Huijun Fan

This article introduces and provides the mathematical foundation of the open string Floer theory of Landau-Ginzburg model viaWitten equation. We introduce the concept of regular tame exact Landau-Ginzburg system on a noncompact Kaehler manifold, and define the notion of Landau-Ginzburg branes, as the objects of the Fukaya category. The study on Witten equation in our context provides the construction of the Fukaya category of Landau-Ginzburg model which was conjectured to be existed in Gaiotto-Moore-Witten's work [22] and Kapranov-Kontsevich-Soibelman's work [30]. This is the first paper in the series of studies on Landau-Ginzburg models in the contexts of the mirror symmetry and other topics.

We prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D_4 with symmetry group and D_4^T with symmetry group G_{max}, respectively, are both tau-functions of the D_4 Kac-Wakimoto/Drinfeld-Sokolov hierarchy. This completes the proof, begun in [FJR], of the Witten Integrable Hierarchies Conjecture for all simple (ADE) singularities.

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.