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34
votes
8answers
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 ...
29
votes
1answer
790 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 $ ...
16
votes
2answers
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 ...
16
votes
3answers
899 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 ...
16
votes
3answers
872 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 ...
15
votes
6answers
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 ...
15
votes
5answers
2k views

Mirror symmetry mod p?! … Physics mod p?!

In his answer to this question, Scott Carnahan mentions "mirror symmetry mod p". What is that? (Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than ...
15
votes
4answers
3k 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 ...
14
votes
2answers
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 ...
13
votes
4answers
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 ...
11
votes
0answers
219 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 ...
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 ...
9
votes
2answers
1k views

Higher genus closed string B-model

The closed string A-model is mathematically described by Gromov-Witten invariants of a compact symplectic manifold $X$. The genus 0 GW invariants give the structure of quantum cohomology of $X$, which ...
9
votes
3answers
391 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 ...
9
votes
1answer
395 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 $$ ...
9
votes
0answers
334 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 ...
8
votes
4answers
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 ...
8
votes
1answer
182 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 ...
7
votes
3answers
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 ...
7
votes
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 ...
7
votes
1answer
791 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 ...
7
votes
2answers
709 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 ...
7
votes
1answer
1k 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, ...
7
votes
1answer
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 ...
7
votes
1answer
371 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 ...
7
votes
1answer
564 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 ...
7
votes
0answers
200 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
2answers
2k views

Picard-Fuchs equations

If I have the periods $$\pi_1(\lambda)=\int_0^1\frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$$ and $\pi_2(\lambda)$ similarly defined of the cubic curve $$y^2z=z(x-z)(x-\lambda z)$$ Such functions will be ...
6
votes
3answers
676 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
1answer
363 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 ...
5
votes
2answers
981 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 ...
5
votes
2answers
490 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 ?
5
votes
2answers
862 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 ...
5
votes
2answers
288 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 ...
5
votes
1answer
1k views

Mirror symmetry for noncompact Calabi-Yau manifolds

In analogy with the Hodge diagram for ordinary de Rham cohomology, we should have some kind of diagram for Alexander-Spanier cohomology. Doing all the relevant duality stuff and assuming that now our ...
5
votes
2answers
1k 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 ...
4
votes
4answers
1k views

Mirror of local Calabi-Yau

What is the mirror manifold of the total space of the bundle $O(-1)\oplus O(-1)$ over $P^1$? I have tried to find the answer on the web but failed. Is there a good reference for this? Thanks.
4
votes
2answers
320 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 ...
4
votes
2answers
789 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 ...
4
votes
3answers
549 views

Which part of physical B model is not rigorous?

Which part of physical B model is not rigorous? the physical theory of B model,if it is not mathematical rigorous just because the Feynman integral,but it looks like for me the space is finite ...
4
votes
1answer
218 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) ...
4
votes
1answer
420 views

Singularity theory references

I am looking for some good references on singularity theory. I'm interested in singularity theory in the context of mirror symmetry, so this means I'm interested in things like Picard-Lefschetz ...
4
votes
0answers
148 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 ...
4
votes
0answers
233 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
654 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 ...
3
votes
1answer
396 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 ...
3
votes
2answers
426 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 ...
3
votes
1answer
189 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
164 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 ...
3
votes
1answer
519 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 ...