# Questions tagged [mirror-symmetry]

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

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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.
...

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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 ...

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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 ...

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### 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 ...

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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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### 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 $\...

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### 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 ...

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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 ...

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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 ...

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### 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 ...

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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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### 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^{...

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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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### 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 ...

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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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### 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 $\...

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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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### 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 :...

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### 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 ...

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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 ...

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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 ...

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### 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 ...

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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 $X$ ...

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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 ...

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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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### 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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### 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 ...

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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 '...

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### 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 ...

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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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### 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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### 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:
...

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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
$$\mathcal F(X)=\mathcal D^b(\check X)$$
...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Hodge theoretic mirror symmetry and DG-BV algebras

Consider two Calabi-Yau manifold $X$ and $\check{X}$ which are meant to be mirror partners. Motivated by "classical MS", In DGBV Algebras and Mirror Symmetry, the following enhancement is proposed:
...

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### SYZ conjecture for varieties of general type or Fano

Let $X$ and $Y$ are Calabi-Yau varieties and mirror to each other. Then from HMS the Fukaya Floer category of Lagrangian intersections in $X$, is equivalent to bounded derived category of coherent ...

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### Large Complex Structure Limit of Calabi-Yau family and uniqueness of limit

Let $\mathcal X$ be a smooth complex manifold of dimension $n+1$. We say $\mathcal X \to ∆$ is a large complex structure limit if and only if it’s maximal unipotent degeneration .
$T: H^n(\mathcal ...

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### Can one relate $K_0$ of an $A_\infty$-category $\mathcal A$ to $K_0(Fun_{A_\infty}(\mathcal A, \mathcal A))$?

For an $A_\infty$-category $\mathcal A$, one defines the group $K_0(\mathcal A)$ by
$$K_0(\mathcal A) := \mathbb Z \operatorname{Ob} \operatorname{Tw} \mathcal A / \left<[A]+[B]-[C]\right>$$
...

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### Is there a hyperkaehler manifold whose mirror is the total space of a tangent/cotangent bundle?

I am looking for an example of a hyperkaehler manifold $Y$ whose mirror is the total space of a tangent bundle $TX$ or a cotangent bundle $T^*X$, where $X$ can be any Riemannian manifold.
Is such a ...

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### Is the mirror of a noncompact hyperkaehler manifold also hyperkaehler?

This is essentially a follow-up question from 'Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?'. Verbitsky's theorem in (https://arxiv.org/pdf/hep-th/9512195.pdf) says that ...

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### Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?

Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?
What I know so far is as follows:
In this paper (https://arxiv.org/pdf/hep-th/9512195.pdf) by Verbitsky, it is claimed that ...