Questions tagged [mirror-symmetry]
Use for questions about mirror symmetry in theoretical/mathematical physics.
66
questions with no upvoted or accepted answers
18
votes
0
answers
1k
views
What is the Hochschild cohomology of the Fukaya-Seidel category?
Let $(Y, \omega)$ be a compact symplectic manifold and let $Fuk(X,\omega)$ be its Fukaya category. The Hochschild cohomology of this category should be given by $HH^\bullet(Fuk(Y,\omega))=H^\bullet(Y, ...
- 6,820
15
votes
0
answers
579
views
Open conjectures on the Fukaya category coming from physics
This is a slightly vague question (for which I apologize in advance): can somebody give examples to open conjectures on the behavior of the $Fuk(M,\omega)$ that come from string theory and can be ...
- 1,971
14
votes
0
answers
531
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 http://arxiv.org/abs/...
- 6,240
12
votes
0
answers
708
views
Any progress on Strominger-Yau and Zaslow conjecture?
In 2002 Hausel - Thaddeus interpreted SYZ conjecture in the context of Hitchin system and Langlands duality. Let briefly explain it
Let $\pi : E \to Σ$ a complex vector bundle of rank $r$ and ...
user21574
11
votes
0
answers
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 ...
user21574
10
votes
0
answers
523
views
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
$$\mathcal F(X)=\mathcal D^b(\check X)$$
...
- 18.3k
8
votes
0
answers
395
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 ...
- 6,240
7
votes
0
answers
220
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 ...
- 502
7
votes
0
answers
255
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 ...
- 2,961
7
votes
0
answers
462
views
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 $S$,...
- 1,971
7
votes
0
answers
387
views
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 ...
- 553
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 ...
- 1,971
7
votes
0
answers
720
views
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 ...
user21574
7
votes
0
answers
324
views
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{...
- 71
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 ...
- 61
5
votes
0
answers
217
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 ...
- 884
5
votes
0
answers
126
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 ...
- 177
5
votes
0
answers
277
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 ...
- 545
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 ...
user74900
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 ...
- 21.1k
5
votes
0
answers
535
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 ...
user21574
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 ?
4
votes
0
answers
146
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 ...
- 233
4
votes
0
answers
204
views
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 ...
- 431
4
votes
0
answers
145
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 ...
user138661
4
votes
0
answers
136
views
Symplectic Hodge Maps and Mirror Symmetry
The notion of Hodge theory for symplectic manifolds seems to be getting more and more attentions in these days. See the series of papers by Yau:
http://arxiv.org/abs/1011.1250
http://arxiv.org/abs/...
4
votes
0
answers
160
views
Mirror Symmetry for Flag Supermanifolds
I recently learned the following two things, and I wish to know how to make them reconciled.
(1) As far as I understand, the flag manifolds serve as a tractable class of examples for the very ...
4
votes
0
answers
164
views
Hodge numbers of non-commutative varieties
Let $(X, \mathcal{A})$ be a non-commutative variety, by this I mean $X$ is a (smooth) algebraic variety and $\mathcal{A}$ is a sheaf of algebra on $X$. One such example in my mind is when $X$ admits ...
- 3,362
4
votes
0
answers
461
views
Towards an enhanced version of homological mirror symmetry for affine varieties
Let $X$ and $X^\vee$ be a mirror pair, homological mirror symmetry relates the symplectic geometry of $X$ to the complex geometry of $X^\vee$ via the equivalence of triangulated categories
$D^\pi\...
- 3,157
4
votes
0
answers
258
views
Construction of mirror quintic family over $\mathbb{A}^{1} \setminus \{0,1\}$
This question is about how to construct a Fermat pencil of quintics and the mirror family over $\mathbb{A}\setminus {0,1}$ as opposed to over $\mathbb{A}^{1}\setminus {0,\mu_{5}}$, where $\mu_{5}$ is ...
- 1,309
4
votes
0
answers
426
views
$A_{\infty}$-categories and mirror symmetry
I have been looking for an electronic version of Kontsevchi's talk in the Arbeitstagung at 1993. In Kontsevich's publications webpage, I found this reference preprint MPI/93-57, but there is no link ...
3
votes
0
answers
167
views
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 ...
- 1,349
3
votes
0
answers
244
views
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 ...
- 163
3
votes
0
answers
104
views
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 ...
- 181
3
votes
0
answers
180
views
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 ...
- 1,410
3
votes
0
answers
198
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 ...
- 205
3
votes
0
answers
116
views
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 ...
- 2,890
3
votes
0
answers
223
views
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 '...
- 11
3
votes
0
answers
261
views
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:
...
- 1,971
3
votes
0
answers
57
views
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
$$K_0(\mathcal A) := \mathbb Z \operatorname{Ob} \operatorname{Tw} \mathcal A / \left<[A]+[B]-[C]\right>$$
...
- 1,873
3
votes
0
answers
145
views
Perfect Complexes on Tangent Bundle
Suppose $X$ is a $k$-variety of dimension $d$, and suppose $TX$ is its tangent bundle. Consider the (triangulated, stable $\infty$-,...) categories of perfect complexes $\text{Perf}(X)$ and $\text{...
- 203
3
votes
0
answers
245
views
A question on the SYZ mirror symmetry
A toy SYZ mirror symmetry is described as follows. Let $X$ be a $n$-dimensional compact manifold. The contangent bundle $T^*X$ has a natural symplectic structure locally given by the $2$-form $\...
- 31
3
votes
0
answers
361
views
genus one Gromov-Witten invariants of Calabi-Yau 3-folds
In
http://arxiv.org/PS_cache/hep-th/pdf/9302/9302103v1.pdf
physicists calculate (predict) genus one GW invariants of quintic (Table 1) and some other cases (Table 2).
Can any body explain to me (...
2
votes
0
answers
42
views
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 ...
- 373
2
votes
0
answers
76
views
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.
...
- 5,384
2
votes
0
answers
91
views
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 ...
- 355
2
votes
0
answers
52
views
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 ...
- 355
2
votes
0
answers
152
views
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^{...
- 121
2
votes
0
answers
431
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 ...
- 884
2
votes
0
answers
169
views
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 :...
- 21