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13
votes
0answers
313 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, ...
11
votes
0answers
296 views

Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s. In the comments of the question, I was directed to the paper http://arxiv.org/abs/...
10
votes
0answers
578 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 ...
7
votes
0answers
282 views

Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s

It is well known (cf. Dolgachev) that there is a beautiful notion of mirror symmetry for lattice-polarized K3 surfaces. That is, if we are given a rank $r$ lattice $M$ of signature $(1, r - 1)$ and a ...
6
votes
0answers
136 views

Mirror Symmetry for Homogeneous Spaces other than Flag Manifolds

Mirror symmetry is (reasonably) well understood for the general flag manifolds, due to the work of Kim, Givental, Rietsch, and others. Do there exist other homogeneous spaces for which mirror symmetry ...
6
votes
0answers
191 views

Reference or explanation: Cup products, deformations of complex structure and Mirror Symmetry

In section 0.3. of their paper "Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields," Barannikov and Kontsevich discuss the fact that Kontsevich's formality morphism (from his paper ...
5
votes
0answers
103 views

Meaning of the support property in the definition of Bridgeland stability condition

Let $(Z,\mathcal{P})$ be a Bridgeland stability condition on a triangulated category $\mathcal{C}$. It is said to satisfy the support property if there exists a norm on $K(\mathcal{C})\otimes_\mathbb{...
5
votes
0answers
290 views

Lie-infinity structure in Lagrangian Floer theory ?

Is there (besides the A-infinity structure) also a L-infinity structure in Lagrangian Floer theory (forming together a G-infinity structure) - like in Hochschild cohomology ?
4
votes
0answers
130 views

Hodge numbers of non-commutative varieties

Let $(X, \mathcal{A})$ be a non-commutative variety, by this I mean $X$ is a (smooth) algebraic variety and $\mathcal{A}$ is a sheaf of algebra on $X$. One such example in my mind is when $X$ admits ...
4
votes
0answers
212 views

Lagrangian fibration on Schoen's Calabi-Yau 3-fold

Schoen's Calabi-Yau 3-fold is the fiber product $X=Y_1\times_{\mathbb{P}^1}Y_2$ of two rational elliptic surfaces $Y_1\rightarrow\mathbb{P}^1$ and $Y_2\rightarrow\mathbb{P}^1$ with $\chi(X)=0$ and $h^{...
3
votes
0answers
90 views

Symplectic Hodge Maps and Mirror Symmetry

The notion of Hodge theory for symplectic manifolds seems to be getting more and more attentions in these days. See the series of papers by Yau: http://arxiv.org/abs/1011.1250 http://arxiv.org/abs/...
3
votes
0answers
104 views

Mirror Symmetry for Flag Supermanifolds

I recently learned the following two things, and I wish to know how to make them reconciled. (1) As far as I understand, the flag manifolds serve as a tractable class of examples for the very ...
3
votes
0answers
198 views

Construction of mirror quintic family over $\mathbb{A}^{1} \setminus \{0,1\}$

This question is about how to construct a Fermat pencil of quintics and the mirror family over $\mathbb{A}\setminus {0,1}$ as opposed to over $\mathbb{A}^{1}\setminus {0,\mu_{5}}$, where $\mu_{5}$ is ...
3
votes
0answers
316 views

genus one Gromov-Witten invariants of Calabi-Yau 3-folds

In http://arxiv.org/PS_cache/hep-th/pdf/9302/9302103v1.pdf physicists calculate (predict) genus one GW invariants of quintic (Table 1) and some other cases (Table 2). Can any body explain to me (...
2
votes
0answers
106 views

holomorphic curves in almost toric fibration and their relation to tropical curves

My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks. We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with ...
2
votes
0answers
254 views

Towards an enhanced version of homological mirror symmetry for affine varieties

Let $X$ and $X^\vee$ be a mirror pair, homological mirror symmetry relates the symplectic geometry of $X$ to the complex geometry of $X^\vee$ via the equivalence of triangulated categories $D^\pi\...
2
votes
0answers
208 views

What is the role of B-field $B \in H^2(X,\mathbb{R}/\mathbb{Z})$ in mathematics?

In mirror symmetry conjecture, we add what is called "B-field" $B \in H^2(X,\mathbb{R}/\mathbb{Z})$ in the Kähler moduli space so that the Kähler moduli space has enough freedom comparable to the ...
2
votes
0answers
276 views

fake Calabi-Yau threefold

(1) What is a "fake Calabi-Yau threefold"? Can I describe as a complex or symplectic manifold with trivial canonical bundle, but no compatible Kahler structure? Some mathematicians actually seem to ...
2
votes
0answers
289 views

$A_{infty}$-categories and mirror symmetry.

I have been looking for an electronic version of Kontsevchi's talk in the Arbeitstagung at 1993. In Kontsevich's publications webpage, I found this reference preprint MPI/93-57, but there is no link ...
2
votes
0answers
217 views

Are the two B model constructions equivalent?

Now we have two B model constructions: Kontsevich-Baranikov (for genus 0) and Costello (for any genus). My question is, are they equal in genus 0? Or now which one is the one we want? Thanks!
1
vote
0answers
190 views

A question on the SYZ mirror symmetry

A toy SYZ mirror symmetry is described as follows. Let $X$ be a $n$-dimensional compact manifold. The contangent bundle $T^*X$ has a natural symplectic structure locally given by the $2$-form $\...
1
vote
0answers
114 views

Maximally unipotent monodromy point of a K3 surface

I have a question on maximally unipotent monodromy point (or large complex structure limit) of the family of polarized K3 surfaces $(X,L)$. It is known that the moduli space of such pair is given by ...
1
vote
0answers
330 views

birational equivalence and mirror CYs

If a CY X is a mirror to Y then any CY Z which is birational to X is also a Mirror of Y. This is the motivation for the Kawamata's "moveable Kahler cone" which includes the Kahler cones of all the CYs ...
0
votes
0answers
138 views

SLAGs on elliptic curves are only lines?

Let $E=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ be an alliptic curve. Define a holomorphic 1-form $dz=dx+idy$ and a Kahler form $\omega=dx\wedge dy$. How can one prove that special Lagrangians on $E$ ...
0
votes
0answers
631 views

a question on Costello's theorem

Costello's theorem,"open TCFT=Calabi Yau A-infinity category",he also mentions when applied to Fukaya category we can recover Gromov-Witten theory, but I see that it needs some assumption,also even if ...