[Paper Review] Fukaya category of Landau-Ginzburg model

Huijun Fan, Wenfeng Jiang+1 more|arXiv (Cornell University)|Dec 31, 2018

Microtubule and mitosis dynamicsBiochemistry, Genetics and Molecular Biology3 citations

TL;DR

This paper constructs the Fukaya category of a Landau-Ginzburg model using Witten's equation and open string Floer theory on a noncompact Kähler manifold equipped with a regular tame exact holomorphic Morse function. It defines Landau-Ginzburg branes as Lefschetz thimbles lifted to the universal abelian cover with pin structures, and establishes an $A_\infty$-structure on the Floer complex via moduli spaces of perturbed Witten maps, proving the existence of the conjectured Fukaya category in the context of homological mirror symmetry.

ABSTRACT

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.

Motivation & Objective

• To provide a mathematical foundation for the open string Floer theory of Landau-Ginzburg models via Witten's equation.
• To define Landau-Ginzburg branes as Lefschetz thimbles with lifted structures and pin structures on the universal cover of the Lagrangian Grassmannian.
• To construct the Fukaya category of a Landau-Ginzburg model as an $A_\infty$-category with $\boldsymbol{G}$-grading, resolving a long-standing conjecture in homological mirror symmetry.
• To establish the $A_\infty$-relations and Floer homology via moduli spaces of perturbed Witten maps with consistent compactifications.

Proposed method

• Introduce a regular tame exact Landau-Ginzburg system on a noncompact Kähler manifold with a holomorphic Morse function $W$.
• Construct Lefschetz thimbles as stable manifolds of $\operatorname{Re}W$'s gradient flow, forming Lagrangian submanifolds in the symplectic form $\omega$.
• Lift each Lefschetz thimble to the universal abelian cover $\widetilde{\mathcal{G}}M$ using a lifting map $i^\#$ and equip with a pin structure $P^\#$ to define Landau-Ginzburg branes.
• Define Hamiltonian chords $S_W(L_0, L_1)$ between branes and perturb the Kähler potential to ensure finiteness (Theorem 3.12).
• Introduce the perturbed Witten equation $\partial_{\bar{z}}u^j = \sum_{\bar{i}} h^{\bar{i}j} \partial_{\bar{i}} \overline{W} \sigma + \mu^j$ on pointed-boundary discs with strip-like ends.
• Use Deligne-Mumford-Stasheff compactifications of the moduli space of stable curves with labeled boundary components and consistent choice of ends.

Experimental results

Research questions

• RQ1 How can one rigorously define the Fukaya category of a Landau-Ginzburg model using open string Floer theory?
• RQ2 What is the correct geometric and topological structure for Lagrangian branes in the context of Landau-Ginzburg models?
• RQ3 How can the $A_\infty$-relations be established for the Floer complex in this setting?
• RQ4 What role does the Witten equation and its perturbation play in constructing the Fukaya category?
• RQ5 How is the $\boldsymbol{G}$-grading and sign structure defined and used in the $A_\infty$-structure?

Key findings

• The paper constructs the Fukaya category $\operatorname{Fuk}(M,h,W)$ of a Landau-Ginzburg model as an $A_\infty$-category with $\boldsymbol{G}$-grading, resolving a conjecture by Gaiotto-Moore-Witten and Kapranov-Kontsevich-Soibelman.
• Landau-Ginzburg branes are defined as triples $L^\# = (L, i^\#, P^\#)$, where $L$ is a Lefschetz thimble, $i^\#$ is a lifting map, and $P^\#$ is a pin structure on the universal cover of the Lagrangian Grassmannian.
• The morphism spaces $\operatorname{Hom}(L_0^\#, L_1^\#)$ are defined as Floer complexes $CF(L_0^\#, L_1^\#)$ generated by Hamiltonian chords in $S_W(L_0, L_1)$.
• Composition maps $\mu^d$ are defined via counting rigid solutions in moduli spaces $\mathscr{W}^{d+1}(L,l;K)$ with signs determined by the orientation and $\sigma$-morphism.
• The $A_\infty$-relations are verified via energy control, $C^0$-estimates, and transversality, ensuring $\mu \circ \mu = 0$ in the $\boldsymbol{G}$-graded sense.
• The first-order map $\mu^1$ defines a Floer differential, and the resulting Floer homology $HF_y(A)$ is isomorphic to $\operatorname{Ker}(\mu^1|_{A_y}) / \operatorname{Im}(\mu^1|_{A_{y-f(1)}})$.