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.

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

References in corpus (8)

• The Extended Bigraded Toda hierarchy
• Phase transitions, double-scaling limit, and topological strings
• Hurwitz numbers, matrix models and enumerative geometry
• Double Bruhat Cells in Kac-Moody Groups and Integrable Systems
• Five-dimensional gauge theories and the local B-model
• Counting Meromorphic Functions with Critical Points of Large Multiplicities
• The number of full exceptional collections modulo spherical twists for extended Dynkin quivers
• The Saito determinant for Coxeter discriminant strata

Cited by in corpus (3)

• Five-dimensional gauge theories and the local B-model
• Dubrovin duality and mirror symmetry for ADE resolutions
• Diagonal invariants and genus-zero Hurwitz Frobenius manifolds