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.

arXiv comment44 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

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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