# Questions tagged [fukaya-category]

For questions about Fukaya categories (as introduced by Fukaya in 1993) and their structure; consider also related tags such as [floer-homology] or [lagrangian-submanifolds].

33
questions

**3**

votes

**0**answers

132 views

### Natural equivalence of Dehn and spherical twist of Fukaya category

We consider the setup of Seidel's book. Let $(M,\omega)$ be an exact symplectic manifold with $2c_1=0$. Seidel defines the Fukaya category $\text{Fuk}(M)$ of $M$.
A Lagrangian sphere $L\subset M$, ...

**49**

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**0**answers

914 views

### What is the current status of derived differential geometry?

I hope you will excuse this naive and general question. I've read from many places (e.g. Dominic Joyce's website, John Pardon's thesis, etc.) that the/a "right" foundations for many ...

**3**

votes

**0**answers

63 views

### Continuation map interpolating two quadratic Hamiltonians with respect to different contact boundaries

Let $(M,\lambda)$ be a Liouville manifold. Consider two different contact boundaries $\partial_{\infty}^1M$ and $\partial_{\infty}^2M$ with respect to the same Liouville flow $Z$. Each of them ...

**8**

votes

**2**answers

634 views

### Geometric Langlands: From D-mod to Fukaya

This post is rather wordy and speculative, but I promise there is a concrete question embedded within. For experts, I'll open with a question:
Question: Given a compact Riemann surface $X$, why ...

**3**

votes

**0**answers

103 views

### Organizing mirror pairs

At a maximally vague and naive level, mirror symmetry asks the following question: given a complex manifold $(X, I)$, is there a symplectic manifold $(M, \omega)$ and an equivalence between the ...

**2**

votes

**0**answers

83 views

### Augmentations of wrapped Floer cochains

Let $M$ be a closed, simply-connected spin manifold and let $F_b \subset T^*M$ be the cotangent fiber over a point $b \in M$. Let $CW^*(L,L)$ be the $A_{\infty}$-algebra of wrapped Floer cochains over ...

**4**

votes

**0**answers

150 views

### Comparing different approaches to HMS for elliptic curves

I am trying to understand homological mirror symmetry for elliptic curves from the article of Zaslow-Polishchuk and from Section 6 of the article of Abouzaid and Smith on homological mirror symmetry ...

**7**

votes

**3**answers

804 views

### Meaning of A-infinity relations

I am learning A-infinity category with Fukaya category in mind, and would like to understand the meaning of A-infinity relations.
In particular, as $N=1$, it means $dd=0$. As $N=2$, it means that $d$ ...

**3**

votes

**1**answer

239 views

### Mirror symmetry for singular Lagrangian torus fibrations

Let $X$ be a closed symplectic manifold equipped with a smooth Lagrangian torus fibration $\pi:X \rightarrow Q$. Assume that $\pi$ admits a Lagrangian section. By work of Kontsevich-Soibelman, one can ...

**9**

votes

**0**answers

439 views

### Orlov equivalence between Fukaya categories

In his famous paper https://arxiv.org/abs/math/0503632, Orlov proves the following theorem (for simplicity, let's just focus on the Calabi-Yau case)
Theorem(Orlov): Suppose that $W: \mathbb{A}^d \...

**3**

votes

**0**answers

59 views

### Reference Request: Central Curvature “Fix”

Context: In Lagrangian-Floer theory, the (an) $\mathbf{A}_\infty$-algebra of a Lagrangian is curved. However, the curvature is central. One consequence of this is that you can get an uncurved $\mathbf{...

**5**

votes

**0**answers

141 views

### Mirror of the autoequivalences of the derived category of del Pezzo surface?

One version of the homological mirror symmetry conjecture states that for every Fano variety $X$ there exists a Landau--Ginzburg model $W$ such that the category of B-branes on $X$ (i.e. the bounded ...

**4**

votes

**1**answer

256 views

### How to understand geometrically, the count of pseudoholomorphic discs by (multi)section perturbation of the kuranish structure on the moduli space?

When defining the $A_\infty$ algebra of a Lagrangian (as done in the book by FOOO) it is done by "counting" (integrating over the moduli space or over the fiber of evaluation map) pseudoholomorphic ...

**17**

votes

**1**answer

915 views

### Categorification of Floer homology

Floer homology associates a vector space $HF^\ast(L_1,L_2)$ to any pair of Lagrangian submanifolds $L_1,L_2$ inside a symplectic manifold $X$. By a categorification of Floer homology, I mean a ...

**3**

votes

**2**answers

581 views

### Generating Fukaya category vs split-generating Fukaya category

I just started learning about Fukaya categories and got slightly confused by the following question. It looks like the statement that a collection of objects generate Fukaya category is stronger than ...

**6**

votes

**0**answers

295 views

### Is there any work on “super Fukaya categories”?

There is a well-established notion of "supermanifold", and in the world of supergeometry it makes sense to talk about symplectic structures. Actually, there are various kinds of symplectic structures:...

**16**

votes

**0**answers

1k views

### What is the Hochschild cohomology of the Fukaya-Seidel category?

Let $(Y, \omega)$ be a compact symplectic manifold and let $Fuk(X,\omega)$ be its Fukaya category. The Hochschild cohomology of this category should be given by $HH^\bullet(Fuk(Y,\omega))=H^\bullet(Y, ...

**3**

votes

**0**answers

232 views

### Fully faithful embedding of the exact Fukaya category

Let $\mathscr{F}(X)$ be the exact Fukaya category of an exact symplectic manifold $(X^{2n},\omega)$, i.e. the objects in $\mathscr{F}(X)$ are all closed exact Lagrangian submanifolds with Maslov index ...

**12**

votes

**1**answer

2k views

### Wrapped Fukaya categories of Stein manifolds

By the work of Abouzaid, we know that the wrapped Fukaya category of $T^\ast Q$ with $Q$ a closed smooth manifold is generated by a cotangent fiber. Basically, this is an application of Abouzaid's ...

**17**

votes

**2**answers

1k views

### What is Known about the $K$-Theory of Fukaya Categories?

Some Background: In Kontsevich and Soibelman's theory of motivic DT-invariants, one is interested in something like the ``number'' of objects in a 3-Calabi-Yau category $\mathcal{C}$ having a fixed ...

**10**

votes

**1**answer

571 views

### The Fukaya category of a simple singularity (reference request)

I have heard that for an ADE singularity $f$,
$ D^b\mathrm{Fuk}(f) \simeq D^b(\mathrm{Rep}\ Q)$
where $Q$ is the corresponding Dynkin quiver. (As one would hope, if $\mathrm{Fuk}$ is some kind of ...

**13**

votes

**2**answers

2k views

### How to relate equivariant symplectic cohomology, Contact Homology, Cyclic Homology and String Topology?

I am trying to understand how all the players in the title relate, but with all the grading shifts,and difficult isomorphisms involved in the subject I am having a hard time being sure that I have the ...

**8**

votes

**2**answers

2k views

### Hochschild homology of Fukaya category in mirror symmetry

Hi
Can one explain to me what is the Hochschild homology of Fukaya category?
I mean the definition.
You can use the notations of FOOO (Fukaya-Oh-Ono-Ohta) if it helps you to explain easier.
I know ...

**15**

votes

**2**answers

2k views

### Comparison between Hamiltonian Floer cohomology and Lagrangian Floer cohomology of the diagonal

Let X be a compact symplectic manifold with a form $\omega$. And $X \times X$ is equipped with the symplectic form $(\omega,-\omega)$. The diagonal $\Delta:X \mapsto X \times X$ is a Lagrangian ...

**5**

votes

**2**answers

1k views

### Generator of a Fukaya category with certain properties

There is an algebraic theory that I'm thinking of trying to develop and I wanted to know if it had any real world prevalence --- I'd like to know an example of a generator L of a Fukaya category on a ...

**14**

votes

**2**answers

2k views

### Deformation quantization and quantum cohomology (or Fukaya category) — are they related?

Good afternoon.
Let $M$ be, say, a compact symplectic manifold. Both deformation quantization (as in Kontsevich) and quantum cohomology yield "deformations" (in the appropriate respective senses) of "...

**11**

votes

**1**answer

2k views

### “Fourier-Mukai” functors for Fukaya categories?

I just skimmed a bit of this fresh-off-the-press paper on homological mirror symmetry for general type varieties.
One thing that intrigued me was statement (ii) of Conjecture 3.3. It suggests that, ...

**9**

votes

**2**answers

2k views

### Are Fukaya categories Calabi-Yau categories?

Let X be a compact symplectic manifold. There is an idea, I think probably originally due to Kontsevich, that we should be able to get Gromov-Witten invariants of X out of the Fukaya category of X. ...

**30**

votes

**6**answers

4k views

### Has anything precise been written about the Fukaya category and Lagrangian skeletons?

At some point in this past year, some Fukaya people I know got very
excited about the Fukaya categories of symplectic manifolds with "Lagrangian skeletons." As I understand it, a
Lagrangian skeleton ...

**18**

votes

**1**answer

3k views

### Hochschild (co)homology of Fukaya categories and (quantum) (co)homology

There is a conjecture of Kontsevich which states that Hochschild (co)homology of the Fukaya category of a compact symplectic manifold $X$ is the (co)homology of the manifold. (See page 18 of ...

**10**

votes

**2**answers

977 views

### Fukaya categories of hyperkahler reductions: general request for information

I'd really like to hear any references or information people have about the Fukaya categories of hyperkahler reductions of vector spaces (for more informations on the varieties, see Nick Proudfoot's ...

**22**

votes

**4**answers

6k views

### Is the Fukaya category “defined”?

Sometimes people say that the Fukaya category is "not yet defined" in general.
What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact ...

**15**

votes

**5**answers

2k views

### How should I think about B-fields?

So, physicists like to attach a mysterious extra cohomology class in H^2(X;C^*) to a Kahler (or hyperkahler) manifold called a "B-field." The only concrete thing I've seen this B-field do is change ...