Mirror symmetry for extended affine Weyl groups
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.
Cite
@article{arxiv.2103.12673,
title = {Mirror symmetry for extended affine Weyl groups},
author = {Andrea Brini and Karoline van Gemst},
journal= {arXiv preprint arXiv:2103.12673},
year = {2023}
}
Comments
44 pages; v2: 54 pages, version accepted for publication on J. Ecol. Polytech (Math). A typo in Corollary 5.2 and Table 3 is fixed compared to the published version