Mirror symmetry for extended affine Weyl groups

TL;DR

This paper constructs a Lie-theoretic mirror symmetry for Frobenius manifolds associated with extended affine Weyl groups, providing explicit formulas and applications across various types including exceptional Dynkin types.

Contribution

It introduces a uniform mirror symmetry construction for these Frobenius manifolds using Landau-Ginzburg models derived from spectral curves, extending previous work to include exceptional types.

Findings

Closed-form flat coordinates and Frobenius prepotentials for all Dynkin types.

Computed the topological degree of the Lyashko-Looijenga mapping for certain Hurwitz spaces.

Constructed integrable hierarchies generalizing the extended Toda hierarchy.

Abstract

We give a uniform, Lie-theoretic mirror symmetry construction for the Frobenius manifolds defined by Dubrovin-Zhang in arXiv:hep-th/9611200 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.