All Questions
Tagged with fukaya-category sg.symplectic-geometry
31 questions
56
votes
1
answer
2k
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 ...
- 2,356
34
votes
6
answers
5k
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 ...
- 44.7k
25
votes
4
answers
7k
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 ...
- 21k
20
votes
1
answer
4k
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 ...
- 21k
18
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, ...
- 6,920
18
votes
1
answer
1k
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 ...
- 18.7k
17
votes
2
answers
3k
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 ...
- 3,758
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 ...
- 2,283
16
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 "...
- 21k
13
votes
2
answers
3k
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 ...
- 3,758
13
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 ...
- 3,187
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, ...
- 21k
10
votes
2
answers
1k
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 ...
- 44.7k
10
votes
1
answer
650
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 ...
- 853
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. ...
- 21k
9
votes
0
answers
629
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 \...
- 915
6
votes
0
answers
208
views
Does this pseudo-holomorphic triangle contribute to the product $\mu_2$ in Lagrangian Floer cohomology?
I'm computing the product map $$\mu_2 : CF(L_0,V)\otimes CF(V,L_1)\to CF(L_0,L_1)$$ in Seidel's exact triangle for this specific case:
This is a genus 2 surface, and I color-coded the three (...
- 2,018
6
votes
0
answers
323
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:...
- 54.6k
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 ...
- 3,758
5
votes
1
answer
292
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 $L\subset M$, ...
4
votes
1
answer
337
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 ...
- 31
4
votes
1
answer
334
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 ...
- 1,893
3
votes
1
answer
307
views
Are all exact Lagrangian spheres, vanishing cycles?
Let $\pi: E \to D$ be an exact Lefschetz fibration with corners (fibers with boundary)over the disk. Fix a point $\theta \in \partial D$ and consider the fiber $F_\theta = \pi^{-1}(\theta)$ over that ...
- 1,409
3
votes
0
answers
102
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 ...
- 461
3
votes
0
answers
119
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 ...
- 3,020
3
votes
0
answers
256
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 ...
- 3,187
2
votes
0
answers
163
views
Compactness of Moduli spaces in Lagrangian Floer Cohomology
I have been reading Denis Aurox lecture notes on Fukaya Categories https://arxiv.org/pdf/1301.7056.pdf , and in page $9$ he starts to discuss the compactness properties of the moduli spaces and how we ...
- 791
2
votes
0
answers
109
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 ...
- 937
1
vote
0
answers
184
views
Structure maps of $\mathcal{A}_\infty$-bimodules
For Fukaya categories there are functors naturally induced by symplectomorphisms. Twisted versions of symplectic homology (fixed point Floer homology), open-closed maps and bimodules can be defined. ...
- 11
1
vote
0
answers
119
views
Seidel's calculation of the Floer cohomology of a cotangent fibre and its Dehn twist
I am reading Seidel's paper on exact Lagrangian submanifolds in $T^*S^n$ and the graded Kronecker quiver, and in Lemma 2 (2) he claims the following fact: if $F_0$ is a cotangent fibre and $F_1$ is $\...
- 143
1
vote
0
answers
241
views
Definition of Floer complex in Floer's "Morse theory for Lagrangian intersections"
I am moving the first steps into Lagrangian Floer theory and I am trying to understand the construction, as in the original paper, for the field $\mathbb{Z}_2$ (no orientations) and $\pi_2(P,L) = 0$.
...
- 178