480 research outputs found
The Witten equation, mirror symmetry and quantum singularity theory
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.Comment: 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 clarification
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