The Witten equation, mirror symmetry and quantum singularity theory

TL;DR

This paper develops a cohomological field theory for non-degenerate, quasi-homogeneous hypersurface singularities, generalizing r-spin curves and resolving key conjectures related to ADE-singularities and integrable hierarchies.

Contribution

It introduces a new framework linking singularity theory, mirror symmetry, and quantum field theory, and proves two major conjectures of Witten.

Findings

ADE-singularities are self-dual.

Total potential functions satisfy ADE-integrable hierarchies.

Generalizes r-spin curves to broader singularity classes.

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.