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Questions tagged [lagrangian-submanifolds]

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9
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3answers
286 views

Why are Lagrangian subspaces in a symplectic vector space interesting?

A subspace in a symplectic vector space could be one of two extremes: either symplectic (meaning the form is nondegenerate there) or Lagrangian. Or it could be something between the two, meaning a ...
5
votes
0answers
75 views

A clarification in the definition of Seidel's absolute Maslov index for a pair of transverse Lagrangians

I'm reading Seidel's paper Graded Lagrangian submanifolds where he introduces the absolute Maslov index of a pair of graded lagrangians as follows: Let $\mathcal{L}(V,\beta)$ be the Lagrangian ...
4
votes
0answers
68 views

Lagrangian subgroup of a nonabelian Lie group

My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory. See a previous post for other background ...
8
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0answers
145 views

Lagrangian subgroups/submanifolds, 2d topological boundary and 3d “non-abelian” Chern–Simons theory

This post is meant to ask for proper references to fill a gap in the literature. My short question is that are there known and precise ways to formulate 2d topological boundary conditions" for ...
3
votes
0answers
50 views

Reference Request: Central Curvature “Fix”

Context: In Lagrangian-Floer theory, the (an) $\mathbf{A}_\infty$-algebra of a Lagrangian is curved. However, the curvature is central. One consequence of this is that you can get an uncurved $\mathbf{...
4
votes
0answers
58 views

GSO projection and $H^d(M, \mathbb{Z}_2)$

This follows up the comment which suggests that asking the later 2nd part of subquestion in "GSO (Gliozzi-Scherk-Olive) projection and its Mathematics" as a new different question GSO (Gliozzi-Scherk-...
6
votes
0answers
155 views

GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?

GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...
2
votes
0answers
86 views

how to understand the manifold with boundary jet bundle and cotangent bundle with boundary

Suppose that $M\subset (W^{2n},\omega)$ is an $n$-dimensional manifold with smooth boundary $\partial M$, where $(W,\omega)$ is a $2n$-dimensional Kähler manifold and boundary with contact type ...
4
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0answers
92 views

Topological cycles with Lagrangian support

For a compact Kähler manifold of dimension $2n$, is there a classification of the homological $n$-cycles which are supported in a compact Lagrangian submanifold? The main example for this question ...
3
votes
0answers
53 views

Symplectic displacement energy for several intersection points?

Let $(X, \omega)$ be a symplectic manifold. For any non-empty subset $Y \subset X$ we may define the displacement energy as $$ e(Y)=\mathrm{inf}\{||\phi||_H \: | \phi \in Ham(X, \omega), \phi(Y) \cap ...
12
votes
1answer
204 views

Symplectic mapping class group and the “Lagrangian sphere complex”

For a genus $g$ surface $\Sigma_g$, the mapping class group $\mathrm{Mod}(\Sigma_g)$ acts on the curve complex $\mathcal C(\Sigma_g)$: vertices being isotopy classes of essential, nonseparating, ...
4
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0answers
80 views

Lagrangian embeddings in prequantizable symplectic manifolds

I'm looking for a reference for a special type of Lagrangian embedding in a prequantizable symplectic manifold. The setting is a symplectic manifold $(M, \omega)$, whose symplectic form is the ...
4
votes
0answers
101 views

Lagrangian homology classes in compact symplectic manifolds?

Let $X$ be a compact symplectic $2n$-fold. Which classes in $H_{n}(X, \mathbb{Z})$ can be realized by embedded (or immersed, if that matters) Lagrangian submanifolds? My question is motivated by ...
2
votes
0answers
100 views

Is every monotone Lagrangian Hamiltonian isotopic to minimal Lagrangian?

Assume we have a closed Lagrangian submanifold $L$ in Kaehler-Einstein manifold of positive scalar curvature (for instance, complex projective space). Dazord has proved that 1-form $\alpha=\omega(\...
3
votes
0answers
91 views

Excessive Lagrangian intersection?

Assume we have a monotone Lagrangian submanifold $L$ in a 'good' symplectic manifold $X$ so that Floer homology can be defined (I am interested in $\mathbb{C}P^n$). Then $L$ and its image under ...
5
votes
0answers
118 views

McLean theorem for Fano varieties?

Well-known McLean theorem states that deformations of special Lagrangian $L$ submanifolds in Calabi-Yau manifold are unobstructed and in bijection with harmonic 1-forms on $L$. The proof relies on the ...
2
votes
2answers
147 views

When is mean curvature flow a Hamiltonian isotopy?

Assume we have a compact immersed Lagrangian $L$ in a Kaehler manifold $X$. Recall that a normal vector field $v \in \Gamma(L, N)$ is called Hamiltonian iff $\omega(v, \bullet)$ is an exact 1-form. My ...
3
votes
1answer
188 views

How to understand geometrically, the count of pseudoholomorphic discs by (multi)section perturbation of the kuranish structure on the moduli space?

When defining the $A_\infty$ algebra of a Lagrangian (as done in the book by FOOO) it is done by "counting" (integrating over the moduli space or over the fiber of evaluation map) pseudoholomorphic ...
1
vote
0answers
149 views

Do A-infinity algebra(in Floer theory)have some kind of intersection theory and Poincare duality?

In Lagrangian Floer theory, we can define an A-infinity algebra. It is by first choosing a subset $X_L$ of chains in the Lagrangian submanifold $L$, and then defining boundary maps on(Actually, sum of ...
4
votes
1answer
445 views

Why is every Hamiltonian system locally integrable?

It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the $2n$-dimensional ...
1
vote
1answer
174 views

Irreducibility of holomorphic symplectic quotients

Let a connected algebraic group $G$ (over $\mathbb C$, say) act Hamiltonianly on an algebraic symplectic variety $M$, with moment map $\Phi: M\to \mathfrak g^*$. In the example I care about, vaguely ...
21
votes
1answer
912 views

Why are Lagrangian submanifolds called Lagrangian?

Much of the terminology in symplectic geometry comes from classical mechanics: the symplectic manifold is modeled on a cotangent bundle $T^*N$ of some configuration space $N$ with local position ...
14
votes
1answer
411 views

What is the motivation behind the characteristic variety of a D-module and what does it's geometry tell me about the D-module?

Given a smooth algebraic variety $X$, and an $\mathcal{M}\in \text{Mod}(D_X)$, there is the characteristic variety of $\mathcal{M}$ defined as $$ \text{Char}(\mathcal{M}):= V\left(\sqrt{Ann(\mathcal{M}...
6
votes
1answer
216 views

In $(\mathbb{R}^4,\omega_{std})$ is positive symplectic area enough to guarantee a pseudoholomorphic disc representative?

I will present my question in the context that I encountered it, although I believe it probably applies in general context. Consider $\mathbb{R}^4 \cong \mathbb{C}^2$ with the standard symplectic form ...
4
votes
0answers
91 views

Maslov class of a diagonal

Let $(M,\omega)$ be a symplectic manifold. Which condition on $M$ guarantees that the diagonal of $(M \times M, (\omega,-\omega))$ has a vanishing Maslov class? $H^1(M,\mathbb{Z})=0$ is enough, but I ...
1
vote
1answer
163 views

Lagrangian foliation

Let $(M,\omega)$ be a sympletic manifold and $\{ \cdot, \cdot \}$ the corresponding Poisson-bracket. Assuming $M$ is completely integrable w.r.t $f=f_1$, so we find $n = \frac{1}{2}\dim M$ functions $...
2
votes
1answer
364 views

Local deformations of Lagrangian submanifolds in holomorphic symplectic manifold and their intersections

Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are ...
1
vote
0answers
94 views

Shape of the bubbling limit of holomorphic discs

I will present my question in the specifics I encountered it, so maybe some of the details are irrelevant for the desired conclusion. Consider $(S^2\times S^2,\omega_{std})$ the product of two ...
3
votes
1answer
201 views

What's the geometric statement of this fibrewise integration on a symplectic manifold with Lagrangian fibration?

I understand this statement from the physics side. Consider an $n-$dimensional manifold $\cal M$ ("configuration space") and its cotangent bundle ${\cal P} = T^*\cal M$ ("phase space"), a symplectic ...
5
votes
0answers
316 views

Lagrangian fibration on Schoen's Calabi-Yau 3-fold

Schoen's Calabi-Yau 3-fold is the fiber product $X=Y_1\times_{\mathbb{P}^1}Y_2$ of two rational elliptic surfaces $Y_1\rightarrow\mathbb{P}^1$ and $Y_2\rightarrow\mathbb{P}^1$ with $\chi(X)=0$ and $h^{...
3
votes
1answer
412 views

What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...
2
votes
1answer
289 views

Special Lagrangians and fat

I am unable to find the MO comments about the first use of the phrase "fat slags" in an article. On page 26 of this we find "these correspond to thickenings of the corresponding special Lagrangian (...
8
votes
2answers
552 views

How many Lagrangian submanifolds?

An $n$-dimensional submanifold $L$ of a symplectic manifold $(M^{2n}, \omega)$ is called Lagrangian if $\omega|_L = 0$. I want to get some feeling about how many Lagrangian submanifolds are. For each ...
0
votes
2answers
546 views

about decomposition of three forms

Patrick D. Baier in his Ph.D. thesis for proving the theorem 2.1.4 used the following non-trivial fact (in chapter 2 on page 14): Let $0\neq X\in V$ (here $V$ is of dimension 6), $W^\ast = Ann(X)$ ...
8
votes
1answer
715 views

For which Calabi-Yau threefolds is SYZ conjecture known to hold?

I would like to know for which Calabi-Yau threefolds SYZ conjecture is known to hold. I am aware of works by Gross-Wilson (Borcea-Voisin CY3s) and Ruan (quintic CY3), but they are quite classical ...
2
votes
1answer
208 views

Image of an isotropic manifold under lagrangian correspondence is isotropic?

Is the following statement well known? Let $M,N$ be symplectic (algebraic) manifolds. Let $L \subset M \times N$ be a (smooth) Lagrangian correspondence. For a subset $X \subset M$ we denote $L(X):=(...
6
votes
4answers
510 views

Lagrangian Kleinian bottles

I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in C^2 for the standard symplectic structure, mentioning that this were the only compact surface for ...
1
vote
1answer
342 views

holomorphic sections on elliptic K3 surface

Hi all, I want to ask something about the holomorphic sections on elliptic K3: Is there any obstruction for an ellptic K3 (as an elliptic fibration) to have holomorphic sections? Is that always some ...
1
vote
0answers
206 views

Co-normal bundle of orthogonal compliment

Is the following fact well known? Let $X$ be a manifold and $V$ be a vector space. Let $E_1$ be a sub-bundle of the constant bundle $X \times V$. Let $E_2$ be its orthogonal compliment in $X \...
4
votes
1answer
1k views

Cohomology theory for symplectic manifolds

Suppose I have a symplectic manifold $(M,\omega)$ and a line bundle $\mathcal L$ with a connection with curvature $\omega$ (or perhaps it's more standard to say $\frac i{2\pi}\omega$; anyway, the ...
4
votes
1answer
328 views

Direct image of Lagrangian subspaces of the co-tangent bundle:

Let p:X \to Y be a map of smooth algebraic varieties. Let $C \subset T^\* X$ be a (locally closed) submanifold. Denote by $p_\*(C) \subset T^\* Y$ the following set: $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \...
10
votes
2answers
1k views

Lagrangian Submanifolds in Deformation Quantization

Suppose I have a symplectic manifold $M$, and have a deformation quantization of it, i.e. an associative product $\ast:C(M)[[\hbar]]\otimes C(M)[[\hbar]]\to C(M)[[\hbar]]$ so that $f\ast g=fg+\{f,g\}\...
5
votes
0answers
528 views

Are there cohomology classes on a hyperkähler manifolds which pull back to the Stiefel-Whitney classes on every Lagrangian submanifold?

This is a bit of a stab in the dark but I was wondering if anyone has defined cohomology classes on a hyperkähler manifold which pull back to the Stiefel-Whitney classes on any submanifold which is ...