Mirror symmetry for extended affine Weyl groups

Brini, Andrea; van Gemst, Karoline

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 907-957.

Nous donnons une construction de symétrie miroir, de façon uniforme et par des méthodes de théorie de Lie, pour les variétés de Frobenius définies par Dubrovin-Zhang sur les orbites des groupes de Weyl affines étendus, y compris les types de Dynkin exceptionnels. Le modèle miroir est donné par un superpotentiel de Landau-Ginzburg construit à partir d’une dégénérescence convenable des courbes spectrales de la chaîne de Toda affine relativiste pour le groupe de Lie-Poisson affine correspondant. Nous fournissons également plusieurs applications de ce théorème miroir. Celles-ci incluent des expressions explicites pour les coordonnées plates pour la métrique de Saito et le prépotentiel de Frobenius en tout type de Dynkin ; le calcul du degré topologique de l’application de Lyashko-Looijenga pour certaines strates des espaces d’Hurwitz en genre supérieur ; et la construction de hiérarchies hydrodynamiques bi-hamiltoniennes (à la fois dans le formalisme de Lax-Sato et hamiltonien) qui donnent des généralisations de la limite de dispersion nulle de la hiérarchie de Toda étendue.

We give a uniform, Lie-theoretic mirror symmetry construction for the Frobenius manifolds defined by Dubrovin–Zhang in [21] on the orbit spaces of extended affine Weyl groups, including exceptional Dynkin types. The B-model mirror is given by a one-dimensional Landau–Ginzburg superpotential constructed from a suitable degeneration of the family of spectral curves of the affine relativistic Toda chain for the corresponding affine Poisson–Lie group. As applications of our mirror theorem we give closed-form expressions for the flat coordinates of the Saito metric and the Frobenius prepotentials in all Dynkin types, compute the topological degree of the Lyashko–Looijenga mapping for certain higher genus Hurwitz space strata, and construct hydrodynamic bihamiltonian hierarchies (in both Lax–Sato and Hamiltonian form) that are root-theoretic generalisations of the long-wave limit of the extended Toda hierarchy.

Classification : 53D45, 14B07, 20H15
Keywords: Frobenius manifolds, mirror symmetry, integrable systems
Mot clés : Variétés de Frobenius, symétrie miroir, systèmes intégrables

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