The Witten equation, mirror symmetry and quantum singularity theory

TL;DR

<3-5 sentence high-level summary> This work develops a comprehensive algebro-geometric framework for the Landau-Ginzburg A-model associated to a nondegenerate quasi-homogeneous singularity $W$, by introducing $W$-curves and their moduli, constructing a state space ${\mathscr H}_{W,G}$ via orbifold Lefschetz thimbles, and formulating a virtual cycle that yields a cohomological field theory. It then analyzes ADE-symmetries, proving mirror-symmetry-type isomorphisms between the resulting state spaces and Milnor rings in many self-mirror cases, and connects the generating potentials to ADE-integrable hierarchies, resolving conjectures for $D_n$ and $E_6, E_7, E_8$ (with a noted exception for certain $D_n$ odd cases). The results hinge on an intricate interaction of W-structures, tautological classes, and wall-crossing of Lefschetz thimbles, and they pave the way for a full ADE-classification of the corresponding hierarchies within the Landau-Ginzburg A-model. The analytic construction of the virtual cycle is addressed in a separate paper, while this work provides the algebro-geometric core and the key consequences for mirror symmetry and integrable systems.

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.

Figures (4)

• Figure 1: The graph ${\Gamma}_1 \in {\boldsymbol{{\Gamma}}}_{0,4,E_7}(J^2 , J^4 , J^7 , J^7)$.
• Figure 2: The graph ${\Gamma}_2$. Two of the three graphs in ${\boldsymbol{{\Gamma}}}_{0,4,E_7}(J^2 , J^4 , J^7 , J^7)$ are decorated as in this figure.
• Figure 3: The graph ${\Gamma}_1 \in {\boldsymbol{{\Gamma}}}_{0,4,D_{n+1}}(J^3 , J^3 , J^{n-1} , J^{n-3})$.
• Figure 4: The graph ${\Gamma}_2$. Two of the three graphs in ${\boldsymbol{{\Gamma}}}_{0,4,D_{n+1}}(J^3 , J^3 , J^{n-1} , J^{n-3})$ are decorated as in this figure.

Theorems & Definitions (144)

• Example 1.0.1
• Theorem 1.0.2
• Conjecture 1.0.3: Witten's ADE-Integrable Hierarchy Conjecture:
• Theorem 1.0.4
• Theorem 1.0.5
• Conjecture 1.0.6: ADE Self-Mirror Conjecture
• Theorem 1.0.7
• Conjecture 1.0.8: Landau-Ginzburg/Calabi-Yau Correspondence
• Definition 2.1.1
• Definition 2.1.2