Questions tagged [lagrangian-submanifolds]
The lagrangian-submanifolds tag has no usage guidance.
26
questions with no upvoted or accepted answers
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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 ...
7
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227
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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 ...
6
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154
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Composition of coproduct and product in Lagrangian Floer (co)homology
Let's take a Riemann surface $\Sigma$ and three Lagrangians $L_0,L_1,L_2$ in general position. let's assume that we can set up Lagrangian Floer (co)homology - Here I'm being vague because I don't want ...
6
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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 ...
5
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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 ...
5
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119
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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 (...
5
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178
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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 ...
5
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568
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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 ...
4
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Can Lagrangian fibrations have multiple fibres in codimension $1$?
I know that if $\pi: S \to \mathbb P^1$ is an elliptic fibration of a K3-surface $S$, then $\pi$ does not have multiple fibers. A proof of this can be found in Huybrechts' Lectures on K3 surfaces, ...
4
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127
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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 ...
4
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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
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153
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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 ...
4
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119
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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 ...
3
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401
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What is the significance of a Lagrangian Submanifold and what are the implications of the symplectic form being zero?
I'd like to understand better the relevance of Lagrangian submanifolds in Hamiltonian Mechanics. A Lagrangian Manifold is defined as a submanifold of a symplectic manifold upon which the restriction ...
3
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91
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Displacing a conormal Lagrangian from the zero section
I was told that the conormal bundle $\nu^*K$ of a knot $K\subset S^3$ can be displaced from the zero section $0_{S^3}$ in $T^*S^3.$ Having no intuition about whether/how often this happens in general, ...
3
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439
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Two possible meanings of "totally real" submanifold
It seems that there are two common meanings for a submanifold of an almost-complex Riemannnian manifold to be "totally real": one says that the almost-complex structure takes the tangent ...
3
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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{...
3
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113
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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 ...
3
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119
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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 ...
2
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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 ...
2
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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(\...
1
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Examples and classification of holomorphic strips in $(\mathbb{C}\mathbb{P}^n,\mathbb{R}\mathbb{P}^n)$
Consider an exact isotopy $\phi_t$ of $\mathbb{C}\mathbb{P}^n$ such that $\phi_1(\mathbb{R}\mathbb{P}^n)\pitchfork \mathbb{R}\mathbb{P}^n$. When trying to compute the Lagrangian Floer cohomology of $(\...
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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 ...
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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 ...
1
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221
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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 \...
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77
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Symplectomorphism and Hamiltonian isotopy
I would like to ask whether a symplectomorphism of a given symplectic manifold respects Hamiltonian isotopy classes of Lagrangian submanifolds. In other words, given two Hamiltonian isotopic ...