The mirror-symmetry tag has no wiki summary.

**2**

votes

**1**answer

85 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 ...

**3**

votes

**0**answers

117 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**

votes

**1**answer

129 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 ...

**7**

votes

**1**answer

172 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 ...

**29**

votes

**1**answer

762 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 $ ...

**4**

votes

**2**answers

243 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 ...

**3**

votes

**1**answer

169 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 ...

**3**

votes

**1**answer

160 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 ...

**4**

votes

**1**answer

206 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) ...

**7**

votes

**1**answer

357 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 ...

**2**

votes

**0**answers

174 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

**0**answers

222 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

**2**answers

177 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 ...

**0**

votes

**0**answers

129 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 $E$ ...

**1**

vote

**0**answers

160 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

**0**answers

99 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 ...

**9**

votes

**3**answers

372 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 ...

**8**

votes

**1**answer

350 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
$$
...

**2**

votes

**1**answer

194 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

609 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 ...

**15**

votes

**2**answers

948 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 ...

**2**

votes

**1**answer

484 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 ...

**7**

votes

**1**answer

517 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 ...

**4**

votes

**0**answers

216 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**

votes

**1**answer

500 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 ...

**10**

votes

**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 ...

**3**

votes

**1**answer

386 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 ...

**5**

votes

**2**answers

818 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 ...

**3**

votes

**0**answers

180 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 ...

**16**

votes

**3**answers

832 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 ...

**2**

votes

**0**answers

258 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 ...

**7**

votes

**1**answer

779 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 ...

**3**

votes

**0**answers

286 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 ...

**7**

votes

**2**answers

688 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 ...

**34**

votes

**8**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 ...

**5**

votes

**2**answers

908 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 ...

**6**

votes

**1**answer

357 views

### Looking for a particular family of C.Y quintics

It is possible to construct (in many ways) a family of Calabi-Yau quintics $\mathcal{X}\rightarrow \Delta$, over disk, such that the fiber over $0$ has a singularities locally given by the equation ...

**15**

votes

**6**answers

2k views

### mirror symmetry with algebraic geometry?

Why is it that mirror symmetry has many relations with algebraic geometry, rather than with complex geometry or differential geometry? (In other words, how is it that solutions to polynomials become ...

**1**

vote

**0**answers

315 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 ...

**6**

votes

**3**answers

643 views

### what is large compex structure limit of CY moduli space

What is the Large Complex Structure limit(LCL) of complex moduli space of a Calabi-Yau 3-fold and why do we need to consider LCL in Mirror symmetry.

**6**

votes

**3**answers

3k views

### Serre's FAC versus Hartshorne as an introduction to sheaves in algebraic geometry

I just found an English translation of Serre's FAC at Richard Borcherds' Algebraic Geometry course web page. I really want to read it sometime. I am beginner in Algebraic Geometry, just started ...

**0**

votes

**4**answers

808 views

### What's dual torus and mirror manifold?

I guess this is a well known fact/definition for many people. It is mentioned in many places that if $\Gamma$ is a lattice of a vector space(vector bundle/affine bundle) $V$, then there is a dual ...

**5**

votes

**2**answers

969 views

### Mirror symmetry for elliptic curves

Lets $E_{\tau}^{\rho}$ be the elliptic curve with complex structure given by $\tau$ in upper half plane and complexified Kahler form $\rho \frac{dz\wedge d\bar{z}}{2}$.( $\rho$ is in upper half plane ...

**13**

votes

**2**answers

2k views

### BRST cohomology

I am reading some work on Mirror Symmetry from Physics perspective,the physicists seem to use some aspects of BRST quantization and BRST cohomology. What is BRST Quantization and BRST cohomology, in ...

**7**

votes

**1**answer

1k views

### Witten's topological twisting

I am reading the Witten's topological twisting for $N = 2$ Superconformal Field Theory(SCFT) http://arxiv.org/abs/hep-th/9112056
In this paper Witten constructed 2 TQFTs i.e. A-model and B-model from ...

**8**

votes

**4**answers

2k views

### Derived categories of coherent sheaves: suggested references?

I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least ...

**13**

votes

**4**answers

2k views

### Roadmap for Mirror Symmetry

I am interested in learning Mirror Symmetry, both from the SYZ and Homological point of view. I am taking a reading course in Mirror Symmetry, which will focus on the SYZ side.
I know basic Complex ...

**1**

vote

**1**answer

417 views

### compute the Kähler moduli of an elliptic curve

Say given elliptic curve $ \{ (x,y) | y^2 = (x^2-1)(x^2-k^2) \}$, what is the right form of the K$\ddot{a}$hler form and how to compute the K$\ddot{a}$hler moduli of this elliptic curve? Thank you.

**5**

votes

**2**answers

477 views

### Mirror of Flop?

If two Calabi-Yau 3-folds are bi-rational to each other via a Flop , then what is the relation between their mirrors ?

**0**

votes

**0**answers

609 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 ...