The mirror-symmetry tag has no usage guidance.

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### Is there a tropical geometric proof for counting genus g curves in any n dimensional projective space?

Consider the following question: Let $X$ be a compact complex manifold
and $\beta \in H_2(X, \mathbb{Z})$ a fixed homology class. Let
$\mu_1, \mu_2, \ldots, \mu_k$ denote certain generic ...

**4**

votes

**1**answer

235 views

### Learning Quantum (Co)Homology and Landau Ginzburg Superpotential

I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer ...

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**0**answers

117 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 ...

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**0**answers

161 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 ...

**4**

votes

**1**answer

232 views

### Localization principle in supersymmetry

In $\S$ 9.3 of the book "Mirror symmetry" (Vafa, Zaslow eds.) the authors formulate the following general localization principle for computation of integrals with respect to both even and odd ...

**1**

vote

**1**answer

255 views

### How to construct (another) Landau-Ginzburg model for a compete intersection Calabi-Yau?

For Calabi-Yau variety $X$ which is a complete intersection
$$
f_1=f_2=\ldots=f_r=0
$$
in ${\mathbb P }^n$ (hence $\mathrm{dim}\,X=n-r$) it is possible to construct a Landau-Ginsburg model in the ...

**7**

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403 views

### Examples of Symplectic Questions Solved by ``Mirror Symmetry Translation'' to Complex Questions

According to the proponents of homological mirror symmetry, when a complex and symplectic manifold are mirror symmetric, we can take difficult questions about the symplectic space and transfer them ...

**0**

votes

**1**answer

147 views

### On understanding Orlov's LG B model

I try to read Orlov's papers on Landau-Ginzburg model, but I am quite puzzled，there are several questions：
1 the method of truncation is used frequently，(that is: using a bounded above complex $Q$ ...

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260 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 ...

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232 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 ...

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**3**answers

978 views

### How mirror of quintic was originally found?

In the 90-91 pager
"A PAIR OF CALABI-YAU MANIFOLDS AS AN EXACTLY SOLUBLE SUPERCONFORMAL THEORY",
Candelas, De La Ossal, Green, and Parkes, brought up a family of Calabi-Yau 3-folds, canonically ...

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**0**answers

61 views

### Conifold singularities on the mirror quintic

It is well-known that conifold transitions go in the reversed direction under mirror symmetry. My question is, what is the mirror picture of a degeneration of the generic quintic CY3 to the one ...

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**0**answers

220 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
...

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votes

**2**answers

521 views

### Mirror Symmetry for Quaternionic-Kähler Manifolds

I take the following quote from Huybrecht's notes on hyperkähler manifolds and mirror symmetry:
Mirror symmetry in a first approximation predicts for any Calabi-Yau manifold (M,g) the existence of ...

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**0**answers

459 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**

votes

**1**answer

166 views

### When will the mirror of a K3 surface be an elliptic K3?

Let $f:Y\rightarrow\mathbb{P}^1$ be an elliptic $K3$ surface, then the holomorphic 2-form $\Omega_Y$ vanishes when restricted to an elliptic fiber $f^{-1}(b)$ with $b\in\mathbb{P}^1$. After a ...

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188 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 ...

**2**

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**1**answer

212 views

### SYZ mirror symmetry for K3 surfaces

My question is essentially related to this post, but let me formulate it again. Let $f:S \rightarrow \mathbb{P}^1$ be an elliptic fibration, then this can be a SLAG fibration with respect to another ...

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**1**answer

195 views

### How is the propagator computed on an elliptic curve?

I've been struggling for a while now understanding why the propagator for the action
$$
S(\varphi) = \int_E \partial \varphi \bar\partial\varphi + \frac{\lambda}{6}(\partial\varphi)^3
$$
on an ...

**31**

votes

**1**answer

869 views

### What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ ...

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votes

**2**answers

357 views

### Why non-compact Calabi-Yau surfaces are not self-mirror?

By the work of Gross and Bernard-Matessi, in dimension 3 $T$-duality should be understood as an exchange of positive and negative local model of Lagrangian torus fibrations, at least in its ...

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votes

**1**answer

217 views

### Question about the Aganagic-Vafa A-brane

According to Aganagic-Vafa (hep-th/0012041) and Fang-Liu (arXiv:1103.0693), for a semi-projective toric Calabi-Yau 3-manifold $X$, the Aganagic-Vafa A-brane $L_{AV}\subset X$ is defined by the ...

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**1**answer

187 views

### Special Lagrangians and fat

I am unable to find the MO comments about the first use of the phrase "fat slags" in an article. On page 26 of this we find "these correspond to thickenings of the
corresponding special Lagrangian ...

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**1**answer

244 views

### Looking for a reference (on GW invariants of quintic)

1) Apparently, physicist can calculate the GW invariants of quintic CY 3-fold up to genus 51.
I am looking for a reference that has a table of these number for some low degrees (say up to degree 5) ...

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**1**answer

389 views

### A question on chiral rings and geometry of the vacuum bundle

I am reading "Mirror Symmetry" by Hori et al, and have a question on Chap.17 (Chiral rings and geometry of the vacuum bundle). On p.425 the authors say
Consider the path-integral on the ...

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**0**answers

192 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 ...

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**0**answers

256 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 ...

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votes

**2**answers

203 views

### A question on the topological change of dualizing a SLAG fibration.

Let $S$ be a K3 surface and $\pi:S\rightarrow B$ be a SLAG $T^2$-fibration. I am struggling with a statement that
Fiberwise dualization does not change the topology of $S$.
Here by fiberwise ...

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**0**answers

135 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 $E$ ...

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**0**answers

175 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 ...

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**0**answers

106 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 ...

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**3**answers

442 views

### Why is the mirror of rigid Calabi-Yau threefold singularity theory?

Mirror symmetry relates two Calabi-Yau threefolds with mirrored Hodge diamonds. Since Calabi-Yau threefold is Kahler, this naive correspondence does not hold for rigid Calabi-Yau threefolds. Here ...

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**1**answer

559 views

### How to understand Givental's I- and J-functions?

I am learning about mirror symmetry and having trouble understanding Givental's I- and J-functions. For example the J-function for the quintic threefold $X$ is defined by the formula
$$
...

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votes

**1**answer

215 views

### Zero and Negative Gromov-Witten invariants in genus 0

I'm working on a project and I've used the Picard-Fuchs equation at a maximally unipotent monodromy point for a certain 1-dimensional family of Calabi-Yau 3-folds to calculate the A-model Yukawa ...

**3**

votes

**1**answer

857 views

### Mirror symmetry for hyperkahler manifold

Hi there,
I have some questions about the mirror symmetry of hyperkahler manifold and K3 surface.
The well-known result said: the mirror symmetry for K3 surface is just given by its hyperkahler ...

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votes

**2**answers

1k views

### Are Donaldson-Thomas invariants “A-model” or “B-model” ?

Donaldson-Thomas invariants are the (virtual) Euler characteristics of moduli spaces of elements of the derived category of coherent sheaves (with some fixed Chern class, satisfying some stability ...

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**1**answer

568 views

### What is known about the Bridgeland stability manifold?

In a fabulous paper (http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf) Bridgeland showed that locally finite stability conditions on a triangulated category form a complex ...

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votes

**1**answer

689 views

### what is the stringy Kähler moduli space?

I saw the stringy moduli space mentioned in a few papers but with little no explanation. I vaguely understand it is supposed to be the moduli space of complex structures on the mirror manifold.
Could ...

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**0**answers

268 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 ?

**3**

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**1**answer

571 views

### Noncommutative Fukaya category?

After reading through part of Victor Ginzburg's notes on Calabi-Yau algebras, I have a question about a principle in mirror symmetry. Let $(X,X')$ be a mirror pair of Calabi-Yau varieties then mirror ...

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**3**answers

1k views

### How far can one get with the Gross-Siebert program?

The Gross-Siebert program is said to be an algebraic analog of the SYZ conjecture and they used toric degeneration to construct a mirror dual of Calabi-Yau varieties. It seems like the singular ...

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**1**answer

407 views

### Mirror to the dualizing sheaf

I wonder - is there a general characterization of the object in the Fukaya category that is mirror to the dualizing sheaf on the other side?
This question has two cases:
1. CY
2. Non-CY
In 1. what ...

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votes

**2**answers

984 views

### Places to learn about Landau-Ginzburg models

Here is what I know about Landau-Ginzburg models:
It is an important player in the story of mirror symmetry.
It involves "potentials" which are functions of complex varibles, which have isolated ...

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votes

**0**answers

193 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 ...

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**3**answers

964 views

### Does the derived category of coherent sheaves determine the hodge theory?

Given two smooth algebraic varieties (proper or not), if the two derived categories of the bounded complexes of coherent sheaves over them are equivalent (if necessary we assume there is a fully ...

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283 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 ...

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822 views

### Is $Sym^g$ of a Riemann Surface of genus $g$ Calabi-Yau?

The $g$-fold symmetric product of a Riemann surface of genus $g$ naturally has both a symplectic structure as well as a complex structure. Is it in fact Calabi-Yau? If so, is anything known about a ...

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308 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 ...

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**2**answers

760 views

### Known Mirror Calabi-Yau pairs

There is a well known class of Calabi-Yau (3 dimensional) pairs constructed by Batyrev. These are resolutions of Calabi-Yau hypersurfaces in reflexive polytops of dimension 4.
Question: Does any body ...

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**9**answers

4k views

### Examples in mirror symmetry that can be understood.

It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So if we consider ...