[Paper Review] Exact Calabi-Yau categories and disjoint Lagrangian spheres

Li Yin|arXiv (Cornell University)|Jul 22, 2019

Geometric and Algebraic Topology4 citations

TL;DR

This paper establishes that exact Calabi-Yau structures on the wrapped Fukaya category of a Weinstein manifold impose strong topological constraints, particularly bounding the number of pairwise disjoint Lagrangian spheres and ensuring their homology classes are non-trivial. It shows that while quasi-dilations imply exact Calabi-Yau structures, the converse is not true, as demonstrated by the Milnor fiber of a 3-fold triple point, which admits such a structure without a quasi-dilation.

ABSTRACT

An exact Calabi-Yau structure, originally introduced by Keller, is a special kind of smooth Calabi-Yau structures in the sense of Kontsevich-Vlassopoulos. For a Weinstein manifold $M$, the existence of an exact Calabi-Yau structure on its wrapped Fukaya category $\mathcal{W}(M)$ imposes strong restrictions on its symplectic topology. Under the cyclic open-closed map constructed by Ganatra, an exact Calabi-Yau structure on $\mathcal{W}(M)$ induces a class $ ilde{b}$ in the degree one equivariant symplectic cohomology $\mathit{SH}_{S^1}^1(M)$. In particular, any Weinstein manifold admitting a quasi-dilation in the sense of Seidel-Solomon has an exact Calabi-Yau wrapped Fukaya category. We prove that for many Weinstein manifolds with exact Calabi-Yau $\mathcal{W}(M)$, there is an upper bound on the number of pairwise disjoint Lagrangian spheres, and the homology classes of these Lagrangian spheres are non-trivial. On the other hand, we show that the wrapped Fukaya category of the Milnor fiber of a 3-fold triple point is exact Calabi-Yau despite that there is no quasi-dilation.

Motivation & Objective

• To understand the topological implications of exact Calabi-Yau structures on the wrapped Fukaya category of a Weinstein manifold.
• To investigate whether the existence of an exact Calabi-Yau structure implies the presence of a quasi-dilation, particularly in the absence of such structures.
• To establish upper bounds on the number of pairwise disjoint Lagrangian spheres in Weinstein manifolds with exact Calabi-Yau wrapped Fukaya categories.
• To analyze the symplectic topology of the Milnor fiber of a 3-fold triple point, showing it supports an exact Calabi-Yau structure despite lacking a quasi-dilation.

Proposed method

• Utilizing the cyclic open-closed map of Ganatra to relate exact Calabi-Yau structures on the wrapped Fukaya category to classes in equivariant symplectic cohomology.
• Applying the theory of exact Calabi-Yau categories, as defined by Keller, to the wrapped Fukaya category of a Weinstein manifold.
• Analyzing the induced class $\tilde{b}$ in $\mathit{SH}_{S^1}^1(M)$ to derive topological constraints on the manifold's Lagrangian submanifolds.
• Using the structure of the Milnor fiber of a 3-fold triple point as a counterexample to test the necessity of quasi-dilations for exact Calabi-Yau structures.
• Establishing homological non-triviality of Lagrangian spheres in the presence of exact Calabi-Yau structures via categorical and symplectic invariants.
• Combining categorical constraints from exact Calabi-Yau structures with symplectic topology to bound the number of disjoint Lagrangian spheres.

Experimental results

Research questions

• RQ1 Does the existence of an exact Calabi-Yau structure on the wrapped Fukaya category of a Weinstein manifold imply the existence of a quasi-dilation?
• RQ2 What topological constraints does an exact Calabi-Yau structure impose on the number and homology classes of pairwise disjoint Lagrangian spheres?
• RQ3 Can a Weinstein manifold admit an exact Calabi-Yau wrapped Fukaya category without supporting a quasi-dilation?
• RQ4 How does the cyclic open-closed map relate exact Calabi-Yau structures to equivariant symplectic cohomology classes in degree one?
• RQ5 What is the symplectic topology of the Milnor fiber of a 3-fold triple point in relation to exact Calabi-Yau structures?

Key findings

• For many Weinstein manifolds with exact Calabi-Yau wrapped Fukaya categories, there exists an upper bound on the number of pairwise disjoint Lagrangian spheres.
• The homology classes of these disjoint Lagrangian spheres are non-trivial in the ambient manifold.
• The Milnor fiber of a 3-fold triple point supports an exact Calabi-Yau wrapped Fukaya category despite not admitting a quasi-dilation.
• The existence of a quasi-dilation implies an exact Calabi-Yau structure, but the converse does not hold, as shown by the Milnor fiber example.
• The cyclic open-closed map induces a class $\tilde{b}$ in $\mathit{SH}_{S^1}^1(M)$, which serves as a key obstruction to the existence of such structures.
• Exact Calabi-Yau structures on $\mathcal{W}(M)$ lead to strong symplectic topological constraints, particularly on Lagrangian configurations.