The Witten equation, mirror symmetry, and quantum singularity theory

Abstract

For any nondegenerate, 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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Authors

Huijun Fan

School of Mathematical Sciences
Peking University
Beijing 100871
China
and
Beijing International Center for Mathematical Research
Beijing 100871
China

Tyler Jarvis

Department of Mathematics
Brigham Young University
Provo, UT 84602

Yongbin Ruan

Yangtz Center of Mathematics at Sichuan University
Chengdu 610064
China
and
Department of Mathematics
University of Michigan
Ann Arbor, MI 48105