Dubrovin duality and mirror symmetry for ADE resolutions

Abstract

We show that, under Dubrovin's notion of ''almost'' duality, the Frobenius manifold structure on the orbit spaces of the extended affine Weyl groups of type $\mathrm{ADE}$ is dual, for suitable choices of weight markings, to the equivariant quantum cohomology of the minimal resolution of the du Val singularity of the same Dynkin type. We also provide a uniform Lie-theoretic construction of Landau-Ginzburg mirrors for the quantum cohomology of $\mathrm{ADE}$ resolutions. The mirror B-model is described by a one-dimensional LG superpotential associated to the spectral curve of the $\widehat{\mathrm{ADE}}$ affine relativistic Toda chain.

26 pages. Ancillary files of this arXiv submission are accessible in the directory anc/ included in the source of this manuscript. The directory contains the Wolfram Language code used to verify Proposition 3.16 for the exceptional series. A Mathematica package verifying the main conjecture in IMRN 19 (2003), 1035-1051 is also included

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