The mirror-symmetry tag has no wiki summary.
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1answer
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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 ...
2
votes
1answer
137 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
1answer
149 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
1answer
169 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
1answer
270 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
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0answers
146 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
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0answers
195 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
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2answers
136 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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0answers
124 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
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0answers
140 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
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0answers
94 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
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2answers
269 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 ...
7
votes
1answer
288 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
1answer
186 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
1answer
506 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
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2answers
861 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
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1answer
442 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
1answer
435 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
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0answers
199 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
1answer
472 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
3answers
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
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1answer
365 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
1answer
655 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
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0answers
175 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
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3answers
777 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
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0answers
245 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
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1answer
744 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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0answers
251 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
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2answers
644 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
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8answers
3k 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
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2answers
859 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
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1answer
355 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
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6answers
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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 ...
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0answers
309 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
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3answers
599 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
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3answers
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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 ...
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4answers
719 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
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2answers
905 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
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2answers
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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
1answer
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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 ...
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4answers
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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
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3answers
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
1answer
408 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
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2answers
472 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 ?
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0answers
606 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 ...
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3answers
2k views
Do you understand SYZ conjecture
The aim of this question is to understand SYZ conjecture ("Mirror symmetry is T-Duality").
I don't expect a full and quick answer but to find a better picture from answers and comments.
The whole ...
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0answers
198 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!
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2answers
706 views
complexified kahler form
In mirror symmetry one usually considers a complexified kahler form $B+iw$ instead of kahler form $w$ itself.(Or their moduli)
Here is the question:
What does $B$ correspond to? what kind of ...
2
votes
1answer
259 views
balanced curves in Calabi-Yau 3-folds
A balanced smooth rational curve in a calabi-Yau X is a smooth rational curve whose normal bundle is $O(-1)\oplus O(-1)$.
We usually like these curves because of their rigidity.
But, Is there any ...
4
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2answers
315 views
Shrinking Fano surfaces to a point in Calabi-Yau 3-folds
Let X be a Calabi-Yau 3-fold, and $D\subset X$, smooth,Fano divisor.
Since $K_X=0$ we have $N_D^X=K_D$. I have seen the following fact in many papers:
By deforming X within Kahler moduli, we can ...
