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 ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 $S$,...
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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 ...
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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 ...
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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{...
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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.
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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 ...
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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 ...
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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(...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ?
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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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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 ...
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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 ...
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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 ...
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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 $\neq 2, 3$. ...
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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 ...
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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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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, ...
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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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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,...
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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 $X$ ...