The Witten equation, mirror symmetry, and quantum singularity theory


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.


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