Questions tagged [mirror-symmetry]

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

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Mirror of a local K3 surface

Is there any description of a mirror manifold of a (non-compact) Calabi-Yau threefold given by the total space of the trivial line bundle on a K3 surface? If yes, in what way is it a mirror? Thanks ...
Cranium Clamp's user avatar
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Enumerative or Gromov-Witten invariants from derived category of coherent sheaves

Let $X$ be a smooth projective toric Fano surface over $\mathbb{C}$. Suppose I have a nice presentation of $D^b_{Coh}(X)$ given by a full, strong exceptional collection $\mathcal{E} = \{E_i\}_{i\in I}$...
locally trivial's user avatar
7 votes
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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
2 votes
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When can GKZ setup encompass HMS?

Are there any instances when the Landau-Ginzburg superpotential describing the mirror of a smooth projective Fano variety $X_\Sigma$ is encompassed by a GKZ hypergeometric system? In some sense I am ...
locally trivial's user avatar
3 votes
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Proof of the existence of a mirror Calabi–Yau manifold

Let $X$ be a Calabi–Yau threefold. Here, Calabi–Yau is understood to a mean a smooth simply connected projective threefold with $h^1(\mathcal{O}_X) = h^2(\mathcal{O}_X)=0$ and holomorphically trivial ...
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Algebraic Fukaya categories and mirror symmetry

Dominic Joyce and collaborators have outlined a programme to construct algebraic Fukaya categories on an algebraic symplectic manifold (“Fukaya categories” of complex Lagrangians in complex symplectic ...
Robert Hanson's user avatar
2 votes
1 answer
96 views

Bridgeland stability to Fukaya stability on elliptic curve; geometric proof of no slope decreasing homs

For a bridgeland stability condition $(P,Z)$ on $\mathcal{C}$ and $a > b$ we know that $Hom^0(A,B)=0$ for $A,B \in P(a), P(b)$ respectively. I would like to see the geometric incarnation of this ...
user135743's user avatar
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Is there any correspondence between Jacobi forms and automorphic forms on the unit ball in $\mathbb{C}^2$?

Apologies in advance if this question is obvious (or obviously false); number theory is far from my area of expertise. Let me state my questions and then I'll explain the motivation for asking them. ...
Gabe K's user avatar
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Explicit Lagrangian fibrations of a K3 surface

I would like to look at the behaviour of the fibres of a Lagrangian fibration (such that at least some fibres are not special Lagrangian) $X\to\mathbb{CP}^1$ under the mean curvature flow (in relation ...
Quaere Verum's user avatar
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What sort of spaces show up as intersection complexes of toric degenerations of Calabi-Yau Varieties?

Roughly, a toric degeneration is a proper flat family $f:\mathcal{X}\to D$ of relative dimension $n$ with the properties that $\mathcal{X}_t$ is an irreducible normal Calabi-Yau and $\mathcal{X}_0$ is ...
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Coordinate free supersymmetric sigma model Lagrangian

I would like to know if there is a coordinate free version of the Lagrangian of the supersymmetric sigma model on a $2$-dimensional spacetime, with target space a Kähler manifold. The action for this ...
Quaere Verum's user avatar
19 votes
1 answer
2k views

What are "branes", and why do they form a category?

I've been trying to read Kapustin–Witten - Electric–Magnetic Duality And The Geometric Langlands Program recently, as someone whose mathematical interests are in the Langlands program. I have some ...
Anton Hilado's user avatar
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6 votes
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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 ...
user489594's user avatar
3 votes
1 answer
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What does does the monodromy weight filtration represent?

I'm trying to understand variations of Hodge structure. I understand that this is a very broad field, and that many of the concepts have been extended to algebraic geometry over fields other than $\...
Quaere Verum's user avatar
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143 views

Bondal-Orlov conjecture on Calabi-Yau varieties

Recently, I am trying to study the various progress made on the Bondal-Orlov conjecture: Birational Calabi-Yau varieties ⟹ Equivalent derived categories. I have started reading the paper by Bridgeland ...
Rio's user avatar
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How much results in Calabi-Yau manifolds and mirror symmetry depends on the existence of a ricci-flat metric?

An important result of CY manifold is the CY theorem, it talks about the existence of a ricci-flat metric. However, this theorem and its proof are highly analytic. There are many results about ...
Reflecting_Ordinal's user avatar
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Cohen-Macaulay modules and connections to Mirror Symmetry

Let $ R $ be a local Noetherian Gorenstein domain. Suppose a module $ M $ fits into an exact sequence $$ 0 \rightarrow K \rightarrow R^n \rightarrow M \rightarrow 0 $$ Then we write $ K = tM $. A ...
Cranium Clamp's user avatar
3 votes
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195 views

Toric degeneration of Kummer Surface

I am wondering if there are any explicit examples of a toric degeneration of a Kummer surface (e.g. as a family of projective varieties say), and what the central fibre can look like? (I am working ...
Evgeny T's user avatar
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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 $\neq 2, 3$. ...
Dimitri Koshelev's user avatar
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Holomorphic anomaly at genus 1

For computing instantons contributions from worldsheet torus to target torus, one can evaluate zero modes contribution of genus 1 partition function given by following expression: $$Tr(-1)^FF_LF_Rq^{...
user44895's user avatar
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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 ...
Markus Zetto's user avatar
2 votes
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426 views

Embedding Calabi-Yau manifolds in projective space

When studying homological mirror symmetry, a lot of work is done not in the setting of complex manifolds, but of smooth (quasi-)projective varieties, see e.g. a paper from Orlov. However, the actual ...
Markus Zetto's user avatar
7 votes
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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
9 votes
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716 views

B-model and Hochschild cohomology

In "On the Classification of Topological Field Theories" in Example 1.4.1, Lurie introduces the B-model with target an (even dimensional) Calabi-Yau variety $X$: The Hochschild cohomology $\...
Markus Zetto's user avatar
5 votes
0 answers
124 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
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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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Tate Curves and SYZ fibrations

I recently looked at some of the work of Nicaise on non-archimedean SYZ, and at the end of this paper arxiv.org/pdf/1708.09637 he constructs $E^{an}$ for $E$ a Tate curve. There is a retraction $\rho :...
English Muon's user avatar
12 votes
3 answers
2k views

Geometric Langlands: From D-mod to Fukaya

This post is rather wordy and speculative, but I promise there is a concrete question embedded within. For experts, I'll open with a question: Question: Given a compact Riemann surface $X$, why ...
Andy Sanders's user avatar
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Organizing mirror pairs

At a maximally vague and naive level, mirror symmetry asks the following question: given a complex manifold $(X, I)$, is there a symplectic manifold $(M, \omega)$ and an equivalence between the ...
Andy Sanders's user avatar
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Comparing different approaches to HMS for elliptic curves

I am trying to understand homological mirror symmetry for elliptic curves from the article of Zaslow-Polishchuk and from Section 6 of the article of Abouzaid and Smith on homological mirror symmetry ...
Paolo Ghiggini's user avatar
2 votes
1 answer
169 views

Lines on a toric cubic surface with a line of nodes

Consider a cubic surface cut out by equations $x^2y - z^2w$ inside $\mathbb{P}^3$. This gives a cubic surface with a line of nodes, it is toric and has normalisation $\mathbb{F}_1$, a Hirzebruch ...
UserUser's user avatar
4 votes
1 answer
372 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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Berglund-Hübsch-Hori-Vafa mirror symmetry is a ring isomorphism?

Let $W = \sum_{i=1}^{m} a_i \prod_{j=1}^{n} x_j^{b_{ij}}$ be a homogeneous polynomial of degree $d$ in $n$ variables. I focus on the $m=n$ case (invertible polynomial in the Berglund-Hübsch ...
Libli's user avatar
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4 votes
1 answer
313 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
5 votes
1 answer
480 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
0 votes
1 answer
303 views

Log Calabi-Yau surfaces without maximal boundaries

Let $X$ be a smooth projective surface over $\mathbb{C}$, $D\subset X$ is an effective divisor. $(X,D)$ is a log Calabi-Yau pair if $K_X+D$ is a principal divisor. The complement $M=X\setminus D$ is a ...
YHBKJ's user avatar
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3 votes
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Locality in Floer theory

There appears to be a dearth of resources and references for the question of 'locality' in Floer theory. In particular, I cannot seem to find any complete statement of what people refer to as '...
John Rached's user avatar
4 votes
0 answers
143 views

Mixed characteristic in symplectic geometry

Are there any mixed-characteristic phenomena in symplectic geometry/mirror symmetry? There are papers on symplectic geometry by Abouzaid (inspired by Kontsevich--Soibelman, I believe) in which there ...
user avatar
7 votes
2 answers
475 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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7 votes
0 answers
253 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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13 votes
2 answers
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Relation between mirror symmetry, homological mirror symmetry, and SYZ conjecture

I'm very new to mirror symmetry, and have a hard time establishing a broad overview of the subject. In particular I do not understand the precise relation between the following three conjectures: ...
user2520938's user avatar
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10 votes
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What is the mirror of an algebraic group?

Background: Kontsevich's homological mirror symmetry conjecture posits the existence of pairs $(X,\check X)$ with an equivalence of dg/$A_\infty$-categories $$\mathcal F(X)=\mathcal D^b(\check X)$$ ...
John Pardon's user avatar
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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 ...
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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
6 votes
1 answer
300 views

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
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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6 votes
2 answers
476 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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6 votes
1 answer
442 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
4 votes
1 answer
446 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
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11 votes
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439 views

K-stability is invariant under D-equivalency

Kawamata conjectured that Let $X$ and $Y$ be birationally equivalent smooth projective varieties. Then the following are equivalent. We denote by $D^b(Coh(X))$ the derived category of bounded ...
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