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Dubrovin duality and mirror symmetry for ADE resolutions

Andrea Brini, Jingxiang Ma, Ian A. B. Strachan

TL;DR

This work connects three Frobenius-manifold realizations associated to marked ADE pairs (R, ω̂): the extended affine Weyl orbit-space (AW), the T-equivariant quantum cohomology of the ADE minimal resolution (GW), and the Landau–Ginzburg mirror on ADE spectral curves (LG). It proves mirror and Dubrovin-duality relations: M_AW is isomorphic to M_LG, and M_GW is the Dubrovin dual of M_AW, with M_LG^flat ≃ M_GW; for A_l and D_l explicit LG mirrors are obtained via residue computations, while E_l cases are settled by reducing to a finite set of initial conditions and performing numerical checks. The results unify ADE mirror symmetry with Dubrovin duality and provide a framework for connecting integrable structures and crepant-resolutions in genus-zero and beyond. Overall, the paper establishes a coherent Lie-theoretic and geometric picture linking orbit-space Frobenius manifolds, equivariant GW theory of ADE resolutions, and LG models through both exact and computational approaches, with potential extensions to higher-genus and integrable hierarchies.

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.

Dubrovin duality and mirror symmetry for ADE resolutions

TL;DR

This work connects three Frobenius-manifold realizations associated to marked ADE pairs (R, ω̂): the extended affine Weyl orbit-space (AW), the T-equivariant quantum cohomology of the ADE minimal resolution (GW), and the Landau–Ginzburg mirror on ADE spectral curves (LG). It proves mirror and Dubrovin-duality relations: M_AW is isomorphic to M_LG, and M_GW is the Dubrovin dual of M_AW, with M_LG^flat ≃ M_GW; for A_l and D_l explicit LG mirrors are obtained via residue computations, while E_l cases are settled by reducing to a finite set of initial conditions and performing numerical checks. The results unify ADE mirror symmetry with Dubrovin duality and provide a framework for connecting integrable structures and crepant-resolutions in genus-zero and beyond. Overall, the paper establishes a coherent Lie-theoretic and geometric picture linking orbit-space Frobenius manifolds, equivariant GW theory of ADE resolutions, and LG models through both exact and computational approaches, with potential extensions to higher-genus and integrable hierarchies.

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 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 resolutions. The mirror B-model is described by a one-dimensional LG superpotential associated to the spectral curve of the affine relativistic Toda chain.
Paper Structure (18 sections, 20 theorems, 165 equations, 1 figure, 2 tables)

This paper contains 18 sections, 20 theorems, 165 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Dubrovin duality and mirror symmetry
  3. Main results
  4. Organisation of the paper
  5. Setup
  6. Generalities on Frobenius manifolds
  7. Dubrovin duality
  8. Frobenius manifolds of type $(\mathcal{R}, \widehat{\omega})$
  9. Extended affine Weyl Frobenius manifolds
  10. Quantum cohomology of ADE resolutions
  11. Landau--Ginzburg mirrors from ADE spectral curves
  12. Dubrovin duality and mirror symmetry
  13. Landau--Ginzburg mirror symmetry
  14. Proof of \ref{['thm:msQH']} for the $\mathrm{A}_l$ series with arbitrary marked weight
  15. Proof of \ref{['thm:msQH']} for the $\mathrm{D}_l$ series
  16. ...and 3 more sections

Key Result

Theorem 1

For all marked ADE pairs $(\mathcal{R}, \widehat{\omega})$, we have \xymatrix{ \mathcal{M}_{\rm AW} \ar[dd]_{E^{-1/2} \circ} \ar[rr]^{\simeq} & & \mathcal{M}_{\rm LG} \ar[dd]^{E^{-1/2} \circ} \\ & \Box & \\ \mathcal{M}^\flat_{\rm AW} \ar[dr]^{\simeq} & & \mathcal{M}^\flat_{\rm L

Figures (1)

  • Figure 1: Marked Dynkin diagrams of pairs $(\mathcal{R}, \widehat{\omega})$. The marked node corresponding to the weight $\widehat{\omega}$ is indicated in black.

Theorems & Definitions (38)

  • Theorem : =\ref{['thm:msQH', 'thm:ddQH']}
  • Proposition 2.1: MR2070050MR2836400
  • Proposition 2.2: MR1606165
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5: Mirror symmetry for $\mathcal{M}_{\rm AW}$
  • Proposition 2.6
  • Theorem 3.1: Dubrovin duality for $\mathcal{M}_{\rm AW}$ and $\mathcal{M}_{\rm GW}$
  • Theorem 3.2: Mirror symmetry for $\mathcal{M}_{\rm GW}$
  • Remark 3.3
  • ...and 28 more