The Witten equation, mirror symmetry and quantum singularity theory

Huijun Fan; Tyler J. Jarvis; Yongbin Ruan

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arXiv · 0712.4021

The Witten equation, mirror symmetry and quantum singularity theory

Huijun Fan·Tyler J. Jarvis·Yongbin Ruan

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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.

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BibTeX RIS

Huijun Fan, Tyler J. Jarvis, Yongbin Ruan. 2012-07-26. The Witten equation, mirror symmetry and quantum singularity theory. https://arxiv.org/abs/0712.4021

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