Questions tagged [mirror-symmetry]

Use for questions about mirror symmetry in theoretical/mathematical physics.

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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 ...
Simon Rose's user avatar
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7 votes
2 answers
478 views

Multiple mirrors phenomenon from SYZ and HMS perspective

There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties ...
paul's user avatar
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2 answers
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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 ...
Juan Corrida's user avatar
7 votes
3 answers
2k 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.
J Verma's user avatar
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7 votes
1 answer
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Kapustin-Witten branes and the derived moduli stack of Higgs bundles

A lot has been discussed on overflow regarding geometric Langlands and the physics of Kapustin and Witten's groundbreaking paper https://arxiv.org/abs/hep-th/0604151. I would like to add my two cents ...
Robert Hanson's user avatar
7 votes
1 answer
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The mirror of the Landau--Ginzburg model given by elliptically fibered K3

Let $f:X\rightarrow \mathbb{P}^1$ be an elliptically fibered K3 surface. Choose a coordinate on $\mathbb{P}^1$ and consider $X\backslash f^{-1}(\infty)\rightarrow \mathbb{C}$ as a Landau--Ginzburg ...
user avatar
7 votes
1 answer
278 views

Fiberwise compactification of a LG model

It is believed that a mirror of $\mathbb{CP}^2$ is a fiberwise compactification of the family $$W \colon (\mathbb{C}^\times)^2 \rightarrow \mathbb{C}, \quad (x,y)\mapsto x+y+\frac{1}{xy}.$$ Is it ...
Misha's user avatar
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1 answer
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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 hemisphere. ...
user2013's user avatar
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7 votes
0 answers
219 views

Physical and mathematical significance of the NS-2 brane

This question is about topological string theory and it was also posted in Physics Stack Exchange. The existence of a new brane called "an NS-2 brane" is predicted in (the second paragraph ...
Ramiro Hum-Sah's user avatar
7 votes
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254 views

Higher genus Gromov-Witten invariants and mirror symmetry

As a physicist, my understanding of mirror symmetry is very limited, and perhaps the most "mathematical" literature I have read on mirror symmetry is the book of M. Gross. In the genus-0 Gromov-Witten ...
Wenzhe's user avatar
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Blow-up/Blow-down correspondence via Hodge Mirror Symmetry?

Let $X$ be a projective variety. Let $S \subset X$ be the nonsingular complete intersection of $k$ nonsingular divisors of $X$ of codimension $2k>2$. Denote $\tilde{X}$ the blow up of $X$ along $S$,...
Nati's user avatar
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Integrality of the mirror map -- non-GKZ examples? Counterexamples?

The mirror map in mirror symmetry is the change-of-variables between the natural coordinatizations on the two mirror sides and is typically a highly-complicated transcendental function (indeed, should ...
Arnav Tripathy's user avatar
7 votes
0 answers
295 views

Automorphism that the Fukaya category is "blind" to

Given a symplectic manifold $(M,\omega)$, there is a natural map $$ Symp(M,\omega) \to Auteq(D^\pi Fuk(M,\omega))$$ which sends a symplectic automorphism to the $A_\infty$-functor it induces on the ...
Nati's user avatar
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Examples of Maximal degeneration of Deligne on Calabi-Yau degeneration

Pierre Deligne in his celebrated paper entitling "Local behavior of Hodge structures at infinity" introduced Maximal degenerations of Calabi-Yau manifolds. Let $\pi:X\to \mathbb C^*$ be a family of ...
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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{...
Kelly's user avatar
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6 votes
4 answers
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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.
Junwu Tu's user avatar
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1 answer
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Question on condition for a sheaf to be locally free in Orlov 2004

In "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models", Orlov twice mentions the following criterion for a sheaf $P_1$ to be locally free: If for all closed points $t:x ...
Marc Besson's user avatar
6 votes
2 answers
2k 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 ...
Mohammad Farajzadeh-Tehrani's user avatar
6 votes
2 answers
478 views

Reference on rigorous formulation of mirror symmetry conjecture

I am looking for a mathematically rigorous formulation of mirror symmetry conjecture in the flavour of the original paper by Candelas, de la Ossa, Green and Parkes https://doi.org/10.1016/0550-3213(...
Wenzhe's user avatar
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Does there exists a Fukaya category with no objects

... and really without even the possibility of having objects, so it's not a matter of just finding the "correct" flavour of Fukaya category to use. Question: Does there exist interesting symplectic ...
Nati's user avatar
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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 ...
Vladhagen's user avatar
6 votes
1 answer
443 views

Birational Calabi-Yau varieties with non-isomorphic cohomological invariants

We know from the work of Kontsevich, for example, that birational Calabi-Yau complex varieties have the same Hodge numbers. I want to understand to what extent the equivalence of cohomological ...
user119047's user avatar
6 votes
0 answers
123 views

Does mirror symmetry require large complex structure limit points?

I would like to understand mirror symmetry, so I have been reading books such as "Mirror symmetry and algebraic geometry" and "Calabi-Yau manifolds and related geometries". In ...
user489594's user avatar
5 votes
1 answer
677 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 ...
user21485's user avatar
5 votes
2 answers
574 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 ?
Mohammad Farajzadeh-Tehrani's user avatar
5 votes
2 answers
408 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 ...
Mohammad Farajzadeh-Tehrani's user avatar
5 votes
1 answer
482 views

Lagrangian torus fibrations and Arnol'd-Liouville theorem

Let $(X, \omega)$ be a closed symplectic manifold of dimension $2n$ and $\pi: X \rightarrow Q$ a Lagrangian torus fibration. Let $F_q$ denote the fiber at $q \in Q$. It is claimed in a paper of ...
John Rached's user avatar
5 votes
1 answer
537 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 ...
Acky's user avatar
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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) ...
Mohammad Farajzadeh-Tehrani's user avatar
5 votes
3 answers
828 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 ...
HYYY's user avatar
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5 votes
1 answer
438 views

Confusion regarding statement of mirror symmetry for elliptic curves

I am a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
user198433's user avatar
5 votes
1 answer
620 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 ...
asv's user avatar
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5 votes
0 answers
216 views

Calabi-Yau structures on dg-categories

A (smooth) dg algebra is called (left) Calabi-Yau if (see for example here) $$ A^! = A[-n]$$ Here we use the inverse dualizing complex $A^!=\mathbf{R}\operatorname{Hom}_{(A^e)^{op}}(A,A^e)$. In ...
Markus Zetto's user avatar
5 votes
0 answers
125 views

Mirror symmetry for $C^*$

The Liouville manifold $T^*S^1$ is said to be "mirror" to the complex variety $C^*$. (see for instance lecture 7 here: http://math.columbia.edu/~topology/Eilenberg_lectures_fall_2016) This is ...
user155668's user avatar
5 votes
0 answers
273 views

Why is a DG-enhancement of the derived bounded category an enhancement?

I asked this question on math.stackexchange with no luck, so I thought I would try here. In order to make mirror symmetry more compatible with homological machinery, I understand it is common to give ...
C.D.'s user avatar
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5 votes
0 answers
178 views

Mirror of the autoequivalences of the derived category of del Pezzo surface?

One version of the homological mirror symmetry conjecture states that for every Fano variety $X$ there exists a Landau--Ginzburg model $W$ such that the category of B-branes on $X$ (i.e. the bounded ...
user avatar
5 votes
0 answers
245 views

Localization principle in integration over supermanifolds

This post is closely related to the post Localization principle in supersymmetry and can be considered as a continuation of it, although independent. In § 9.3 of the book "Mirror symmetry" (K. Hori ...
asv's user avatar
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534 views

connectedness of moduli space of Calabi-Yau 3-folds by symplectic surgery theory

"Motto" Moduli space of Calabi-Yau varieties can be connected by using Symplectic surgery theory. Miles Reid’s Fantasy:“There is only one Calabi-Yau space” i.e "All CY connected through conifold ...
user avatar
5 votes
0 answers
377 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 ?
Oliver Fabert's user avatar
4 votes
2 answers
385 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 ...
Tobias's user avatar
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4 votes
2 answers
2k 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 ...
Mohammad Farajzadeh-Tehrani's user avatar
4 votes
1 answer
314 views

Mirror symmetry for singular Lagrangian torus fibrations

Let $X$ be a closed symplectic manifold equipped with a smooth Lagrangian torus fibration $\pi:X \rightarrow Q$. Assume that $\pi$ admits a Lagrangian section. By work of Kontsevich-Soibelman, one can ...
John Rached's user avatar
4 votes
1 answer
354 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 ...
Mohammad Farajzadeh-Tehrani's user avatar
4 votes
1 answer
218 views

Mirror partners of some Calabi-Yau threefolds

I don't have experience in mirror symmetry, hence I am not sure that my question is of research level. Sorry in advance. Let $k$ be an algebraically closed field of characteristic $\neq 2, 3$. ...
Dimitri Koshelev's user avatar
4 votes
1 answer
448 views

Mirror symmetry for blowups of the projective plane

Let $S$ be a blowup of the projective plane $\mathbb{CP}^2$ at $n$ points. When $n\le 9$, Auroux, Katzarkov and Orlov showed that them a mirror Landau-Ginzburg model is given by a certain elliptic ...
Lee's user avatar
  • 41
4 votes
1 answer
1k 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 ...
Jeremy Pecharich's user avatar
4 votes
1 answer
228 views

Gromov-Witten invariant $\langle p, p, \ell\rangle_{0, 1}$ counting degree $1$, genus $0$ curves in $\mathbb{CP}^2$?

Let $p \in H^4(\mathbb{CP}^2)$ and $\ell \in H^2(\mathbb{CP}^2)$ be the cohomology classes Poincaré dual to a point and a line respectively. Question. What is the Gromov-Witten invariant $\langle p, ...
user102036's user avatar
4 votes
1 answer
944 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 ...
temp's user avatar
  • 1,990
4 votes
1 answer
602 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 theory,...
4 votes
1 answer
374 views

Comparison of Hochschild homology in Mirror Symmetry

Given a triangulated category $D$, there is a Chern character from the Grothendieck group to the Hochschild homology: $$ch:K_0(D) \to HH_0(D).$$ Consider a pair of projective Calabi-Yau threefolds $X$ ...
François's user avatar
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