---
title: "Fukaya category"
aliases: [Fukaya categories, Fukaya, wrapped Fukaya category, split generators]
created: 2022-02-23T18:45
updated: 2023-04-02T12:58
---

---

- Tags 
	- #geomtop/symplectic-topology 
- Refs:
	- [A general overview of the construction due to Kontsevich](https://arxiv.org/pdf/alg-geom/9411018.pdf#page=16&zoom=160,-136,626)
- Links:
	- [mirror symmetry](Unsorted/mirror%20symmetry.md)
	- [A_infty categories](A_infty%20categories)
	- [homological mirror symmetry](homological%20mirror%20symmetry.md)
	- [Heegard-Floer homology](Heegard-Floer%20homology)
	- [integrable system](Unsorted/integrable%20system.md)
	- [symplectic field theory](symplectic%20field%20theory)
	- [Massey product](Massey%20product)

---

# Fukaya category

Categorifies [Lagrangian Floer homology](Unsorted/Lagrangian%20Floer%20homology.md).
![](attachments/Pasted%20image%2020220430184718.png)
![](attachments/Pasted%20image%2020220430184821.png)

A warning about the [A_infty](Unsorted/A_infty.md) products:

![](attachments/Pasted%20image%2020220430191031.png)

Idea:
![](attachments/Pasted%20image%2020220424200451.png)

![](attachments/Pasted%20image%2020220424165923.png)

![](attachments/Pasted%20image%2020220422113309.png)

Relation to [homological mirror symmetry](Unsorted/homological%20mirror%20symmetry.md) using [Hochschild homology](Unsorted/HH.md) of [derived categories](Unsorted/derived%20category.md):
![](attachments/Pasted%20image%2020220424170014.png)

 
Description of a certain wrapped Fukaya category $\OO$: take the objects to be [Lagrangian](Lagrangian.md) embedded curves, the morphisms are the graded abelian groups $\hom_\OO \definedas \qty{\bigoplus_{L_0 \transverse L_1} \ZZ/2\ZZ, \bd}$ wher $\bd$ is given by counting holomorphic strips, localize along small isotopies.


![](attachments/Pasted%20image%2020220325223659.png)
![](attachments/Pasted%20image%2020220325223812.png)
![](attachments/Pasted%20image%2020220326013458.png)

# Wrapped Fukaya

![](attachments/Pasted%20image%2020220326014621.png)![](attachments/Pasted%20image%2020220326014639.png)
![](attachments/Pasted%20image%2020220326014701.png)

# Twisting

![](2023-04-02-5.png)

![](2023-04-02-6.png)

Use of [Picard-Lefchetz theory](Picard%20Lefchetz%20theory.md):
![](2023-04-02-7.png)

Theorem 3. $S_1, \ldots, S_m$ are split-generators for $D^\pi(\mathcal{F}(M))$. This means that any object of $T w^\pi(\mathcal{F}(M))$ can be obtained from them, up to quasi-isomorphism, by repeatedly forming mapping cones and idempotent splittings.

Deformations defined in terms of [Hochschild homology](HH.md), receives a map from [symplectic cohomology](symplectic%20cohomology): ![](2023-04-02-9.png)