The Witten equation, mirror symmetry and quantum singularity theory

2007/12/24 by Huijun Fan, Fan, Huijun, Tyler J. Jarvis +3 · 3 citations
Mathematics · Physics and Astronomy · #14B05 #14N35 #32S40 #32S55 #53D45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.0712.4021

openalex publication_date 2007/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity A_r-1. We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual; and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

Citations

• Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves
• The Witten equation and its virtual fundamental cycle
• Three constructions of Frobenius manifolds: a comparative study

Cited by

• Open Hurwitz numbers and the mKP hierarchy
• Genus zero Whitham hierarchy via Hurwitz--Frobenius manifolds
• Generalized Frobenius Manifold Structures on the Orbit Spaces of Affine Weyl Groups I

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• The Witten equation and its virtual fundamental cycle
• Quantum Singularity Theory for A_r-1 and r-Spin Theory
• SYZ mirror symmetry from Witten-Morse theory
• Virtual cycles of gauged Witten equation
• Landau-Ginzburg Mirror Symmetry Conjecture