Calabi-Yau structures on topological Fukaya categories

Abstract

We develop a local-to-global formalism for constructing Calabi-Yau structures for global sections of constructible sheaves or cosheaves of categories. The required data - an isomorphism of the sheafified Hochschild homology with the topological dualizing sheaf - specializes to the classical notion of orientation when applied to the category of local systems on a manifold. We apply this construction to the cosheaves on arboreal skeleta arising in the microlocal approach to the A-model.

Figures (15)

• Figure 1: Arboreal singularity $\overline{\mathbb{A}_2}$. For simplicity we use the notation described above for each correspondence.
• Figure 2: Arboreal singularity $\overline{\mathbb{A}_3}$. This singularity is homeomorphic to a union of three 2-discs, along half-discs; in the figure above the gray disc is horizontal and the two white half-discs discs are glued to it along two perpendicular diameters, one to the top and another to the bottom. We only labeled the 0-simplices; the labels on all other simplices can be deduced from their vertices. The link $\mathbb{A}_3^\mathrm{{link}}$ can be seen to be homeomorphic to the 1-skeleton of a tetrahedron.
• Figure 3: Stratification of the arboreal singularity $\overline\mathbb{A}_2$, by the strata $\mathbb{A}_2(\mathfrak p)$.
• Figure 4: Gluing of $\overline\mathbb{A}_3$ from the discs $\mathbb{A}_3(\bullet)$. $\mathbb{A}_3(\alpha)$ and $\mathbb{A}_3(\gamma)$ are shown creased, with half-disc flaps pointing up and down, respectively.
• Figure 5: The subsets $\mathbb{A}_3(\bullet,\bullet)$ where $\mathbb{A}_3$ has the quiver structure $\alpha \rightarrow \beta \leftarrow \gamma$. The subsets $\mathbb{A}_3(\alpha),\mathbb{A}_3(\beta),\mathbb{A}_3(\gamma)$ are homeomorphic to closed discs, and the arboreal singularity is obtained by gluing them appropriately; on the left $\mathbb{A}_3(\alpha),\mathbb{A}_3(\gamma)$ are shown with folded flaps up and down, and are glued along the horizontal parts to $\mathbb{A}_3(\beta)$. Note that the subsets $\mathbb{T}(\lambda_1,\lambda_2)$ depend on the directions of the arrows in $T$, and moreover as in the proof above for any vertices $\lambda_1,\lambda_2$, the difference between $\mathbb{T}(\lambda_1,\lambda_2)$ and $\mathbb{T}(\lambda_1) \cap \mathbb{T}(\lambda_2)$ is at most deletion of some boundary strata.

Theorems & Definitions (236)

• Definition 1.1
• Example
• Example
• Example
• Definition 1.2
• Example
• Example
• Example
• Theorem 1.3
• Corollary 1.4