b [[x]].

Canonical coordinates system

Let us introduce an alternative de nition to the nite exponents [START_REF] Kostant | The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group[END_REF]. De ne the nite principal nilpotent element

I = -E 0 = E 1 + • • • + E r .
Since I ∈ g1 , then for all j ∈ {1 -h, . . . , h -1}, we have ad I g j-1 ⊂ gj . Moreover, for all j ∈ {1, . . . , h -1}, the map ad I : g-j-1 → g-j is injective and an integer j ∈ {1, . . . , h -1} is a nite exponent of g if and only if the map ad I : g-j-1 → g-j is not surjective; if so with multiplicity m j = dim g -j-1dim g -1 [START_REF] Kostant | The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group[END_REF].

Therefore for any j a nite exponent of g, since ad I : g-j-1 → g-j is injective and not surjective, we can choose a subspace V j ⊂ g-j such that g-j =

V j ⊕ [I, g-j-1 ]. Now since b = g≤0 and n = g<0 , we get b = V ⊕ I , n , V = j ∈F(g) V j , (4.38)
where F(g) are the nite exponents of g. Since b = h ⊕ n and since ad

I : n → b is injective, it follows that dim V = dim b -dim n = dim h, that is dim V = rk g = r .
Such a subspace V ⊂ b is called a Drinfeld-Sokolov gauge for reasons that become clear with the next proposition.

L = ∂ + Λ + Q, Q ∈ V [[x]] .
Moreover, the functions Q and S are di erential polynomials in the components of Q. The functions { Q j | j ∈ F(g)} are called the canonical coordinates of L .

Proof. We want to equate ∂ + Λ +Q = e ad S (∂ + Λ + Q). Since [e 0 , n] = 0, then e ad S Λ = e 0 + e ad S I , so that it amounts to solving

I + Q = e ad S (I + Q) -∇ S S.
Let us decompose Q and S following the gradation of b, i.e.

Q = Q0 + Q1 + • • • + Qh-1 , S = S 1 + • • • + S h-1 ,
with Qi , S i ∈ g-i , possibly Qi = 0 (if i is not a nite exponent). It is not hard to see that by equating each term of the above equation in each component g-j , we nd that Q j + [I, S j+1 ] can be expressed in terms of Q0 , . . . , Q j-1 and S 1 , . . . , S j and Q 0 , . . . , Q j-1 . For example, the rst equation, in g0 and g-1 , read

Q 0 = Q0 + [S 1 , I ], Q 1 = Q1 + [S 2 , I ] + [S 1 , Q0 ] + 1 2 [S 1 , [S 1 , I ]] -∂S 1 .
Now since g-j = V j ⊕ [I, g-j-1 ],then Q j and S j+1 are unique and can be found recursively as di erential polynomials in Q 0 , . . . , Q j .

To sum up, the choice of a gauge V ⊂ b such that b = V ⊕ [I , n] entirely determines the gauge class of a connection L ∈ op g through unique r canonical coordinates Q, associated to V, indexed by the nite exponents of g.

Example 4.3.5 (with g = sl(2)). We treat the sl(2) case; we recall the classical Weyl generators of this algebra:

E = 0 0 1 0 , F = 0 1 0 0 , H = 1 0 0 -1 .
∂ + Λ + Q with Q, S ∈ n [[x]] (since V = n).
We decompose the functions Q, Q and S through the root space gradation:

Q = H ⊗ q 0 + F ⊗ q 1 , Q = F ⊗ q1 , S = F ⊗ s 1 ,
where q 0 , q 1 , q1 ,

s 1 ∈ C [[x]
]. (In the proof of Proposition 4.3.4, H ⊗ q 0 is what we denoted Q 0 , etc.) Since I = E, the two rst equations read

H ⊗ q 0 = F ⊗ s 1 , E , F ⊗ q 1 = F ⊗ q1 + 1 2 F ⊗ q 0 , F ⊗ q 0 , E -F ⊗ s 1 (1) ,
where, as before, s 1 (1) = ∂ x (s 1 ). The rst equation yields s 1 = -q 0 , so that the second equation becomes F ⊗ q 1 = F ⊗ ( q1 -(q 0 ) 2 + q 0 (1) ), which yields q1 = q 1 + q 0 2 -q 0 (1) .

In the end, we obtain

L = ∂ + q 0 λ + q 1 1 -q 0 = e -ad S ∂ + 0 λ + q1 1 0 , s 1 = -q 0 , q1 = q 1 + q 0 2 -q 0 (1) .

Hamiltonian expansion

Let X be any subspace of g and X = k ∈Z X k , with X k ⊂ g k , be its decomposition through the principal gradation of g. We de ne

X := X + ⊕ X -, X + = k ≥0 X k , X -= k <0 X k .
In particular, X + = X ≥0 . We de ne X + and X -similarly for the homogeneous gradation.

Elements of X consists of formal power series ξ = k ∈Z ξ k , with ξ k ∈ X k , with nitely many nonzero terms of positive degree. We de ne ξ + as the projection of ξ on X + and ξ -= ξξ + ; we de ne ξ + and ξ -similarly.

Remark 4.3.6. Importantly, X + = X ≥0 , but X -X <0 , since X <0 is a direct sum, while X -is a direct product. For example,

g = g λ -1 ⊕ Cc,
i.e. a central extension of g-valued formal Laurent series in λ. The Lie bracket on g extends to g; similarly for the action of G on g

[[x]].
We recall that g = s + Im(ad Λ ) and s ∩ Im(ad Λ ) = Cc, where s = Ker (ad Λ ) is the principal abelian subalgebra. These spaces decompose through the principal gradation:

s = k ∈Z s k , Im(ad Λ ) = k ∈Z Im(ad Λ ) k ,
where s k = s ∩ g k and Im(ad Λ ) k = Im(ad Λ ) ∩ g k . They decompose through the homogeneous gradation as well. Now, in comparison to the the group N of elements of the form e ad S with S ∈ n [[x]], we want to de ne an action on g [[x]] of elements of the form e ad U , with U ∈ g -[[x]]. It is not obvious that one can de ne such an endomorphism e ad U , however, since U is assumed to be of negative degree, it is not hard to see that for any A ∈ g [[x]], the expression e ad U A contains nitely many terms for each degree w.r.t. the principal gradation. Hence, the action is well de ned. The following theorem, which plays a major role in the building of the Drinfeld-Sokolov hierarchies, rst appeared in [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] in its version based on g; the following version is due to C.-Z. Wu.

Theorem 4.3.7 ([Wu17]). Let L = ∂ + Λ + Q ∈ op g be a connection. There exists a unique function U ∈ g -[[x, t]] such that the following holds true. 6(i) The connection L 0 = e -ad U L has the form7

L 0 = ∂ + Λ + H, H ∈ s -[[x]] (4.39) (notice that L 0 op g ). The coe cients U i , H i ∈ g -i [[x]
] are di erential polynomials in the component of Q.

(ii) For every ξ ∈ s + , the following relation holds,

(e ad U ξ ) c = 0. (4.40)
Here, for η ∈ g, the complex number η c denotes the component of η along the center Cc in the decomposition

g 0 = C α1 ⊕ • • • ⊕ C αr ⊕ Cc.
Vocabulary. The connection L 0 = e -ad U L we call a Hamiltonian expansion of L , for reasons that will become clear later (cf. Proposition 4.3.21).

Proof. We rst prove point (i); we proceed in the same fashion as the proof of Proposition 4.3.4. We rst need to assume that

U ∈ g -[[x]] , H ∈ g 0 ⊕ g -[[x]] .
Equating L 0 = e -ad U L amounts to

Λ + Q = e ad U (Λ + H ) -∇ U U .
Denoting the components

H i , U i ∈ g -i [[x]], one nds that H i + [U i+1
, Λ] can be expressed in terms of H 0 , . . . , H i-1 and U 1 , . . . , U i . We then obtain

H 0 ∈ g 0 ∈ C [[x]
] and, for i > 0,

H i ∈ s -i [[x]] , [U i+1 , Λ] ∈ Im(ad Λ ) -i [[x]]
But since g -i = s -i + Im(ad Λ ) -i , then H i and [U i+1 , Λ] can be recursively determined and are di erential polynomials in the coordinates of Q. Finally, s 0 = {0} so that H 0 = 0 and

H ∈ s -[[x]].
We now prove point (ii). Since only [U i , Λ] is determined, two cases must be distinguished. Either i is an a ne exponent, in which case U i is entirely determined; or i is not an exponent, in which case U i is determined up to the addition of a multiple of Λ -i . So let i be a positive exponent of g. Since c ∈ g 0 , then for ξ = Λ i , Equation (4.40) reads

0 = (e ad U Λ i ) c = [U i , Λ i ] c + 1 2 [U 1 , [U i-1 , Λ i ]] c + • • • (4.41)
where the remaining terms involve only the functions U 1 , . . . , U i-1 . Let us assume that U 1 , . . . , U i-1 are uniquely determined; choose U i and consider the transformation

U i → U i + µ i Λ -i (which does not a ect Equation (4.56)). Since [Λ -i , Λ i ] = ic, then Equation (4.41) gives iµ i = -[U i , Λ i ] c -1 2 [U 1 , [U i-1 , Λ i ]] c -• • • which determines µ i .
e ad Ũ L = e -ad U L = ∂ + Λ + H .
Third, recall that the bracket on g is given by

[X λ p + ac, Y λ q + bc] = [X , Y ]λ p+q + qδ p+q,0 (X , Y ) 0 • c, for X , Y ∈ g, p, q ∈ Z and a, b ∈ C. So for S ∈ n [[x]],
we have (ad S (g)) c = 0 and also ad S c = 0. Therefore, ad S commutes with the projection onto Cc. Consequently, for any ξ ∈ s + , (e ad Ũ ξ ) c = e ad S (e ad U ξ ) c = 0.

Therefore, the function Ũ satis es the condition.

The matrix Lax pairs of the Drinfeld-Sokolov equations

From now on we will assume that our functions Q, U , H , etc. depend not only on x but also on an "evolution variable" t (or even multiple variables, depending on the dynamics considered).

In

t := t j | j ∈ E + . De nition 4.3.12 (Drinfeld-Sokolov hierarchy). Let Q ∈ b [[t]
] be a function and a connection L = ∂ + Λ + Q ∈ op g . We call the Drinfeld-Sokolov hierarchy of type g (or simply the g-DS hierarchy) the family of equations

∂ t j L = (e ad U Λ j ) + , L j ∈ E + (4.45)
modulo gauge transformation of L . We call ∂ t j L = (e ad U Λ j ) + , L the j-th g-DS equation.

We make a few remarks.

Remark 4.3.13. The function Q = 0 is always a solution. Indeed, then the connection L ∈ op g has the form L = ∂ + Λ, and is already in the form

L 0 = ∂ + Λ + H , with H = 0 ∈ s -[[t]].
The function U = 0 also does the job of Theorem 4.3.7. The j-th g-DS equation then becomes

∂ t j L = [Λ j , ∂ + Λ]. Since Q = 0, then ∂ t j L = 0,
) + = Λ 1 +[U 1 , Λ 1 ], while [U 1 , Λ] = Q (by construction of U ). Therefore [L , (e ad U Λ 1 ) + ] = ν[∂ + Q + Λ, Q + Λ] = ν ∂ x Q.
Remark 4.3.15. In most cases, for a simple Lie algebra g, the subspace s j of principal degree j is one-dimensional. However, it might not be the case (e.g. in the D n case with n even, cf. Section A.1, or for a nonsimple Lie algebra). If so, then we may distinguish several ows: t j , t j , t j , etc., one for each generator of s j .

Equivalence between Drinfeld-Sokolov and Wu's constructions

Before continuing, let us have a few words on the equivalence between Drinfeld-Sokolov's construction, which is based on the loop algebra g = g[λ, λ -1 ], and the one of Wu, described above, which is based on the (derived) a ne algebra g = g ⊕ Cc. In [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF], the authors give the following theorem (in comparison to Wu's theorem 4.3.7):

Theorem 4.3.16 ([DS84]). Let L = ∂ + Λ + Q ∈ op g be a connection. There exists a non unique function U ∈ g -[[x, t]
] such that the connection L 0 = e -ad U L has the form

L 0 = ∂ + Λ + H , H ∈ s -[[x]] (4.47)
The coe cients

U i , H i ∈ g -i [[x]] are di erential polynomials in the component of Q. Moreover, the function U is unique if we impose U ∈ ( s ⊥ ) -
, where the orthogonal is taken with respect to the standard bilinear form on g.

Then Drinfeld and Sokolov de ne the following equations:

∂L ∂t = (e ad U ξ ) + , L , ∂L ∂t = (e ad U ξ ) + , L .
We prove that the two constructions de ne the exact same equations.

Proof. First of all, the function U ∈ g -[[x, t]] of Theorem 4.3.7 also belongs to g -[[x, t]], as a set, since c g -. Then since (e ad U ξ ) c = 0 for all ξ ∈ s + , then in particular, (e ad U Λ) c = 0; moreover, (e ad U Q) c = 0 since Q ∈ g 0 . It follows that the action of e ad U on L within g is identical to its action within the quotient g = g/Cc. Similarly, since as a set s -= s -, it follows that the function H ∈ s -[[x, t]] also belongs to s -[[x]] as a set. Therefore, the function U is also a functions as in Theorem 4.3.22. Finally, since (e ad U ξ ) c = 0, it follows that the two constructions de ne the same equations.

Consistency of the Drinfeld-Sokolov equations

The rst thing we need to prove is that the DS equations are consistent (well de ned). Since

L = ∂ + Λ + Q, it follows that ∂ t L = ∂ t Q ∈ b [[x, t]].
Hence it is necessary, for Equation (4.43) to be consistent, that the following proposition holds.

Proposition 4.3.17 (Consistency). Let L ∈ op g be a connection and let ξ ∈ s + . Then

L , (e ad U ξ ) + ∈ b [[x, t]] .
Proof. Here we adapt the proof given in [DS84] ( §6.2 and Proposition 3.7)-where the construction is on the loop g g/Cc-to the construction of [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF]-based on the a ne algebra g. We will proceed in several steps. First, let us prove that9

L , e ad

A ∈ C [[x, t]] such that [L , M] = A • c, so that [L , M + ] = [M -, L ] + A • c.
To show Equation (4.49), we rst prove that

[M -, L ] ∈ g 0 ⊕ g -[[x, t]] . (4.50) On the one hand, M -∈ g -[[x, t]] and g -⊂ g -, so that ∂M -∈ g -[[x, t]]. On the other hand, Λ ∈ g 1 and Q ∈ b [[x, t]], so [Λ+Q, M -] ∈ (g 0 ⊕ g -) [[x,
(∂M + ) c + [Λ + Q, M + ] c = 0. (4.52)
By Theorem 4.3.7, M c = 0, so (∂M + ) c = 0 as well. for the remaining part, let us recall the formula for the bracket in g: for any X , Y ∈ g, p, q ∈ Z and a, b ∈ C,

[X λ p + ac, Y λ q + bc] = [X , Y ]λ p+q + qδ p+q,0 (X , Y ) 0 • c.
Clearly, the bracket produces a nonzero component in c only if p > 0 and q < 0 (or the reverse).

In other words, for any η, ζ ∈ g, we have

[η, ζ ] c 0 only if η ∈ g >0 and ζ ∈ g <0 (or the reverse). Yet Λ = E -θ λ + I ∈ g 1 ⊕ g 0 and Q ∈ g 0 [[x, t]], while M + ∈ g + [[x, t]]. So [Λ + Q, M + ] c = 0,
which concludes the proof of Proposition 4.3.17.

Integrability of the DS hierarchies

Commutativity of the flows

In this section, we prove that the ows de ned by the DS equations are integrable in the sense of the following theorem.

Theorem 4.3.18 (Drinfeld-Sokolov [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF]). Let L ∈ op g be a connection. Let ξ , η ∈ s + and consider the associated DS equations, namely,

∂L ∂t = (e ad U ξ ) + , L , ∂L ∂s = (e ad U η) + , L , where U ∈ g -[[x, t]
] is a function as in Theorem 4.3.7. Then the ows de ned by the above equations commute pairwise, that is,

∂ 2 L ∂t ∂s = ∂ 2 L ∂s∂t .
We will proceed by a series of lemmas, as is done in [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF]. Let us introduce the following vector space,

S L = A ∈ g [[x, t]] A c = 0, ∂ t L = [A, L ] .
Lemma 4.3.19. Let L ∈ op g be a connection and let U ∈ g -be a function as in Theorem 4.3.7, then

S L ⊂ e ad U s [[x, t]] . Proof. Let A ∈ g [[x, t]] be a function such that ∂ t L 0 = [A, L 0 ], which is equivalent to ∂H ∂t + ∂A ∂x = [A, Λ] + [A, H ].
Let us decompose A and H through the principal gradation:

A = i ≥m A i (for some m ∈ Z) and H = i ≥1 H i , with A i , H i ∈ g -i [[x, t]].
Then since H contains only negative terms, we get the following system of equations:

∂H i ∂t + ∂A i ∂x = [A i+1 , Λ] + i-1 j=m [A j , H i-j ] ∈ g -i .
The rst equation, within g -m , reads

∂H m-1 ∂t = [A m , Λ].
possibly H m-1 = 0 if m ≤ 1 or is not an exponent. If m ≤ 1 or is not an exponent, then the equation simply becomes [Λ, A m ] = 0. If m > 1 and is an exponent, the above implies that

[A m , Λ] ∈ (s ∩ Im(ad Λ )) -[[x, t]]. Yet (s ∩ Im(ad Λ )) -= {0} (since the center Cc ∈ g 0 ), so that [A m , Λ] = 0. In both cases, A m ∈ s -m [[x, t]]. Similarly, if A m , A m+1 , . . . , A i ∈ s [[x, t]], then the equation in g -i becomes ∂H i ∂t + ∂A i ∂x = [A i+1 , Λ],
If i > 0 and is an exponent, the above implies that [A i+1 , Λ] = 0. If i ≤ 0 or is not an exponent, then the equation becomes

∂ x A i = [A i+1 , Λ], implying that [A i+1 , Λ] ∈ (s ∩ Im(ad Λ )) + [[x, t]]. Now, (s ∩ Im(ad Λ )) + = Cc, yet A c = 0, so that that [A i+1 , Λ] = 0. In both case, A i+1 ∈ s -i-1 [[x, t]]. By induction, A ∈ s [[x, t]]. Lemma 4.3.20. Let ξ ∈ s + , let U ∈ g -[[x, t]
] be a function as in Theorem 4.3.7 and let us denote M = e ad U ξ . For any A ∈ S L the following equation holds,

∂M ∂t = [A, M].
Proof. It is given that

∂ t L = [A, L ], or equivalently, [∂ t -A, L ] = 0. Since L 0 = e -ad U L
, by applying e -ad U to the latter, we nd On the other hand, we want to prove that [∂ t -A, M] = 0. Since M = e ad U ξ , by applying e -ad U to the latter, we nd ∂ t -Ã, ξ = 0. Yet ξ ∈ s + and, in particular, is constant so that it reduces to Ã, ξ = 0. Now, by Remark 4.3.11, we can assume that à ∈ s + [[x, t]], so that Ã, ξ = 0 holds true by commutativity of s + . 10We can now prove that the ows of the Drinfeld-Sokolov hierarchies commute pairwise, that is, are integrable. Still, the proof rst appeared in [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF].

∂ t -Ã, L 0 = 0, Ã = e -ad U A -∇ U U , ( 4
Proof of Theorem 4.3.18. Let us denote M = e ad U ξ and N = e ad U η. Then by de nition,

∂ s ∂ t L = ∂ s [M + , L ] = [∂ s M + , L ] + [M + , [N + , L ]] . By Lemma 4.3.20, we know that ∂ s M = [N + , M], so that ∂ s ∂ t L = [[N + , M] + , L ] + [M + , [N + , L ]] (4.54) = [[N + , M] + , L ] -[N + , [L , M + ]] -[L , [M + , N + ]] ,
where we have used Jacobi's identity to obtain the second equation. We can then use the antisymmetry of the bracket to rewrite the above as

∂ s ∂ t L = [[N + , M] + , L ] + [N + , [M + , L ]] + [[M + , N + ] , L ] .
In the same way we obtained Equation (4.54), on nds that

∂ t ∂ s L = [[M + , N ] + , L ] + [N + , [M + , L ]] .
Subtracting the two equations, we obtain

∂ s ∂ t L -∂ t ∂ s L = [[N + , M] + + [M + , N + ] -[M + , N ] + , L ] .
(4.55)

Yet since [M, N ] = e ad U [ξ , η] = 0 (by commutativity of s + ), we have

[N + , M] + = [N -N -, M] + = [N -, M] + = [M + , N -] + ,
where we have used the fact that [M -, N -] + = 0. Finally, we obtain that in Equation (4.55),

[N + , M] + + [M + , N + ] -[M + , N ] + = [M + , N -] + + [M + , N + ] -[M + , N ] + = -[M + , N + ] + + [M + , N + ] = 0.
Consequently, the two derivatives commute.

Conserved densities

Similarly to the construction of local functionals in Section 3.1, we naturally de ne an analogue for our g-valued functions: we set

g [[x, t]] := g [[x, t]] /(Im(∂) ⊕ C1).
We call the elements of this quotient local functionals as well and alternatively denote

F = ∫ F = ∫ F (x)dx ∈ g [[x, t]] the class of an element F ∈ g [[x,
∂L ∂t = (e ad U ξ ) + , L .
Then the H i 's, i ∈ E <0 , are conserved densities in the sense that ∂ ∂t

∫ H i = 0.
The above proposition justi es the denomination Hamiltonian expansion for the operator L 0 = e -ad U L , although we still need to prove that the conserved densities H i commute with respect to a local Poisson bracket, see Section 5.1.

Proof. Let us denote M = -e ad U ξ so that we start with [∂ t + M, L ] = 0, i.e. M ∈ S L . As in the proof of Lemma 4.3.20, it amounts to M ∈ S L 0 , where M is given by Equation (4.53) (the formula does not matter here). Yet since, by Lemma 4.3.20,

S L 0 ⊂ s [[x, t]], then 0 = ∂ t + M, L 0 = ∂H ∂t - ∂ M ∂x -[ M, Λ + H ] = ∂H ∂t - ∂ M ∂x .
Consequently,

∂ t H = ∂ x M, meaning that ∂ t H = 0.

Example: From the A 1 -DS hierarchy to the KdV equation

In this section we show explicitly how to obtain the KdV equation from the rst nontrivial equation of the DS hierarchy of type A 1 (i.e. sl(2)). However, this example is developed from the de nition of the DS hierarchies given in the original paper [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF], not the version given in [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF] (which is the one described in the previous section). We recall the following theorem (we drop the bar over symbols for this section speci cally).

Theorem 4.3.22 ([DS84]). Let L = ∂ + Λ + Q ∈ op g . There exists a non unique function U ∈ g -[[x, t]
] such that the connection L 0 = e -ad U L has the form

L 0 = ∂ + Λ + H , H ∈ s -[[x]] (4.56)
The coe cients

U i , H i ∈ g -i [[x]] are di erential polynomials in the component of Q. Moreover, the function U is unique if we impose U ∈ ( s ⊥ ) -[[t]],
where the orthogonal is taken with respect to the standard bilinear form on g.

Conveniently, we start with a connection L = ∂ + Λ + Q ∈ op g in its canonical form L = L , i.e.

L = ∂ + 0 λ + q 1 0 .
The principal Heisenberg subalgebra of sl(2) is generated by the Λ k 's with k ∈ E(sl(2)) = Z odd . We are interested in the second equation of the hierarchy (the rst describing the ow along t 1 as a translation in x), i.e for Λ 3 = λΛ. More precisely, we want to show that the equation

∂L ∂t 3 = (e ad U Λ 3 ) + , L , Λ 3 = e 1 λ + f 1 λ 2 = 0 λ 2 λ 0 , reads nothing but the KdV equation on the function q ∈ C [[x, t 3 ]].
The subscript + denotes the projection onto the subspace g ≥0 of nonnegative homogeneous degree (i.e. with only nonnegative powers of λ).

Recall that, in the above equation, the U ∈ (s ⊥ ) -[[x, t 3 ]] is such that the connection de ned by L 0 = e -ad U L has the form

L 0 = ∂ + Λ + H, H ∈ s -[[x]] .
Finding the components of U that will contribute

Since U ∈ (s ⊥ ) -[[x, t 3 ]] and (s ⊥ ) -= k <0 (s ⊥ ) k , the rst thing we need to gure out is how many terms U i ∈ (s ⊥ ) -i we need to compute. The easiest way to compute the components of U is through solving the equation L 0 = e -ad U L for each degree w.r.t. the principal gradation (as in the proof of Theorem 4.3.7).

On the other hand, we need to compute (e ad U Λ 3 ) + , where

Λ 3 = e 1 λ + f 1 λ 2 ∈ g 1 ⊕ g 2 ,
so that we only to compute U down to the homogeneous degree -2. Therefore, it is worth rst nding the homogeneous subspaces (s ⊥ ) -i , i = 0, -1, -2, and then decompose them though the principal gradation. To do so, we rst notice that the Heisenberg generators Λ -1 ∈ s -1 and Λ -3 ∈ s -3 decompose through the homogeneous gradation as

Λ -1 = f 1 + e 1 λ -1 ∈ g 0 ⊕ g -1 , Λ -3 = f 1 λ -1 + e 1 λ -2 ∈ g -1 ⊕ g -2
Therefore, it su ces to subtract the vectors e 1 λ -1 and e 1 λ -2 (for instance) from g 0 ⊕ g -1 ⊕ g -2 to get a decomposition of (s ⊥ ) 0 ⊕ (s ⊥ ) -1 ⊕ (s ⊥ ) -2 . In other words,

(s ⊥ ) 0 = C α1 ⊕ Cf 1 , (s ⊥ ) -1 = C α1 λ -1 ⊕ Cf 1 λ -1 , (s ⊥ ) -2 = C α1 λ -2 ⊕ Cf 1 λ -2 , is one possible decomposition of (s ⊥ ) 0 ⊕ (s ⊥ ) -1 ⊕ (s ⊥ ) -2
. This tells us that we only need to compute the components of U along each of the vectors in the above decomposition. Now we need to decompose these spaces through the principal gradation. To do so, notice the following:

α1 ∈ g 0 , α1 λ -1 = [f 0 , f 1 ] ∈ g -2 , α1 λ -2 = f 0 , f 1 λ -1 ∈ g -4 , f 1 ∈ g -1 , f 1 λ -1 = 1 2 f 1 , α1 λ -1 ∈ g -3 , f 1 λ -2 = 1 2 f 1 , α1 λ -2 ∈ g -5 .
Consequently, there U has no component along α1 (since

U ∈ (s ⊥ ) -[[x, t 3 ]]). Moreover, [f 1 λ -1 , Λ 3 ] = -α1 λ -1 ∈ g -1
, so that the component along f 1 λ -2 does not contribute to (e ad U Λ 3 ) + .

To sum up, we can decompose the function U ∈ (s ⊥ ) -[[x, t 3 ]] through the principal gradation as follows,

U = f 1 ⊗ u 1 + α1 λ -1 ⊗ u 2 + f 1 λ -1 ⊗ u 3 + α1 λ -2 ⊗ u 4 + O(λ -2 ),
where the component proportional to u i ∈ C [[x, t 3 ]] belongs to (s ⊥ ) -i ; the above are exactly the components that will contribute to (e ad U Λ 3 ) + .

Computing the components of U

We can now compute the functions u i via the equation L 0 = e -ad U L , i.e.

Λ + Q = e ad U (Λ + H ) -∇ U U .

We decompose the function H in the basis

{Λ c | k ∈ Z odd } of s, i.e. H = Λ -1 ⊗ h 1 + Λ -3 ⊗ h 3 + • • •

The tau functions of the DS hierarchies

In this section we de ne the tau functions of the Drinfeld-Sokolov hierarchy in the sense of Dubrovin and Zhang [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF], that is, through tau symmetric Hamiltonian densities (we de ne these notions below). Then we express the tau function through a zero-curvature expression of the Drinfeld-Sokolov hierarchies. This universal construction, which applies to the DS hierarchy of any type g (for g an untwisted a ne algebra) is due, again, to C.-Z. Wu [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF]. In the latter article, Wu actually de nes a tau function even for twisted a ne algebra, but we will not use that in the sequel.

De nition of the tau function

Let L = ∂ + Λ + Q ∈ op g and let functions U ∈ g -[[t]] and H ∈ s -[[t]] be as in Theorem 4.3.7. Let us consider a function Ω ∈ s -[[t]
], which we decompose in the basis {Λ j | j ∈ E} as

Ω = j ∈E + Ω j j Λ -j ,
and such that

∂Ω ∂x = -H, (4.57)
As a formal power series, the function Ω is well de ned as a primitive of the function -H . However, Ω may no longer be a di erential polynomials in the components q 1 , . . . , q r of Q (as is H ). Nonetheless, we ask Equation (4.57) to hold within the ring of di erential polynomials in the q i 's. For that reason, the function Ω is de ned up to the addition of some constant term in s -; equivalently, its coe cients Ω j are de ned up to the addition of some constant complex number.

Lemma 4.3.23 ([Wu17]). Each function ∂ t i Ω j , for i, j ∈ E + , is a di erential polynomial in the components of the function Q; they satisfy the symmetry condition

∂Ω j ∂t i = ∂Ω i ∂t j . (4.58)
Proof. Recall that L = e ad U (∂ + Λ + H ). It can be veri ed straightforwardly that

∂L ∂t j = e ad U ∂H ∂t j + ∇ t j ,U (U ), L , (4.59)
where, similarly to Equation (4.34),

∇ t j ,U (U ) = e ad U -1 ad U ∂U ∂t j = k ≥0 1 (k + 1)! (ad U ) k ∂U ∂t j . Now using ∂ t j L = [(e ad U Λ j ) + , L ],
we can rewrite Equation (4.59) as e ad U ∂H ∂t j + ∇ t j ,U (U ) + (e ad U Λ j ) -, L = + e ad U Λ j , L . (4.60)

Now, notice that

[e ad U Λ j , L ] = e ad U [Λ j , ∂ + Λ + H ] = -e ad U ∂Ω j ∂x • c,
where we have used [Λ j , Λ k ] = jδ j+k,0 • c and H = -∂ x Ω (4.57). Therefore, Equation (4.60) becomes

∂H ∂t j + e -ad U ∇ t j ,U (U ) + (e ad U Λ j ) -, ∂ + Λ + H = - ∂Ω j ∂x • c. (4.61)
Now, since g = s + Im(ad Λ ) with s ∩ Im(ad Λ ) ∈ g 0 , it follows that there exists two functions

G [j] ∈ s -[[t]] and G[j] ∈ (Im(ad Λ )) -[[t]] such that e -ad U ∇ t j ,U (U ) + (e ad U Λ j ) -= G [j] + G[j] ∈ g -. (4.62)
Equation (4.60) then splits into three parts:

∂ t j H -∂ x G [j] = 0, (4.63) [G [j] , Λ] c + ∂ x Ω j = 0, (4.64)
[

G[j] , Λ + H ] -∂ x G[j] = 0.
From the de nition of Ω, Equation (4.63) reads

∂ x ∂ t j Ω = -∂ x G [j] , so that ∂Ω ∂t j = -G [j] . (4.65)
There would actually be an integration constant K ∈ C, yet for the particular solution Q = 0 of the hierarchy, we have U = H = 0 so that Ω is constant and G [j] = 0 as well by Equation (4.62) (since (Λ j ) -= 0). Therefore, the integration constant vanishes. It follows from Equation (4.65) that

[Λ i , G [j] ] c = -[Λ i , ∂ t j Ω] = - ∂Ω i ∂t j .
Since U is made of di erential polynomials in Q, it follows that the ∂ t j Ω i is as well. Taking i = 1 in the above and using Λ 1 = ν Λ and (4.64), we nd

∂Ω 1 ∂t j = -ν [Λ, G [j] ] c = -ν ∂Ω j ∂x .
By symmetry of the second derivatives, we get

∂ 2 Ω j ∂x ∂t i = ∂ 2 Ω i ∂x ∂t j .
Since the ∂ t j Ω i 's are di erential polynomials in Q, it follows that ∂ t j Ω i = ∂ t i Ω j up to a constant. Yet, again, for Q = 0, the Ω i 's vanish and so does the constant. Hence the result.

By Poincaré lemma, Equation (4.58) implies the existence of a function τ ∈ C [[t]] such that Ω j = ∂ t j log τ . We can now de ne the tau function.

De nition 4.3.24 (Tau function). Let L ∈ op g and let H ∈ s -[[t]] be as in Theorem 4.3.7.

Let Ω be such that ∂ x Ω + H = 0, with Ω = j >0 j -1 Ω j Λ -j . We call a tau function of the g-DS hierarchy any function

τ ∈ C [[t]] such that ∂ log τ ∂t j = -Ω j , (4.66)
or equivalently, writing

H = j ∈E + h j Λ -j , ∂ 2 log τ ∂x ∂t j = h j .
We say that two tau functions τ , τ coincide if their logarithm di er from a linear function of the variables t j , i.e. if there exist c j ∈ C such that

log τ = log τ + j ∈E + c j t j .

Tau symmetric conserved densities

We now establish the link between the tau function and the coe cients

H i ∈ s -i [[t]
] of the function H as in Theorem 4.3.7. First recall that the functions H i are di erential polynomials in the q i 's and are conserved densities in the sense that for any i, j ∈ E + , we have

∂ ∂t j ∫ H i = 0
(Proposition 4.3.21). By the above we mean that there exists functions

F i ∈ s -i [[t]
]-which are also di erential polynomial in the q i 's-such that ∂ t j H i = ∂F i (with ∂ = ∂ x ). Therefore, in the sequel, we will consider the formal expression

∂ -1 (∂ t j H ) ∈ s -[[t]] ,
as the only di erential polynomial expression such (∂ -1 (∂ t j H )) i = F i , with the integration constant xed to 0.

Proposition 4.3.25. The function H as in Theorem 4.3.7 is related to the tau function through the relation

∂ 2 log τ ∂t i ∂t j = j ∂ -1 (Λ j , ∂ t i H ) (Λ j , Λ -j ) . (4.67)
The above does not depend on the normalization factor h (in that case, the Coxeter number of g) in (Λ i , Λ j ) 0 = hδ i+j,0 , and can be rewritten as

∂ 2 log τ ∂t i ∂t j = -∂ -1 ∂ t i H, Λ j c .
It is proved straightforwardly by noticing that, by the de nition ∂ x Ω = -H , we have [H , Λ j ] = ∂(Ω j ) • c and then using Ω j = -∂ t j log τ . We naturally de ne the following di erential polynomials:

hj = -j (Λ j , H ) (Λ j , Λ -j ) ∈ A q .
They are conserved densities as well and satisfy the tau symmetry property in the sense of the following proposition.

Proposition 4.3.26 (Tau symmetric densities). The conserved densities hj satisfy the tau symmetry property: for any i, j ∈ E + ,

∂ hj ∂t i = ∂ hi ∂t j .
It su ces to notice that

∂ t i hj = j(Λ j , Λ j ) -1 (Λ j , ∂ t i ∂ x Ω) = -∂ x ∂ t i Ω j and then use ∂ t i Ω j = ∂ t j Ω i (see Equation (4.58)).

Zero-curvature representation and dressing of the connection

In this section, we construct a dressing operator for the connection L as well as a zerocurvature representation of the g-DS hierarchy. We follow Since both U and Ω have only negative degrees w.r.t. the principal gradation, T ∈ g -[[t]] is well de ned. In general, the function T may not be a di erential polynomials in the q i 's, for Ω my not as well. Notice that in [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF] (which we still follow here), the author introduces the power series Θ = e U e Ω which belongs to the formal Kac-Moody group of g. However, everything can be formulated using only T which need not introducing the formal Kac-Moody group.

[Wu17], §3.3. Let functions U ∈ g -[[t]] and H ∈ s -[[t]] be as in Theorem 4.3.7. Let Ω ∈ g -[[t]]
In the study of the KP hierarchy (Section 3.2) we have seen that there always exists an operator M such that the scalar Lax operator Q can be written as Q = M ∂M -1 (Proposition 3.2.3); the operator M is what we call a dressing operator for the operator Q. The proposition below is an adaptation of this notion of dressing to the case of connections of op g ; the operator e ad T plays the role of of the dressing M. However we will not dress up the connection L ∈ op g , but the following connection:

L 1 := ν L -Ω 1 • c, where ν ∈ C is such that Λ 1 = ν Λ and ∂ t 1 = ν ∂ as well.
Proposition 4.3.27 (Dressing operator). Let L ∈ op g and let the functions U , H be as in Theorem 4.3.7. Recall the function

Ω = j ∈E + Ω j j Λ -j ∈ s -[[t]] such that ∂Ω = -H . Let T ∈ g -[[t]] be such that e ad T = e ad U e ad Ω . Let L 1 = ν L -Ω 1 • c, then L 1 = e ad T (∂ t 1 + Λ 1 ). (4.68)
We call the automorphism e ad T a dressing operator for the connection L 1 ; or we call T a dressing function.

As advertised for, this naturally agrees with a notion of dressing since, if we consider the formal series Θ = e U e Ω , the equation becomes

L 1 = Θ(∂ t 1 + Λ 1 )Θ -1 . Proof. First, via Λ 1 = ν Λ and ∂ t 1 = ν ∂, it amounts to proving that L = e ad U e ad Ω ∂ + Λ + ν -1 Ω 1 • c .
In turns, from L = e ad U (∂ + Λ + H ), it amounts to proving that e -ad Ω (∂

+ Λ + H ) = ∂ + Λ + ν -1 Ω 1 • c. (4.69)
So we compute: using (4.37), ∂Ω = -H and the commutativity of s -, we have

e -ad Ω ∂ = ∂ + k ≥0 (-1) k (k + 1)! (ad Ω ) k ∂Ω = ∂ -H .
On the other hand, using again the commutativity of s -along with the Heisenberg relations (4.26), we have

e -ad Ω (Λ + H ) = Λ + H -[Ω, Λ] = Λ + H -ν -1 Ω 1 [Λ -1 , Λ 1 ] = Λ + H + ν -1 Ω 1 • c.
Hence Equation (4.69), hence the proposition holds.

Let S ∈ n [[t]] and consider the gauge transformation L → L = e ad S L . Since H = H is invariant under gauge transformation (and so is Ω) while e ad Ũ = e ad S e ad U , then e ad T = e ad S e ad T .

The following lemma holds.

Lemma 4.3.28. There exists

S ∈ n [[t]] and T ∈ g -[[t]]
such that e ad S e ad T = e ad T .

Proof. The problem here is that if

T ∈ g -[[t]],
there can be some component

T 0 ∈ g 0 [[t]] (namely, of the form [e i 1 , [• • • , e i k ] • • • ] with i 1 , . . . , i k 0)
. But if we nd a gauge transformation e ad T = e ad S e ad T such that T0 = 0, then by de nition T ∈ g - [[t]]. From e ad S e ad T = e ad S •T , since the BCH formula (4.3) is a homomorphism, then (S • T ) 0 = S 0 • T 0 . We easily see that choosing S 0 = -T 0 , we get S 0 • T 0 = 0. Now recall that n = g 0 ∩ g <0 and therefore T 0 ∈ n [[t]] so that S 0 = -T 0 is consistent. That proves the lemma.

From now one we will assume that we chose a slice L of the gauge class [L ] such that the function T belongs to g - [[t]]. We can rewrite the g-DS hierarchy using the dressing operator T as follows.

Lemma 4.3.29. The g-DS hierarchy (4.43) is equivalent to the following system of equations,

∂ t j L 1 = (e ad T Λ j ) + , L 1 j ∈ E + . (4.70) Proof. First notice that e ad T Λ j = e ad U Λ j -Ω j • c. Indeed, [Ω, Λ j ] = -Ω j • c from which we get that e ad Ω Λ j = Λ j -Ω j • c. (4.71) Therefore, L 1 , (e ad T Λ j ) + = νL -Ω 1 • c, (e ad U Λ j ) + -Ω j • c = ν L , (e ad U Λ j ) + -ν L , Ω j • c = -ν ∂ t j L -ν ∂Ω j • c.
On the other hand,

∂ t j L 1 = ν ∂ t j L + ∂ t j Ω 1 • c.
We conclude via ∂ t j Ω 1 = ν ∂Ω j (Equation (4.58)).

The following proposition, which we take from [Wu17] (Lemma 3.7), already appeared in [HM93, HMSG94, Mir99].

Proposition 4.3.30. The dressing function T ∈ g -[[t]], which is such that e ad T = e ad U e ad Ω , satis es the following system of consistent equations,

∇ t j ,T (T ) = -(e ad T Λ j ) -j ∈ E + , (4.72)
where

∇ t j ,T (T ) = k ≥0 1 (k + 1)! (ad T ) k ∂T ∂t j .
Moreover, the system (4.72) is equivalent to the g-DS hierarchy (4.72) on the connection L .

Remark 4.3.31. If we take Θ = e T in the Kac-Moody group of g, then it is not hard to see from the relation e ad

T ∂ t j = ∂ t j -∇ t j ,T (T ) (4.37) that ∇ t j ,T (T ) = (∂ t j Θ)Θ -1
, so that the equations of (4.72) become

∂ t j Θ = -(ΘΛ j Θ -1 ) -Θ,
as in [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF], Lemma 3.7. Compare with Sato's equation (Theorem 3.2.4) on the dressing of the scalar Lax operator in the study of the KP hierarchy.

Proof. First, these equations are consistent since we chose T in g -[[t]] thanks to Lemma 4.3.28. Second, notice that Equation (4.70) is equivalent to

∂L 1 ∂t j = L 1 , (e ad T Λ j ) -.
Indeed, it su ces to see that [L 1 , e ad T Λ j ] = e ad T [∂ t 1 + Λ 1 , Λ j ] = 0 by Equation (4.68) and commutativity of s + . Substituting this into Equation (4.68), we get

0 = ∂ t j e ad T (∂ t 1 + Λ 1 ) + (e ad T Λ j ) -, L 1 = e ad T ∂ t j (∂ t 1 + Λ 1 ) + ∇ t j ,T (T ), L 1 + (e ad T Λ j ) -, L 1 = ∇ t j ,T (T ) + (e ad T Λ j ) -, L 1 .
Let us denote, for j ∈ E + ,

∆ [j] = e -ad T ∇ t j ,T (T ) + (e ad T Λ j ) -.
Then

∆ [j] ∈ g -[[t]],
and satis es

∆ [j] , ∂ t 1 + Λ 1 = 0. Since g <0 = s <0 ⊕ Im(ad Λ ) <0 , it follows that ∆ [j] = i ∈E + c i Λ -i ,
with c j ∈ C constants. Now notice that for the special solution Q = 0, which implies U = H = Ω = 0 and therefore T = 0 as well, we get ∆ [j] = 0 (since (Λ j ) -= 0). Therefore, ∆ [j] = 0 identically. Moreover, we easily derive Equations (4.70) from (4.72).

We can now formulate the zero-curvature representation of the g-DS hierarchy.

Proposition 4.3.32 (Zero-curvature representation). Let j ∈ E + be a positive exponent and de ne the following connection,

L j := e ad T (∂ t j + Λ j ).
They satisfy the zero-curvature equation:

L i , L j = 0. (4.73)
Proof. The proof is immediate: by construction, for any i, j ∈ E + , we have

[L i , L j ] = e ad T [∂ t i + Λ i , ∂ t j + Λ j ],
and by commutativity of s + , this yields Equation (4.73).

Relation with the tau function

Using Proposition 4.3.30, we can rewrite the operator L j as

L j = ∂ t j + Λ j + Q [j] , Q [j] = (e ad T Λ j ) + -Λ j ∈ g + [[t]] .
Now recall that the tau function of the g-DS hierarchy was de ned as any function τ ∈ C [[t]] such that

Ω j = - ∂ log τ ∂t j ,
where

Ω = j ∈E + j -1 Ω j Λ -j is such that ∂Ω = -H and H ∈ s -[[t]] is as in Theorem 4.3.7.
From this, we get the following proposition which rede nes the tau function in terms of the dressing function T .

Proposition 4.3.33. The tau function τ ∈ C [[t]
] of the g-DS hierarchy satis es the following system of equations,

∂ log τ ∂t j = (e ad T Λ j ) c , j ∈ E + , (4.74)
where the subscript c denotes the component along Cc in the decomposition h = Cc ⊕ r i=1 C αi of the a ne Cartan subalgebra.

Proof. The proof is straightforward:

Q [j] = (e ad T Λ j ) + -Λ j = (e ad U (Λ j -Ω j • c)) --Λ j = (e ad U Λ j ) + -Ω j • c -Λ j .
Since (e ad U Λ j ) c = 0 (4.40), we get that (Q [j] ) c = -Ω j . Yet by de nition of Q [j] , we also have (Q [j] ) c = (e ad T Λ j ) c , hence the result.

Tau functions as limits of block Toeplitz determinants

In this section, we follow [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF][START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF]. First of all, we x an n-dimensional faithful and traceless matrix realization of g that we denote π : g → gl(n, C). For the classical simple Lie algebras, we choose π as in Appendix A. This representation extends to

g = g[λ, λ -1 ] ⊕ Cc by π (X λ p + a • c) = π (X )λ p + a • c.
In the sequel, we will identify g or g with their representation. The goal of this section is to prove the following theorem, due to Cafasso and Wu:

Theorem 4.4.1 ([CW15a]). Let X ∈ g -and let Ξ(t; λ) = j ∈E + t j Λ j . Then 11 τ X := det e Ξ(t;λ) e X κ , is well de ned (provided the context developed below) and is a tau function of the g-DS hierarchy.

Of course, an important part of the present section will be devoted to providing the context in which the determinant de ned above is well de ned. The number κ depends on the representation of π : g → gl(n, C); it is given by

(X , Y ) 0 = κ • tr(π (X )π (Y )),
where (X , Y ) 0 is the normalized invariant form (see Section 4.1). In the cases we will study and given the representation in Appendix A, κ = 1 for the A n and C n cases and κ = 1/2 for the B n and D n cases. Before getting into the subject, we make an important observation. Since we have xed a representation, given any X ∈ g -, the power series e

X = n ≥0 X n n! = id + O(λ -1 )
is well de ned: each term in λ k is a nite sum; the product of elements of g is de ned by the matrix product. Similarly, e X is well de ned for all X ∈ g + . Following Remark 4.3.31, we can then de ne the following function,

Θ := e U e Ω ∈ G -[[t]] ,
where ∂ x Ω = -H and where U , H are functions as in Theorem 4.3.7. The function Θ we call the (matrix) dressing function and is obviously equivalent to the (matrix) dressing operator e ad T = e ad U e ad Ω . Then Equations (4.72), which are equivalent to the g-DS hierarchy, become

∂Θ ∂t j = -ΘΛ j Θ -1 -Θ. (4.75)
Let us denote Θ| t=0 (t 1 = νx = 0 included) the initial data of a solution Θ of the g-DS hierarchy.

The following lemma holds.

Lemma 4.4.2. For any X ∈ g -, there exists a solution Θ of the g-DS hierarchy such that

Θ| t=0 = e X .
11 Notice that in [CW15a, CW15b], = exp( t j Λ j ) is replaced with -1 = exp(-t j Λ j ). This is due to the fact that in [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF][START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF], the authors use the convention to write the g-DS hierarchy as ∂ t j L = [(e ad U Λ j ) + , L ]; while in this thesis, we use the convention ∂ t j L = [L , (e ad U Λ j ) + ] (as in [START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF]), which amounts to replacing t j with -t j and therefore with -1 .

To show that, it su ces to write e X = e Y Θ| t=0 with e Y = e X Θ| -1 t=0 . Then

e Y ΘΛ j Θ -1 e -Y -e Y Θ = e ad Y ΘΛ j Θ -1 -e Y Θ = e Y ΘΛ j Θ -1 -Θ,
where in the second equation, we have used the fact that Y ∈ g -. Hence, if Θ satis es Equations (4.75), then so does e Y Θ. Yet (e Y Θ)| t=0 = e X , so the lemma is proved. We can rephrase this result as the group {e X | X ∈ g -} acts transitively on the space of dressing operators.

The Grassmannian approach and Toeplitz determinants

Preliminaries on topology and the formal loop group

In what follows, we want to de ne a structure such that something like e Ξ(t;λ) e X converges for some topology. This structure we call the formal loop group associated to g. To do so, we will consider formal power series of the form

γ = I ∈Z γ I λ I , γ I = A ≥0 γ I,A ζ A , γ I,A ∈ gl(n) [[t]] .
In other words,

γ ∈ gl(n) [[λ, λ -1 , ζ , t]].
Here the problem is that the Cauchy product of two power series of this form may not always be well de ned, some coe cients in λ I may give in nite sums. To solve this problem, we de ne a topology given by the valuation distance. For this section, let us denote by R the ring

R = gl(n) [[λ, λ -1 , t]],
which is not a commutative ring nor is it an integral domain. The Cauchy product of

f , ∈ R [[ζ ]
] is given by

f • := A ≥0 ζ A A B=0 f A-B • B ,
where f A-B • B is de ned by the usual matrix product. Similarly to the case of pseudodi erential operators (see De nition 3.1.3), the valuation of

f ∈ R [[ζ ]], denoted val ζ f , is de ned as the smallest V ≥ 0 such that f V 0, i.e. val ζ A ≥0 f A ζ A = inf {A ≥ 0 | f A 0} . If we set val ζ 0 = +∞, then val ζ (f + ) ≥ min{val ζ f , val ζ } and val ζ (f • ) ≥ val ζ f + val ζ
(we would have had an equality if R were an integral domain). Then the valuation distance can be de ned by

dist (f , ) = 2 -val(f -)
The topology induced by this distance is such that a sequence

f N = a ≥0 f N ,A ζ A converges to f = a ≥0 f A ζ A if
and only if for all B ≥ 0, there exists N ≥ 0 such that for all A ≤ B, we have

f N ,A = f A . This construction makes R [[ζ ]
] into a complete topological ring (see e.g. [Bou88] §4). Below we de ne the notion of admissible series in a way that generalizes to the matrix case the one given by Borodin and Okounkov in [START_REF] Borodin | A Fredholm determinant formula for Toeplitz determinants[END_REF] in the scalar case. In [START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF] a similar de nition is given, although it makes use of the principal gradation rather than the homogeneous gradation; this applies well to the computation of the topological solution.

De nition 4.4.3 (Admissible series). Let γ = I ∈Z γ I λ I be a formal power series with entries

γ I ∈ gl(n) [[t, ζ ]].
We say that γ is admissible if for any I ∈ Z, if the valuation of the entries satisfy

val ζ (γ I ) ≥ |I |, (4.76)
This de nition seems arbitrary, but below we will see that it naturally applies to the computation of exp (Ξ(t; λ)) exp (X ). The following elementary lemma holds.

Lemma 4.4.4. If γ , γ are two admissible series, then their Cauchy product (over Z),

γ • γ := I ∈Z λ I ∈Z γ I -γ , (4.77)
is well de ned and is also an admissible series.

Proof. Let γ , γ be two admissible series, then

γ • γ = I ∈Z λ I ∈Z γ I -γ = I ∈Z λ I γ I γ0 + >0 γ I -γ + γ I + γ-.
On the one hand, Clearly, the Cauchy product (4.77) is associative and the element id ∈ gl(n) is the identity element for this law. This makes the set of all admissible series into a monoid. Naturally, we say that an admissible series

val ζ γ I -γ ≥ |I -| + | | ≥ 2| | -|I |,
γ ∈ R [[ζ ]] is invertible if there exists an admissible series γ -1 ∈ R [[ζ ]] such that γ • γ -1 = γ -1 • γ = id. (4.78)
This leads us to the following de nition.

De nition 4.4.5 (Formal loop group). We call a (formal) admissible loop any power series

γ ∈ R [[ζ ]
] which is invertible for the Cauchy product (4.77). We denote LG(n), and call the (formal) loop group, the multiplicative group of admissible loops, i.e.

LG(n

) = {γ ∈ R [[ζ ]] | γ is admissible (4.76) and invertible (4.78)} , where R = gl(n)[[λ, λ -1 , t]].
We now explain how this relates to computing e Ξ(t;λ) e X for X ∈ g -and Ξ(t; λ) = j ∈E + t j Λ j . First, we de ne a map Φ : g → g [ζ ], de ned by

Φ : X = I ∈Z X I λ I -→ X ζ := I ∈Z X I λ I ζ |I |
(where the sum over Z is nite). Now for any X = I ∈Z X I λ I (where the sum is nite), we can de ne the exponential of

X ζ by exp(X ζ ) = k ≥0 1 k! I ∈Z X I λ I ζ |I | k .
In the above equation, the product X I X is de ned as the matrix product within the representation we chose. Then e X ζ ∈ LG(n), i.e. is an admissible loop. Indeed, it su ces to write

exp(X ζ ) = I ∈Z λ I k ≥0 1 k! I 1 , ..., I k ∈Z I 1 +•••+I k =I ζ |I 1 |+•••+ |I k | X I 1 • • • X I k . Since |I 1 | + • • • |I k | ≥ |I |, it
follows that e X ζ is an admissible loop; its inverse is e -X ζ . We now apply that same construction to

Ξ(t; λ) = j ∈E + t j Λ j . The map Φ : g → g[ζ ] naturally extends to a map Φ : g [[t]] → g [[t]] [ζ ]
. The Λ j 's are not homogeneous, although, since g -∩ g + = , we can decompose each Λ j though the homogeneous gradation with only positive powers of λ; we write Λ j = Λ j, I λ I . Then

Ξ(t; λ) = I ≥0 F I (t)λ I , F I (t) = j ∈E + t j Λ j, I .
We assume F I (t) to be a nite sum, meaning that for a xed I ≥ 0, there is only nitely many Λ j 's such that val λ Λ j ≤ I . This will be realized by the matrix representations we choose (see Appendix A). We de ne

Ξ ζ (t; λ) := I ≥0 F I (t)λ I ζ |I | = Φ(Ξ(t; λ)).
Then there is no obstruction to de ning

exp Ξ ζ (t; λ) = k ≥0 1 k! I ≥0 F I (t)λ I ζ |I | k = I ∈Z λ I k ≥0 1 k! I 1 , ..., I k ≥0 I 1 +•••+I k =I ζ |I 1 |+•••+ |I k | F I 1 (t) • • • F I k (t).
It is not hard to see that the coe cient in λ I is a well de ned power series in t. Therefore, e Ξ ζ (t;λ) is an admissible loop with inverse e -Ξ ζ (t;λ) . Now let X ∈ g -, i.e. with only negative powers of λ. Let us denote γ = e X ζ and (t) = e Ξ ζ (t;λ) ; these loops have the form

γ = I ≥0 γ -I λ -I = id + O(λ -1 ), (t) = I ≥0 I λ I = id + O(λ),
with val ζ γ -I ≥ I and val ζ I ≥ I . The Cauchy product of (t) and γ can be written as

γ (t) := (t) • γ = I ∈Z λ I ≥max{0,-I } I + γ - = I ≥0 λ I ( I γ 0 + I +1 γ -1 + • • • ) + I <0 λ I ( 0 γ -I + 1 γ -I -1 + • • • ) .
We easily see, once again, that this is indeed an admissible loop in the sense of Equation (4.76). Moreover, if we de ne

+ γ = I ≥0 γ -I λ I = id + O(λ), - γ (t) = I ≥0 I λ -I = id + O(λ -1 ),
then γ (t) admits two factorization:

γ (t) = (t) • γ = - γ (t) • + γ . (4.79)
The above equation will be important in the sequel for the Borodin-Okounkov formula. In what follows, we will write X = X ζ and Ξ(t; λ) = Ξ ζ (t; λ) when there is no ambiguity.

Laurent block matrices and formal loops

Following the de nition of the formal loop group, let H n be the subspace of formal power series over Z in a parameter λ, say = I ∈Z I λ I , with entries

I ∈ C n [[ζ ]] such that val ζ I ≥ |I |, i.e. H n = = I ∈Z I λ I I ∈ C n [[ζ ]] , val ζ I ≥ |I | ⊂ C n [[λ, λ -1 , ζ ]].
The space H n is spanned by the standard family of vectors {e α λ I | α ∈ {1, . . . , n}, I ∈ Z}, where e α is the column vector with 1 at the α-th position and 0 elsewhere. With respect to this basis, a vector = I ∈Z I λ I ∈ H n can be identi ed with its coordinates in the topological basis {λ I | I ∈ Z} (w.r.t. the topology induced by the gradation):

∼ . . . -1 0 1 . . . I = A ≥ |I | ζ A n α =1 α I,A e α ∈ C n [[ζ ]] .
We denote by H + and H -the subspaces consisting of vectors with only nonnegative and negative powers of λ respectively, i.e.

H + = I ≥0 I λ I ⊂ C n [[λ, ζ ]], H -= I <0 k λ I ⊂ λ -1 C n [[λ -1 , ζ ]].
We denote

pr + : H → H + , pr -: H → H -
the projections on H + and H -respectively. Given an admissible loop γ = I ∈Z γ I λ I ∈ LG(n) we can de ne an action of γ on by a Cauchy product:

γ = I ∈Z λ I ∈Z γ I - ∈ H n (4.80)
Thanks to the assumption of admissibility (val ζ γ I , val ζ I ≥ |I |), this action is well de ned (the expansion in ζ contains only nitely many terms for each power λ I ζ A ). It follows from Equation (4.80) that γ admits an in nite-dimensional "Z × Z" matrix representation as

L(γ ) := γ I -I, ∈Z = . . . . . . . . . . . . γ 0 γ -1 γ -2 . . . γ 1 γ 0 γ -1 . . . γ 2 γ 1 γ 0 . . . . . . . . . . . . . (4.81)
The matrix L(γ ) we call the Laurent matrix of γ . We have γ = L(γ ) , where the second product is the standard product of matrices. It follows that

γ • γ = L(γ ) • L( γ ),
making the Laurent matrix map into a homomorphism

L : LG(n) → (γ I -) I, ∈Z γ K ∈ C [[ζ ]] , val ζ γ K ≥ |K |
Notations. We shall emphasize the important distinction, hereafter, between capital indices I , and small indices i, j. Capital indices I , denote "block indices" in the sense that the coefcient γ I of a loop γ = I ∈Z γ I λ I is a matrix in γ I ∈ GL(n). On the other hand, small indices i, j will denote "usual indices" of a matrix, e.g. (γ I ) i, j ∈ C. For the Laurent matrix of a loop, the relation between the block indices and the usual indices is given in the following relation:

L(γ ) I n+α, n+β = (γ I -) α +1, β +1 , I, ∈ Z, α, β ∈ {0, . . . , n -1}.
For any admissible loop γ ∈ LG(n) of the form

γ = id + O(λ -1 ),
we associate the subspace W γ ⊂ H n de ned by W γ = γ (H n + ). The following lemma is the basis for the Grassmannian interpretation of the sequel. Lemma 4.4.6. For any admissible loop of the form γ = id + O(λ -1 ), the subspace W γ = γ (H + ) of H satis es the following, (i) The map pr

+ | W γ : W γ → H + is a linear isomorphism; (ii) λζ • W γ ⊂ W γ . 12
Proof. Point (ii) is direct: For any = I ≥0 I λ I ∈ H + , we have

λζ • = I ≥0 λ I +1 ζ I = I ≥1 λ I ζ I -1 .

Yet by de nition val

ζ I -1 ≥ |I -1|, therefore val ζ ζ I -1 ≥ |I -1| + 1 ≥ |I |
(by the reverse triangular inequality). Therefore, λζ • ∈ H + too and so λζ • γ ( ) = γ (λζ • ) ∈ W γ . We now prove point (i). To do so, we will prove that the map pr + | W γ is injective and then surjective. Let w = γ ( ) ∈ W γ be a nonzero element and let us assume that pr + w = 0. The vector w = γ ( ) can be written as

w = I ∈Z λ I ≥0, ≥I γ I -.
Thus we get the equation

pr + w = I ≥0 λ I ≥I γ I - = 0.
For any I ≥ 0, and since γ 0 = id, this expands as

I + γ -1 I +1 + γ -2 I +2 + • • • = 0. (4.82)
12 Notice that here we must include ζ in the factor λζ because of our de nition of admissible loops and vectors. (As we can see in the proof.) But that can be simply seen as a consequence of the morphism γ I λ I → γ I λ I ζ [I | that allows us to treat things in a purely algebraic setup. This is a di erence with other article dealing with this topic (e.g. And so on: applying the same argument recursively we get that A I = 0 for all A ≥ I , meaning that = 0. Thus pr + | W γ is injective.

We now prove that it is surjective. We consider an arbitrary u = I ≥0 λ I u I ∈ H + and we want to nd some = I ≥0 λ I I ∈ H + such that pr + γ ( ) = u; i.e. we want to nd such that

I ≥0 λ I ≥I γ I - = I ≥0 λ I u I .
For each value of I ≥ 0, this expands as

I + γ -1 I +1 + γ -2 I +2 + • • • = u I
Now with very similar arguments as for the injectivity, we can expand both sides of the above in powers of zeta and ne that

I +2 I +1 = u I +3 I , i.e. I +2 I = u I +2 I -γ -1 u I +1 I +1 , I +3 I = u I +3 I -γ -1 u I +2 I +1 .
And so on, this determines (and therefore w = γ ( )) recursively. Thus pr + | W γ is surjective. This concludes the proof.

Conversely, one associate such a loop γ = id + O(λ -1 ) to any subspace W ⊂ H n which satis es the condition

λζ • W ⊂ W , pr + | W : W ∼ -→ H n + .
The set of all such subspaces is what we call the "big cell" associate to H n ; we denote Gr 0 . The notion of big cell was introduced by [SW85], although the idea was already present in [START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF]. Hereafter we will identify any admissible loop γ ∈ LG(n) of the form γ = id + O(λ -1 ) with the element W γ ∈ Gr 0 such that W γ = γ (H + ). We identify W γ with the Z ≥0 × Z matrix that represents the map γ | H n : H n → W γ , namely:

W γ ∼ γ I -I ∈Z, ≥0 = . . . . . . . . . . . . id γ -1 γ -2 • • • 0 id γ -1 • • • 0 0 id • • • . . . . . . . . . . . . .
In what follows, we will drop the superscript n and simply denote H = H n .

The Sato-Segal-Wilson tau function

In this section, we give a de nition of a tau function associated to a loop γ = id + O(λ -1 ), or equivalently to a point W γ ∈ Gr 0 , called the Sato-Segal-Wilson tau function, following [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF] 13 . It originally appeared in a certain form in [START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF] and [START_REF] Segal | Loop groups and equations of KdV type[END_REF]. One of the aimes is to identify this tau function with the limit of certain Toeplitz determinant and then with a tau function of the g-DS hierarchy. We de ne the following maps:

Ω + = pr + • γ | H + : H + → H + , Ω -= pr -• γ | H + : H + → H -,
along with

h W = Ω -• (Ω + ) -1 = pr -| W • (pr + | W ) -1 : H + → H -.
These maps are well de ned since, by hypothesis, pr + | W is bijective.

Recall the power series

(t) = exp (Ξ (t; λ)) = id + O(λ).
Consider the following submatrices of the Laurent matrix of ,

a = I -I, ≥0 , b = I -I ≥0, <0 , d = I -I, <0 .
They are the matrices of the following maps,

a = pr + • | H + , b = pr + • | H -, d = pr -• | H -.
Notice that actually a = | H + since by the form of , we see that (H + ) ⊂ H + . The latter also implies that the map pr -• | H + vanishes identically. Consequently, the loop admits the following block matrix representation, = a b 0 d .

De nition 4.4.8 (Sato-Segal-Wilson tau function [START_REF] Segal | Loop groups and equations of KdV type[END_REF]). Let W γ ∈ Gr 0 be a subspace and let = exp (Ξ(t; λ)). We call the Sato-Segal-Wilson the formal power series

τW γ = det id + a -1 • b • h W γ = det id + b • h W γ • a -1 .
For now, it is unclear why the determinant that de nes τW γ (t) is well de ned. This will be solved by the following lemma.

Lemma 4.4.9. For any X ∈ g -and associated loop γ = e X , the following identity holds,

bh W γ a -1 = -H( -1 γ ) H(γ -1 )
Proof. First, we write

a -1 = pr + -1 | H + , b = pr + | H -, Ω -1 + = pr + γ -1 | H + , Ω -= pr -γ | H + . Therefore, bh W γ a -1 = pr + pr -γ pr + γ -1 -1 | H + .
On the other hand, since ι 2 = id, we have

H( γ ) H(γ -1 -1 ) = pr + γ pr -γ -1 -1 | H + .
Therefore, its su ces to prove that (hereafter we underline the summands for clarity)

pr + γ pr -γ -1 -1 + pr + pr -γ pr + γ -1 -1 | H + = 0.
Then it su ces to use pr -= idpr + and the fact that

Ω + = pr + γ | H + and Ω -1 + = pr + γ -1 | H + : pr + γ pr -γ -1 -1 + pr + (id -pr + )γ pr + γ -1 -1 | H + = pr + γ pr -γ -1 -1 + pr + γ pr + γ -1 -1 -pr + pr + γ pr + γ -1 -1 | H + = pr + γγ -1 -1 -pr + Ω + Ω -1 + -1 | H + = id -id | H + = 0.
Now thanks to Lemma 4.4.9, the Sato-Segal-Wilson tau function can be written as

τW γ = det 1 -H( γ ) H(γ -1 -1 ) .
Notice that because the loops (t) and γ are admissible, the entries of H( γ ) I, and H(γ -1 -1 ) I, have valuation greater than |I + + 1|. In particular, both these Hankel matrices are Hilbert-Schmidt operators in the sense that the norm

||H( γ )|| 2 = I, ≥0 |H( γ ) I | 2
converges for the topology induced by the valuation (it su ces to see that the partial sums form a Cauchy sequence). Then the product of two Hilbert-Schmidt operators is of trace class and the determinant det(id +T ) with T trace-class is a well de ned Fredholm determinant. We refer to [START_REF] Simon | Trace Ideals and Their Applications: Second Edition[END_REF] for more details.

The Borodin-Okounkov formula and a formal Szegő-Widom theorem

In this section, we identify the Sato-Segal-Wilson with the limit of the Toeplitz determinant of the loop γ by means of the Borodin-Okounkov formula and a formal version of the Szegő-Widom theorem; we still follow [START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF].

Besides the in nite Z ≥0 × Z ≥0 Toeplitz matrix, we associate to a loop ϕ = I ∈Z ϕλ I the following matrix:

T N (ϕ) = ϕ I - N I, =0 = ϕ 0 ϕ -1 • • • ϕ -N ϕ 1 ϕ 0 • • • ϕ -N +1 . . . . . . . . . . . . ϕ N ϕ N -1 • • • ϕ 0
, called the N -truncated block Toeplitz matrix. We denote its determinant

D N (ϕ) := det T N (ϕ).
Moreover, we denote p N : H + → H + the projection on the subspace of power series of the form N I =0 I λ I and q N = id -p N . Finally, if ϕ is admissible, we denote

Z(ϕ) = det id -H(ϕ) H(ϕ -1 ) ,
which, as noted an the end of the previous section, is a well de ned Fredholm determinant. We can now present the celebrated Borodin-Okounkov formula as it appeared in [START_REF] Baso | On a Toeplitz determinant identity of Borodin and Okounkov[END_REF].

Theorem 4.4.10 (Borodin-Okounkov formula [START_REF] Borodin | A Fredholm determinant formula for Toeplitz determinants[END_REF]). Let ϕ ∈ LG(n) be an admissible loop such that ϕ = ϕ + ϕ -= ψ -ψ + admits two distinct factorizations such that

ψ -(∞) = ϕ -(∞) = id, ψ + (0) = id + N + , ϕ + = id + M + ,
where N + and M + are strictly lower triangular matrices. Then for any N ≥ 1,

D N (ϕ) = Z(ϕ) det (id -K N (ϕ)) ,
where K N (ϕ) = q N K(ϕ)q N and

K(ϕ) = H(ϕ -ψ -1 + ) H(ψ -1 -ϕ + ).
Notice that K N (ϕ) is an in nite matrix but its determinant is a well de ned Fredholm determinant for the same reasons Z(ϕ) is. We refer to [CW15b] (Theorem 2.3) for a proof of Theorem 4.4.10 which is adapted to the present case of formal matrix-valued power series; the proof follows that of [START_REF] Baso | On a Toeplitz determinant identity of Borodin and Okounkov[END_REF]. As a corollary, we obtain a formal version of the Szegő-widom theorem (see [START_REF] Widom | Asymptotic behavior of block Toeplitz matrices and determinants II[END_REF] for example). Then, it is not hard to see that the entries of K N (ϕ) have a valuation greater or equal to Nh, so that val (log Z(ϕ)log D N (ϕ)) ≥ Nh, which implies that Equation (4.83) holds.

Now to identify the Sato-Segal-Wilson tau function with the limit of the Toeplitz determinant of γ = -1 γ , it su ces to show that γ satis es the assumptions of the Borodin Okounkov theorem. First, as noted at the beginning of Section 4.4.1,

γ = γ = - γ γ +
Therefore, γ admits two di erent factorizations, with , + γ ∈ id + O(λ) and γ , - γ ∈ id + O(λ -1 ). So γ satis es the assumptions of Theorem 4.4.10.

Corollary 4.4.12 ([CW15b]). For any loop γ = id + O(λ -1 ) and γ (t) = (t) • γ , the following identity holds,

τW γ (t) = lim N →∞ D N ( γ (t)) =: D ∞ ( γ (t)).

Relation with the tau functions of the DS hierarchies

The matrix wave function

We start this section by de ning the notion of matrix wave (or Baker-Akhiezer) function associated to a loop γ = id + O(λ -1 ) and the power series (t) = exp(Ξ(t)) as de ned in [START_REF] Dickey | Soliton Equations and Hamiltonian Systems[END_REF] (see De nition 12.1.3) and given again in [START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF] (see De nition 2.9). De nition 4.4.13. Let γ = id + O(λ -1 ) and the associated point of the big cell Gr 0 W γ = γ (H + ). We say that a loop w(t) ∈ LG(n) is a wave function (or Baker-Akhiezer) function associated to γ if (i) For any ∈ H + , we have

w(t) ∈ W γ ; (ii) pr + • (t) • w(t) = id H .
This de nition is to be compared with the notion of wave function in KP theory and the bilinear identity developed in Section 3.3.

Lemma 4.4.14. For any loop γ = id+O(λ -1 ), the following function is a wave function associated to γ :

w(t) = -1 (t) - γ (t).
Proof. For point (i), it amounts to proving that for any ∈ H + , we have γ -1 w(t) ∈ H + . Indeed,

γ -1 w(t) = γ -1 -1 (t) - γ (t) = ( - γ (t) + γ (t)) -1 - γ (t) = ( + γ (t)) -1 ,
the latter belongs to

H + since ( + γ (t)) -1 = id + O(λ). Point (ii) is immediate since pr + • (t)w(t) = pr + • -1 γ (t) = id.
Then in [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF], and in analytical context which adapts to our formal context, the authors related the Sato-Segal-Wilson tau function with the wave function via the following theorem (see Theorem 2.11 and Corollary 2.12 of [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF]).

Theorem 4.4.15 [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF]). Given a point W γ ∈ Gr 0 and its associated wave function w(t), the following identity holds, for any j ∈ E + ,

∂ log τW γ ∂t j = - ∮ dλ 2iπ tr ∂w ∂λ w -1 Λ j , (4.84)
where the contour integral is to be understood as the sum of all residues; the trace is to be understood as the trace of the representation we chose.

Theorem 4.4.15 can be understood as a generalization of 3.3.7 in KP theory. Indeed, in the latter the wave function is scalar and related to the tau function by

w(t; λ) = τ (t -[λ -1 ]) τ (t) -1 (t), (t) = exp i ≥1 t i λ i .
Let us verify that in this scalar case the KP wave function satis es Equation (4.84). First, Equation (4.84) is equivalent to

∂ log τW γ ∂t j = - ∮ dλ 2iπ tr ∂w ∂λ w -1 ∂ ∂t j -1
Then, the trace is the identity, so it reduces to

∮ dλ 2iπ ∂w ∂λ w -1 ∂ -1 ∂t j = - ∮ dλ 2iπ ∂τ (t -[λ -1 ]) ∂λ 1 τ (t -[λ -1 ]) λ j = - ∮ i ≥1 ∂ log τ (t -[λ -1 ]) ∂t i λ j-i-1 = - ∂ log τ ∂t j

Sato-Segal-Wilson vs. Drinfeld-Sokolov tau functions

We now make the link between the Sato-Segal-Wilson tau function and the Drinfeld-Sokolov tau function as de ned by Wu [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF] in (4.74): given a dressing operator Θ,

∂ log τ Θ ∂t j = ΘΛ j Θ -1 c .
Recall that the dressing operator Θ = e U e Ω = id + O(λ -1 ) ∈ g -[[t]]. The following lemma connects the wave function w(t) and the dressing operator Θ(t) in the case where the initial data γ = id + O(λ -1 ) is taken as

γ = Θ -1 | t=0 [CW15a]. Lemma 4.4.16. Let Θ ∈ g -[[t]
] be an arbitrary solution of the g-DS hierarchy and let γ = Θ -1 | t=0 = id + O(λ -1 ). Then the corresponding wave function is given by

w(t) = -1 (t)Θ -1 (t).
Moreover, if τ W γ (t) is the Sato-Segal-Wilson tau function associated to γ , then Equation (4.84) becomes the following,

∂ log τW γ ∂t j = ∮ dλ 2iπ tr ∂Θ ∂λ Λ j Θ -1 (4.85)
Proof. First, we clearly have pr + (t)w(t) = pr + Θ -1 (t) = id. So it remains to prove that for any ∈ H + , we have γ -1 w(t) ∈ H + . Let us denote ŵ(t) = γ -1 w(t), then

∂ t j w = --1 Λ j Θ -1 --1 Θ -1 (∂ t j Θ)Θ -1 = --1 Θ -1 (ΘΛ j Θ -1 ) + -1 Θ -1 (ΘΛ j Θ -1 ) - = -w(ΘΛ j Θ -1 ) + . (4.86) Therefore, ŵ-1 ∂ t j ŵ = w -1 ∂ t j w = -(ΘΛ j Θ -1 ) + ∈ g + [[t]].
Moreover the initial condition of ŵ(t) reads ŵ(t)| t=0 = γ -1 (t) Θ -1 (t) | t=0 = γ -1 γ = 1, (4.87) since (t)| t=0 = e 0 = 1 and γ = Θ -1 (t)| t=0 . Now, Equations (4.86) and (4.86) together imply that ŵ(t) contains only nonnegative powers of λ, hence the result. Then to prove Equation (4.85) it su ces to use the analyticity of , the commutativity within the trace and an integration by part:

∂ log τW γ ∂t j = - ∮ dλ 2iπ tr ∂ -1 Θ -1 ∂λ Θ Λ j = - ∮ dλ 2iπ tr ∂ -1 ∂λ Λ j + -1 ∂Θ -1 ∂λ Θ Λ j = - ∮ dλ 2iπ tr ∂Θ -1 ∂λ ΘΛ j = ∮ dλ 2iπ tr ∂Θ ∂λ Λ j Θ -1
We can now state the main theorem of this section.

Theorem 4.4.17 [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF]). Let Θ be a solution of the g-DS hierarchy and let τ Θ be the associated tau function. Let γ = Θ -1 | t=0 = id + O(λ -1 ) and let τW γ be the associated Sato-Segal-Wilson tau function. Then

log τ Θ = κ log τW γ ,
where κ is de ned by (Y , Z ) 0 = κ • tr(π (Y )π (Z )) for all Y , Z ∈ g.

We proceed by two lemmas; we give the proofs of [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF].

Lemma 4.4.18. For any A, B ∈ g, the following identity holds,

[A, B] c = κ ∮ dλ 2iπ tr ((∂ λ A)B) .
Proof. It su ces to prove the identity on elements of the form X λ p , Y λ q ∈ g with X , Y ∈ g and p, q ∈ Z. By the de nition (4.15), we have X λ p , Y λ q c = κpδ p+q,0 tr(XY ).

On the other hand,

∮ dλ 2iπ tr (∂ λ X λ p )Y λ q = ∮ dλ 2iπ
pλ p+q-1 tr(XY ) = pδ p+q,0 tr(XY ).

Lemma 4.4.19. For any X ∈ g -and Y ∈ g, the following identity holds,

e X Y e -X c = κ ∮ dλ 2iπ tr (∂ λ e X )Y e -X .
Proof. To simplify notations, we write X = ∂ λ X and X = ∮ dλ 2iπ trX . First, we have

∂ λ e X = p ≥0 1 p! p-1 q=0 X q X X p-q-1 . Let us denote (∂ λ e X )Y e -X = m ≥1 R m with R m = m-1 k =0 (-1) r (m -k)!k! m-k-1 j=0 X j X X m-k-j-1 Y X k . We rewrite R m as R m = m-1 k=0 (-1) k (m -k)!k! X m-k-1 j=0 X j Y X k = 1 m! X m-1 j=0 m-j-1 k =0 (-1) k m k X j Y X m-1-j = 1 m! X m-1 j=0 (-1) m-1-j m -1 m -1 -j X j Y X m-1-j = 1 m! X (ad X ) m-1 Y ,
where in the third equation we have used the recursion relation m-1 k -m k+1 = -m-1 k+1 . By Lemma 4.4.18, it follows that

(∂ λ e X )Y e -X = 1 κ m ≥1 1 m! X , (ad X ) m-1 Y c = 1 κ m ≥1 1 m! ((ad X ) m Y ) c = 1 κ (e ad X Y ) c = 1 κ (e X Y e -X ) c
Proof of Theorem 4.4.17. Now it su ces to use the de nition (4.74) of Wu's tau function for the g-DS hierarchy, Lemma 4.4.19 and Equation (4.85), for all j ∈ E + ,

∂ log τ Θ ∂t j = ΘΛ j Θ -1 c = ∮ dλ 2iπ tr ∂Θ ∂λ Λ j Θ -1 = κ ∂ log τW γ ∂t j .
Therefore, log τ Θ and log τW γ di er from a linear function of the variables t j , which, in our framework, amounts to their coincidence.

It remains to derive Theorem 4.4.1 (which we stated at the beginning of Section 4.4) from Theorem 4.4.17 (which we just proved). Let us rst ecall Theorem 4.4.1; notice that from now on we denote det = D ∞ .

Theorem 4.4.20 [START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF]). Let X ∈ g -and let Ξ(t; λ) = j ∈E + t j Λ j . Then τ X := det e Ξ(t;λ) e X κ , is a tau function of the g-DS hierarchy.

First, since we chose γ = γ , by Corollary 4.4.12, we have

τW γ = lim N →∞
D N e Ξ(t;λ) e X =: det e Ξ(t;λ) e X .

Moreover, γ = Θ -1 | t=0 and, as noted at the beginning of Section 4.4, for any X ∈ g -there exists Θ such that Θ| t=0 = e -X . Therefore, if we denote Θ X the solution associated to the initial data Θ X | t=0 = e -X , we can write log τ X := log τ Θ X = κ log τ W γ = κ det e Ξ(t;λ) e X .

In particular, τ X = det e Ξ(t;λ) e X κ .

4.5 Polynomial tau functions of the DS hierarchies and Schur polynomials

Young diagrams expansion of the tau function

Preliminaries on notations

We denote Y the set of integer partitions ('Y' for Young diagrams; see below), i.e. the set of decreasing nite sequences of positive integers, i.e.

Y = {µ = (µ 1 , . . . , µ r ) ⊂ Z ≥1 | r ≥ 0, µ i ≥ µ i+1 }
We denote (µ) = r and call it the length of µ. If (µ) = 0 the partition is denoted µ = and called the empty partition. We denote by

|µ | = µ 1 + • • • + µ (µ)
the weight of µ, i.e. the integer of which µ is a partition.

To any partition we associate a Young diagram as follows. Given a partition µ = (µ 1 ≥ • • • ≥ µ (µ) ), its Young diagram is a table of squares, all ushed to the left, with (µ) rows and where row i contains µ i squares; see Equation (4.88) for an example.

Using the Young diagram of a partition, we de ne its Frobenius notation as follows. First, denote d(µ) is the number of squares in the main diagonal of the Young diagram of µ (i.e. the squares with coordinates (i, i), i = 1, . . . , d(µ)). Denote k i the number of squares in row i that are strictly to the right of square (i, i); denote l i is the number of squares in column i that are strictly below square (i, i). The integers k i , l i and d(µ) determine µ; we denote

µ = (µ 1 , . . . , µ (µ) ) = (k 1 , . . . , k d (µ) | l 1 , . . . , l d (µ) )
For example, the partition µ = (5441) has length (µ) = 4, weight |µ | = 14, and can be written in the Frobenius notation as µ = (421|310), as illustrated below.

ν = (5441) ↔ (4) (2) (1) (1) (0) (3) ↔ ν = (421|310) (4.88)
Let X ∈ g -and recall Ξ(t; λ) = j ∈E + t j Λ j . Let γ = e X and (t) = e Ξ(t;λ) . Let their Laurent matrices be denoted by

r := L(γ ), s(t) := L( (t)).
We introduce the following determinants.

De nition 4.5.1. Let a partition µ ∈ Y. We call the Plücker coordinates of γ with respect to the representation π : g → gl(n) the determinant

r µ := det r i-µ i -1; j-1 i, j ∈[1, (µ)] ∈ C. (4.89)
We call the generalized Schur polynomials of type (g, π ) the determinant

s µ (t) := det s i-1; j-µ j -1 i, j ∈[1, (µ)] ∈ C[t]. (4.90)
We also set r = s = 1. For the adjoint representation π (X ) = ad X ∈ End(g), we call s µ (t) the intrinsic Schur polynomial of type g.

Moreover, we de ne matrices D I (t) and Z I , for I, ≥ 0, by ide Ξ(t;λ) e -Ξ(t;µ) λµ

= ∞ I, =0 D I (t) λ I +1 µ +1 , (4.91) id -e X (λ) e -X (µ) λ -µ = ∞ I, =0 Z I λ -I -1 µ --1 . (4.92)
The above equations are to be understood as formal expansion in the parameters λ, µ. De ne s (i |j) , r (i |j) , for i, j ≥ 0, via

(D I ) ab = s (nI +a-1|n +n-b) , (Z I ) ab = r (I +n-a |n +b-1) ,
where a, b ∈ {1, . . . , n}. We call Z I the matrix-valued a ne coordinates and r X (i |j) the a ne coordinates. These matrices were introduced in [BY17] for case of sl(2).

For a partition µ ∈ Y (m) , we de ne determinants R µ and S µ in the same fashion as in Equation (4.89). Then the following identity holds,

det SR = µ ∈Y (m) S µ R µ .
Notice that because of the speci c de nition of the matrices R and S, the determinants R µ and S µ can only be de ned for µ ∈ Y (m) , hence the proposition. For example, if m = 1, one easily computes that det S 0,-1 1 0

S 1,-1 S 1,0 1 • R -1,0 R -1,1 1 R 0,1 0 1 = S R + S R + S R .
Proof. It is a particular case of the well known Cauchy-Binet formula (see e.g. [START_REF] Gantmacher | The Theory of Matrices, vols I and II[END_REF]). The latter states that if A is an n × n matrix and B is an n × n matrix, then the determinant of AB is the sum of the product of the minors over all minors of size n; in other words,

det AB = 1≤i 1 < •••<i n ≤n A 1,i 1 • • • A 1,i n . . . . . . A n,i 1 • • • A n,i n • B i 1 ,1 • • • B i n ,n . . . . . . B i 1 ,1 • • • B i n ,n . So let a sequence of m + 1 integers -m ≤ i 1 < i 2 < • • • < i m+1 ≤ m.
We can assume without loss of generality that (i 1 , . . . , i p , i p+1 , . . . i m+1 ) = (i 1 , . . . , i p , p, . . . , m), (4.95)

for some p ∈ {0, . . . , m + 1}. If p = 0, the sequence reads (0, . . . , m); if p = m + 1, the sequence is simply such that i m+1 < m. We de ne

µ k = k -i k -1 k ∈ {1, . . . , p}
Then we automatically have µ k +1 ≤ µ k from i k+1 > i k and it is not hard to see that µ k ≥ 0 starting from i p+1 = p and then applying a backward induction to i p+1 > i p > i p-1 > . . . Therefore, the µ k 's form a partition µ = (µ 1 ≥ • • • ≥ µ p ). If p = 0, the partition is the empty partition µ = . Moreover, by construction we have µ 1 = -i 1 ≤ m and (µ) = p ≤ m +1, so that µ ∈ Y (m) . Reciprocally, any partition of µ ∈ Y (m) of length (µ) = p ∈ {0, . . . , m+1} determines a unique sequence (i 1 , . . . , i m+1 ) satisfying Equation (4.95). This explains the expansion over partitions of Y (m) . It remains to prove that given a sequence -m ≤ i 1 < i 2 < • • • < i m+1 ≤ m satisfying Equation (4.95) for some 0 ≤ p ≤ m + 1, and given µ = (µ 1 ≥ • • • ≥ µ p ) ∈ Y (m) the partition it determines, the minors of R and S are respectively given by Equation (4.89). We prove it for the matrix S.

If p = m + 1, it is immediate: the partition µ has length (µ) = m + 1 so that S µ given in Equation (4.89) is indeed the minor of size m + 1 associated to the sequence (i 1 , . . . , i m+1 ).

If p = 0, the sequence is given by (0, . . . , m) for which the minor is the determinant of a lower triangular matrix with 1's on the diagonal and therefore equals 1; accordingly, the associated partition is the empty partition µ = , for which we set S = 1.

If 1 ≤ p ≤ m, the minor of S associated to the sequence (i 1 , . . . , i m+1 ) has the form

S 0,i 1 • • • S 0,i p 0 • • • 0 . . . . . . . . . . . . S p-1,i 1 • • • S p-1,i p 0 • • • A A 0 S p,i 1 • • • S p,i p 1 0 • • • 0 . . . . . . . . . . . . . . . 1 0 S m,i 1 • • • S m,i p S m,p • • • S m,m-1 1 = S 0,i 1 • • • S 0,i p . . . . . . S p-1,i 1 • • • S p-1,i p . (4.96)
By de nition, the right-hand side of the above equation is S µ as de ned in Equation (4.89); which concludes the proof.

We now come back to the Laurent matrices r = L(γ ) and s(t) = L( (t)). For m ≥ 1, we de ne the following nite-size rectangular matrices:

s m (t) = s i, j (t) | i ∈ [0, m], j ∈ [-m, m] , r m = r i, j | i ∈ [-m, m], j ∈ [0, m] .
µ ∈Y s µ (t) • r µ -u m = µ ∈Y\Y (m) s µ (t) • r µ . Now consider a partition µ ∈ Y \ Y (m) with minimal length (µ) = m + 2, minimal µ 1 = m + 1 and minimal weight |µ | = µ 1 + • • • + µ (µ)
. This is realized by µ = (m + 1, 1, . . . , 1). The minor associated to µ reads To nally prove Theorem 4.5.2, we make the following observation.

s µ = s 0,-m-1 1 0 • • • 0 s 1,-m-1 s 1,0 1 . . . . . . . . . . . . . . . . . . 0 s m+1,-m-1 s m+1,0 • • • s m+1,m 1 = s 0,-m-1 s 1,0 • • • s m+1,m -1 .
If m = N n, then the matrices s N n (t) and r N n admit a block decomposition:

s N n (t) = N • • • 1 id 0 • • • 0 . . . . . . . . . . . . . . . 0 2N • • • 1 id r N n = γ -N • • • γ -2N . . . . . . γ -1 id . . . 0 . . . γ -1 . . . id 0 • • • 0
Consequently, if we denote θ nN = s nN (t) • r nN , then in terms of blocks,

(θ nN ) I, = N K =-N I -K γ K -, (T N (ϕ)) I, = K ∈Z I -K γ K -.
Therefore,

(T N (ϕ)) I, -(θ nN ) I, = +∞ K =N I -K γ K -+ I +K γ -K -. Yet val ζ I -K γ K -≥ 2N -|I |-| |;
this holds for I +K γ -K -as well. Now this computation was done block-wise; we let the reader convince themselves that it implies that component-wise,

lim m→∞ (θ m ) i, j = T (ϕ) i, j
for all i, j ≥ 0. This concludes the proof of Theorem 4.5.2.

Polynomial tau functions and bilinear equations

Now consider the case where X ∈ g -is such that π (X ) is a nilpotent matrix. Then γ = e X admits a nite expansion in negative powers of λ. It follows that for su ciently large partitions, the Plücker coordinates r µ of X vanish. As a consequence, we get the following corollary to Theorem 4.5.2.

Corollary 4.5.8. For any X ∈ g -such that π (X ) is a nilpotent matrix, the function (τ X ) 1 κ is polynomial.

Let us condense the above statement in what will hold as the leitmotiv of this chapter's remainder:

∀ X ∈ g -, π (X ) is a nilpotent matrix =⇒ τ X (t) 1 κ is polynomial.
Although this has not been proven, there is strong indication that the above procedure, in th A 1 case, recovers all tau functions of KdV. For other cases though, we do not know if we would recover all polynomial tau function. In what follows we compute the rst few polynomial tau functions of the DS hierarchy of type g for g = A 1 , A 2 , B 2 and D 4 .

We will use the particular tau functions to deduce possible bilinear equations of small degrees. Note that each Drinfeld-Sokolov hierarchy has in nitely many solutions. The usual question is to nd particular solutions (and their tau-functions) to the DS hierarchy (e.g. to solve all PDEs in this hierarchy together). Here, as we mentioned above, we will also consider the reverse: deduce possible bilinear equations associated to a hierarchy from particular solutions. Remark 4.5.9. If κ = 1, then the tau function itself is a polynomial. Interestingly enough, in the computations that we performed, even when κ = 1/2, we obtained some polynomial tau functions; in other words, the nite sum in (4.93) is a perfect square.

Bilinear derivatives

The bilinear derivatives are operators

D i 1 • • • D i k : C [[t]] ⊗ C [[t]] → C [[t]
], for i 1 . . . , i k ∈ E + , de ned by the following identity: for any

f , ∈ C [[t]], 14 exp i ∈E + h i D i (f , ) = f (t + h) (t -h).
For example,

D i (f , ) = ∂ f ∂t i -f ∂ ∂t i , D i D j (f , ) = ∂ 2 f ∂t i ∂t j + f ∂ 2 ∂t i ∂t j - ∂ f ∂t i ∂ ∂t j - ∂ f ∂t j ∂ ∂t i .
The bilinear derivatives give a way to translate the g-DS hierarchy on the solution into equations on the tau function itself.

Example 4.5.10. In the A n case, the equations of the DS hierarchy, when translated onto the tau function τ , are equivalent to the following two chains of bilinear equations (these can be deduced from results of [START_REF] Date | Transformation groups for soliton equations[END_REF], they also appear in [START_REF] Adler | Non-linear PDEs for gap probabilities in random matrices and KP theory[END_REF], Lemma 4.1):

p k +1 ([D]) -1 2 D 1 D k (τ , τ ) = 0, D 1 D k -1 2 D 2 D k-1 -D 1 p k ([D]) (τ , τ ) = 0.
(4.98)

In the above, [D] := (D 1 , 1 2 D 2 , 1 3 D 3 , . . .) while the p k 's are the elementary Schur polynomials which can be de ned by

exp i ≥1 t i λ i = k ≥1 p k (t)λ k .
They were already introduced in Chapter 3, Equation (3.49). Notice that since we reduced to the A n case, all variables t k(n+1) = 0 because h = n + 1 (the Coxeter number of g).

The bilinear derivatives have a natural gradation given by deg

D i 1 • • • D i k = i 1 + • • • +i k .
Denote by HDS g the linear space of bilinear equations15 satis ed by the Drinfeld-Sokolov hierarchies of type g, which decomposes into homogeneous subspaces

HDS g = i ≥0
HDS [i] g .

The gradation allows us to list all possible bilinear equations up to certain degree.

The A 1 case

Let us chose the standard matrix realization g = sl(2; C). Consider the following two elements in λ -1 g λ -1

1 λ F = 1 λ 0 0 1 0 , 1 λ E = 1 λ 0 1 0 0 . (4.99)
The associated polynomial tau functions are

τ 1 = 1 + t 1 , τ 2 = 1 + t 3 - t 3 1 3 (4.100)
respectively. Similarly, one computes polynomial tau functions corresponding to elements of the form λ -k F , λ -k E, k ≥ 2. For example, for k = 2, we obtain

τ 3 = 1 + 2t 3 -t 5 t 1 + t 2 3 + t 3 1 3 + 1 3 t 3 t 3 1 - 1 45 t 6 1 , (4.101)
τ 4 = 1 -t 3 t 7 + 2t 5 + t 2 5 + t 3 3 t 1 -t 3 t 5 t 2 1 -t 3 t 2 1 + 1 3 t 7 t 3 1 - t 5 1 15 - 1 15 t 5 t 5 1 + 1 105 t 3 t 7 1 - t 10 1 4725 , (4.102)
corresponding to λ -2 F and λ -2 E, respectively.

Remark 4.5.11. Up to a certain a ne transformation of the variables, the above polynomials are the Adler-Moser polynomials studied in Chapter 1. This is no surprise since the Adler-Moser polynomials recover all the polynomial tau functions KdV. Although we do not have a proof of this statement, we believe that the procedure we use here allows to recover all the Adler-Moser polynomials. A rst clue in this direction is by identifying the degree in t 1 (which characterizes each Adler-Moser polynomial). Now consider all bilinear equations up to degree 4:

(β + α 0 D 2 1 + α 1 D 4 1 + α 2 D 1 D 3 )(τ , τ ) = 0 (4.103)
where β, α 0 , α 1 , α 2 are complex constants. Requiring that τ 1 , τ 2 satisfy the above ansatz (4.103), we nd that up to a multiplicative constant there is only one possible choice of coe cients:

(D 4 1 -4D 1 D 3 )(τ , τ ) = 0. (4.104)
Similarly up to degree 6, we nd out only two more possible linearly independent bilinear equations that are satis ed by τ 1 , τ 2 , τ 3 , τ 4 The A 2 case

(D 6 1 + 20D 3 1 D 3 -96D 1 D 5 )(τ , τ ) = 0, (4.105) (D 3 1 D 3 + 2D 2 3 -6D 1 D 5 )(τ , τ ) = 0, ( 4
We still chose the standard matrix realization g = sl(3; C). Consider for example the following two elements in λ -1 g λ -1 :

X 1 = 1 λ 0 0 0 a 1 0 0 a 2 a 3 0 , X 2 = 1 λ 0 a 1 a 2 0 0 a 3 0 0 0 , (4.107)
where a 1 , a 2 , a 3 are arbitrary constants. The corresponding polynomial tau functions will be denoted by τ 1 , τ 2 , respectively. We have

τ 1 = 1 + a 2 t 1 + 1 2 a 1 t 2 1 -1 2 a 3 t 2 1 + 1 8 a 1 a 3 t 4 1 -1 160 a 2 1 a 3 t 6 1 + 1 160 a 1 a 2 3 t 6 1 -1 1792 a 2 1 a 2 3 t 8 1 + a 1 t 2 + a 3 t 2 + 1 16 a 2 1 a 3 t 4 1 t 2 + 1 16 a 1 a 2 3 t 4 1 t 2 + 3 2 a 1 a 3 t 2 2 -1 8 a 2 1 a 3 t 2 1 t 2 2 + 1 8 a 1 a 2 3 t 2 1 t 2 2 + 1 32 a 2 1 a 2 3 t 4 1 t 2 2 + 1 4 a 2 1 a 3 t 3 2 + 1 4 a 1 a 2 3 t 3 2 + 1 16 a 2 1 a 2 3 t 4 2 -1 4 a 2 1 a 3 t 2 1 t 4 -1 4 a 1 a 2 3 t 2 1 t 4 -1 2 a 2 1 a 3 t 2 t 4 + 1 2 a 1 a 2 3 t 2 t 4 -1 4 a 2 1 a 2 3 t 2 1 t 2 t 4 -1 4 a 2 1 a 2 3 t 2 4 + 1 2 a 2 1 a 3 t 1 t 5 -1 2 a 1 a 2 3 t 1 t 5 + 1 4 a 2 1 a 2 3 t 1 t 7 ,
Requiring that τ 1 satis es this ansatz we nd that there is only one possible choice:

(D 4 1 + 3D 2 2 )(τ , τ ) = 0.
Similarly, requiring that τ 1 and τ 2 both satisfy the ansatz of bilinear equation of degree 6, we nd that there are only two linearly independent bilinear equations of degree 6:

(D 6 1 + 45D 2 1 D 2 2 + 90D 2 D 4 -216D 1 D 5 )(τ , τ ) = 0, (D 6 1 + 15D 2 1 D 2 2 + 60D 2 D 4 -96D 1 D 5 )(τ , τ ) = 0,
which are identi ed to two of the well-known bilinear equations for the hierarchy of type A 2 (the Boussinesq hierarchy) (see Example 4.5.10).

The B case

We choose the matrix realization of the B 2 simple Lie algebra as in [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] (cf. page 2032 therein). We consider two explicit examples given respectively by the following matrices16

X 1 = 1 λ 0 0 0 0 0 a 2 0 0 0 0 a 3 a 5 0 0 0 a 4 0 a 5 0 0 0 a 4 -a 3 a 2 0 , X 2 = 1 λ 0 0 a 3 a 4 0 0 0 0 0 a 4 0 0 0 0 -a 3 0 0 0 0 0 0 0 0 0 0 a 2 2 a 4 5 t

τ 2 = 1 + 1 288 a 3 t 6 1 -1 2016 a 4 t 7 1 - 1 11612160 a 2 3 t 12 1 -1 12 a 3 t 3 1 t 3 + 1 48 a 4 t 4 1 t 3 -1 69120 a 2 3 t 9 1 t 3 + 1 2 a 3 t 2 3 -1 2 a 4 t 1 t 2 3 -1 1920 a 2 3 t 6 1 t 2 3 -1 48 a 2 3 t 3 1 t 3 3 + 1 16 a 2 3 t 4 3 + 1 4032 a 2 3 t 7 1 t 5 -1 96 a 2 3 t 4 1 t 3 t 5 + 1 4 a 2 3 t 1 t 2 3 t 5 + a 4 t 7 + 1 160 a 2 3 t 5 1 t 7 -1
2 a 2 3 t 5 t 7 . Consider all bilinear equations up to degree 4

(α 0 + α 1 D 2 1 + α 2 D 4 1 + α 3 D 1 D 3 )(τ , τ ) = 0
where α 0 , . . . , α 3 are constants. Requiring that τ 1 satis es this ansatz of bilinear equations we nd that there is no solution. Similarly, up to degree 8, we nd that there are only two possible homogeneous equations (one is of degree 6 and the other is of degree 8). We arrive at Proposition 4.5.12. The following dimension estimates hold true

dim C HDS [deg≤4] B 2 = 0, dim C HDS [deg≤6] B 2 ≤ 1, dim C HDS [deg≤8] B 2 ≤ 2.
Moreover, the only possible elements in HDS

[deg≤8] B 2
are linear combinations of the following,

(D 6 1 -5D 3 1 D 3 -5D 2 3 + 9D 1 D 5 )(τ , τ ) = 0, (D 8 1 + 7D 5 1 D 3 -35D 2 1 D 2 3 -21D 3 1 D 5 -42D 3 D 5 + 90D 1 D 7 )(τ , τ ) = 0.
Remark 4.5.13. As far as we know, explicit bilinear equations for the DS hierarchy of type B 2 are not pointed out in the literature, except that there is a super-variable version given in [START_REF] Kac | Exceptional hierarchies of soliton equations[END_REF]. However, the relationship between the super bilinear equations of Kac-Wakimoto [START_REF] Kac | Exceptional hierarchies of soliton equations[END_REF] and the DS hierarchy of type B 2 is not known. Finding explicit generating series of bilinear equations for the DS hierarchy of type B 2 remains an open question. It is also interesting to remark that the very same equations are contained in [START_REF] Date | Transformation groups for soliton equations II: Vertex operators and tau functions[END_REF], as the rst two equations of the BKP hierarchy.

The D 4 case Take the matrix realization of g as in [START_REF] Bertola | Correlation functions of the KdV hierarchy and applications to intersection numbers over M ,n[END_REF] (cf. Example 4.4 therein). Consider the particular point of the Sato Grassmannian of type D 4 given by

γ = 1 + λ E θ .
We put t 11 = 0. It follows from Theorem 4.5.2 that the corresponding tau function is given by

τ = 1 - 1 2 s (7 |6) - 1 2 s (6 |7) - 1 4 s (7,6|7,6) 1 2
where s (7,6|7,6) = s (7|7) s (6|6) -s (7 |6) s (6|7) , s (6|6) = s (7|7) = 0, and

s (6|7) = s (7|6) = t 11 1 1900800 - 1 480 t 5 t 6 1 + 1 160 t 2 3 t 5 1 + 1 120 t 2 3 t 5 1 + 1 80 t 3 t 3 t 5 1 - 1 8 t 3 3 t 2 1 - 1 4 t 3 t 2 3 t 2 1 - 3 8 t 2 3 t 3 t 2 1 + 1 2 t 2 5 t 1 + 3 4 t 2 3 t 5 + t 2 3 t 5 + 3 2 t 3 t 3 t 5 .
Hence we have

τ = 1 - 1 2 s (7|6) = 1 - t 11 1 3801600 + 1 960 t 5 t 6 1 - 1 320 t 2 3 t 5 1 - 1 240 t 2 3 t 5 1 - 1 160 t 3 t 3 t 5 1 + 1 16 t 3 3 t 2 1 + 1 8 t 3 t 2 3 t 2 1 + 3 16 t 2 3 t 3 t 2 1 - 1 4 t 2 5 t 1 - 3 8 t 2 3 t 5 - 1 2 t 2 3 t 5 - 3 4 t 3 t 3 t 5 .
Proposition 4.5.14. The following dimension estimates hold true

dim C HDS [deg≤4] D 4 = 0, dim C HDS [deg≤6] D 4 ≤ 3.
Moreover, the only possible elements in HDS

[deg≤6] D 4
are linear combinations of

(2D 3 1 D 3 + 4D 3 D 3 -3D 2 3 )(τ , τ ) = 0, (4.108) (D 3 1 D 3 -D 3 1 D 3 + D 3 D 3 -D 2 3 )(τ , τ ) = 0, (4.109) (D 6 1 + 9D 1 D 5 -10D 3 1 D 3 + 5D 3 1 D 3 -5D 3 D 3 )(τ , τ ) = 0. (4.110)
Our last remark is that under the following linear change of time variables

∂ t 1 → 2 -1/6 ∂ T 1 , ∂ t 3 → 2 1/2 ∂ T 3 + 2 1/2 3 -1/2 ∂ T 3 , ∂ t 3 → 2 1/2 ∂ T 3 , ∂ t 5 → 2 7/6 ∂ T 5
the bilinear equations (4.108)-(4.110) in the new time variables T 1 ,T 3 ,T 3 ,T 5 coincide with those of Kac and Wakimoto [START_REF] Kac | Exceptional hierarchies of soliton equations[END_REF]. Essentially speaking such a change of times is simply a renormalization of ows.

List of generalized Schur polynomials

Take π as in [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF][START_REF] Bertola | Correlation functions of the KdV hierarchy and applications to intersection numbers over M ,n[END_REF] (cf. page 2032 of [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] and Example 4.4 of [START_REF] Bertola | Correlation functions of the KdV hierarchy and applications to intersection numbers over M ,n[END_REF]). We list in Table 4.4 the rst several Schur polynomials of type (g, π ) for simple Lie algebras of low ranks.

Type A 1 A 2 B 2 B 3 C 2 D 4 s 1 t 1 t 1 0 0 t 0 s 2 1 2 t 2 1 1 2 t 2 1 + t 2 1 2 t 1 1 2 t 1 1 2 t 1 2 t 1 s 1 2 1 2 t 2 1 1 2 t 2 1 -t 2 -1 2 t 1 -1 2 t 1 1 2 t -1 2 t 1 s 3
In general, we will denote the sequence of indeterminates u * * := (u α k | α ∈ {1, . . . , N }, k ≥ 0). We de ne a derivation ∂ x : A → A by

∂ x := N α =1 k ≥0 u α k+1 ∂ ∂u α k .
In particular, u α k = ∂ k x u α ; we denote u α := u α 0 . Elements of the algebra A we call di erential polynomials. It is equipped with a natural gradation given by deg u α k = k. The indeterminates u * * are not functions of x but abstract variables. The notation ∂ x we explain as follows: For arbitrary functions w1 , . . . , w N ∈ C ∞ (R, C) de ne w α k (x) := ∂ k w α /∂x k , then the evaluation of a di erential polynomial f ∈ A on the functions w * * (x) we de ne as f → f (w * * (x)). We then de ne the space of local functionals 12

F := A/(Im(∂ x ) ⊕ C1),
(5.1) whose elements are classes of di erential polynomials modulo the addition of an exact derivative or a constant term. The class of a polynomial f ∈ A we denote alternatively by

f = ∫ f = ∫ f (x)dx .
This comes with a integration by part formula f ∂ x = -∂ x f , or more generally

∫ f ∂ k x = (-1) k ∫ ∂ k x f . (5.2)
It is easy to check that for any α ∈ {1, . . . , N } and any k ≥ 1,

∂ ∂u α k ∂ x -∂ x ∂ ∂u α k = ∂ ∂u α k -1 , ∂ ∂u α ∂ x -∂ x ∂ ∂u α = 0.
(5.3)

In particular, the derivatives ∂/∂u α are well de ned on F . We also introduce the so-called variational derivatives which are linear endomorphisms denoted δ /δu α : A → A, for α ∈ {1, . . . , N }, de ned by, for f ∈ A,

δ f δu α := k ≥0 (-1) k ∂ k x ∂ f ∂u α k .
(5.4)

The following proposition is a fundamental and classical result. A purely algebraic proof was in given in [GKMZ70] (Lemma 2), which we follow here.

Proposition 5.1.1. Let f ∈ A with no constant term. Then f is an exact derivative if and only if all the variational derivatives of f vanish; i.e.

∃ , f = ∂ x ⇔ ∀ α, δ f δu α = 0. Proof.
In what follows, we use Einstein's notation and omit the sum over indices α, β. Let us rst prove that if f ∈ Im(∂ x ) (and in particular has no constant term), then each variational derivative δ f /δu α vanish. Let f = ∂ x ∈ A, then using Equation (5.3),

δ f δu α = k ≥0 (-1) k ∂ k x ∂ ∂u α k ∂ x = k ≥1 (-1) k ∂ k x ∂ ∂u α k -1 + k ≥0 (-1) k ∂ k+1 x ∂ ∂u α k = 0.
For the converse, let us introduce the following operator:

D 0 := k ≥0 u α k ∂ ∂u α k .
Then it is not hard to see that for any f ∈ A with no constant term, there exists a unique ∈ A with no constant term such that f = D 0 ; we denote = D -1 0 f . Moreover,

D 0 ∂ x = ∂ x D 0 .
(5.5) So let f ∈ A with no constant term such that δ f /δu α = 0 for all α. Then using an integration by parts, we get that

0 = u α δ f δu α = u α k ≥0 (-1) k ∂ k x ∂ f ∂u α k = u α ∂ f ∂u α -u α ∂ x k ≥1 (-1) k -1 ∂ k -1 x ∂ f ∂u α k = u α ∂ f ∂u α -∂ x u α k ≥1 (-1) k-1 ∂ k -1 x ∂ f ∂u α k + u α 1 k ≥1 (-1) k -1 ∂ k-1 x ∂ f ∂u α k .
We let the reader convince themselves that using the same procedure successively, we obtain that

0 = n k=0 u α k ∂ f ∂u α k -∂ x       n k=0 u α k j ≥k+1 (-1) k -j-1 ∂ f ∂u α k       + u α n+1 k ≥n+1 (-1) k-j-1 ∂ f ∂u α k .
Yet since f ∈ A is a polynomial, then the rightmost term in the above equation vanishes for higher values of n. It follows, by Equation (5.5), that for any f ∈ A with no constant term and such that δ f /δu α = 0 for all α,

f = ∂ x D -1 0       k ≥0 u α k j ≥k+1 (-1) k -j-1 ∂ f ∂u α k       .
This concludes the proof of Proposition 5.1.1.

Local Poisson structures and Hamiltonian equations

De nition 5.1.2 (Local Poisson structure). A local Poisson structure is a matrix-valued di erential operator over di erential polynomials K = (K α β ),

K α β = j ≥0 K α β j ∂ j x , K α β j ∈ A,
where the sum is nite, along with the action K : F ⊗ F → F de ned by

{ f , } K := ∫ N α, β =1 δ f δu α K α β δ δu β ,
and which satis es the following properties, for any f , , h ∈ F , (i) Skew-symmetry:

{ f , } K + { , f } K = 0; (ii) Jacobi identity: { f , { , h} K } K + cyclic permutations = 0.
An important example is given by the so-called standard Poisson structure associated to a constant nondegenerate symmetric bilinear form η α β , namely

K α β = η α β ∂ x , η α β ∈ C.
The bracket reads

{ f , } K := ∫ N α, β =1 δ f δu α η α β ∂ x δ δu β . (5.6)
It is easy to show that this is indeed a local Poisson structure. Notice that since F = A/Im(∂ x ) is not a ring (because ∂ x is not a ring homomorphism), we cannot de ne Leibniz's rule, the latter being included in the de nition of a Poisson bracket on a nite-dimensional manifold. Therefore, a local Poisson structure K makes F into a Lie algebra (F , {-, -} K ), but not into a Poisson algebra.

However, a local Poisson structure K comes with a natural action K : A ⊗ F → A de ned by

{ f , } K := N α, β =1 k ≥0 ∂ f ∂u α k ∂ k x K α β δ f δu β , (5.7)
which, by the integration by part formula (5.2), satis es

{ f , } K = ∫ { f , } K .
Now the action (5.7) clearly satis es Leibniz's rule w.r.t. to the rst variable: for any di erential polynomials f , ,

h ∈ A, { f , h} K = f { , h} K + { f , h} K .
A similar construction can be done w.r.t the second variable. This is as close as we can get to a Poisson algebra structure.

We can now de ne an in nite dimensional system of Hamiltonian PDEs.

De nition 5.1.3 (Hamiltonian structure). A system of Hamiltonian PDEs on A is a system of PDEs of the form

∂u α ∂t i = {u α , h i } K = N β =1 K α β δh i δu β ,
where {h i | i ≥ 0} is a family of di erential polynomials called Hamiltonian densities which satisfy, for all i, j ≥ 0,

{h i , h j } K = 0.

Poisson algebra of Fourier expansion

Let B be the subalgebra of

C [[ p α n | α ∈ {1, . . . , N }, n ∈ Z * ]],
where Z * = Z \ {0}, which consists of formal power series of the form

f = r ≥0 α 1 , ...,α r ∈ {1, ..., N } n 1 , ...,n r 0 n 1 +•••+n r =0 f n 1 , ...,n r α 1 , ...,α r r j=1 p α j n j , f n 1 , ...,n r α 1 , ...,α r ∈ C.
Given a nondegenerate symmetric bilinear form η α β , we endow the algebra B with a Poisson bracket, de ned on the indeterminates by (in) k p α k e inx , (5.9) and extended on A by linearity and by imposing T (f ) = T (f )T ( ). The above can be thought of as a formal Fourier expansion of the "functions" u α . For a di erential polynomial f ∈ A, we write

{p α m , p β n } η = imη α β δ m+n,0 , (5.8)
T (f ) = m ∈Z T m (f ) e imx , T m (f ) ∈ C[[p * * ]],
in particular, we consider the map T 0 : f → T 0 (f ). Via the homomorphism property and multiple Cauchy product, the Fourier expansion of an arbitrary monomial u

α 1 k 1 • • • u α r k r ∈ A reads T (u α 1 k 1 • • • u α r k r ) = n ∈Z e inx n 1 ∈Z n 1 n n 2 , ...,n r 0 n 1 +•••+n r =n 1 (i(n -n 1 )) k 1 p α 1 n-n 1 r j=2 (in j ) k j p α j n j .
Then taking the coe cient in n = 0 and changing n 1 → -n 1 in the above sum, we have

T 0 (u α 1 k 1 • • • u α r k r ) = n 1 , ...,n r 0 n 1 +•••+n r =0 r j=1 (in j ) k j p α j n j .
(5.10) Therefore, we are naturally brought to consider the subalgebra B pol ⊂ B which consists of power series of the form

f = r ≥0 n 1 , ...,n r 0 n 1 +•••+n r =0 f α 1 , ...,α r (n 1 . . . , n r ) r j=1 p α j n j , f α 1 , ...,α r ∈ C ≤d [z 1 , . . . , z r ], (5.11)
for some d ≥ 0, meaning that the polynomials f α 1 , ...,α r have a degree that is bounded from above independently of α 1 , . . . , α r . From Equation (5.10), we have

T 0 (A) ⊂ B pol .
On the other hand, we easily see that the map T

• ∂ x : A → C[[p * * , x]] is given by T (∂ x f ) = m ∈Z im f m e imx .
(5.12) Indeed, because T is a homomorphism of algebras, the map T • ∂ x is a derivation; therefore it su ces (5.12) to check the relation on the indeterminates u α k , for which (5.12) comes from the de nition of T (5.9). It follows that T 0 • ∂ x ≡ 0, which implies that the following map, which we still denote T 0 , is well de ned:

T 0 : F -→ B pol f -→ T 0 (f ).
We give two important lemmas concerning the map T 0 which we take from [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF] (see also [START_REF] Eliashberg | Introduction to symplectic eld theory[END_REF] § 2.2.3 and 2.9.2); proofs are given in the appendix of [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF].

Lemma 5.1.4. The map T 0 : F → B pol is surjective. Its kernel is given by

Ker (T 0 ) = Span u 1 , . . . , u N ⊂ F .
Consequently, for any power series f ∈ B pol , there exists a unique local functional h ∈ F such that T 0 (h) = f and which is such that for all α ∈ {1, . . . , N }, ∂h ∂u α u * k >0 =0 = 0.

Lemma 5.1.5. The pullback of the bracket {-, -} η on B pol (5.8) by the homomorphism T 0 is the standard bracket {-, -} η ∂ x (5.6) on F .

Extended spaces and Miura transforms

We now consider generalizations of di erential polynomials and local functionals as follows. We de ne the algebra of extended di erential polynomials by

A N := C u α , ε u α k >0 .
We de ne a grading on it by deg ε = -1 while deg u α k = k still. We get a decomposition

A N = k ∈Z A [k ] ,
where A [k] is the subspace of polynomials of homogeneous degree k. Within the subspace A [0] , the degree in ε counts the number of derivatives which appear in a monomial; in particular, each u α ∈ A [0] . For instance,

u 3 ε 3 + uu 1 ε ∈ A [0] . The derivation ∂ x = k ≥0 u α k +1 ∂ ∂u α k
naturally acts on A as well so that we generalize local functionals to the space of extended local functionals by

F := A/(Im(∂ x ) ⊕ C[[ε]]).
This induces the same gradation

F = k ∈Z F [k] .
We can now introduce the group of Miura transforms which plays a key role in the theory we present here. They are named after American mathematician Robert M. Miura, of whom we already cited several major contributions to the domain.

De nition 5.1.6 (Miura transform). A Miura transform is any map φ : A → A which is such that ũα := φ(u α ) ∈ A [0] and which satis es

det ∂ ũα ∂u β ε =0 α, β ∈[1, N ] 0.
The group of such maps we call the Miura group.

Miura transforms will play the role of change of coordinate on A. We shall emphasize here the importance of the parameter ε. Indeed, for example, the following simple transformation is a Miura transform:

ũ = u + u 1 ε + u 2 ε 2 + u 3 ε 3 + • • • Its inverse is u = ũ -ũ1 ε -ũ2 ε 2 -ũ3 ε 3 -• • •
As we can see, by solving the equation for each power of ε, we nd the inverse of the Miura transform. Such transformations would have obviously been impossible to encode in the ring A. Thus we get a much broader de nition of "change of variables. " We can motivate the introduction of such transformation through the applications to the DR hierarchy as we will explain in the next sections.

We now generalize what was done on Fourier expansion to the extended case. For any d ≥ 0, consider the subalgebra B

f = i ≥0 ε i f i , f i ∈ B pol i .
Then with the exact same procedure we can de ne a map

T 0 : F [0] → B pol
which is also surjective and which kernel is still spanned by the functionals u α .

Extended Hamiltonian and local Poisson structures

We generalize the notion of local Poisson structure and Hamiltonian densities to the extended context as follows.

De nition 5.1.7 (Poisson and Hamiltonian extended structures). A local Poisson structure on the space of extended local functionals F is a matrix-valued di erential operator K = (K α β ) of the form

K = j ≥0 K α β ∂ j x , K α β j ∈ A [1-j] .
where the sum is nite, along with the action, as before, given by

{ f , } K = ∫ N α, β =1 δ f δu α K α β δ δu β ,
where the variational derivatives are given by the same Equation (5.4). A system of Hamiltonian PDEs on extended di erential polynomials A is a system of PDEs of the form

∂u α ∂t i = {u α , h j } K = N β =1 K α β δh j δu β , h i ∈ F [0] ,
where {h i | i ≥ 0} is a family of extended local functionals h i ∈ F [0] which satisfy, for any i, j ≥ 0,

{h i , h j } = 0.
The skew-symmetry and Jocobi identity conditions on K are non trivial. In particular, the ordinary di erential polynomial K α β | ε =0 ∈ A N must have the form

K α β | ε =0 = α β (u)∂ x + Γ α β γ (u)u γ 1 , (
(Γ • φ) γ α β = ∂u µ ∂ ũα ∂u ν ∂ ũβ Γ σ µν ∂ ũγ ∂u σ + ∂ ũγ ∂u τ ∂ 2 u τ ∂ ũα ∂ ũβ ;
(ii) The skew-symmetry condition { f , } + { , f } = 0 implies that α β be symmetric and that ∇ be compatible with in the sense that ∇ γ α β = 0 for all γ ; (iii) Jacobi identity implies that the torsion and curvature tensors of the connection ∇ vanish.

Moreover, ∇ is the Levi-Civita connection associated to the metric .

We now introduce the formal Dirac delta notation (see e.g. [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF]): for any f , ∈ A and k ≥ 0, we denote

∫ f (x)δ (k) (x -) ( )d := f ∂ k x , (5.14) ∫ f (x)δ (k) (x -) ( )dx := (-1) k ∂ k x f .
(5.15)

The di erence in the two equation above lies in the variable w.r.t. which we "integrate. " In other words, we introduce a symbol δ (k ) (x -), where x and are actually purely notational, which we see as two di erent bilinear maps A → A.

We shall emphasize here that the integral symbol, in the above, is entirely di erent from the one used so far. Indeed, in Equations (5.14) and (5.15), the integral symbol is purely notational and does not represent the class modulo derivative. However, if we denote by π : A → F the projection on local functionals, then we get a double integration formula which, thanks to the integration by parts, reads

∬ f (x)δ (k ) (x -) ( )d dx := π ∫ f (x)δ (k) (x -) ( )d = π ∫ f (x)δ (k ) (x -) ( )dx .
Now the above equation is somehow very strange since we have two integral symbols: one for the action of the delta symbol, the other for the projection onto F . But since the two operations commute, it makes sense not to distinguish the two integral symbols.

the so-called tau symmetry (which amounts to the closedness of a 1-form). The tau function then naturally arise via Poincaré lemma. As for the most part of what preceeds, the following content is taken from [START_REF] Rossi | Integrability, Quantization and Moduli Spaces of Curves[END_REF].

De nition 5.1.10 (Tau structure). Let K : F ⊗ F → F be a local Poisson structure. Let {h α,p | α ∈ {1, . . . , N }, p ≥ -1} ⊂ A [0] be a family of di erential polynomials, such that4 {h α,p , h β,q } = 0, p, q ≥ -1, α, β ∈ {1, . . . , N }.

Consider the Hamiltonian system of PDEs it generates, namely

∂u α ∂t β p = {u α , h β,p } = K α µ δh β,p δu µ . (5.19)
We say that the Hamiltonian densities h α,p form a tau structure of the system (5.19) if the following holds.

(i) The functional h 1,0 is compatible with spatial translation in the sense that

∂u α ∂t 1 0 = u α 1 ;
(5.20)

(ii) The functionals h β,-1 generate trivial ows, i.e. for all β ∈ {1, . . . , N },

∂u α ∂t β -1 = K α µ δh β,-1 δu µ = 0;
(iii) The functionals h β,-1 are linearly independent;

(iv) The densities h α,p satisfy the tau symmetry property, for all α, β ∈ {1, . . . , N }, p, q ≥ 0, ∂h α,p-1 ∂t β q = ∂h β,q-1 ∂t α p .

(5.21)

Tau symmetry (5.21) is simply asking that the 1-form h α,p-1 dt α p is closed. It can also be written as {h α,p-1 , h β,q } = {h β,q-1 , h α,p }.

Since the functionals h β,-1 are linearly independent and of degree 0, it follows that the map φ : u α → η α β h β,-1 is a Miura transform, where η α β is the unique constant symmetric invertible matrix of Getzler's theorem 5.1.9. We call the coordinates

ũα := η α β h β,-1 (u * * ; ε) (5.22)
the normal coordinates associated to the tau structure. In the normal coordinates, the local Poisson structure takes the following form [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF] (see §3.3),

(K • φ) α β = η α β ∂ x + O(ε).
We now explain how the existence of a tau structure allows to de ne a tau function in a way entirely similar to what as done in §4.3.4. The commutativity condition {h α,p-1 , h β,q } = 0 amounts to

∫ ∂h α,p-1 ∂t β q = 0.
Since ∂h α,p-1 /∂t β q has no constant term, it follows that there exists a unique di erential polynomial Ω α,p; β,q ∈ A [0] with no constant term either and such that5

∂ x Ω α,p; β,q = ∂h α,p-1 ∂t β q
.

(5.23)

In particular, thanks to the spatial translation compatibility condition (5.20)

h α,p-1 = Ω α,p;1,0 .
The following proposition holds. .

In particular, by identifying x = t 1 0 , if ũα = η α β h β,-1 are the normal coordinates of the system (5.19), then

ũα = η α β ∂ 2 log τ ∂x ∂t β 0 .
Notice that, depending on the literature, either τ is called the tau function or F = exp(τ ) is. Oftentimes though, F is called the potential or the free energy of system (5.19).

Proof. We will proceed by applying Poincaré lemma twice. We start from the de ning equation of the Omega function (5.23) which, using the commutativity of the ows

[∂ x , ∂ t γ r ] = 0, implies ∂ x ∂ ∂t γ r Ω α,p; β,q = ∂ ∂t γ r ∂ ∂t β q h α,p-1 .
(5.24)

Now the commutativity of the ows allows to swap t γ r and t β q , while the tau symmetry condition (5.21) allows to swap h α,p-1 and t β q . All in all, (5.24) is completely symmetric in permutation of the indices ((α, q), (β, q), (γ , r )). Moreover, since we chose Ω α,p; β,q such that it contains no constant term, then ∂ t γ r Ω α,p; β,q is itself completely symmetric. Let us x the indices (α, p). Then we have

∂ ∂t γ r Ω α,p; β,q = ∂ ∂t β q Ω α,p; γ ,r .
This means that the 1-form Ω α,p; β,q dt γ r is closed. Therefore, by Poincaré lemma, there exists a function A α,p such that Ω α,p; β,q = ∂A α,p ∂t β q

. Now the tau symmetry property, again, means that Ω α,p; β,q = Ω β,q; α,p so that the 1-form A α,p dt α p is also closed. Therefore, there exists another function τ such that

Ω α,p; β,q = ∂ 2 log τ ∂t α p ∂t β q
.

Finally, we state the following proposition; a proof will be found in [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF] (see Theorem 3.1.9 and §3.3). Notice however that checking point (ii) of Proposition 5.1.12 is immediate.

The double rami cation hierarchy of type D 4

The goal of this section is to compute the local Poisson structure K α β , the Hamiltonian functional h 1,1 and the Hamiltonian densities h 0,1 to h 0,4 (in normal coordinates) of the DR hierarchy of type D 4 . Due to results of [START_REF] Buryak | Towards a description of the double rami cation hierarchy for Witten's r -spin class[END_REF], these are the only data that we need to compute in order to con rm the strong DR/DZ equivalence (see Theorem 5.2.24). In rst three sections, we give the de nitions of cohomological eld theories (CohFT) [START_REF] Manin | Gromov-Witten classes, quantum cohomology, and enumerative geometry[END_REF]; the de nition of FJRW theory [FJR11, FJR13] (a special case of CohFT which we use for the hierarchy of type D 4 ); we describe the construction of the DR hierarchy [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF] and nally, in section 5.2.4 we describe the Hamiltonian structure of the DR hierarchy of type D 4 .

Cohomological eld theories, FJRW theory

In this section we introduce the notion of cohomological eld theories. We mostly follow the survey [START_REF] Rossi | Integrability, Quantization and Moduli Spaces of Curves[END_REF] and refer to [START_REF] Zvonkine | An introduction to moduli spaces of curves and its intersection theory[END_REF] for more details and proofs. We then introduce Fan-Jarvis-Ruan-(inspired by )Witten (FJRW) theory, taking material from introductions given in [START_REF] Francis | A Brief Survey of FJRW Theory[END_REF][START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF][START_REF] Guéré | Hodge Integrals in FJRW Theory[END_REF].

Moduli spaces of stable curves

We recall the de nition and some properties of the (Deligne-Mumford compacti cation of the) moduli spaces of stable curves M ,n ; we will not give much details on the topic and refer to [START_REF] Zvonkine | An introduction to moduli spaces of curves and its intersection theory[END_REF]. First consider the set of all connected compact Riemann surfaces (or equiv. connected projective complex curves) C such that C has genus ; C is smooth but for a nite number of transverse nodes, say { 1 , . . . , r } ⊂ C; C has n distinct non nodal marked point {x 1 , . . . , x n } ⊂ C \ { 1 , . . . , r }; where by transverse we mean that each node has local model x = 0. We call a special point any marked point or node. We say that two n-marked genus curves (C, (x 1 , . . . , x n )) and (C , (x 1 , . . . , x n )) are isomorphic if there exists a bi-holomorphism ϕ : C → C such that image of a node in C is a node of C and such that ϕ(x i ) = x i . We say that a curve C is stable if its automorphism group Aut(C) is nite. The stability condition is equivalent to the following constraint: 2 -2 + n > 0.

(5.25)

The moduli space M ,n , for , n ≥ 0 satisfying Equation (5.25), is then the set of all equivalences classes [C] of stable curves isomorphic to C,

M ,n = {(C; (x 1 , . . . , x n ))}/ ∼ .
It is a compact complex orbifold (or a smooth Deligne-Mumford stack) of complex dimension dim C M ,n = 3 -3 + n.

(5.26) On M ,n there is a morphism of orbifolds π : C ,n → M ,n called the universal curve, whose bre over a point x ∈ M ,n is homeomorphic to the curve C of which the point x is the isomorphism class, i.e. π -1 (x) C. So there are n marked points on x1 , . . . , xn on π -1 (x); this gives us n sections

s i : M ,n → C ,n
which associate to a point x ∈ M ,n the marked point xi ∈ π -1 (x).

Forgetful, loop and tree morphisms

There are three natural morphisms de ned on the moduli spaces of stable curves. The forgetful morphism, for any m ≥ 1,

π m : M ,n+m → M ,n ,
forgets the last m marked points on a curve and contracts all components of the curve that might have become unstable in the process. For example, for a curve C ∈ M 0,4 consisting of two P 1 's intersecting at a node with two marked points on each P 1 component, the forgetful morphism π 1 : M 0,4 → M 0,3 contracts the whole P 1 component containing the 4th marked point on the node, the latter becomes the third marked point on the resulting P 1 . This is illustrated below; labelled dots represent marked points, while unlabelled dots represent nodes. glues one curve onto itself by identifying the two last marked points to a node of the resulting curve. The new node does not separate the resulting curve into two components. Hence, the loop morphism is sometimes called the non separating gluing map while the tree morphism is sometimes called the separating gluing map. This is illustrated below.

τ :

-→

Tautological bundles

On the universal curve C ,n we de ne a line bundle ϱ : Ω ,n → C ,n as follows. On the nonnodal points x ∈ C ,n , the bre ϱ -1 ( x) ⊂ Ω ,n is de ned as the relative cotangent space w.r.t. the projection π : C ,n → M ,n . Then it extends canonically to the singular points and de nes an actual line bundle over C ,n . We recall the de nition of the relative cotangent bundle though. Let x be a nonsingular point in C ,n . Consider the pullback bundle π * T * M ,n → C ,n de ned by

π * T * M ,n = ( x, α) ∈ C ,n × T * π ( x ) M ,n (5.27)
Then consider the map π : π * T * M ,n → T * C ,n de ned on bres by

π x : T * π ( x ) M ,n -→ T * x C ,n α π ( x ) -→ α π ( x ) • π * ,
x where π * , x : T x C ,n → T π ( x ) M ,n is the pushforward associated to π . Then the relative cotangent bundle ϱ : Ω ,n → C ,n has bres de ned by

ϱ -1 ( x) := T * x C ,n Im( π x ).
Then one can prove that if x ∈ M ,n and x ∈ π -1 (x), then ϱ -1 ( x) is isomorphic to the cotangent to the curve π -1 (x) ⊂ C ,n at the point x, that is, ϱ -1 ( x) T * x π -1 (x). Now recall the sections s i : M ,n → C ,n which map a class of curve x ∈ M ,n to the i-th marked point xi = s i (x) on the bre π -1 (x). We can pullback the bundle Ω ,n to a bundle

L i := s * i Ω ,n → M ,n
whose bre at x ∈ M ,n is de ned by

L i | x = ϱ -1 ( xi ) T * xi π -1 (x).
The bre bundle L i we call the i-th tautological bundle. Here is the situation in terms of diagrams:

Ω ,n L i = s * i Ω ,n C ,n M ,n ϱ π s i
We denote the rst Chern class of L i by

ψ i = c 1 (L i ) ∈ H 2 (M ,n , Q).
We call the classes ψ 1 , . . . ,ψ n the psi classes. (The rst Chern class establishes an isomorphism between isomorphism classes of complex line bundles over M ,n and the homology group

H 2 (M ,n , Q), in particular c 1 (L ⊗ L ) = c 1 (L) + c 1 (L ).)
The Hodge bundle

In a similar way we can de ne the Hodge bundle

H → M ,n
as the rank-vector bundle over M ,n whose bre over x ∈ M ,n is the space of abelian di erentials on the curve π -1 (x). We recall that an abelian di erential on a nodal curve C is a meromorphic di erential on C of which the poles satisfy (i) all poles are at a node, (ii) all poles are at most simple (i.e. have order 0 or 1) and (iii) the residue at a node on two branches meeting at this node are opposite to each other. We denote by

λ i = c i (H) ∈ H 2i (M ,n , Q),
for i ∈ {1, . . . , }, the i-th Chern class of H; we also denote

Λ(s) = i=0 s i λ i ∈ H (M ,n , Q)[s].
More formally, we de ne H as the pushforward bundle

H := π * Ω ,n → M ,n .
Recall that on nonsigular point x ∈ M ,n , the bre of π * Ω ,n is de ned as

π * Ω ,n = (x, α) ∈ M ,n × Ω ,n π • ϱ(α) = x .
Then we clearly have, on nonsigular points x ∈ M ,n , that H| x = (π * Ω ,n )| x = T * π -1 (x). In the end, the situation is as follows:

Ω ,n H = π * Ω ,n L i = s * i Ω ,n C ,n M ,n ϱ π s i
We end this section with the de nition of the tautological ring. The tautological ring R * (M ,n ) can be de ned as the smallest Q-subalgebra of H * (M ,n , Q) that is closed under the pushforward along any of the forgetful, tree and loop morphisms π , σ , τ de ned previously. In particular, it contains the psi classes ψ 1 , . . . ,ψ n and the lambda classes λ 1 , . . . , λ . It is generated by yet other classes; we refer to [START_REF] Zvonkine | An introduction to moduli spaces of curves and its intersection theory[END_REF] for more details.

Cohomological eld theories

De nition 5.2.1 (Cohomological eld theory). Let , n ≥ 0 be such that 2 -2+n > 0. Let V be an N -dimensional complex vector space with basis (e 1 , . . . , e N ) and equipped with a symmetric nondegenerate bilinear form η = (η α β ). A cohomological eld theory (CohFT) is a system of linear maps

c ,n : V ⊗n → H * (M ,n , C)
satisfying the following axioms. (i) c ,n is equivariant w.r.t. to simultaneous permutation of copies of V in V ⊗n and of the marked point on the curves; (ii) For the speci c case = 0, n = 3, c 0,3 (e 1 , e α , e β ) = η α β ;

(iii) For the forgetful map π 1 = π : M ,n+1 → M ,n ; π * c ,n (e α 1 , . . . , e α n ) = c ,n+1 (e α 1 , . . . , e α n , e 1 );

(iv) For the tree morphism σ : M 1 ,n 1 +1 ×M 2 ,n 2 +1 → M ,n , with = 1 + 2 and n = n 1 +n 2 , σ * c ,n (e α 1 , . . . , e α n ) = c 1 ,n 1 +1 (e 1 , . . . , e n 1 , e µ )η µν c 2 ,n 2 +1 (e ν , e 1 , . . . , e n 2 );

(v) For the loop morphism τ : M ,n+2 → M +1,n , τ * c +1,n (e α 1 , . . . , e α n ) = η µν c ,n+2 (e µ , e ν , e α 1 , . . . , e α n ).
Given a CohFT {c ,n | , n ≥ 0}, we de ne the intersection numbers (or correlators) associated to the CohFT by

τ d 1 (e α 1 ) • • • τ d n (e α n ) := ∫ M , n c ,n (e α 1 , . . . , e α n ) • ψ d 1 1 • • •ψ d n n ∈ C.
These numbers are 0 if the degree of c ,n (e α 1 , . . . , e

α n ) • ψ d 1 1 • • •ψ d n n is not 3 -3 + n.
We then de ne the so-called potential F of a CohFT as the generating series of the intersection numbers of the CohFT, namely,

F (t * * ; ε) = ≥0 ε 2 F (t * * ),
where

F (t * * ) := n ≥0 2 -2+n >0 1 n! d 1 , ...,d n ≥0 τ d 1 (e α 1 ) • • • τ d n (e α n ) • t α 1 d 1 • • • t α n d n .
(5.28)

Example 5.2.2 (Trivial CohFT, Hodge CohFT). The trivial CohFT is the one for which the vector space V = C is one-dimensional, the metric is η = 1 and the CohFT is simply c ,n = 1. In a similar way, the Hodge CohFT is the one for which, as above, V = C and η = 1, yet c ,n = Λ(s) = i=0 λ i s i .

Example 5.2.3 (Gromov-Witten theory). Let us rst recall the de nition of the moduli spaces of stable maps M ,n (X , β). Let X be a smooth projective variety and let β ∈ H 2 (X , Z). Consider a stable n-marked genus curve (C, (x 1 , . . . , x n )) together with a morphism f : C → X such that f * [C] = β. We say that two such triplets (C, (x i ), f ) and (C , (x i ), f ) are isomorphic if there exists a biholomorphism ϕ : C → C such that ϕ(x i ) = ϕ(x i ) (i.e. the two stable curves are isomorphic) and such that f •ϕ = f . We say that (C, (x i ), f ) is a stable map if it has a nite number of automorphism for the relation we just de ned. The moduli space of stable maps is then the set of isomorphism classes of stable maps,

M ,n (X , β) := {(C; (x 1 , . . . , x n ); f : C → X ) | f * [C] = β }/ ∼ .
It is a singular stack and carries a virtual complex dimension

vdim C M ,n (X , β) = 3 -3 + n + (1 -) dim C X + ∫ β c 1 (TX ).
We have two natural maps M ,n (X , β): the rst one,

p : M ,n (X , β) → M ,n
takes a stable map (C, (x i ), f ) and maps it to the stable curve (C, (x i )) ∈ M ,n ; the second, ev : M ,n (X , β) → X n takes a stable map (C, (x i ), f ) and maps it to (f (x 1 ), . . . , f (x n )) ∈ X n . The associated CohFT is de ned as follows. Take the vector space V = H * (X , C) equipped with the pairing de ned by We refer to [START_REF] Behrend | The intrinsic normal cone[END_REF] for more details on the virtual fundamental class and how to de ne the Gromov-Witten classes.

Example 5.2.4 (Witten's r -spin classes). For this example, we give a very brief description; the next one, Fan-Jarvis-Ruan-Witten theory generalizes Witten's r -spin classes. Let r ≥ 2 and consider the vector space V = C r -1 . We equip V with the metric η α β = δ α +β,r . In [START_REF] Polishchuk | Matrix factorizations and cohomological eld theories[END_REF],

Polischuk and Vaintrob constructed classes

c ,n (e α 1 +1 , . . . , e α n +1 ) ∈ H * (M ,n , Q) that have degree d α 1 , ...,α r = 1 r (r -2)( -1) + 1 r n i=1 α i
if the a i ∈ {0, . . . , r -2} are such that d α 1 , ...,α r is a nonnegative integer, and vanish otherwise. These classes are de ned via pushingforward to M ,n Witten's virtual class on the moduli space of curves with r -spin structures. An r -spin structure on a smooth n-marked curve (C, (x 1 , . . . , x n )) is an r -th root L of the twisted canonical bundle K( α i x i ) of the curve. This notions will be clari ed in the next section.

The last example we give is the FJRW theory associated to a pair (W , G) where W is a quasihomogeneous polynomial and G is a certain group acting on polynomials and leaving W invariant. The computations performed in Section 5.3 relate to the FJRW theory associated to the Dynkin diagram D 4 so we will discuss FJRW theory in a whole section.

Fan-Jarwis-Ruan-Wi en theory

In this section we recall the main ideas underlying the Fan-Jarvis-Ruan-(inspired by )Witten (FJRW) theory introduced in [FJR11, FJR13]. We refer, for instance, to the survey [FJ16] by T. J. Jarvis and A. Francis, but also to [START_REF] Guéré | Hodge Integrals in FJRW Theory[END_REF] for a brief review. First we have the following de nition.

De nition 5.2.5 (Quasihomogeneity, nondegeneracy). A polynomial W ∈ C[z 1 , . . . , z m ] is called quasihomogeneous if there exist q 1 , . . . , q m ∈ Q, called weights, such that for any λ ∈ C * , W (λ q 1 z 1 , . . . , λ q m z m ) = λW (z 1 , . . . , z m ).

A quasihomogeneous polynomial W is called nondegenerate if the weights q i are unique and if W has an isolated singularity at the origin.

By an isolated singularity at the origin we mean that the di erential of W vanishes at (0, . . . , 0) and that (0, . . . , 0) is a connected component of the zero locus the di erential. For any quasihomogeneous polynomial W we de ne a group called the maximal diagonal symmetry group of W de ned by

G W max := { (γ 1 , . . . , γ m ) ∈ (C * ) m | γ • W := W (γ 1 z 1 , . . . , γ n z m ) = W (z 1 , . . . , z m )} .
We will often drop the superscript W when there is no ambiguity. The maximal diagonal symmetry group G max always contains and special element called the exponential grading element, := (e 2iπ q 1 , . . . , e 2iπ q m ) ∈ G max . (1 -2q i ).

If ĉW < 1 thenW is said to have a simple singularity at the origin. The latter have been classi ed into the ADE singularities, as described in Table 5.2.1

) := { (z 1 , . . . , z n ) ∈ C n | γ • (z 1 , . . . , z n ) = (z 1 , . . . , z n )} .
We denote

W γ := W | Fix(γ )
We can now de ne the vector space underlying the CohFT of the FJRW theory of a pair (W , G).

In this introduction, we de ne the vector space of the FJRW CohFT associated to a pair (W , G) as follows.

De nition 5.2.7 (State space). The vector space of the FJRW CohFT associated to a pair (W , G), which we call the state space, we de ne by

H W ,G = γ ∈G H W ,G γ := γ ∈G Q W γ ⊗ dz γ G ,
where the superscript G denotes the invariant part, dz γ = z i ∈Fix(γ ) dz i and where Q W γ is the Jacobian ring of W γ , i.e.

Q W γ = C[z i ∈ Fix(γ )] ∂ z i W | z i ∈ Fix(γ ) .
The state space H W ,G is equipped with a nondegenerate symmetric bilinear form η : H W ,G ⊗ H W ,G → C; we refer to [START_REF] Polishchuk | Matrix factorizations and cohomological eld theories[END_REF] (Equation (4)) and [START_REF] Chiodo | Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence[END_REF] (Equation (5.12)) for the latter.

Remark 5.2.8. For the special case γ = , which always belong to G by de nition, it is not hard to see that

dim C H = 1.
Indeed, since = (e 2iπ q 1 , . . . , e 2iπ q m ) since each q i < 1, then B = . Therefore, dim C Q W = 1 and dz = 1. Consequently, we will denote 1 ∈ H W ,G the sole generator of H W ,G .

We now brie y de ne the moduli space S W ,G ,n of (W , G)-spin curves as it appeared in [START_REF] Fan | The Witten equation and its virtual fundamental cycle[END_REF][START_REF] Fan | The Witten equation, mirror symmetry, and quantum singularity theory[END_REF]. First, we start with a nodal orbicurve (orbifold curve) C with genus and n marked points (x 1 , . . . , x n ). We de ne on C a line bundle Ω log,C → C by

Ω log,C := Ω C (x 1 + • • • + x n ) ,
where Ω C is the canonical bundle over C. Roughly speaking, Ω log,C consists of meromorphic 1-forms with poles of order 1 at the marked points.

De nition 5.2.9 ((W , G)-spin structure). Let W ∈ C[z 1 , . . . , z m ] be a quasihomogeneous nondegenerate polynomial and G be an admissible symmetry group. Let us decompose W as a sum of monomials

W = r k=1 b k W k = r k=1 b k m i=1 z a i, k i .
Let C be an orbicurve of genus with n marked points together with m orbifold line bundles L 1 , . . . , L m → C such that there exist r isomorphisms of orbifold bundles ϕ 1 , . . . , ϕ r such that

ϕ k : W k (L 1 , . . . , L m ) := k L ⊗a i, k i ∼ -→ Ω log,C .
(5.29)

The data

(C; (x 1 , . . . , x n ); (L 1 , . . . , L m ); (ϕ 1 , . . . , ϕ r )) is called an (W , G)-spin structure.
We say that two (W , G)-spin structures (C; (x i ); (L j ); (ϕ k )) and (C ; (x i ); (L j ); (ϕ k )) (both of genus and with n marked points) are isomorphic if there exists an isomorphism of orbicurves ψ : C → C such that ψ (x i ) = x i and satisfying the condition

ψ * (L i ) L i (5.30)
for all i ∈ {1, . . . , m}. Notice that because ψ : C → C is an isomorphism, we automatically have Ω log,C Ω log,C which, together with (5.30), implies that the following diagram commutes,

W k (L) Ω log,C W k (L ) Ω log,C ϕ k ϕ k where W k (L) = W k (L 1 , .
. . , L m ) and accordingly for W k (L ). Then we de ne the moduli space S W ,G ,n of (W , G)-spin structures as the set of equivalence classes modulo isomorphism as described above, S W ,G

,n := {(C; (x 1 , . . . , x n ); (L 1 , . . . , L m ); (ϕ 1 , . . . , ϕ r ))}/ ∼ . This a compact complex orbifold (or a smooth Deligne-Mumford stack).

There is a natural forgetful morphism which forgets both the orbifold structure of the orbicurve and the (W , G)-spin structure; we denote it

π : S W ,G
,n → M ,n . In [START_REF] Fan | The Witten equation and its virtual fundamental cycle[END_REF] it is shown that, as a consequence of Equation (5.29), the orbifold structure of a (W , G)-spin structure on an orbicurve C at its marked points (x 1 , . . . , x n ) is characterized by a n-tuple γ = (γ (1), . . . , γ (n)) ∈ G ×n . This in turns induces a a decomposition

S W ,G ,n = γ ∈G ×n S W ,G ,n (γ ), which is such that S W ,G ,n (γ ) only if γ (1) • • • γ (n) = 2 -2+n .
(5.31)

Still in [START_REF] Fan | The Witten equation and its virtual fundamental cycle[END_REF], Fan, Jarvis and Ruan constructed a virtual fundamental class for each γ = (γ (1), . . . , γ (n)) ∈ G ×n such that Equation (5.31) holds (we do not de ne virtual class here)

[S W ,G ,n (γ )] vir ∈ H * (S W ,G ,n , Q). We can now de ne the CohFT classes c ,n associated to the FJRW theory of a pair (W , G) we just described.

Theorem 5.2.10 (FJRW CohFT [START_REF] Fan | The Witten equation, mirror symmetry, and quantum singularity theory[END_REF]). Let W ∈ C[z 1 , . . . , z n ] be a quasihomogeneous nondegenerate polynomial and G be an admissible symmetry group. Let γ = (γ (1), . . . , γ (n)) ∈ G ×n be such that Equation (5.31) holds. For each i ∈ {1, . . . , n} let α i ∈ H W ,G γ (i) . De ne the following cohomology classes,

c W ,G ,n (α 1 , . . . , α n ) = |G | deg(π ) P • π * [ S W ,G ,n (γ ) ] vir ∩ α 1 ∩ • • • ∩ α n ,
where |G | is the cardinal of G and P is the Poincaré duality. These classes extend by linearity to maps c W ,G ,n : H W ,G → H * (M ,n , Q) which de ne a CohFT in the sense of De nition 5.2.1 with e 1 = 1 ∈ H W ,G (see Remark 5.2.8).

The double rami cation hierarchies

The double rami cation cycles

Here we de ne the double rami cation cycles, introduced in [GV05], following the article [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF]. Let integers a 1 , . . . , a n ∈ Z be such that n i=1 a i = 0.

(5.32)

To such integers, we will assign a space of rubber stable maps with target variety P 1 relative to 0, ∞ ∈ P 1 (stable maps in the sense of Gromov-Witten theory, see Example 5.2.3). First recall the de nition of the moduli space of stable maps associated to a pair (X , β), where X is a smooth compact complex manifold and β ∈ H 2 (X , Q):

M ,n (X , β) := {(C; (x 1 , . . . , x n ); f : C → X ) | f * [C] = β }/ ∼,
where (C; (x 1 , . . . , x n )) is a stable curve and f : C → X is holomorphic map. Notice that in the particular case X = P 1 , we have dim H 2 (P 1 , Q) = 1, hence we can consider M ,n (P 1 ) without ambiguity. Let (C; (x 1 , . . . , x n ); f ) be the representative of a point in M ,n (P 1 ); we say that f : C → P 1 has rami cation pro le (a 1 , . . . , a n ) w.r.t. (x 1 , . . . , x n ) if for any i ∈ {1, . . . , n},

a i > 0 ⇒ x i is a pole of f of order a i ; a i < 0 ⇒ x i is a zero of f of order -a i .
(5.33)

In particular, if a i > 0 (resp.

a i < 0) then f (x i ) = 0 (resp. f (x i ) = ∞). 7 If a i = 0, then x i is just a regular marked point and f (x i ) 0, ∞.
In what follows, we will denote a := (a 1 , . . . , a n ) with vanishing sum when there is no ambiguity on n. We de ne the following moduli space of rubber stable maps, Notice that in genus 0, and for n ≥ 3, the DR cycle coincides with the fundamental class of the moduli space of stable curves [START_REF] Goulden | The Moduli Space of Curves, Double Hurwitz Numbers, and Faber's Intersection Number Conjecture[END_REF]:

M ;a := (C; (x 1 , . . . , x n ); f ) ∈ M ,
DR 0 (a) = [M 0,n ] ∈ H 2(n-3) (M 0,n , C).

Polynomiality and Hain's formula

Let us introduce the moduli space M ct ,n ⊂ M ,n of nodal stable curves of which every node is separating (again, it means that removing any node breaks the curve into two disjoint components); such curves are called of compact type. Then Hain [START_REF] Hain | Normal functions and the geometry of moduli spaces of curves[END_REF] showed that, restricted to the moduli space M ct ,n , the (Poincaré dual of the) DR cycles is tautological and is a polynomial in the a i 's which is homogeneous of degree 2 . Namely

P(DR (a 1 , . . . , a n )) M ct , n = (-1) ! 4       ⊂ {1, ...,n } h=0 (a ) 2 δ h      
, where a = j ∈ a j . The divisor δ h ∈ H 2 (M ,n , C) is the class of curves C with only one node, say ∈ C, which are such that separates C into two smooth components, i.e. the normalization of C \ { } is a disjoint union C 1 C 2 , and such that C 1 has genus h and contains the marked points {x j } j ∈ , while C 2 has genus -h and contains the remaining marked points. Besides we set δ {i } 0 = -ψ i . Because the top lambda class λ vanishes outside M ct ,n , it follows that for any cohomology class α ∈ H * (M ,n , C), the pairing

∫ DR (a) λ α ∈ C 2 [a 1 , . . . , a n ],
(5.34) by which we mean that it is polynomial in a 1 , . . . , a n of degree 2 . In general, outside of the moduli space M ct ,n , the DR are also polynomial but they are not homogeneous. However, they have degree 2 and terms of lower degree (e.g. [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF]). Equation (5.34) is crucial in the construction of the DR hierarchy.

The double rami cation hierarchy: the Hamiltonian structure

We now de ne the so-called double rami cation hierarchy (or DR hierarchy) introduced by Alexander Buryak in [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF]; we follow the approach of [START_REF] Rossi | Integrability, Quantization and Moduli Spaces of Curves[END_REF] though. As already mentioned, the input of the DR hierarchy is a cohomological eld theory c ,n : V ⊗n → H * (M ,n , C), where V is an N -dimensional complex vector space equipped with a nondegenerate symmetric bilinear form (a metric) η : V ⊗V → C. The classes c ,n must satisfy axioms of De nition 5.2.1. Let integers a = (a 1 , . . . , a n ) with vanishing sum and let α = (α 1 , . . . , α n ) ⊂ {1, . . . , N }. We introduce the following speci c intersection numbers, for any α ∈ {1, . . . , N } and any d ≥ 0, (5.37)

Because of the constraint b j = 2 in the above sum, it is clear that α,d ∈ A [0] . These di erential polynomials will will be the Hamiltonian densities of the DR hierarchy. Now it su ces to apply the de nition T (u α k ) = a 0 p α n e inx to nd that

T ( α,d ) = ˆ α,d .
Now, as we will see in the next theorem, besides the somehow simple Hamiltonian densities, we equip the DR hierarchy with the natural standard Poisson bracket

{ f , } η ∂ x = ∫ δ f δu α η α β ∂ x δ δu β ,
where η α β is the metric carried by the CohFT. Before giving the main theorem of this sectionthe integrability of the DR hierarchy-we make an important remark.

Remark 5.2.12. The original construction of Buryak [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF] was actually based not on the series ˆ α,d but on ˆ B α,d := ˆ α,d ; (0) which contains only the terms in ˆ α,d for which a j = 0. On the other hand, the homomorphism T 0 : F → B pol is surjective and its kernel is spanned by the u α 's. Let us denote B α,d the functional such that T 0 ( B α,d ) = ˆ B α,d and which does not depend linearly in the u α 's. Since we obviously have T 0 ( α,d ) = ˆ B α,d , it follows that α,d = B α,d modulo a linear dependence in the u α 's. Yet the latter are clearly Casimir for the standard Poisson structure and hence, such a dependence does not a ect the Hamiltonian structure.

Theorem 5.2.13 (DR hierarchy [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF]). Let an arbitrary CohFT along with the di erential polynomials α,d (5.37). For any α, β ∈ {1, . . . , N } and p, q ≥ 0, { α,p , α,q } η ∂ x = 0.

The Hamiltonian system generated by the above we call the double rami cation integrable hierarchy associated to said CohFT, namely,

∂u α ∂t β q = η α β ∂ x δ β,q δu β .
We refer to [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF] for the proof. As an important special case, we have

1,0 = 1 2 ∫ η α β u α u β .
Consequently, the compatibility with spatial translation holds:

∂u α ∂t 1 0 = u α 1 .

Recursion relations

Here we give two strong recursion relations of the DR hierarchy. Now we make the following observations. The local Poisson bracket is determined by the CohFT: {u α (x), u β ( )} = η α β ∂ x . One the other hand, the level -1 Hamiltonian densities α,-1 are given by α,-1 = η α β u β . Therefore, the following corollary holds.

Corollary 5.2.17. The sole data of 1,1 entirely determines the DR hierarchy.

We shall emphasize that, assuming the DR/DZ conjecture holds true, the above reconstruction technique was an unknown feature of the DZ, a fortiori DS, hierarchies. We have a second recursion theorem. (5.39)

Strong Double Rami cation/Dubrovin-Zhang conjecture

The Dubrovin-Zhang hierarchies

In this section, we brie y describe the construction of the Dubrovin-Zhang (DZ) hierarchies for these are the subject of the DR/DZ conjectures, the weak and the strong. We will not give proofs and refer to [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF]. In the speci c cases of a CohFT constructed as the FJRW theory of a simple singualrity of type ADE, the DZ hierarchy is known to coincide with the DS hierarchy counterpart [LRZ15, [START_REF] Givental | Simple singularities and integrable hierarchies[END_REF][START_REF] Frenkel | Soliton equations, vertex operators, and simple singularities[END_REF]. The introduction we give is taken from [START_REF] Rossi | Integrability, Quantization and Moduli Spaces of Curves[END_REF]. The Dubrovin-Zhang hierarchies are hierarchies of PDEs which are de ned for any semisimple CohFT. So, to start with, a CohFT with potential F (see Equation (5.28)) is said to said to be semisimple if the N -dimensional complex associative algebra de ned by the structure constants

f α βγ := η α µ ∂ 3 F 0 ∂t µ 0 ∂t β 0 ∂t γ 0 t * p >0 =0
is semisimple. Now let us denote Ω α,p; β,q (t * * ; ε) =

∂ 2 F ∂t α p ∂t β q = ≥0 Ω [2 ]
α,p; β,q ε 2 .

Then we de ne Hamiltonian densities and local Poisson structure:

h ,(0) α,p ( * ) := Ω [0] α,p+1; 1,0 (t * 0 = * , 0, 0, . . .), K α β ,(0) := η α β ∂ x .

These Hamiltonians indeed commute w.r.t. to the above bracket [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF]. They form the socalled genus 0 Dubrovin-Zhang hierarchy, or the principal hierarchy of the CohFT. By construction, they automatically form a tau structure.

To construct the full genera DZ hierarchy, we make use of the so-called topological solution; it is the one solution α of the DZ hierarchy that is determined by the initial data α (x, t * * = 0) = δ α,1 x, called topological. Then there exist rational functions P α p such that This change of variable is invertible and normal (preserves the tau structure), although it is not a Miura transform because P * * are not di erential polynomials. The full genera DZ hierarchy is then obtained by applying the inverse transform φ -1 to the genus 0 hierarchy. Namely, h α,p (w * * ) := h ,(0) α,p • φ -1 (w * * ), K α β (w * * ) = ((φ -1 ) * K ,(0) ) α β (w * * ),

where (φ -1 ) * K ,(0) is given in Equation (5.18). By construction, the potential of the CohFT is a tau function of the DZ hierarchy and is associated to the topological solution.

Our presentation of the DZ hierarchies ends here. We retain, as was already stated, that for the ADE CohFTs, the DZ hierarchy is Miura equivalent to the corresponding DS hierarchy [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF].

Tau structure of the DR hierarchy and the strong DR/DZ equivalence conjecture

In [START_REF] Buryak | Tau-structure for the Double Rami cation Hierarchies[END_REF], the authors de ned a very simple tau structure for the DR hierarchies which is actually only due to the simple standard Poisson structure it is equipped with. First, we add to the Hamiltonian densities α,d Casimir functionals α,-1 := η α β u β , where η α β denotes the inverse matrix. Now take the normal coordinates ũα = η α β h β,-1 and consider the special solution given by the initial data ũα (x, t * * = 0; ε) = δ α,1 x. It is the same condition as for the topological solution of the DZ hierarchy, however, for the DR hierarchy it does not carry the same geometrical interpretation (since it is not constructed as the potential of the CohFT). This special solution gives a tau function τ which we write as

F (t * * ; ε) := log τ = ≥0 ε 2 F (t * * ).
We also denote its expansion as if it were the generating series of some intersection numbers:

F =: n ≥0 2 -2+n >0 1 n! d 1 , ...,d n ≥0 τ d 1 (e α 1 ) • • • τ d n (e α n ) n i=1 t α i d i .
We still call the above the genus partition function of the DR hierarchy and the complex numbers τ d 1 (e α 1 ) • • • τ d n (e α n ) we also call correlators. Even though it is not clear whether the DR partition function carries a geometric interpretation, the following surprising selection rule holds. The above upper bound shall be compared with the well known selection rule on of the DZ hierarchy which states that the correlators τ d 1 (e α 1 ) • • • τ d n (e α n ) (the actual intersection numbers of the CohFT) vanish whenever d i > 3 -3 + n. The latter is only due to the limit on the number of psi classes one integrates. But the lower bound has no equivalent in the DZ hierarchies! And as it turns out, the lower bound plays a key role in re ning the DR/DZ conjecture into its strong form which we state below. First, we have the following theorem.

Theorem 5.2.22 ([BDGR18]). Let c ,n be a semisimple CohFT and let w α be the coordinates of the DZ hierarchy in its full form (5.40) and let F be the generating function of the CohFT (also a tau function of the DZ hierarchy). Then there exists a unique di erential polynomial f ∈ A [-2] such that the normal Miura transform ũα = w α + η α β ∂ x { f , h β,0 } , gives rise to a potential F red := F + f | x =0 such that F red satis es the selection rules (5.42).

By requiring that the Miura transform if normal, the above entirely determines the Miura transform that is conjectured to hold between the DR and the DZ hierarchy. Hence the strong DR/DZ equivalence conjecture: Conjecture 5.2.23 (Strong DR/DZ equivalence [START_REF] Buryak | Tau-structure for the Double Rami cation Hierarchies[END_REF]). For any semisimple CohFT, the DR and DZ hierarchies coincide up to the normal Miura transform generated by f ∈ A [-2] of Theorem 5.2.22.

In particular, for the CohFT arising from the FJRW of a simple singularity ADE, the Miura transform is conjectured to be the identity.

We will not prove here that the expected Miura transform reduces to the identity for the ADE cases. The strong DR/DZ equivalence has been con rmed in a number of cases [Bur15a, BG16, BDGR18]:

The trivial CohFT;

The full Hodge class CohFT; Witten's r -spin classes for r = 3, 4, 5 (equivalent to the A 3 , A 4 , A 5 FJRW theories);

The Gromov-Witten theory of P 1 ; Up to genus 5 for any CohFT of rank 1; Up to genus 2 for any semisimple CohFT.

In the next section, we describe the DR hierarchy for the FJRW of type D 4 . In Section 5.3 we describe the DS hierarchy of type D 4 (which is known to coincide with the DS hierarchy [LRZ15, GM05, FGM10]). We will observe that in normal coordinates, the DS and DR hierarchies coincide, con rming the strong DR/DZ equivalence in the D 4 case. Theorem 5.2.24 ([BG16]). Suppose there exists a Miura transform φ : u α → ũα (u * * ; ε), such that ∂ ũα ∂u 1 = δ α,1 , φ * (η α β ∂ x ) = K α β , 1,1 • φ = h 1,1 , (5.43) thesis we described it only for untwisted algebras). These hierarchies have the form of matrix Lax pairs (4.43). But as we can see with the example of the KdV hierarchy 4.3.3 (i.e. the DS hierarchy of type A 1 ), it becomes quickly cumbersome to actually compute the ows of the hierarchy using the matrix Lax pairs. Already in [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF], Drinfeld and Sokolov gave a scalar Lax pair for the hierarchies of type A (1) n , B (1) n , C (1) n and D (1) n (and others), along with a bi-Hamiltonian representation. Yet for the hierarchy of type D (1) n precisely, only part of the hierarchy was represented. The part that has no scalar Lax pair representation we call the negative ows and we insist on the fact that it does have a matrix Lax pair though. To make this statement clearer, we need to give a brief description of the untwisted a ne algebra D (1) n .

Hamiltonian

The untwisted a ne algebra of type D The matrices in o(2n) are not necessarily traceless and cannot be nilpotent. We take the usual faithful traceless representation of o(2n) (see [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF], but also [START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF][START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF]):

o(2n) M ∈ gl(2n, C) | MS + S(M at ) = 0 =: g, where (M at ) i, j = M n+1-j,n+1-i is the antitranspose of M and S = diag 1, -1, . . . , (-1) n-1 , (-1) n-1 , . . . , -1, 1 .

A direct computation shows indeed that tr (MS + SM at ) = 2tr M so that g ⊂ sl(2n). As before, we denote h, b, n the Cartan, negative Borel and negative nilpotent subalgebras of g. We choose the following Weyl generators of the associated a ne algebra g = g ⊗ C[λ, λ -1 ] ⊕ Cc:

e i = E i+1,i + E 2n+1-i,2n-i , e n = 1 2 (E n+1,n-1 + E n+2,n ), f i = E i,i+1 + E 2n-i,2n+1-i , f n = 2(E n-1,n+1 + E n,n+2
),

h i = E i+1,i+1 -E i,i + E 2n+1-i,2n+1-i -E 2n-i,2n-i , h n = E n+1,n+1 -E n,n + E n+2,n+2 -E n-1,n-1 and e 0 = 1 2 (E 1,2n-1 + E 2,2n ) ⊗ λ, f 0 = 2(E 2n-1,1 + E 2n,2 ) ⊗ λ -1 , h 0 = E 1,1 + E 2,2 -E 2n-1,2n-1 -E 2n,2n .
The a ne algebra g has a ne exponents E = Z odd (n -1)Z odd . It is often condensed in the following form: E = {1, 3, 5, . . . , 2n -3} ∪ {(n -1) } + (2n -2)Z, where the (n -1) means that if, n is even, every integer that is congruent to n -1 modulo 2n -2 has multiplicity 2; such integers are of the form a = (n -1)(2b + 1), with b ∈ Z. The principal Heisenberg subalgebra admits the following family of generators [START_REF] Kac | In nite dimensional Lie algebras[END_REF]:

-Λ k ∈ g k , Γ k ∈ g k(n-1) k ∈ Z odd ,
where Λ j := Λ j if j > 0, (λ -1 Λ 2n-3 ) -j if j < 0, Γ j := Γ j if j > 0, (λ -1 Γ) -j if j < 0, (5.54)

where Λ = n i=0 e i is the principal cyclic element and

Γ := µ E n,1 -1 2 E n+1,1 -1 2 E n,2n ⊗ λ + 1 4 E n+1,2n ⊗ λ (5.55) + (-1) n E 2n,n+1 -1 2 E 2n,n -1 2 E 1,n+1 ⊗ λ + 1 4 E 1,n ⊗ λ , (5.56)
where µ = 1 if n is even and µ = √ -1 when n is odd.

The positive and negative flows of the hierarchy of type D n where P = L 1 2n-2 , which exists in virtue of Theorem 3.1.8. The precise statement is that there exists a gauge slice L within the gauge class of the connection L such that Equation (5.59) is exactly Equation (5.57) on the connection L . The latter implies that the coordinates s α are di erential polynomials in the coordinates q α of the function Q.

Let L = ∂ + Λ + Q ∈ op g , that is, Q ∈ b [[t]]. Let U ∈ g -[[t]]
Now the problem attacked in [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF] is that the negative ows cannot be written using the (2n -2)-th root P of L. To solve this problem, the authors introduced a new class of pseudodi erential operators they called of the second type. Doing so, they were able to produce a square root for the operator L. We brie y describe their construction in the next section. ], but for the present construction, they are not needed. They will appear when we will compare the DS and the DR hierarchies. The ring A is graded by deg s α k = k, producing a decomposition

A = k ≥0 A k , A k • A ⊂ A k+ ,
where A k is the subspace of homogeneous elements of degree k. For this gradation, the derivation ∂ has degree 1, i.e. ∂A k ⊂ A k+1 . Notice that Liu-Wu-Zhang's construction [START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF] works for an arbitrary commutative ring as long as it is graded, topologically complete and equipped with degree-1 derivation. Now we consider the following additive group:

D = +∞ k=-∞ a k ∂ k a k ∈ A
Of course, the traditional PDOs (the one we will now call of the rst type) form a subgroup

D = m k =-∞ a k ∂ k a k ∈ A, m ∈ Z ⊂ D.
Clearly, the ring structure (3.1) on D cannot be extended to D. In what follows, we describe how to produce a new ring of whose elements are allowed to contain in nitely positive and negative powers of ∂.

Let A ∈ D be a PDO of the rst type. We say that A is homogeneous of degree k if A has the form

A = k i=-∞ a i ∂ i , a i ∈ A k-i .
We denote by D k the subspace of homogeneous operators of degree k. Then clearly

D = k ∈Z D k .
We introduce the following subgroups of D: By the gradation on A, the general form of an element A ∈ D reads A = i ∈Z j ≥0 a i, j ∂ i .

D (d ) =
(5.61)

The following lemma gives a nice characterization of D and D.

Lemma 5.3.2. Let A be of the general form of Equation (5.61), then the following characterization holds true. (i) A ∈ D k i a i, j = 0 outside of the diagonal {(i, j) ∈ Z × Z ≥0 | i + j = k };

(ii) A ∈ D i there exists m such that a i, j = 0 outside of the the set {(i, j) ∈ Z×Z ≥0 | j ≥ m -i};

(iii) A ∈ D i there exists n such that a i, j = 0 outside of the the set {(i, j) ∈ Z × Z ≥0 | i ≤ n}.

Proof. Point (i) is the de nition of D k . Point (ii) follows from the decreasing ltration property D (d ) ⊂ D (d -1) . Point (iii) is the de nition of D. Similarly to the case of PDOs of the rst type, we can de ne the positive and negative parts, the residue as well as the adjoint:

A + = i ≥0 j ≥0 a i, j ∂ i , A -= i <0 j ≥0 a i, j ∂ i , A * = i ∈Z j ≥0 (-1) i ∂ • a i, j , res A = j ≥0
a -1, j . such that P 2n-2 = L. Moreover, the operator P satis es [P, L] = 0 and

We know that if

P * = -∂P ∂ -1 .
The existence and uniqueness of the (2n -2)-th root P of L was already established since it involves only PDOs of the rst type; see Theorem 3.1.8. However, the additional equations satis ed by P was not established here. We refer to [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF] for the proof. The proposition below is the main tool in building the scalar Lax pair of the negative ows.

Proposition 5.3.4 ([LRZ15]

). There exists a unique PDO of the second type Q ∈ D of the form

Q = ∂ • ϱ + m ≥0 Q m • ∂,
(5.62) such that Q 2 = L and where Q m ∈ D b of order 2m. Moreover, Q satis es Bi-Hamiltonian structure of the DS hierarchy of type D n

Q * = -∂Q ∂ -1 Q * + (ϱ) = -
We now describe the bi-Hamiltonian structure of the DS hierarchy of type D n . Notice that the local Poisson structure we will de ne were already given in [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF]. What was not, is the Hamiltonian densities for the negative ows. First de ne the following operator L := ∂•L ∈ D, which we write in the following form:

L = ∂ • L =: ∂ 2n-1 + n-1 µ=1 s µ ∂ 2µ-1 + ∂ 2µ-1 s µ + ϱ∂ -1 • ϱ.
It satis es the condition L = -(L ) * . We can represent the operator the L in an alternative form:

L = ∂ 2n-1 + n-1 µ=1 µ ∂ 2µ-1 + ˜ µ ∂ 2µ-2 + ϱ∂ -1 • ϱ.
(5.65)

Then the functions µ are related to the s µ by an Miura transform (within di erential polynomials) while the ˜ µ are related to the functions µ via a non invertible transform thanks to the condition L = -(L ) * . Given a local functional f ∈ F , we can de ne its variational derivative w.r.t. L (or its variational di erential) by

δ f δL := δ f δ n + 1 2 n-1 µ=1 ∂ -2µ • δ f δ µ + δ f δ µ ∂ -2µ .
(5.66) Notice that the above expression is well de ned since the variational derivatives do not depend on the choice of the representative in the class modulo Im(∂). Besides, this choice of variational di erential is speci c to the algebra of type D n ; for other algebras, see [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] but also [START_REF] Dubrovin | Frobenius manifolds and central invariants for the Drinfeld-Sokolov bihamiltonian structures[END_REF] for a particularly clear exposé. The bi-Hamiltonian structure of the DS hierarchy of type D n is given by the following local Poisson structures: let two local functionals f , ∈ F and their variational di erentials

X = δ f δ L , Y = δ δ L , then { f , } 1 = ∫ res X [(∂ x Y + L ) --(L Y + ∂ x ) --(∂ x Y -L ) + + (L Y -∂ x ) + ] dx, { f , } 2 = ∫ res X [(L Y ) + L -L (Y L ) + ] dx .
These brackets are compatible in the sense that for any λ, µ ∈ C, the map λ{•, •} 1 + µ{•, •} 2 still satis es Jacobi identity. In [START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF], the authors prove the following theorem. where the Hamiltonian functionals are given by

H k = 2n -2 k ∫ res P k dx, Ĥk = 2 k ∫ res Q k dx

Tau structure

Finally, we describe the tau structure given in [START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF] by means of these PDOs of the second type. The authors prove that the Hamiltonian functionals H 1 , H 3 , . . . , H 2n-3 and Ĥ1 are linearly independent and are Casimir of the rst Poisson bracket {-, -} 1 .

Next we need to introduce the so-called topological variables: for all µ ∈ {1, . . . , n -1}, t µ,p := (2n -2)Γ(p + 1 + 2α -1 2n-2 ) Γ( 2α -1 2n-2 ) t (2n-2)p+2α -1 , t n,p := 2Γ(p + 3 2 ) Γ( 1 2 )

t2p+1 .

Then de ne the following di erential polynomials: for all µ ∈ {1, . . . , n -1}, h µ,p-1 := Γ( 2α -1 2n-2 ) (2n -2)Γ(p + 1 + 2α -1 2n-2 ) res P (2n-2)p+2α -1 , h n,p-1 := 2Γ( 1 2 ) Γ(p + = ∂h β,q-1 ∂t α p .

The content described in the present section is all we need to actually compute the rst Hamiltonian structure of the DS hierarchy of type D 4 in view of con rming the strong DR/DZ conjecture. This is the goal of the next and last section. The explicit tau-symmetric Hamiltonian densities.

In this section, we give the explicit fomulae for the Hamiltonian densities h α,-1 , h α,0 , for 1 ≤ α ≤ 4, and h 1,1 . From now, in order to compare the DZ hierarchy with the DR hierarchy, we perform the following substitution:

∂ k f ∂x k -→ ε √ 2 k ∂ k f ∂x k
Therefore, we will now work again in the extended spaces. When we reduce to the case of the DS hierarchy of type D 4 , the Lax operator becomes

L = ∂ 6 x + ∂ -1 x 3 µ=1 s µ ∂ 2µ-1 x + ∂ 2µ-1 x s µ + ∂ -1 x ϱ∂ -1 x ϱ, (5.69)
Recall that the normal coordinates are given by ũα = η α β h β,-1

where the matrix (η α β ) is given by (see e.g. [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF]) (η α β ) = 0 0 6 0 0 6 0 0 6 0 0 0 0 0 0 2 By Equation (5.68), the normal coordinates read, for µ ∈ {1, 2, 3}, ũµ = 6 7 -2µ res P 7-2µ , ũ4 = 2 res Q.

Because of the form of the operator Q (5.62), its residue is straightforwardly res Q = ϱ, which means that ũ4 = 2ϱ = 2 √ s 4 . For the remaining normal coordinates, we compute the residues in the coordinates s α and then inverse the system. We nd (5.70) This agrees with the expressions found in [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF], p. 751. Then the Hamiltonian densities h α,0 are given by: for µ ∈ {1, 2, 3}, h µ,0 = 6 (2µ -1)(2µ + 5) res P 2µ+5 , h 4,0 = 2 3 res Q 3

                
The computation of res P 2µ+5 can be done straightforwardly by programming. However, the computation res Q 3 requires a little bit more. To do so, we write res

Q 3 = res QL = res ∂ -1 x ϱL + res m ≥0 Q m ∂ x L , = res ∂ -1 x ϱL + res m ≥0 Q m ϱ∂ -1 x ϱ ,
where in the last equation we used the fact that ∂ x L = L = L + + ϱ∂ -1 x ϱ. The term res ∂ -1

x ϱL can also be easily computed by programming. For the remaining part let us rst write

Q m = k ≤2m q m,k ∂ k x , then res m ≥0 Q m ϱ∂ -1 x ϱ = res m ≥0 k ≤2m q m,k ∂ k x ϱ∂ -1 x ϱ = res m ≥0 k ≤2m q m,k k =0 k ε √ 2 ϱ ∂ k -1- x ϱ = m ≥0 k ≤2m q m,k ε √ 2 k ϱ k ϱ = ϱ m ≥0 Q m (ϱ).
In the above equations, all matters of convergence are resolved by the grading of A and the fact that Q ∈ D + . Now thanks to Equation (5.63), it follows that

m ≥0 Q m (ϱ) = 1 2 L + (1) = 3 µ=1 s µ 2µ-2 ε 2µ-2 2 µ-1 .
(Note that this precise computation does not depend on the integer n of D n , meaning that similarly, in the D n case, { ũn (x), ũn ( )} 1 = 2δ (x -).) It is easy to see that for any µ ∈ {1, 2, 3}, { ũµ (x), ũ4 ( )} 1 = 0. The remaining components are computed in a similar fashion, this time using

{ ũµ (x), ũν ( )} 1 = ε -1 ∫ res X µ (z) [(L (z)Y ν (z)∂ z ) + -(∂ z Y ν (z)L (z)) + ] dz,
for µ, ν ∈ {1, 2, 3}. Now we equivalently write ∂ k x instead of δ (k) (x -), for their action is identical. We write down the components { ũα (x), ũβ ( )} 1 in the matrix K α β := { ũα (x), ũβ ( )} 1 .

Proposition 5.3.9. The rst local Poisson bracket of the DS hierarchy of type D 4 , in normal coordinates reads

(K α β ) = 1 6 ũ3 2 ∂ x + 1 2 ũ3 1 ∂ 2 x + 1 3 ũ3 ∂ 3 x ε 2 + 4 15 ∂ 5 x ε 4 0 6∂ x 0 0 6∂ x 0 0 6∂ x 0 0 0 0 0 0 2∂ x .
(5.77)

In particular, K α β = η α β ∂ x + O(ε).

Remark 5.3.10. Let ψ : ũα → α be the inverse of the Miura transform given by Equation (5.74). Then one can also compute the matrix K α β in normal coordinates using Equation (5.18) which we recall here:

K α β = (M * ) α µ • (K • ψ ) µν • M β ν , (5.78)
where

(M * ) α µ = s ≥0 ∂ ũα ∂ µ s ∂ s x , M β ν = s ≥0 (-1) s ∂ s x • ∂ ũβ ∂ ν s ,
and where, by construction, (K • ψ ) α β = { α (x), β ( )} 1 . But then we need to compute the bracket in the coordinates α and then use Equation (5.78). It is probably a bit easier in all honesty.

To conclude, comparing (1) the Poisson structures of the DR hierarchy and the DS hierarchy of type D 4 , both in their normal coordinates (Propositions 5.2.25 and 5.3.9 respectively) and

(2) the Hamiltonian functional h 1,1 of the DR hierarchy with the Hamiltonian functional h 1,1 of the DS hierarchy (Propositions 5.2.27 and 5.3.8 respectively), we obtain the theorem below. Notice that it hold true in virtue of Theorem 5.2.24.

Remark A.1.1. For the D 2 case, the diagram is disconnected as it reads D 2 : . Therefore, D 2 A 1 ⊕ A 1 is semisimple rather than simple. However, the above representation still hold (see [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF]).

A 2 : CM (A 2 ) = 2 -1 -1 2
Weyl generators of our realization:

X 1 = 0 1 0 0 X 2 = 0 0 1 0 Y 1 = 0 1 0 0 Y 2 = 0 0 1 0 H 1 = -1 1 0 H 2 = 0 -1 1 Type B 2
The simple Lie algebras of type B 2 are isomorphic to so(5, C). They have the following features.

Coxeter number: h = 4. Finite exponents: F = {1, 3}. Dynkin diagram and nite Cartan matrix of size 2 × 2:

B 2 : CM (B 2 ) = 2 -1 -2 2
A.1. Classical simple Lie algebras v Weyl generators of our realization:

X 1 = 0 1 0 0 0 1 0 X 2 = 0 0 1 0 1 0 0 Y 1 = 0 1 0 0 0 1 0 Y 2 = 0 0 2 0 2 0 0 H 1 = -1 1 0 -1 1 H 2 = 0 -2 0 2 0 Type C 2
The simple Lie algebras of type C 2 are isomorphic to sp(4, C). They have the following features.

Coxeter number: h = 4. Finite exponents: F = {1, 3}. Dynkin diagram and nite Cartan matrix of size 2 × 2:

C 2 : CM (C 2 ) = 2 -2 -1 2
Weyl generators of our realization:

X 1 = 0 1 0 0 1 0 X 2 = 0 0 1 0 0 Y 1 = 0 1 0 0 1 0 Y 2 = 0 0 1 0 0 H 1 = -1 1 -1 1
H 2 = 0 -1 1 0 Type D 2
The simple Lie algebras of type D 2 are isomorphic to so(4, C). They have the following features.

Coxeter number: h = 2. In Chapter 3, we establish an explicit transformation between the polynomial tau functions of the Korteweg-de Vries hierarchy (associated to the algebra sl(2,C)) and the Adler-Moser polynomials (1978). The latter form a sequence of polynomials satisfying a certain differential recursion relation.

Chapter 4 is dedicated to the computation of tau functions via Toeplitz determinants; a method introduced by Cafasso and Wu (2015). In collaboration with Cafasso and Yang, we obtained an expansion of the tau function as a sum over all integer partitions. It follows a simple criterion for the polynomiality of the tau function; we give some nontrivial examples.

In Chapter 5, in collaboration with Paolo Rossi, we confirm the so-called 'strong DR/DZ conjecture' for the algebra o(8,C) (D4). The latter states an equivalence between, in particular, Drinfeld-Sokolov hierarchies and another kind of hierarchies called 'the double ramification hierarchies' introduced by [START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF] and constructed from the cohomology of the moduli spaces of stables complex curves Mg,n.

  1.6) où [L, P] = L • P -P • L est le commutateur entre opérateurs di érentiels. La composition d'opérateurs di érentiels est donnée par la règle de Leibniz, dé nie sur les monômes par 15 b (k) := ∂ k x (b), et étendue par linéarité. Nous omettrons le signe • lorsqu'il n'y a pas d'ambiguïté. La dérivation ∂ x est dé nie sur les opérateurs di érentiels comme agissant sur les coe cients : ∂ t a∂ n

x + 3 2 uu x 0 0 .

 0 Dans l'équation ci-dessus, [L , M] := ∂ x M + [N , M ] où N = L -∂ x et où la dérivée ∂ x M est dé nie comme agissant sur les composantes de la matrice. Ainsi, ∂ t L = [L , M ] si et seulement si u t = 1 4 u x x x + 3 4 uu x ,i.e. si et seulement si l'équation KdV (1.5) est satisfaite. Autrement dit, l'équation KdV (1.5) peut être réécrite comme ∂L ∂t = [L , M ].

  .50) for any f , ∈ F , 34 where X = δ f δ L , Y = δ δ L are given in Equation (5.66). This allows to de ne a scalar Lax pair along with a Hamiltonian representation 35 of Equation (2.47) by[START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF]

  Theorem 2.4.2 (Strong DR/DZ for D 4 [DR19]). The strong DR/DZ equivalence holds true for the hierarchy of type D 4 under the normal Miura transform (2.46). More precisely, in the normal coordinates ũα , (i) The local Poisson structures are the same: K α β = K α β ;

  .1) where b (k ) (sometimes just b k ) denotes the k-th derivative w.r.t. x of the function b and b (0) = b.

Figure 3 . 1 :

 31 Figure 3.1: Generalized binomial coe cients in an extended Pascal's "triangle"

  .11) and extended on D by linearity. It is an anti-homomorphism, i.e. for any A, B ∈ D, (AB) * = B * A * .

  Lemma 3.3.4. For any operators A, B ∈ D, ∮ dλ 2iπ A(e λx ) • B(e -λx ) = res ∂ (AB * ) . Proof. Let us denote A = k ≤n a a k ∂ k and B = k ≤n b b k ∂ k . The left hand-side of the above equality reads ∮ dλ 2iπ A(e λx ) • B(e -λx ) = res λ k ≤n a a k λ k ≤n b b (-1) λ = ≤n b (-1) a --1 b . The left-hand side reads res ∂ (AB * ) = res ∂ k ≤n a a k ∂ k • ≤n b (-1) ∂ b = ≤n b (-1) a --1 b . Lemma 3.3.5. If Q satis es the KP hierarchy (3.18) (i.e. if w satis es (3.23)), then w satis es the bilinear identity (3.26).

  Theorem 3.3.7 (Tau function (KP)). Let M = 1 + k <0 m k ∂ k be an operator satisfying Sato's equation (3.21) an de ne functions w, w * ∈ W by w(t; λ) = Me ξ (t;λ) , w * (t; λ) = (M * ) -1 e -ξ (t;λ) . If w and w * satisfy the bilinear identity (3.26), then there exists a (non unique, nonzero) function τ ∈ C[[t]], called the tau function, such that

  Equation (3.42) holds true. Then setting χ = 1 in Equation (3.41), one nds Equation (3.40).

  Remark 4.3.1. By a connection we simply mean an linear endomorphism of g [[x]] which satises Leibniz's rule; i.e. a derivation. The action of L on g [[x]] that makes it into a connection we de ne by

  Proposition 4.3.3 (Gauge classes). The group N acts on op g , i.e. for any connection L ∈ op g and any S ∈ n [[x]], the derivation e ad S L = ∂ + e ad S (Λ + Q) -∇ S S belongs to op g . We say that L and e ad S L are gauge equivalent, or that L and e ad S L belong to the same gauge class denoted [L ].

  Proposition 4.3.4 (Canonical coordinates). Let V ⊂ b be a gauge and let L = ∂ +Λ+Q ∈ op g . There exists a unique function S ∈ n [[x]] such that the connection L = e -ad S L has the form

  The root space gradation assigns deg E =deg F = 1 and deg H = 0. The Cartan, negative Borel and principal nilpotent subalgebras are given by n = CF , h = CH, b = h ⊕ n = CH ⊕ CF . The principal nilpotent element I is simply I = E. Since n = CF and [E, F ] = H , it follows that b = CF ⊕ [I , n]. Hence, we naturally choose the gauge V = CF = n. We start a connection L = ∂ + Λ + Q, with Q ∈ b [[x]] and we want to represent it in the form L = e ad S L =

  be such that ∂Ω = -H . We now introduce a new function, T = U • Ω, where • denotes the Baker-Campbell-Hausdorf product (4.3), i.e. T is characterised by e ad T = e ad U e ad Ω .

  and similarly val ζ γ I + γ-≥ 2| |-|I |, so that both val ζ γ I -γ and val ζ γ I + γare strictly increasing functions of > 0. In particular, for any A ≥ 0, only nitely many terms appear in the coe cient in ζ A of (γ • γ ) I . This shows that the Cauchy product is well de ned. On the other hand, val ζ (γ I γ0 ) ≥ |I |, but also val ζ γ I -γ ≥ |I -| + | | ≥ |I |, and similarly val ζ γ I + γ-≥ |I |. This proves that γ • γ is an admissible series.

  [BO00, CW15a]) Now notice that in terms of ζ valuations, val ζ I ≥ I while val ζ γ -1 I +1 ≥ I + 2 (since I ≥ 0). Therefore, if we denote I = A ≥I ζ A A I , Equation (4.82) implies that I I = I +1 I = 0. But, applied to order λ I +1 , it also implies that I +1 I +1 = I +2 I +1 = 0, so that val ζ I +1 ≥ I + 3 now, so that val ζ γ -1 I +1 ≥ I + 4. But then looking back at Equation (4.82), we must have I +2 I

  I and I +1 I . But applied to λ I +1 , it also xes I +1 I +1 = u I +1 I and I +2 I +1 = u I +2 I +1 . Therefore, the equation in λ I ζ I +2 and λ I ζ I +3 reads I +2 I

  Corollary 4.4.11. Under the conditions of Theorem 4.4.10, we have lim N →∞ D N (ϕ) = Z(ϕ) = det id -H(ϕ) H(ϕ -1 ) . (4.83) Proof. Using the Borodin-Okounkov formula, log Z(ϕ)log D N (ϕ) = log (det(id -K N (ϕ)) .

  These matrices have entries in C [[t, ζ ]]. Since γ = id+O(λ -1 ) and (t) = id+O(λ), the matrices s m (t) and r m have the form assumed in Lemma (4.5.6). Lemma 4.5.7. The sequence (det (s m (t) • r m )) m ≥1 converges w.r.t. the topology induced by the valuation in ζ . Moreover, lim m→∞ det (s m (t) • r m ) = µ ∈Y s µ (t) • r µ . (4.97) Proof. Let us denote u m := det (s m s m (t)). Let N = m/n (the oor integer part of m/n, where n is the dimension of the representation π : g → gl(n) we chose). Then by Lemma 4.5.6, we have

  Therefore, val ζ s µ ≥ val ζ s 0,-m-1 ≥ |N | since N = m/n . Clearly, lim m→∞ val ζ s µ = ∞. Moreover, any partition in Y \ Y (m) contains terms of higher valuation in ζ , so that lim m→∞ val ζ µ ∈Y s µ (t) • r µ -u m = ∞; this concludes the proof. Now let ϕ = (t) • γ and let T N (ϕ) be the associated N -truncated Toeplitz block matrix, which we recall the de nition below, T N (ϕ) = ϕ I -I, ∈[0, N ] .

  .106) which are identi ed to two of the well-known bilinear equations for the hierarchy of type A 1 (the KdV hierarchy) (see Example 4.5.10). Consequently, we have shown that dim C HDS 104)-(4.106) are the three only possible choices of homogeneous basis (up to constant factors) of HDS [deg≤6] g .

  and extended on B by imposing Leibniz's rule, i.e. and so on. Now consider the homomorphism of algebras T : A → C[[p * * , x]] de ned on the indeterminates by 3 T (u α k ) := n ∈Z *

  pol d ⊂ B pol consisting elements of the form of Equation (5.11) such that the polynomials f α 1 , ...,α n ∈ C[z 1 , . . . , z n ] are homogeneous of degree d. Then de ne the algebra B := B ⊗ C[[ε]] and let B pol be the subalgebra of B consisting of power series of the form

  5.13)and be such that the following hold[START_REF] Dubrovin | The Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method[END_REF].(i) Under Miura transforms ũα := φ(u α ) of the type ∂ ũα /∂u β k >0 = 0, the matrix α β transforms like a (2,0) tensor, i.e.( • φ) α β = ∂ ũα ∂u µ ∂ ũβ ∂u ν µν , while the matrix Γ γ α β :=α µ Γ µγ β, where ( α β ) := -1 , transforms like a connection matrix ∇ = d + Γ, i.e.

  It can be shown that the forgetful map π 1 : M ,n+1 → M ,n coincides with the universal curve π : C ,n → M ,n ; hence in what follows we consider indistinctly π = π 1 .The tree morphismσ : M ,n+1 × M ,n +1 → M + ,n+nglues two (classes of) stable curves ∈ M ,n+1 and ∈ M ,n +1 by identifying their respective last marked points, say x n+1 and x n +1 , to a single point in σ ( , ) which becomes a stable node. The new node separates 6 the resulting curve into two components. This is illustrated below.τ : M ,n+2 → M +1,n

  any , w ∈ H * (X , C) where P : H * (X ) → H * (X ) is the Poincaré duality which gives a ring isomorphism H k (X , C) → H d -k (X , C), where d = dim R X . Then de ne the linear maps c ,n (e α 1 , . . . , e α n ) = p * ev * (e α 1 , . . . , e α n ).

∫

  DR (-a i ,a) λ ψ d 1 c ,n+1 (e α , e α 1 , . . . , e α n ). (5.35) Thanks to Equation (5.34), P ;a α,d ;α is a homogeneous polynomial in a i 's of degree 2 . Hence, let us write it as such:P ;a α,d ;α = b 1 , ...,b n ≥0 b j =2 P ;b α,d ;α a b 1 1 • • • a b n n ∈ C 2 [a 1 , . . . , a n ],where, similarly to previous notations, b = (b 1 , . . . , b n ). From this we de ne the following generating series, still for any α ∈ {1, . . . , N } and any d ≥ 0 (the sum over1 ≤ α 1 , . . . , α N ≤ N is understood), 8ˆ α,d := j e ix a j .(5.36) Now consider the following power series (still, the sum over 1 ≤ α 1 , . . . , α N ≤ N is understood):α,d := ,n ≥0 2 -2+n >0 ε 2 n! b 1 , ...,b n ≥0

  Theorem 5.2.14 ([START_REF] Buryak | Recursion Relations for Double Rami cation Hierarchies[END_REF]). The Hamiltonian densities α,d of the Double rami cation hierarchy satisfy the following recursion:(D -1) α,d +1 = ∂ -1 x α,d , 1,1 η ∂ x , α ∈ {1, . . . , n}, d ≥ -1.(5.38)where D = ε∂ ε + k ≥0 u α k ∂ u α k .Remark 5.2.15. Since the Hamiltonian functionals α,d commute, meaning that { α,d , β,e } = 0, we have { α,d , β,e } ∈ Im(∂ x ). Therefore, Equation (5.39) is meaningful. Remark 5.2.16. Equation (5.39) allows to compute the density α,d +1 . Indeed, by de nition, Du α k = (k + 1)u α k . So let us write α,d +1 as a plain combination α,d +1 = <∞ m ≥0 m i=1 1≤α i ≤n k i ≥0 C α, ...,α m k 1 ...,k m i ≤n k i ≥0C α, ...,α m k 1 ...,k m -1 +

  Theorem 5.2.18[START_REF] Buryak | Recursion Relations for Double Rami cation Hierarchies[END_REF]). The Hamiltonian densities α,d of the Double rami cation satisfy the following recursion:∂ α,p+1 ∂u β = ∂ -1 x α,p , β,0 η ∂ x .

F

  [START_REF] Buryak | On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket[END_REF] (t * 0 , t * 1 , t * 2 , . . .) = F (P * 0 ( * 0 , . . . , * 3 -2 ), . . . , P * 3 -2 ( * 0 , . . . , * 3 -2 ), 0, 0, . . .) | x =0 .Now consider the change of coordinates φ( α ) = w α := α +

  Theorem 5.2.19. The following Hamiltonian densities are tau symmetric: h α,p := δ α,p+1 δu 1 , α ∈ {1, . . . , N }, p ≥ -1. (5.41) A proof is given in [BDGR18], see Propositions 3.1 and 4.1. Already in [Bur15a], A. Buryak, guided by the examples of the trivial and Hodge hierarchies he computed, made the so-called weak DR/DZ equivalence conjecture. Conjecture 5.2.20 (Weak DR/DZ equivalence). Given a semisimple CohFT, the associated DR and DZ hierarchies are equivalent though a Miura transform.

  Proposition 5.2.21([BDGR18]). The DR correlators τ d 1 (e α 1 ) • • • τ d n (e α n )are nonzero only if the following inequalities are satis ed2 -3 ≤ n i=1 d i ≤ 3 -3 + n(5.42)

  structure of the double rami cation hierarchy of type D 4 In [BG16] (Theorem 5.3), A. Buryak and J. Guéré proved the following theorem which is of great help in proving the DR/DZ conjecture. First, let us x some notations. Let a semisimple CohFT and consider the associated Dubrovin-Zhang hierarchy. Let h α,d ∈ A( ũ * * ; ε) be the Hamiltonian densities of the DZ hierarchy and let K α β be the local Poisson structure of the DZ heirarchy.

n

  The simple Lie algebras of type D n are isomorphic to the algebra o(2n) of 2n-dimensional complex skew-Hermitian matrices, i.e.o(2n) = M ∈ gl(2n, C) (M t ) * + M = 0These algebras have Coxeter number h = 2n -2.

  be a function as in Theorem 4.3.7. Following[START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF], we introduce the following Drinfeld-Sokolov equations of type D n :∂L ∂t k = L , (e ad U Λ k ) + ,(5.57)∂L ∂ tk = L , (e ad U Γ k ) + ,(5.58)with k ∈ Z odd >0 . By construction, they generate the whole DS hierarchy. The ows along ∂ t k we call the positive ows, while the ows along ∂ tk we call the negative ows.In [DS84] ( §7.1), Drinfeld and Sokolov introduced the following PDO 9L = ∂ 2n-2 + ∂ -1 n-3 µ=1 s µ ∂ 2µ-1 + ∂ 2µ-1 s µ + ∂ -1 ϱ∂ -1 ϱ, s n = (ϱ) 2 , with the condition ∂ • L = L * • ∂.Drinfeld and Sokolov proved that the positive ows of the hierarchy admit the following scalar Lax pair representation: ∂L ∂t k = L, (P k ) + , (5.59)

.

  5.3.1 Scalar Lax pair and bi-Hamiltonian structure of the DS hierarchy of type D nPseudo-di erential operators of the second typeIn this section we describe the construction pseudo-di erential operators (PDOs) of the second type, as done in[START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF]. We start the algebra of (un-extended) di erential polynomials A with yet another set of indeterminates s 1 k , . . . , s n k , with k ≥ 0, namelyA := A s 1 , . . . , s n = C s α k α ∈ {1, . . . , n}, k ≥ 0 ,In the previous sections, we introduced the extended algebra of di erential polynomialsA = C[[u * 0 ]][[ε]][u * k >0

  Notice that the above induces a decreasing ltration D (d ) ⊂ D (d -1) . Recall that the product of PDOs of the rst type is given, for a, b ∈ A and n, m ∈ Z, bya∂ n • b∂ m = k ≥0 n k ab (k) ∂ n+m-k , n k = (n -k + 1) • • • (n -1) • n k! . (5.60) Clearly, if A ∈ D k and B ∈ D than AB ∈ D k+ , so the product can be extended to D, which makes it into an algebra. De nition 5.3.1. We call PDOs of the second type the elements of the algebra D. 10 The elements of the subalgebra D b = D ∩ D are called bounded operators.

Figure 5 . 1 :

 51 Figure 5.1: Graphical interpretation of Lemma 5.3.2. Each gure shows the lattice Z × Z ≥0 ; each dot on the lattice indicates the points (i, j) where a i, j needs not vanish.

  A ∈ D, then A + , A -, A * ∈ D; moreover, A -∈ D as well so that A + ∈ D b . Similarly, if A ∈ D, then we easily see that A + , A -, A * ∈ D and that A -∈ D so that A -∈ D b .In the next section, we describe how the algebra D of PDOs of the second type were used by Liu Ruan and Zhang to construct a scalar Lax pair for the negative ows of the DS hierarchy of type D 4 .Scalar Lax pair of the DS hierarchy of type D nFirstly, recall the following operator of the rst type,L = ∂ 2n-2 + ∂ -1 n-1 µ=1 s µ ∂ 2µ-1 + ∂ 2µ-1 s µ + ∂ -1 ϱ∂ -1 ϱ, s n = (ϱ) 2 with the condition ∂ • L = L * • ∂.The following proposition holds.Proposition 5.3.3 ([LWZ10]). There exists a unique PDO of the rst type P ∈ D of the formP = ∂ + k <0 p k ∂ k 10 In [LWZ10], the algebra D is denote D -while the algebra D is denoted D + . Here we made a di erent choice since D + and D -were already used in Chapter 3.

  L + )(1). (5.63) Now the (2n -2)-th root P ∈ D and the square root Q ∈ D together allows to de ne a hierarchy represented by a scalar Lax pair. The latter is equivalent to the full matrix Lax pairs of the Drinfeld-Sokolov hierarchy of type D n . Theorem 5.3.5 ([LWZ10]). The hierarchy of equations de ned by the following ows is consistent and commute pairwise:for k ∈ Z odd >0 , ∂L ∂t k = L, (P k ) + , ∂L ∂ tk = L, (Q k ) + (5.64)Moreover, it coincides with the ows de ned by the matrix Lax pair equations (5.57) and (5.58) respectively.

Theorem 5.3. 6 .

 6 The hierarchy (5.64) admits the following bi-Hamiltonian representation: for any local functional f ∈ F ,∂s α ∂t k = {s α , H k +2n-2 } 1 = {s α , H k } 2 , ∂s α ∂ tk = {s α , Ĥk+2 } 1 = {s α , Ĥk } 2 ,(5.67)

  rewrite the Hamiltonian ows (5.67) as∂s α ∂t β,p = {s α , h β,p } 1 = p + 1 2 + (1 -δ β,n ) 2β -n 2n -2 -1 {s α , h β,p-1 } 2These Hamiltonian ows are tau symmetric in the sense that ∂h α,p-1 ∂t β q

  5.3.2 FirstHamiltonian structure of the DS hierarchy of type D 4 and strong DR/DZ equivalence

A. 1

 1 .1 Examples: Rank 2 simple Lie algebras Type A 2 The simple Lie algebras of type A 2 are isomorphic to sl(3, C). They have the following features. Coxeter number: h = 3. Finite exponents: F = {1, 2}. Dynkin diagram and nite Cartan matrix of size 2 × 2:

Titre:

  Fonctions tau polynomiales et topologique des hiérarchies de Drinfeld-Sokolov Mots clés : systèmes intégrables, algèbres de Lie affines, hiérarchies de Drinfeld-Sokolov, fonctions tau, hiérarchie de double ramification. Résumé : Cette thèse traite du calcul et des applications des fonctions tau des hiérarchies de Drinfeld-Sokolov introduites en 1984. Les hiérarchies de Drinfeld-Sokolov sont des suites d'équations aux dérivées partielles intégrables que l'on associe à n'importe quelle algèbre de Lie semisimple. La fonction tau est une fonction associée à toute solution d'une hiérarchie donnée et qui contient toute l'information de la solution. Les fonctions tau sont au coeur des liens qui unissent les hiérarchies de Drinfeld-Sokolov et la géométrie algébrique. Au chapitre 3, nous établissons une transformation explicite entre les fonctions tau polynomiales de la hiérarchie de Korteweg-de Vries (associée à l'algèbre sl(2,C)) et les polynômes d'Adler-Moser (1978). Ces derniers forment une suite de polynômes satisfaisant une certaine relation de récurrence différentielle. Le chapitre 4 traite du calcul des fonctions tau polynomiales par les déterminants de Toeplitz ; une méthode introduite par Cafasso et Wu (2015). En collaboration avec Cafasso et Yang, nous avons obtenu une expansion de la fonction tau en une somme sur les partitions d'entiers. Nous en déduisons un critère de polynomialité de la fonction tau et donnons quelques exemples non triviaux. Au chapitre 5, en collaboration avec Paolo Rossi, nous confirmons la conjecture dite « DR/DZ forte » dans le cas de l'algèbre de Lie simple o(8,C) (D4). Elle prévoit l'équivalence, en particulier, entre les hiérarchies de Drinfeld-Sokolov et d'autres hiérarchies dites de « double ramification, » introduite par Buryak (2015) et construites à partir de la cohomologie de l'espace de modules des courbes complexes stables Mg,n. Title: Polynomial and topological tau functions of the Drinfeld-Sokolov hierarchies Keywords: Integrable systems, Affine Lie algebras, Drinfeld-Sokolov hierarchies, Tau functions, Double ramification hierarchies.Abstract: This thesis deals with the computation and applications of tau functions of the Drinfeld-Sokolov hierarchies introduced in 1984. The Drinfeld-Sokolov hierarchies are sequences of integrable partial differential equations which one associates to any semisimple Lie algebra. The tau function is a function associated to any solution of a given hierarchy and which contains all the information of the solution. Tau functions are at the heart of the bonds between Drinfeld-Sokolov hierarchies and algebraic geometry.

  Hiérarchie de type D 4 et l'équivalence DR/DZ forte . . . . . . . . . . . . . . . Les hiérarchies de double rami cations . . . . . . . . . . . . . . . . . . 1.4.3 Hiérarchie de type D 4 et l'équivalence DR/DZ forte . . . . . . . . . . . The Korteweg-de Vries equation . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Korteweg-de Vries hierarchy . . . . . . . . . . . . . . . . . . . . . 2.2.3 Polynomial tau functions of the KdV hierarchy and the Adler-Moser polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Polynomial tau functions of the Drinfeld-Sokolov hierarchies . . . . . . . . . 2.3.1 The Drinfeld-Sokolov hierarchies . . . . . . . . . . . . . . . . . . . . . 2.3.2 Tau functions as Toeplitz determinants . . . . . . . . . . . . . . . . . . 2.3.3 Polynomial tau functions and applications . . . . . . . . . . . . . . . . 2.4 Hierarchy of type D 4 and the strong DR/DZ equivalence . . . . . . . . . . . .

1 Introduction en français 1.1 Aperçu historique de la théorie et de nos résultats . . . . . . . . . . . . . . . . 1.2 La hiérarchie de Korteweg-de Vries : un exemple canonique . . . . . . . . . . 1.2.1 L'équation de Korteweg-de Vries . . . . . . . . . . . . . . . . . . . . . 1.2.2 La hiérarchie de Korteweg-de Vries . . . . . . . . . . . . . . . . . . . . 1.2.3 Fonctions tau polynomiales de la hiérarchie KdV et les polynômes d'Adler-Moser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Fonctions tau polynomiales des hiérarchies de Drinfeld-Sokolov . . . . . . . . 1.3.1 Les hiérarchies de Drinfeld-Sokolov . . . . . . . . . . . . . . . . . . . 1.3.2 Fonctions tau comme déterminants de Toeplitz . . . . . . . . . . . . . 1.3.3 Fonctions tau polynomiales et applications . . . . . . . . . . . . . . . . 1.4 1.4.1 Théories cohomologiques des champs et les hiérarchies de Dubrovin-Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 2 Introduction in English 2.1 Historical overview of the theory and of our results . . . . . . . . . . . . . . . 2.2 The Korteweg-de Vries hierarchy: a canonical example . . . . . . . . . . . . . 2.2.1

  sur les fonctions tau polynomiales des hiérarchies de Drinfeld-Sokolov et des applications ; c'est l'objet du chapitre 4, §4.5. Nous commençons par décrire la construction des hiérarchies de Drinfeld-Sokolov dans l'approche la plus directe, bien que technique : la paire de Lax matricielle. Dans cette approche, elles généralisent largement la représentation (1.10) de l'équation KdV. Mais tout d'abord, il nous faut introduire du matériel concernant les algèbres de Lie semisimples et a nes.1.3.1 Les hiérarchies de Drinfeld-SokolovPoint de départ : une algèbre de Lie semisimple. Nous démarrons avec une algèbre de Lie nie-dimensionnelle semisimple (voir §4.1), i.e. une algèbre de Lie qui est une somme directe d'algèbres de Lie simple (i.e. qui ne contiennent pas d'idéaux non triviaux). Ces algèbres se décomposent ainsi,

  1.31) et telle que (e ad U ξ ) c = 0 pour tout ξ ∈ s + , où (-)

c désigne la composante le long de Cc dans la décomposition (1.27) de g. Les composantes des fonctions U et H sont calculables récursivement. Le point clé ici est que, grâce aux relations de Heisenberg (1.30), l'algèbre s -[[t]] est abélienne ! On peut à présent formuler la paire de Lax matricielle des hiérarchies de Drinfeld-Sokolov. Soit une connexion L ∈ op g , la hiérarchie de Drinfeld-Sokolov de type g (ou la hiérarchie g-DS) consiste en le système d'équations suivant :

  Ici nous expliquons la construction d'une fonction tau générique pour toute hiérarchie de Drinfeld-Sokolov par C.-Z. Wu[START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF]. Puis nous expliquons comment réaliser ces fonctions tau comme la limite de déterminants de Toeplitz par Cafasso et Wu[START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF][START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF]. Finalement, nous décrivons les résultats obtenus dans[START_REF] Cafasso | Drinfeld-Sokolov hierarchies, tau functions, and generalized Schur polynomials[END_REF] sur le calcul et les applications des fonctions tau polynomiales à partir des travaux de Cafasso et Wu.Fonctions tau des hiérarchies de Drinfeld-Sokolov. Dans[START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF], Wu dé nit une fonction tau générique pour n'importe quelle hiérarchie de Drinfeld-Sokolov. Tout d'abord nous introduisons une fonction

  .33) L'équation ci-dessus est appelée la condition de symétrie tau pour les densités conservées hi . Par le lemme de Poincaré, il s'en suit que l'on peut dé nir une fonction τ ∈ C [[t]] telle que27

	∂ log τ ∂x ∂t j	= hj .	(1.34)

On appelle τ la fonction tau. La fonction tau de Wu est générique dans le sens où sa construction marche pou toute algèbre a ne (les dé nitions précédentes étaient quelque peu au cas par cas). Maintenant en considérant la série formelle Θ = e U e Ω (nous montrons qu'elle est bien dé nie), on peut réexprimer la fonction tau par la relation

∂ log τ ∂t j = (ΘΛ j Θ -1 ) c (1.35)

Fonctions tau comme déterminants de Toeplitz. Dans [CW15a, CW15b], Cafasso et Wu montrèrent comment exprimer la fonction tau τ associée à la fonction Θ comme limite de déterminants de Toeplitz. Tout d'abord, nous xons une représentation dèle de trace nulle 28 π : g → gl(n) comme à l'appendice A. Les matrices de Teoplitz sont des matrices de taille « Z ≥0 × Z ≥0 » constantes le long des diagonales, i.e. de la forme

  Bien entendu, ce n'est pas clair que le déterminant de T(γ ) peut être dé ni et c'est un aspect technique de cette approche. Pour ce faire, on considère le matrice de Toeplitz tronquée T N (γ ) = γ I -avec I , ∈ {1, . . . , N }.À la section 4.4 nous expliquons le résulta suivant de[START_REF] Cafasso | Tau functions and the limit of block Toeplitz determinants[END_REF][START_REF] Cafasso | Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies[END_REF]. Considérons une élément quelconque X ∈ g -et la fonction suivante que nous appelons la fonction Xi matricielle,

	γ 2 γ 1 γ 0	. . .
	. . .	. . .	. . .

, où les γ I sont certains coe cients que, dans notre cas, nous choisissons dans gl(n) [[t]]. On rassemble ces éléments dans une série formelle γ = I ∈Z γ I λ I ∈ gl(n)[[λ, λ -1 , t]] (même λ que précédemment).

  1.4 Hiérarchie de type D 4 et l'équivalence DR/DZ forte +1,n , qui recolle une courbe sur elle-même en identi ant ses deux derniers points marqués.Soit un point x ∈ M ,n , i.e. la classe d'isomorphisme d'une courbe, disons C. Alors il peut être montré que π -1 (x) ∈ M ,n+1 est en réalité isomorphe à la courbe C elle-même (avec l'action de son groupe d'automorphisme). On peut alors dé nir des sections s

	Finalement, nous décrivons les résultats concernant la con rmation de l'équivalence Double
	Rami cation / Dubrovin-Zhang forte (DR/DZ) ; un travail accepté pour publication [DR19].
	C'est l'objet du chapitre 5. Tout d'abord, nous expliquons brièvement le concept de théorie
	cohomologique des champs (CohFT, pour cohomological eld theory), l'exemple de la théorie
	de Fan-Jarvis-Ruan-Witten (FJRW) et la construction de Dubrovin-Zhang.
	1.4.1 Théories cohomologiques des champs et les hiérarchies
	de Dubrovin-Zhang

Espaces de module des courbes stables. Essentiellement, l'espace de module des courbes stables M ,n est l'espace qui paramétrise les classes d'isomorphisme de courbes complexes C, compactes, possiblement nodales, de genre , avec la donnée de n points x 1 , . . . , x n ∈ C appelés des marques. Les marques ne peuvent être des noeuds et doivent être distinctes. Cela permet de dire que deux courbes marquées (C, (x 1 , . . . ,

x n )) et et (C , (x 1 , . . . , x n )) sont isomorphes s'il existe un biholomorphisme ϕ : C → C tel que ϕ(x i ) = x i .

On dit qu'une telle courbe est stable si son groupe d'automorphisme est ni. L'hypothèse de stabilité est nécessaire a n d'obtenir un espace de module bien dé ni ; elle donne la condition suivante, 2 -3 + n ≥ 0.

On dé nit alors l'espace de module des courbes stables de genre avec n marques comme l'espace quotient suivant, 31 M ,n := { (C, (x 1 , . . . , x n ))}/ ∼ C'est une orbifold complexe compact, i.e. grossièrement, un espace topologique qui est localement isomorphe à un espace a ne Euclidien quotienté par l'action d'un groupe ni. En tant qu'orbifold, elle a la dimension complexe suivante,

dim C M ,n = 3 -3 + n.

Cette orbifold vient avec plusieurs applications et brés importants. Les détails peuvent être trouvés à la section §5.2.1 ; nous référons à

[START_REF] Zvonkine | An introduction to moduli spaces of curves and its intersection theory[END_REF]

. Tout d'abord, le groupe symétrique S n agit sur M ,n en permutant les points marqués. Puis on a trois applications naturelles : L'application d'oubli, π : M ,n+1 → M ,n , qui oublie le dernier point marqué ; L'application de recollement séparante, σ : M 1 ,n 1 +1 × M 2 ,n 2 +1 → M 1 + 2 ,n 1 +n 2 , qui recolle deux courbes sur leur dernier point marqué ; L'application de recollement non séparante, τ : M ,n+2 → M

  Les classes ψ 1 , . . . ,ψ n sont appelées de classes psi et les classes λ 1 , . . . , λ sont appelées les classes lambda.

Théories cohomologiques des champs. Les théories cohomologiques des champs (CohFT) furent introduites par Kontsevich et Manin

[START_REF] Manin | Gromov-Witten classes, quantum cohomology, and enumerative geometry[END_REF]

. Une CohFT consiste en la donnée d'un espace vectoriel complexe V , avec un vecteur spécial e 1 , équipé d'une métrique η = (η α β ), ainsi que d'applications

c ,n : V ⊗n → H * (M ,n , C)

qui satisfont les propriétés suivantes (on choisit une base (e 1 , e 2 , . . . , e N ) et utilisons la notation d'Einstein) :

  Pour ce faire, nous avons utilisé la paire de Lax scalaire et la structure Hamiltonienne trouvée par Liu, Wu et Zhang[START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF]. En e et, les hiérarchies DS sont construites génériquement comme des hiérarchies de paires de Lax matricielles. Mais précisément pour le cas des hiérarchies de type D n , Drinfeld et Sokolov ne donnèrent pas une paire de Lax scalaire pour toutes les équations de la hiérarchie. A n d'énoncer le problème, il nous faut discuter l'algèbre a ne de type D(1) 4 .

	46)
	C'est en accord avec la première condition de l'équation (1.45). Les densités Hamiltoniennes
	α,0 et 1,1 dont données à la section §5.2.4.
	Il reste donc à expliquer comment nous avons calculer la hiérarchie de Drinfeld-Sokolov
	de type D 4 .

  6 .

	Cette dernière est relativement similaire à celle de la hiérarchie KdV (1.20), (1.14). Pour les ots
	par rapport aux variables tk , il n'est pas possible d'utiliser la racine 6-ième L 1/6 .
	Ce que rent Liu, Wu et Zhang [LWZ10] c'est qu'ils construisirent une nouvelle algèbre D
	qui intersecte D (équation (1.19)) mais qui ne la contient pas. Essentiellement, ils utilisèrent la
	topologie sur A induite par la gradation A =	k ∈Z A k dé nie par deg s α k = k, pour dé nir
	une structure d'anneau (avec le même produit que (1.7)) sur l'espace

  where the derivative∂ x M is de ned component-wise. Then ∂ t L = [L , M ] if and only if u t = 1 4 u x x x +

	3 4 uu x , i.e. if and only if the KdV equation (2.5) holds. In other words, the KdV equation (2.5)
	can be rewritten as

  Finally we describe the bilinear, or Hirota's, representation of the KdV equation. Contrary to the previous representations, the bilinear one acts on the tau function τ rather than on the solution u. As mentioned above, there are di erent ways of de ning the tau function, but in the case of the KdV equation, it always comes down to de ning τ by the relation21

	Representation (2.14) was found simultaneously in 1971 by Gardner [Gar71] and Fadeev
	and Zakharov [FZ71] (although Fadeev and Zakharov attribute it to Gardner); it is called the
	rst-or the Gardner-Zakharov-Fadeev Hamiltonian structure of KdV. Representation (2.15)
	is due to Magri [Mag78] and is called the second-or Magri's Hamiltonian structure. 20
	Bi-Hamiltonian representations were much studied by Dubrovin and Zhang (e.g. [DZ08,
	DLZ08]), especially in view of classifying integrable hierarchies. Moreover, the double rami -
	cation hierarchies, which we study in Chapter 5, naturally arise a Hamiltonian systems.
	(d) The bilinear representation. u = -2	∂ 2 log τ ∂x 2 .	(2.16)
	In particular, given a solution u, tau function τ is not unique. We now introduce the bilinear,
	or Hirota's, derivatives D t i , with i ≥ 1. These derivatives are de ned by the following relation:
	consider two smooth functions f , of parameters t := {t i | i ≥ 1}, we set 22
	exp
	i ≥1
		.15)
	The local functionals h 1 and h 2 are called Hamiltonian functionals and the above equations
	are called Hamiltonian representations of the KdV equation. The fact that KdV admits two
	Hamiltonian representations with two compatible local Poisson structures is what we call a
	bi-Hamiltonian representation.

Table 4

 4

			.1: Finite Dynkin diagrams
	Type	Lie algebra Dynkin diagram Type Type Dynkin diagram
	A n , n ≥ 1	sl(n + 1)	E 6
	B n , n ≥ 3 o(2n + 1)	E 7
	C n , n ≥ 2	sp(2n)	E 8
	D n , n ≥ 4	o(2n)	F 4
			G 2

{α 1 , . . . , α r } a set of simple roots. Let θ ∈ ∆ be the highest root w.r.t. Π and let b 0 be the normalized bilinear form, i.e. such that (θ, θ ) 0 = 2. We choose Weyl generators {E

Table 4

 4

		.2: Untwisted a ne Dynkin diagrams (or extended Dynkin diagrams)
	Type	A ne algebra Dynkin diagram Type Type Dynkin diagram

  For type D n , the parity of n matters: If n is odd, then n -1 is even and therefore n -1 {1, 3, . . . , 2n -3}, thus n -1 has multiplicity 1; if n is even, then n -1 ∈ {1, 3, . . . , 2n -3}, thus n -1 has multiplicity 2. This makes a huge di erence as one can see in Section 5.3.2. Table 4.3: Finite exponents and Coxeter number of simple Lie algebras It is not hard to see that sl(3) has Coxeter number equal to 3. Therefore, we can de ne the elements E 0 ∈ g-2 , F 0 ∈ g2 and H 0 ∈ g0 by

	Type	Finite exponents	Type Type	Finite exponents
	A n	{1, 2, 3, . . . , n}	E 6	{1, 4, 5, 7, 8, 11}
	B n	{1, 3, 5, . . . , 2n -1}	E 7	{1, 5, 7, 9, 11, 13, 17}
	C n	{1, 3, 5, . . . , 2n -1}	E 8	{1, 7, 11, 13, 17, 19, 23, 29}
	D n	{1, 3, 5, . . . , 2n -3} ∪ {n -1}	F 4	{41, 5, 7, 11}
			G 2	{21, 5}
	Remark.
				.24)

  .30) Equation (4.30) is a key fact in the building of the Drinfeld-Sokolov hierarchies.Example 4.2.4 (A ne exponents of sl(n)). Very similar computations to those of Example 4.2.3 allow to compute the a ne exponents of sl(n). Again, we treat the case of sl(3), the other cases are computed in the same fashion. Recall that

  Therefore, Equation (4.41) entirely determines the function U . Moreover, since the Λ j 's form a basis of s, then Equation (4.40) is satis ed for any ξ ∈ s + . The next proposition tells us what happens to the functions U and H of Theorem 4.3.7 when the connection L undergoes a gauge transformation. Proposition 4.3.8 (Wu, 2016). The function H of Theorem 4.3.7 is invariant with respect to gauge transformation. More precisely, let U ∈ g -[[x, t]] be the unique function such that (e ad U ξ ) c = 0 for any ξ ∈ s + and e -ad U L = ∂ + Λ + H , where H ∈ s -[[x, t]]. Then for any function S ∈ n [[x]], there exists a unique function Ũ ∈ g -[[x, t]] such that (e ad Ũ ξ ) c = 0 for any ξ ∈ s + and e -ad Ũ L = ∂ + Λ + H . (4.42) Proof. We prove that the function Ũ ∈ g [[x, t]] such that e ad Ũ = e ad S e ad U satis es the conditions of the proposition. First, by the homomorphism property of the BCH formula, and since n ⊂ g <0 , it follows that Ũ ∈ g -[[x, t]]. Second,

  + , we consider the following equations:8 Notice that since Λ is constant, then ∂ t L = ∂ t Q. Before proving that the above equations are consistent (well de ned) or integrable, we give two important propositions. Proofs of these propositions can be found in[START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] (Proposition 3.7). Proposition 4.3.9 (Gauge symmetry). Let L ∈ op g be a connection and let ξ ∈ s + . Assume that L satis es Equation (4.43) with ow ∂ t . Then for any function S ∈ n [[x, t]], the operator L = e ad S L also satis es Equation (4.43), i.e. ad Ũ ξ ) + , L , where e ad Ũ = e ad S e ad U . Proposition 4.3.10. If L is a solution of Equation (4.43), then there exists a gauge transformation L = e ad S L , for some S ∈ n [[x, t]], such that L is a solution of Equation (4.44). Remark 4.3.11. An important consequence of Proposition 4.3.10 is that both Equations (4.43) and (4.44) are invariant under the transformation ξ → ξ + η where η ∈ s -. This is obvious when considering Equation (4.44), and it holds true for Equation (4.43) by Proposition 4.3.10. Therefore, we can assume that ξ ∈ s + without loss of generality. With the last two propositions, we can consider Equation (4.43) modulo gauge transformations. We call the Drinfeld-Sokolov equation of type g (or g-DS equation) associated to ξ ∈ s + the Equation (4.43) modulo gauge transformation of L . In analogy to the KP hierarchy (3.18), in order to de ne the Drinfeld-Sokolov hierarchies, we need to introduce in nitely many time variables, one for each a ne exponent of g:

	∂ ∂t	L	= (e
			∂L ∂t	= (e ad U ξ ) + , L ,	(4.43)
			∂L ∂t	= (e ad U ξ ) + , L .	(4.44)

other words, hereafter, Q, U , H , etc. ∈ g [[x, t]] (or otherwise speci ed). That clearly does not a ect the previous results and will allow us to produce a dynamics.

Let L ∈ op g be a connection and let U ∈ g -[[x, t]] be a function as in Theorem 4.3.7. For ξ ∈ s

  and by constancy of Λ j and commutativity of s + , it follows that [Λ j , ∂ + Λ] = 0 as well. Hence Q = 0 is a solution.Remark 4.3.14. Similarly to the KP hierarchy (3.18), the rst equation of the g-DS hierarchy, i.e. for ξ = Λ 1 = ν Λ (with associated ow ∂ t 1 ) simply reads

	∂Q ∂t 1	= ν	∂Q ∂x	(4.46)
	Indeed, considering Equation (4.44), we clearly have (e ad U Λ 1

  Let M = e ad U ξ , then [L , M] = e ad U [L 0 , ξ ], and since ξ is constant, [L 0 , ξ ] = [Λ + H , ξ ]. The latter belongs to Cc [[x, t]] since all terms involved belong to the Heisenberg subalgebra. Since c is central, it follows that [L , M] ∈ Cc [[x, t]] as well. We easily see that by adjusting ξ , we recover all elements of the center.

		(4.48)
	Next, we prove that
	[L , M + ] ∈ g 0 ∩ g ≤0 [[x, t]] .	(4.49)

U s = Cc [[x, t]] .

(Recall that b consists of elements of g 0 ∩ g ≤0 that have no component in c.) By Equation (4.48), there exists

  De nition 5.2.6 (Admissible symmetry group). Let W ∈ C[z 1 , . . . , z m ] be a quasihomogeneous nondegenerate polynomial. A subgroup G ⊂ G max of the maximal diagonal symmetry group of W is called admissible if ∈ G.

	We call the central charge of W the rational number
	m
	ĉW :=
	i=1

Table 5 .

 5 1: Simple singularities and related simply-laced Dynkin diagrams

	Type	Dynkin diagram Singularity Type Type Dynkin diagram Singularity
	A n , n ≥ 1	x n+1	E 6	x 3 + 4
	D n , n ≥ 4	x 2 + n-1	E 7	x 3 + 3
			E 8	x 3 + 5

Now let W ∈ C[z 1 , . . . , z m ] be a quasihomogeneous nondegenerate polynomial and let G be an admissible symmetry group of W . Let γ ∈ G and let us denote Fix(γ

  The moduli space of rubber stable maps carries a virtual fundamental class [M ;a ] vir ∈ H 2(2 -3+n) (M ;a , C). nition 5.2.11 (Double rami cation cycle). Let a 1 , . . . , a n ∈ Z be integers with vanishing sum. We call the double rami cation (DR) cycle associated to the rami cation pro le a and denote DR (a) the pushforward of the virtual fundamental class of of the moduli space M ;a by the forgetful morphism π : M ;a → M ,n , i.e.DR (a):= ϕ * [M ;a ] vir ∈ H 2(2 -3+n) (M ,n , C).The DR cycles were introduced by Graber and Vakil[START_REF] Graber | Relative virtual localization and vanishing of tautological classes on moduli spaces of curves[END_REF]. It follows that the Poincaré dual P(DR (a)) of the DR cycle belongs to H 2 (M ,n , C). Moreover, it is known that P(DR (a)) lies in the tautological ring of M ,n[START_REF] Faber | Relative maps and tautological classes[END_REF]. As an immediate consequence of its de nition, the pullback of the DR cycle DR (a) by the forgetful map π 1 : M ,n+1 → M ,n reads π * 1 DR (a 1 , . . . , a n ) = DR(a 1 , . . . , a n , 0).

	De

n (P 1 ) f : C → P 1 satis es (5.33) C * , where C * acts on the target P 1 . There is a natural forgetful morphism ϕ : M ;a → M ,n , which simply returns the underlying stable curve (C; (x 1 , . . . , x n )).

J'utilise ici et là des terminaisons non standards et non genrées. Je me le permets, ce sont après tout mes remerciement. C'est assez simple, le x est muet.

Un doux mélange inclusif entre celles et ceux.

Notons que l'algèbre de Kac-Moody a ne associée à l'algèbre de Lie nie-dimensionnelle semisimple g est en réalité une extension co-centrale de l'algèbre g de (1.1) (voir [Kac90] §7). Cependant, les hiérarchies de Drinfeld-Sokolov n'impliquent pas l'extension co-centrale ; appeler g un algèbre a ne est assez standard.

La hiérarchie de Kadomtsev-Petviashvili contient toutes les hiérarchies de Gelfand-Dickey dans le sens où toute solution d'une hiérarchie de Gelfand-Dickey est aussi solution de la hiérarchie de Kadomtsev-Petviashvili (l'inverse étant faux).

Il doit être noter que la même école de Kyoto inventa, durant la même période, la théorie des fonctions tau pour les équations de Painlevé et les problèmes isomonodromiques[START_REF] Miwa | Painlevé property of monodromy preserving equations and the analyticity of τ -function[END_REF][START_REF] Miwa | Monodromy preserving deformations of linear ordinary di erential equations with rational coe cients i[END_REF]. Mais ce n'est pas la théorie que nous discutons ici.

Pour la clari er la chronologie : cet article fut originellement soumis à l'arXiv le 23 août 2001.

Pour clari er la chronologie : cet article fut originellement soumis à l'arXiv le 26 mars 2012.

Dans certain cas, il se peut qu'en réalité une certaine puissance entière τ a de la fonction tau soit polynomiale (dans les cas que nous avons calculés, a = 1 ou 2). Mais, de façon surprenante, dans tous les exemples que nous avons calculés, la fonction tau elle-même était polynomiale ; i.e. τ était un carré parfait.

Ce résultat di ère des travaux originaux de Sato[START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF] en ce qu'ici, on n'a plus besoin de considérer des combinaisons linéaires de polynômes de Schur.

La semisimplicité signi e qu'une certaine algèbre associée est semisimple, i.e. somme direct d'algèbres simples.

Ces cas sont précisément ceux pour lesquels la hiérarchie de Drinfeld-Sokolov est construite à partir de l'algèbre a ne associée à une algèbre de Lie simple de type A n ≥1 , D n ≥4 ou E 6 , E 7 , E 8 (la classi cation ADE). Dans les cas restant, les hiérarchies de Dubrovin-Zhang et de Drinfeld-Sokolov sont aussi équivalentes dans le sens d'une transformation quasi-Miura (i.e. rationnelle).

Parmi ceux-là, les exemples de correspondance « CohFT / Hiérarchie int. » que nous avons cités jusque là : Nombres d'intersection sur M ,n / Hiérarchie KdV; Classe r -spin de Witten / Hiérarchies Gelfand-Dickey ; Théorie

En e et, si u est vu comme un potentiel en mécanique quantique, alors l'équation de Schrödinger normalisée sur la fonction d'onde ψ devient i ψ t = -2 ψ x x + uψ .

Pour les non spécialistes, la composition des opérateurs di érentiels donnée à l'équation (1.7) vient simplement de l'action sur des fonctions test. Par exemple, ∂x (f ∂ x (ψ )) = ∂ x (f ψ x ) = f x ψ x + f ψ x x ,ce qui est en accord avec la formule ∂ x • f ∂ x donnée par l'équation (1.7).

Le facteur (-2) est expliqué à la proposition 3.4.5 ; il ne peut être éliminé puisque KdV est non linéaire.

Notons que lorsqu'on développe l'identité (1.17) le symbole D t i 1 • • • D t in est un opérateur en soi et non un produit d'opérateurs.

Elle fut en réalité découverte pour la première fois par Andrew Lenard durant l'été 1967, bien qu'il n'est fameusement jamais écrit de papier sur le sujet. Voir[START_REF] Praught | A Mystery Unraveled[END_REF] pour les détails de cette découverte et comment cela fut reconnu par Lax, Miura et al.

L'équation (1.8) donne une exemple de ces connexions L avec g = sl(2, C), bien que nous ayons même choisi Q ∈ n+ [[t]] puisque pour sl(2, C), on a n+ = CE.

L'équation (1.34) est équivalente au fait de poser ∂ t j log τ = -Ω j . Nous insistons sur le fait que puisque les fonctions Ω j ne sont pas des polynômes di érentiels, cela n'implique pas que les hj soient des dérivées exactes de polynômes di érentiels. En particulier, l'équation (1.34) n'implique pas que ∫ hj = 0.

Par gl(n) on entend Mat(n, C) (matrices n ×n complexes) équipée du crochet de Lie dé nie par le commutateur.

Les problèmes de convergences sont traités de façon purement algébrique via la topologie induite par la valuation. C'est expliqué au début de la section 4.4

Il est important de noter que, dans l'équation (1.37), les indices i, j représentent les vraies composantes complexes des matrices de Laurent r et s(t). Ils ne représentent pas les indices de block qui étaient noté avec des indices capitaux I, jusque là.

La bar sur M ,n vient du fait qu'il s'agit en réalité de la compacti cation de l'espace M ,n qui paramétrise les mêmes courbes mais non nodales. Ce dernier n'est pas compact malheureusement.

Il s'agit bien d'une correspondance « CohFT / Hiérarchie int. » dans le sens où le potentiel de la CohFT semisimple est une fonction tau de la hiérarchie DZ, mais permet aussi de construire la structure de Poisson locale ainsi que toutes les densités Hamiltoniennes. En retour, la fonction tau topologique, vue comme le potentiel, contient par construction toute les classes de cohomologie et donc, toute l'information de la CohFT.

 33 Pour des raisons apparentes dans cet exemple, on appelle souvent F la fonction tau.

The fact that we only consider ows w.r.t. variables t i with i 0 mod n simply comes from the fact that for i = jn, the ow is null since then (L jn/n ) + = L j commutes with L.

The de nition of the n-th root necessitates the introduction of pseudo-di erential operators; see §2.2.2.

The Kadomtsev-Petviashvili hierarchy contains all the Gelfand-Dickey hierarchies in the sense that any solution of any of the Gelfand-Dickey hierarchies is also a solution of the Kadomtsev-Petviashvili hierarchy (the converse being wrong).

It should be noted that the same Kyoto school invented, around the same time, the theory of tau functions for Painlevé equations and isomonodromic problems[START_REF] Miwa | Painlevé property of monodromy preserving equations and the analyticity of τ -function[END_REF][START_REF] Miwa | Monodromy preserving deformations of linear ordinary di erential equations with rational coe cients i[END_REF]. But this not the theory we discuss here.

To clarify the chronology: this article was originally submitted to the arXiv on August 23 2001.

To clarify the chronology: this article was originally submitted to the arXiv on March 26 2012.

In certain cases, it can actually be that some integer power τ a of the tau function τ is polynomial (in the cases we computed, a = 1 or 2). But interestingly enough, in all cases we computed, the tau function itself was polynomial; i.e. when a = 2, τ was a perfect square. We have not proven this fact and only observed it on examples.

This result di ers from the original work of Saot[START_REF] Sato | Soliton Equations as Dynamical Systems on an In nite Dimensional Grassmann Manifolds[END_REF] for Kac and van de Leur they did not have to consider linear combinations of Schur polynomials.

Among them, the examples of mirror symmetry "CohFT / Int. hierarchy" we have cited so far: Intersection numbers of M ,n / Korteweg-de Vries hierarchy; Witten's r -spin classes / Gelfand-Dickey hierarchies; Gromov-Witten theory of P 1 / Extended Toda hierarchy.

 13 Dubrovin-Zhang construction actually applies to very much related objects called Frobenius manifolds, but we will not focus on this approach.

Indeed, if u is seen as a potential in quantum mechanics, then the normalized Schrödinger equation on a wave function ψ reads i ψ t = -2 ψ x x + uψ .

For the unaccustomed reader, the composition of di erential operator given in Equation (2.7) simply comes from the action on test functions. For example, ∂x (f ∂ x (ψ )) = ∂ x (f ψ x ) = f x ψ x + f ψ x x ,which agrees with the formula for ∂ x • f ∂ x given by Equation (2.7).

In Equation (2.12), the di erential polynomials f , ∈ A are any representative of the classes f , ∈ F (i.e. f = f + Im(∂ x )). This is well de ned since δ /δu vanishes on Im(∂ x ).

Since F is not a ring, we cannot state Leibniz's rule for local Poisson structures. Therefore, (F , {-, -} a ), for a = 1, 2, is a Lie algebra, but not a Poisson algebra. Nonetheless, for any local functional ∈ F , the maps f → { f , } a of Equation (2.13) satisfy Leibniz's rule w.r.t. f .

It was actually rst discovered by Andrew Lenard during the summer 1967, although he famously wrote no paper on the subject. See[START_REF] Praught | A Mystery Unraveled[END_REF] for the details of that discovery and how it was acknowledged by Lax, Miura et al.

Equation (2.8) gives an example of the connection L with g = sl(2, C), although for this one we even chose Q ∈ n+ [[t]] since for sl(2, C), we have n+ = CE.

All issues of convergence are treated in a purely algebraic way using the valuation topology. This is explained at the beginning of §4.4

Very much importantly, in Equation (2.37), the indices i, j refer to the actual complex entries of the Laurent matrices r and s(t). They do not refer to the matrix entries which were denoted by capital indices I, so far.

The bar on M ,n comes from the fact that it is actually the compacti cation of another space, M ,n , which parameterizes the same curves yet not nodal. The latter is not compact unfortunately.

It is a correspondence "CohFT / Int. hierarchy" in the sense that the potential of the semisimple CohFT is a tau function of the DZ hierarchy, but also allows to construct the local Poisson structure as well as all the Hamiltonian densities. On the other hand, the topological tau function, seen as the potential, contains by construction all the cohomology classes and therefore, all the information of the CohFT.

Note that here,F = A/Im(∂ x ) where A = C[[s α 0 ; ε]][s α k >0 ].

Notice that here we only need to compute one of the two Hamiltonian representations of the hierarchy. This is because what we want is to compare the Hamiltonian structures of two a priori distinct hierarchies. The bi-Hamiltonian property has many interest, but we do not need it here.

The notation a i for i-th coe cient of X is not the most standard. But in the sequel, and especially in Chapter 5, we will denote u α k := ∂ k x u α (as is customary in [DZ08,[START_REF] Dubrovin | Frobenius manifolds and central invariants for the Drinfeld-Sokolov bihamiltonian structures[END_REF][START_REF] Buryak | Recursion Relations for Double Rami cation Hierarchies[END_REF] to cite a few). Therefore, in what follows, it is the subscript that will denote the order of di erentiation.

In their original paper[START_REF] Adler | On a Class of Polynomials Connected with the Korteweg-De Vries Equation[END_REF] the authors use the variables τ i instead. But since then, the letter τ has been used rather for the tau functions. Moreover, we use odd indices so that we can later complete the set into the variables of KP.

We emphasize the fact that η(t; λ) does not depend on t 1 = x, contrary to ξ (t; λ).

We denote g the semisimple Lie algebra; the notation g wille be used for the associated a ne Lie algebra, as is done in[START_REF] Kac | In nite dimensional Lie algebras[END_REF][START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF].

For some authors, the simplicity condition does not require nite-dimensionality. In our context, this requirement is rather customary and convenient.

Notice that the actual a ne algebra is a co-central extension g = g[λ, λ -1 ] ⊕ Cc ⊕ Cd, i.e. such that its derived algebra reads g = g[λ, λ -1 ] ⊕ Cc[START_REF] Kac | In nite dimensional Lie algebras[END_REF]. In other words, here we call a ne algebra what is actually its derived algebra. This denomination is convenient here since the co-central element d would play no role.

Again, following Remark 4.2.1, a ne Cartan matrices are actually more general than the ones that arise in the above construction; here we only discuss the case of untwisted a ne Cartan matrices.

Note that in[START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] the function U is not unique.

We chose the convention L 0 = e -ad U L instead of L 0 = e ad U L (as is done in[START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] for the latter) so that the Drinfeld-Sokolov equations take the form (4.43).

The fact that we denote the ows of Equations (4.43) and (4.44) both by ∂ t is an anticipation of Proposition 4.3.10; the latter justi es that the notation is consistent.

Compare (4.48) with Lemma 6.3 of[START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF], which states that [L , s ] = {0}.

Notice that since [ Ã, ξ ] = 0, we also have [∂ t + Ã, ξ ] = 0. So it seems as though we could have proved that [∂ t + A, M] = 0 as well. Yet that is not true because the relation between A and à is precisely given by ∂ t + à = e -ad U (∂ t + A), so that à cannot be replaced by -Ã. The relation [∂ t + Ã, ξ ] = 0 holds true as well, but does not imply [∂ t + A, M] = 0.

We shall emphasize that the terminology tau function here anticipates the fact that it will indeed be identiti ed with a tau function of the DS hierary, see Section 4.4.2.

Notice that each expressionD i 1 • • • D i k is abilinear operator in itself is not the product of D i 1 to D i k (this would actually not make sense).

The "H" in "HDS" stands for Ryugo Hirota who rst introduced the bilinear derivative, also called Hirota derivatives[START_REF] Hirota | Direct method of nding exact solutions of nonlinear evolution equations[END_REF].

X 2 is not the most general upper triangular element of homogeneous degree -1, as the tau function for the most general case is too big.

The traditional notation for F is rather Λ, but we wanted to avoid confusion with the principal cyclic element Λ of an a ne algebra, which we will use in the current chapter as well. The notation F is used in[START_REF] Liu | On the Drinfeld-Sokolov hierarchies of D type[END_REF].

It is also frequent to do not divide by C1 in the quotient (5.1); for our purpose is rather customary and convenient.

In Equation (5.9), the expression e inx could have been replaced by z n for some indeterminate z and therefore does not need to introduce e inx = (inx) k /k!. The reason why we write speci cally e inx is because of the factor (in) k which agrees with the action of ∂ x on A.

The choice of having two indices α, p for the Hamiltonian densities h α,p seems arbitrary and is not necessary in order to de ne the tau symmetry property. However, it is actually intrinsicly linked to the de nition of a tau structure. Moreover, such a way of ordering Hamiltonian densities makes sense when working with the Drinfeld-Sokolov hierarchies since the set of a ne exponent has a natural periodic structure modulo the Coxeter number (see Appendix A).

This Omega function should of course be compared with the Omega function introduced by Wu; see Equation (4.57). The latter however was matrix-valued.

We say that a node separates the curve if its removal disconnects the curve.

Notice that this explains the vanishing sum condition a i = 0 (5.32). Indeed, for f : C → P 1 , Cauchy's argument principles imposes that the sum of the order of the zeros of f equals the sum of the order of its poles (see e.g.[START_REF] Miranda | Algebraic Curves and Riemann Surfaces[END_REF], Corollary 3.18, p. 124).

In[START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF] and other references on the topic (e.g.[START_REF] Buryak | Recursion Relations for Double Rami cation Hierarchies[END_REF][START_REF] Rossi | Integrability, Quantization and Moduli Spaces of Curves[END_REF]), the series ˆ α,d is actually denoted α,d . However, we denote it ˆ α,d while α,d will be kept for the di erential polynomial we associate to ˆ α,d through the map T -1 0 (although both representations are equivalent in virtue of the previous sections' content).

Hereafter, we will use µ, ν to denote the coordinates s µ , with µ ∈ {1, . . . , n-1} and α, β to denote the coordinates s α , with α ∈ {1, . . . , n}.

Remerciements

Remerciements

Since Λ ∈ s 1 and Q = f 1 ⊗ q ∈ g -1 , the equation in principal degree 0 reads 0 = f 1 ⊗ u 1 , Λ , so that f 1 ⊗ u 1 ∈ Ker(ad Λ ) ∩ Im(ad Λ ). Therefore, u 1 = 0. Then, the equation in principal degree -1 reads

from we get that u 2 = -1 4 q and also h 1 = 1 2 q. Keeping on with the computations, we nd that the components of U we are interested in read u 1 = 0, u 2 = -1 4 q, u 3 = 1 4 q (1) , u 4 = 1 8 q (2) -5 16 (q) 2 .

We sum up these in the matrix form of U :

Computing the KdV equation

Now that we have the matrix U ∈ (s) -[[x, t 3 ]], we need to compute (e ad U Λ 3 ) + . Since U ∈ (s ⊥ ) ≤1 [[x, t 3 ]] and Λ 3 ∈ g 1 ⊕ g 2 , we get that (e ad U Λ 3 ) + = Λ 3 + U , Λ 3 + 1 2 U , U , Λ 3 .

We nd (e ad U Λ 3 ) + = 1 4 q (1) λ 2 -1 2 qλ + 1 4 q (2) -1 2 (q) 2 λ -1 2 q -1 4 q (1)

Together with the connection L = ∂ + Λ + Q, the above is a well known matrix Lax pair of the KdV equation; indeed, ∂L ∂t 3 -(e ad U Λ 3 ) + , L = 0 ∂ t 3 q -1 4 q (3) + 3 2 qq (1) 0 0 .

So, the above equation holds if and only if ∂ t 3 q = 1 4 q (3) -3 2 qq (1) . Substituting q = -u, we get that the equation along t 3 of the A 1 -DS hierarchy holds i.f.f.

which is the KdV equation.

Toeplitz and Hankel block matrices

Besides the Laurent matrix of a loop γ = I ∈Z γ I λ I , we introduce the two following block matrices; all of them are Z ≥0 × Z ≥0 matrices:

The matrix T(γ ) we call the block Toeplitz matrix associated to γ the matrices H(γ ) and H(γ ) we call the Hankel block matrices. These matrices can be visualized as follows:

γ 1 γ 0 γ -1 . . .

Is is clear that ι 2 = id and that the restrictions ι : H + → H -and ι : H -→ H + are bijective. Then it is not hard to show that the Toeplitz and Hankel matrices are the matrix representations of the following endomorphisms:

Now the Toeplitz and Hankel matrices are not homomorphisms, contrary to the Laurent matrix, yet the following lemma holds [START_REF] Böttcher | Introduction to Large Truncated Toeplitz Matrices[END_REF].

Lemma 4.4.7. Given two loops γ 1 , γ 2 , the following identity holds,

In particular, T(γ 1 ) • T(γ 2 ) = T(γ 1 • γ 1 ) whenever T(γ 1 ) is upper triangular or T(γ 2 ) is lower triangular.

Proof. It su ces to use the de nition of these matrices:

Young diagram expansion

The aim of this section is to prove the following theorem. (i.e. the integer of which µ is a partition), then Equation (4.93) is a well de ned power series for this grading: each term of a given degree is a nite sum of s µ r µ 's and each s µ r µ is a nite determinant. Therefore, besides all that was said in the previous sections, Equation (4.93) gives an intrinsic formal expansion of the tau function.

Remark 4.5.4. In the A n case, if one chooses the fundamental representation A n sl(n + 1), then the Schur polynomials s µ (t) are exactly the traditional Schur polynomials [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF] but with the constraint t k (n+1) = 0.

Remark 4.5.5. Once Theorem 4.5.2 is proven, formula (4.94) is due to Giambelli type formula, see e.g. [START_REF] Balog | Geometric interpretation of Zhou's explicit formula for the Witten-Kontsevich tau function[END_REF][START_REF] Enolskii | Schur function expansions of KP tau-functions associated to algebraic curves[END_REF][START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF].

To prove Theorem 4.5.2 we will proceed in several lemmas. The rst one is a well known result, although we wanted to present a rigorous proof to the reader.

Lemma 4.5.6. Let Y (m) be the following subset of Y,

Let two rectangular matrices, S of size (m + 1) × (2m + 1) and R of size (2m + 1) × (m + 1), of the speci c form Chapter 5

Hamiltonian structure of the hierarchy of type D 4 and the strong DR/DZ equivalence

The goal of this chapter is to con rm the strong double rami cation / Dubrovin-Zhang (DR/DZ) equivalence in the case of the hierarchy of type D 4 . To do so, we describe the Hamiltonian structure of the DR hierarchy associated to the FJRW theory of type D 4 . Then we describe the Hamiltonian structure of the Drinfeld-Sokolov hierarchy of type D 4 , knowing that it coincides with the corresponding Dubrovin-Zhang hierarchy [START_REF] Givental | Simple singularities and integrable hierarchies[END_REF][START_REF] Frenkel | Soliton equations, vertex operators, and simple singularities[END_REF]. Then using results of [START_REF] Buryak | Towards a description of the double rami cation hierarchy for Witten's r -spin class[END_REF], we conclude on the equivalence (actually coincidence in this case) between the DR and DS (hence DZ) hierarchy of type D 4 . The st section is dedicated to the framework of in nite-dimensional Hamiltonian structures.

In nite dimensional Hamiltonian structures

In this section we describe the purely algebraic approach to de ning in nite-dimensional Hamiltonian structures. Di erent authors contributed to constructing this framework, among which Lax [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF][START_REF] Lax | Periodic Solutions of the KdV Equation[END_REF], Gardner, Kruskal, Miura and Zabusky [GKMZ70], Gelfand and Dickey [START_REF] Gelfand | Fractional powers of operators and Hamiltonian systems[END_REF][START_REF] Dickey | Soliton Equations and Hamiltonian Systems[END_REF], played major roles. However, important contemporary de nitions and notations were introduced by Dubrovin and Zhang [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF]. Here we take material from [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF][START_REF] Buryak | Double rami cation cycles and integrable hierarchies[END_REF][START_REF] Rossi | Integrability, Quantization and Moduli Spaces of Curves[END_REF]. Some notions, such as the algebra of di erential polynomials and the space of local functionals, were already introduced in the previous chapters. However, we will generalize these notions.

Di erential polynomials, local functionals

Let us x an integer N ≥ 1 and consider the following commutative algebra of polynomials in in nitely many variables,

We can then rewrite the Poisson bracket as

We denote

It is easy to see, using the rules de ned for the symbol δ , that

Notice the following useful identity, which is only a consequence of Leibniz's rule,

(5.17)

Proposition 5.1.8. Let φ : A → A, ũα := φ(u α ) be a Miura transform. Given a local Poisson structure K : F ⊗ F → F , the Miura transform φ acts on K via the following formula,

where the matrices M and M * are given by

where the composition of di erential operators (in particular, with a function) is still the one dened in Section 3.1.

Proof. Let us start by developing the rst variational derivative:

We can now integrate w.r.t. the variable z and use Equation (5.17) to nd

where

and accordingly for (M * ) α µ (x, z ). Now by developing the second variational derivative and using Equation (5.17) again, we nd (notice that we can swap and z in δ ( -z ))

Hence the result.

In classifying all possible local Poisson structures modulo Miura transforms, the following powerful Darboux-like theorem is due to Erza Getzler.

Theorem 5.1.9 ([Get02]). Let K be a local Poisson structure with genus 0 part

If α β (u) is nondegenerate, than K is Miura equivalent to a standard Poisson structure. More precisely, there exist a unique constant symmetric invertible matrix η α β ∈ C and a Miura transform φ : A → A such that

Tau symmetry and general notion of tau function

We end this section with the de nition of tau structure for Hamiltonian densities as given in [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF]. We have already encountered tau symmetry in Chapter 2, §4.3.4, where we followed the approach of Wu [START_REF] Wu | Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchies[END_REF] to de ne a tau function for any Drinfeld-Sokolov hierarchy. His achievement was in particular to o er a universal construction (rather than a case by case study). Wu's construction is then a speci c case of a much broader de nition of tau function.

Let us make clear the vocabulary thereafter. In the context we are about to describe, a tau structure is a speci c choice of Hamiltonian densities for a Hamiltonian system which satisfy where φ * (η α β ∂ x ) is given in (5.18). Then the DR hierarchy, under the Miura transform φ, coincides with the DZ hierarchy.

It what follows we give the standard Poisson structure η α β ∂ x and the rst Hamiltonian densities α,0 and 1,1 of the DR hierarchy constructed from the CohFT associated to the FJRW theory of the singularity

with symmetry group G = = (e 2iπ /3 , e 2iπ /3 ) . Then we give the Poisson structure and the Hamiltonian densities after the Miura transform ũα . The details of the computations are given in the next section.

Hamiltonian densities and Poisson structure in the coordinates u α of the DR hierarchy of type D 4

The local Poisson structure of the DR hierarchy is the standard one and reads

Using Equation (5.37), we computed the Hamiltonian densities β,0 , for β ∈ {1, . . . , 4}. The latter, through the tau structure, will allow us to compute the normal coordinates ũα . We obtain the following densities.

Thanks to Theorem 5.2.24, it su ces to compute the Hamiltonian functional 1,1 to check if the DZ and the DR hierarchy coincide (if the condition of the theorem are satis ed, but we will see that is the case in the next section). We nd

Hamiltonian densities and Poisson structure in normal coordinates ũα of the DR hierarchy of type D 4

Now, since the tau structure is given by h α,p = δ α ,p+1 δ u 1 (5.41), the normal coordinates associated to said tau structure are given by ũα = η α β δ β,0 δu 1 .

We nd the following Miura transform.

(5.45)

In particular, the above satis es hypothesis ∂ ũα /∂u 1 = δ α,1 of Theorem 5.2.24. The latter is necessary, yet of course not su cient for the theorem to apply. Using to Equation (5.18), we get the following proposition.

δ u 1 and the Miura transform (5.45), we can compute the Hamiltonian densities h α,0 in normal coordinates. Notice that the Hamiltonian densities h α,0 are not necessary to apply Theorem 5.2.24 (in the view of con rming the DR/DZ conjecture); we give them here for the sake of describing the DR hierarchy of type D 4 .

Proposition 5.2.26. The level 0 tau symmetric Hamiltonian densities h 1,0 , . . . , h 4,0 of the DR hierarchy of type D 4 in normal coordinates are given by the following. Proposition 5.2.27. The tau symmetric Hamiltonian functional h 1,1 of the DR hierarchy of type D 4 in normal coordinates is given by the following. From the data of the level 0 Hamiltonian densities h 0, β , we can compute the level 0 equations of motion in normal coordinates. The computations below agrees with those of [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF] (see §5.1 on the hierarchy of type D 4 ). First, as we already know, ũα

Then we have the following.

Details on the computation of the Hamiltonian density 1,1

Here we describe how we computed explicitly the fundamental Hamiltonian density 1,1 , which is the key object in con rming the DR/DZ conjecture here. And the computation of 1,1 has to do with the intrinsic grading of the CohFT. We refer to [START_REF] Du Crest De Villeneuve | Quantization of the D 4 Drinfeld-Sokolov hierarchy and Fan-Jarwis-Ruan-Witten theory[END_REF] for more details. Consider an arbitrary CohFT with vectors space V and suppose that V is graded with deg e 1 = 0. We say that the CohFT is homogeneous if there exists δ ∈ C such that the maps c ,n :

In this case the formal variables u α k and ε acquire a grading too and the Hamiltonian densities of the resulting DR hierarchy are homogeneous:

Notice importantly that this grading | • | is not related to the di erential grading on A and F .

Integrable systems of DR type

We now need introduce the concept of integrable hierarchies of double rami cation type [START_REF] Buryak | Integrable systems of double rami cation type[END_REF]. Let {•, •} denote the Poisson structure associated to the Hamiltonian operator η µν ∂ x . For a local functional h ∈ F [0] consider the operator

Suppose there exist

Then a new vector of solutions in the same class can be found by the following transformation

where

The following result constitutes the main technical tool in the computation the the Hamiltonian density 1,1 of DR hierarchy of the D 4 CohFT.

Theorem 5.2.28 ([BDGR16]). Assume that h ∈ F [0] has the following properties:

(a) there exist

Then, up to a transformation of type (5.47), we have

hence in particular h is part of an integrable tau-symmetric hierarchy.

We call a system of densities originating from an Hamiltonian h = 1,1 as in the theorem above an integrable system of double rami cation type. As proven in [START_REF] Buryak | Integrable systems of double rami cation type[END_REF], the DR hierarchy of any given cohomological eld theory, is always and integrable system of DR type. This fact implies in particular that the entire hierarchy of DR Hamiltonian densities can be reconstructed from 1,1 ∈ F [0] alone by means of the DR recursion equation [BR16a]

(5.49)

The above result will be used here to e ectively compute the Hamiltonian 1,1 itself, starting from a limited amount of information on the CohFT, as we will show in the next section.

Computation via the Frobenius manifold of type D 4

In [START_REF] Dubrovin | Di erential geometry of the space of orbits of a Coxeter group[END_REF], based on the work of Kyoji Saito [Sai81, Sai83a, Sai83b], Boris Dubrovin constructed a structure of Frobenius manifold [START_REF] Dubrovin | Geometry of 2D topological eld theories[END_REF] on the space of orbits of any nite irreducible Coxeter group. This Frobenius manifold is generically semisimple and conformal, so Givental-Teleman [START_REF]Semisimple Frobenius structures at higher genus[END_REF][START_REF] Teleman | The structure of 2D semi-simple eld theories[END_REF] (see also [START_REF] Pandharipande | Relations on M ,n via 3-spin structures[END_REF]) theory can be applied to produce a uniquely de ned homogeneous cohomological eld theory.

As already stated, we are interested in this construction for the case of the D 4 simple singularity, i.e. W = x 3 + x 2 . The resulting CohFT c ,n : V ⊗n → M ,n has vector space V = Span(e 1 , . . . , e 4 ) with cohomological degrees given by

In genus 0 the corresponding Frobenius structure can described by the Frobenius potential (see for instance [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF]) which, in particular, gives the metric

Notice that this CohFT was proved to be isomorphic to the quantum singularity theory of Fan-Jarvis-Ruan-Witten [START_REF] Fan | The Witten equation, mirror symmetry, and quantum singularity theory[END_REF] for the simple singularity W = x 3 + x 2 , with respect to the non-maximal diagonal symmetry group = Z/3Z [START_REF] Fan | Witten's D 4 Integrable Hierarchies Conjecture[END_REF]. In [START_REF] Givental | Simple singularities and integrable hierarchies[END_REF][START_REF] Frenkel | Soliton equations, vertex operators, and simple singularities[END_REF] it was proved that the Dubrovin-Zhang hierarchy for the Dubrovin-Saito CohFT associated to a Coxeter group coincides with the Drinfeld-Sokolov hierarchy of the corresponding semisimple Lie algebra. As explained above, the ε → 0 limit of both the DZ and DR hierarchies of any (semisimple) CohFT coincides with the principal hierarchy of the Frobenius manifold which is, consequently, completely determined, thanks to the recursion equation (5.39), by the Hamiltonian [0] 1,1 = 1,1 | ε =0 . To compute the latter in the D 4 case we can use (see [START_REF] Dubrovin | Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants[END_REF]) the dilaton equation of the DZ principal hierarchy (which coincides here with the dispersionless DR hierarchy), namely

(5.52)

Moreover, thanks to homogeneity of the CohFT, we know that, for any k ≥ 0,

(5.53)

In Equation (5.44), we stated that the dispersive expansion of bar 1,1 is given by

We nish this section by explaining how we computed the above started from the data of

. By de nition, only even powers of ε appear in the Hamiltonian densities of any DR hierarchy. Now remark from (5.53) that all the variables have positive degree, guaranteeing the polynomiality of 1,1 and, in fact, of all the Hamiltonian densities α,d , for all 1 ≤ α ≤ N and d ≥ -1. In particular, since |ε | = 1 6 and since in each monomials there are as many xderivatives as powers of ε, we see that any term where the power of ε is bigger than 10 is either trivial or a total x derivative.

One can then verify, by direct computation, that up to rescaling of the variable ε, there exist a unique local functional 1,1 ∈ F [0] such that the following are satis ed:

1,1 is given by (5.52); 1,1 contains only even powers of ε; the cohomological degrees satisfy | 1,1 | = 7 3 ; 1,1 satis es Equation (5.39). In practice one writes down the most general polynomial deformation of (5.52) with the given degree and notices that imposing conditions (a) and (b) of theorem 5.2.28 determines, up to rescalings of ε, all the coe cients.

Finally, to determine the correct normalization of ε, it is su cient to compute the coecient of a single monomial of 1,1 containing ε. We can do this by recalling (see for instance [START_REF] Buryak | Tau-structure for the Double Rami cation Hierarchies[END_REF]) that, for any CohFT, Coef[(u 1 1 ) 2 ε 2 ]( 1,1 ) = -1 24 dim V .

The Drinfeld-Sokolov hierarchy of type D 4

In Section 4.3, we describe the celebrated Drinfeld-Sokolov procedure [START_REF] Drinfeld | Lie algebras and equations of Korteweg-de Vries type[END_REF] which associate an integrable hierarchy of PDEs to an arbitrary a ne Lie algebra (although in the present Using ũ4 = 2ϱ, we nally nd that

The results below where obtained using the program Wolfram Mathematica. For these programmed computations, the estimates given in Propositions 3.1.7 and 3.1.8 are of particular interest.

Proposition 5.3.7. The level 0 tau symmetric Hamiltonian densities h 1,0 , . . . , h 4,0 of the DS hierarchy of type D 4 in normal coordinates are given by the following.

Finally, the density h 1,1 is given by

Again, the result was obtained using the program Wolfram Mathematica.

Proposition 5.3.8. The Hamiltonian density h 1,1 of the DS hierarchy of type D 4 in normal coordinates is given by the following.

+ 977 10080 u 3 3 u 3 5 u 3 + 15103 272160 u 3 2 u 3 6 u 3 + 1831 90720 u 3 1 u 3 7 u 3 + 19 9720 u 3 8 u 3 2 ε 8 + 340200 u 3 5 2 + 1301 25200 u 3 4 u 3 6 + 347 10080 u 3 3 u 3 7 + 4427 272160 u 3 2 u 3 8 + 89 18144 u 3 1 u 3 9 + 1 1296 u 3 10 u 3 ε 10 + 41 393120 u 3 12 ε 12

The explicit form of the rst Poisson structure.

Finally, we compute the components of the rst Poisson structure given, for two local functionals f , ∈ F , by

where

We aim to express this bracket as bi-vector using Dirac's notation of the components:

We will use the following identity to compute the Poisson structure (see Equation (5.16)):

Chapter 5. Hierarchy of type D 4 and the strong DR/DZ equivalence Now the Poisson bracket (5.71) is given in terms of the variational di erentials, so that we rst need to compute those of the coordinate functionals. To do so, we need to transform the normal coordinates ũα into the coordinates α of Equation (5.65). The coordinates ˜ α are related to the α 's by the condition L * + L = 0; they read

(5.73)

Then by identifying the two operators in Equation (5.71) and inverting Equation (5.70), we nd the following Miura from the normal coordinates ũα to the coordinates α :

(5.74)

Now recall that the variational di erentials are given by

It follows that the variational di erentials of the coordinates functionals are given by, for µ ∈ {1, 2, 3},

δ (x -z).

(5.76)

Let us denote, for α ∈ {1, 2, 3, 4},

As we can see in Equations (5.75) and (5.76), for any µ ∈ {1, 2, 3}, the pseudo-di erential operators (of the rst type) Y µ and Y 4 satisfy

Knowing this, it is therefore not hard to show that the component { ũ4 (x), ũ4 ( )} 1 of the rst Poisson structure reads (ii) The Hamiltonian densities h 1,1 and h 1,1 di er by a total x-derivative, i.e. h 1,1 = h 1,1 . Moreover, the level 0 Hamiltonian densities coincide: h α,0 = h α,0 for α ∈ {1, . . . , 4}.

We recall that, speci cally for the hierarchies of type ADE, say of type X n , the DS hierarchy constructed from the untwisted a ne algebra X (1) n coincides with the DZ hierarchy constructed from the FJRW of the simple singularity of type X n ; see e.g. [START_REF] Liu | BCFG Drinfeld-Sokolov hierarchies and FJRWtheory[END_REF]. Besides, we recall that two local functionals coincide if and only if all the variational derivatives δ /δ ũα of their di erence vanish; see Proposition 5.1.1.

Appendix A

Matrix realizations of classical Kac-Moody algebras

In this appendix, given a n × n matrix A, we denote by A at the anti-transpose of A, that is, the transpose of A with respect to the antidiagonal (or secondary diagonal). It is given by the formula

Besides, in accordance with previous notations, we denote by E i, j the matrix with 1 at the intersection of row i and column j and 0 elsewhere. When we say that the Lie algebra with (generalized) Cartan matrix A has Weyl generators {X i , Y i , H i | 1 ≤ i ≤ r }, we mean that they satisfy the relations

A.1 Classical simple Lie algebras

Here we give the main features and our chosen matrix realization of the classical simple Lie algebras. By that we mean the algebras whose Dynkin diagram is either of type A, B, C or D. The semisimple Lie algebras correspond exactly to the Kac-Moody algebras of nite type, hence the title of this appendix.

Type A n , n ≥ 1

The simple Lie algebras of type A n are isomorphic to sl(n + 1, C). They have the following features.

Coxeter number:

Finite exponents: F = {1, 2, . . . , n}.

i Dynkin diagram and nite Cartan matrix of size n × n:

(The trivial one.) Weyl generators of the representation: For i ∈ {1, . . . , n},

The simple Lie algebras of type B n are isomorphic to o(2n + 1, C). They have the following features.

Coxeter number: h = 2n. Highest root:

Finite exponents: F = {1, 3, . . . , 2n -1}. Dynkin diagram and nite Cartan matrix of size n × n:

where S = diag(1, -1, 1, . . . , -1, 1). Weyl generators of the representation: For i ∈ {1, . . . , n -1},

),

The simple Lie algebras of type C n are isomorphic to sp(2n). They have the following features.

Coxeter number: h = 2n. Highest root:

Finite exponents: F = {1, 3, . . . , 2n -1}. Dynkin diagram and nite Cartan matrix of size n × n:

Weyl generators of the representation:

The simple Lie algebras of type D n are isomorphic to o(2n). They have the following features.

Coxeter number:

If n is even: F = {1, 3, . . . , 2n -3} ∪ {(n -1) }. (We mean that when n is even, exponent n -1 has multiplicity 2.) Dynkin diagram and nite Cartan matrix of size n × n:

where S = diag(1, -1, . . . , (-1) n-2 , (-1) n-1 , (-1) n-1 , (-1) n , . . . , -1, 1). Weyl generators of the representation:

),

Finite exponents: F = {1, 1 }. Dynkin diagram and nite Cartan matrix of size 2 × 2:

(As we stated, this implies that D 2 A 1 ⊕ A 1 and is therefore semisimple rather than simple.) Weyl generators of our realization:

Except for the A (1) 1 case, which reads

Weyl generators of the representation: For i ∈ {1, . . . , n} (the ⊗1 is implicit),

To which we add the highest root triplet:

Normalization constant of the standard bilinear form: κ = 1, i.e., (X , Y ) = tr(π (X )π (Y )).

Generators of the principal Heisenberg subalgebra:

),

viii

Chapter A. Matrix realizations of classical Kac-Moody algebras

To which we add the highest root triplet:

Normalization constant of the standard bilinear form: κ = 1 2 , i.e., (X , Y ) = 1 2 tr(π (X )π (Y )).

Generators of the principal

To which we add the highest root triplet:

),

To which we add the highest root triplet:

x Chapter A. Matrix realizations of classical Kac-Moody algebras

Normalization constant of the standard bilinear form: κ = 1 2 , i.e., (X , Y ) = 1 2 tr(π (X )π (Y )). Generators of the principal Heisenberg subalgebra: Λ j = √ 2Λ j and Λ ((n-1)j) = √ 2n -2Γ j for j ∈ Z odd , where