The Witten equation, mirror symmetry and quantum singularity theory
arXiv:0712.4021
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.
To appear in Annals of Mathematics. Includes resolution of the Witten ADE integrable hierarchies conjecture and Witten's ADE self-mirror conjecture. Several corrections and clarifications
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- FJRW-Rings and Landau-Ginzburg Mirror Symmetry in Two Dimensions
- On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
- On genus-0 invariants of Calabi-Yau hybrid models
- LG/CY correspondence: the state space isomorphism
- On the Genus Two Free Energies for Semisimple Frobenius Manifolds
- The modular group for the total ancestor potential of Fermat simple elliptic singularities
- Random Matrix, Singularities and Open/Close Intersection Numbers
- Conditions for the vanishing of the genus-2 G-function
- Quantum Singularity Theory for A_{r-1} and r-Spin Theory