Dubrovin duality and mirror symmetry for ADE resolutions (2501.05753v1)

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.

• The paper provides explicit computations showing how Dubrovin duality embeds Frobenius structures in ADE resolutions, unifying quantum cohomology and LG mirror symmetry.
• It formulates extended affine Weyl Frobenius manifolds via invariant Fourier polynomials, detailing their symmetries and algebraic properties in marked ADE pairs.
• The research validates mirror symmetry theorems with rigorous symbolic and numerical verifications, offering practical insights into enumerative geometry and moduli problems.

Overview of "Dubrovin Duality and Mirror Symmetry for ADE Resolutions"

The paper "Dubrovin Duality and Mirror Symmetry for ADE Resolutions" authored by Andrea Brini, Jingxiang Ma, and Ian A. B. Strachan offers a comprehensive analysis of three interconnected Frobenius manifold structures related to marked ADE pairs: the extended affine Weyl Frobenius manifolds, the quantum cohomology of ADE resolutions, and their Landau–Ginzburg mirrors. The authors provide explicit calculations and proofs, confirming intricate mathematical relationships and embeddings within the framework of these geometric and algebraic structures.

Key Contributions and Results

1. Extended Affine Weyl Frobenius Manifolds: These manifolds are constructed through the polynomial ring of invariant Fourier polynomials under the extended affine Weyl group action. The structure functions, symmetries, and algebraic properties are meticulously detailed. The construction hinges on Dubrovin and Zhang's canonical Frobenius manifold structures applied to the orbits within these groups and their Dynkin diagrams.
2. Quantum Cohomology of ADE Resolutions: The quantum cohomology of the minimal resolutions of ADE singularities is analyzed within the context of $\mathsf{T}$-equivariant theories. For the classical and exceptional series, the connections are formulated through weighted homogeneous polynomials and structure constants that intertwine within the algebraic identities, governed by the group $\mathsf{T}$ actions. Explicit connections by invoking the degree axiom articulate how the equivariant intersection forms manifest in these spaces.
3. Landau–Ginzburg Mirrors: The authors derive Landau–Ginzburg mirror symmetry models from ADE spectral curves. By elaborating on the Hurwitz spaces, they reveal the process of extending Dubrovin duality to these remarkable low-dimensional and explicitly computable superpotential models. This section explores the moduli space of branched covers and establishes connections with mirror structures for quantum cohomologies of du Val singularities.
4. Dubrovin Duality and Mirror Theorems: Establishing Dubrovin duality, the authors exhibit how Frobenius structures can be transformed under the "dualization" process where the structure constants of one manifold are mapped to another, revealing hidden symmetries and dualities. This novel insight empowers the realization of mirror symmetry theorems across the ADE category, showcasing deep connections between enumerative geometry and string-theoretical models.
5. Closed-form and Numerical Verifications: Throughout the paper, theoretical explorations are supplemented with comprehensive symbolic and numeric calculations across several SEO configurations and matrices. This rigorous approach strengthens the veracity of throughlines between extended affine Weyl groups and equivariant quantum cohomologies.

Implications and Future Directions

• Theoretical Implications: This research papers a path for further inquiries into Frobenius structures and their applications in complex algebraic geometry and theoretical physics. The mirror theorems and duality constraints offer new potential insights into moduli problems and symplectic geometry.
• Computational Advances: Explicitly embedding computational routines for verifying symmetry constraints and dual structures could broaden accessibility. The methodologies used for initial conditions verification and tensor calculations are foundational advancements for computational algebra software toolkits.
• Quantization and Topological Models: Integrating R-matrix quantisation with these findings could extend higher genus Gromov–Witten theories, exploring their potential in string theory. The incorporation of quantum field theories and low-dimensional topology offers new theoretical landscapes to explore.

Conclusion

The paper stands as a crucial addition to the growing body of work on Frobenius manifolds, especially in understanding their role within ADE resolutions. By combining intricate mathematical proofs with comprehensive numerical computations, Brini, Ma, and Strachan have elucidated critical relationships within algebraic geometry, effectively setting the stage for future mathematical explorations and applications in broader theoretical contexts. The collaborative synthesis of traditional algebraic techniques with modern computational approaches enriches both the theoretical landscape and the field of practical applications in mirror symmetry and geometric representation theory.