Dubrovin duality and mirror symmetry for ADE resolutions

TL;DR

This paper establishes a duality between Frobenius manifolds from ADE Weyl groups and the quantum cohomology of ADE surface resolutions, also constructing Landau-Ginzburg mirror models via Lie theory.

Contribution

It demonstrates a duality under Dubrovin's framework linking Frobenius structures to quantum cohomology and provides a uniform Lie-theoretic method to construct mirror Landau-Ginzburg models for ADE resolutions.

Findings

Frobenius manifold structures are dual to quantum cohomology of ADE resolutions.

Constructs Landau-Ginzburg mirror models using spectral curves of affine Toda chains.

Establishes a uniform Lie-theoretic approach to mirror symmetry for ADE singularities.

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.