Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
1,427
questions
4
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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 ...
2
votes
0
answers
199
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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(\...
6
votes
1
answer
290
views
Every contractible smooth loop has a neighbourhood with $H^2=0$
Let $c: S^1 \to M$ be a smooth contractible loop (not necesarily an embedding, or even an immersion) on the connected, compact symplectic manifold $(M,\omega)$ (if this helps somehow, $c$ is a $1$-...
12
votes
3
answers
1k
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Do contact and CR structures have corresponding $G$-structures?
For an $n$-dimensional manifold $M$, almost complex and almost symplectic structures on $M$ correspond to reductions on the structure group of the tangent bundle, introducing a $\operatorname{GL}(n/2,\...
3
votes
0
answers
118
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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 ...
22
votes
2
answers
684
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$(M,\omega)$ not symplectomorphic to $(M,-\omega)$
Looking for an example of a symplectic manifold $(M,\omega)$ that is not symplectomorphic to $(M,-\omega)$.
In particular this means that $M$ must be chiral (i.e. doesn't admit an orientation-...
3
votes
1
answer
362
views
Disconnecting the Lagrangian Grassmannian
Let $(V, \omega)$ be a symplectic vector space of dimension 2n. This has a Lagrangian Grassmannian $\Lambda(V)$ of Lagrangian subspaces of $V$. Now consider the following subvariety: Fix a half-...
27
votes
2
answers
1k
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Intuition for symplectic groups
My question essentially breaks down to
How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...
3
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2
answers
264
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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 ...
21
votes
2
answers
2k
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Infinite dimensional symplectic geometry
Could anyone comment on possible references concerning infinite dimensionsal symplectic manifolds?. I am mainly concerned with hilbert spaces, so i am not interested in the convenient analysis ...
2
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0
answers
189
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Are rational varieties symplectically rationally connected?
Was it proven already that smooth rational complex projecitve varieties are symplectically rationally connected? I.e. some GW invariant with two point insertions is non zero. What about smooth toric ...
5
votes
0
answers
292
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The structure of Banach manifolds in symplectic geometry
Let $M$ be a symplectic manifold, and let $L_0$ and $L_1$ be Lagrangian submanifolds which transverse to each other. In Floer theory, we need to consider a Banach manifold $\mathcal B$ of maps $u:\...
6
votes
1
answer
264
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More pseudoholomorphic curves than complex valued functions
A lecture I heard had a remark - "There is a rich class of pseudohoplomorphic curves to a symplectic manifold with an almost complex structure (tamed by the symplectic structure). On the other hand, ...
3
votes
1
answer
206
views
Why is the matrix in Dirac's bracket formula invertible?
I am reading the book "Introduction to mechanics and symmetry" by J.Marsden and T.Ratiu and am experenced a problem.
Let $(P,\Omega)$ be a symplectic manifold, a submanifold $S\subset P$ is called a ...
6
votes
0
answers
114
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Is every contractible open bounded domain in $\mathbb R^{2n}$ symplecomorphic to a star-shaped domain?
In Hofer & Zehnder's book "Symplectic Invariants and Hamiltonian Dynamics" (Page 99) they present an example of a star shaped domain (bounded, with smooth boundary) in the shape of a "Bordeaux ...
4
votes
1
answer
222
views
Contradiction between fixed points of a hamiltonian diffeomorphism of a torus and quasi-periodic motion on a torus
Again a very simple question. I currently hold two contradictory ideas in my head
1) A hamiltonian diffeomorphism of a torus necessarily has fixed points
2) most hamiltonian actions on a torus in an ...
2
votes
0
answers
116
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Embeddings of the configuration space into the phase space of integrable systems
As always, I'm not sure if I'm about to ask a very stupid question, and I apologise if that is the case.
Most systems from physics come from classical Hamiltonians, defined on the phase space of ...
4
votes
1
answer
313
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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 ...
11
votes
0
answers
532
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Third cohomology of symplectic $6$-manifolds
Suppose that $A$ is a finitely generated abelian group with even rank and without $2$-torsion. Does there exist a compact symplectic $6$-manifold $(M,\omega)$ such that $A$ Is isormorphic to $H^{3}(M,\...
6
votes
0
answers
156
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A functional on paths in a symplectic vector space
I'm running into a functional associated to a piecewise smooth curve $\gamma: [0,1] \to V$, where $V$ is a real vector space with a symplectic form $\omega$:
$$ \int_{0 \leq x \leq y \leq 1} \omega(\...
2
votes
0
answers
117
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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 ...
2
votes
1
answer
387
views
A Lagrangian connection and its algebraic interpretation
Assume that $M$ is a $n$ dimensional Riemannian manifold. So $TM$ has a natural structure of a symplectic manifold.
A Lagrangian connection $D$ on $M$ is a $n$ dimensional distribution for $TM$ such ...
2
votes
0
answers
240
views
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 ...
7
votes
1
answer
279
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Trying to prove one of C.Taubes' theorems gauge-theory-freely
One of C.Taubes' theorems says that for a symplectic 4-manifold $X$ with $b^2_+>1$ (where $b^2_+$ denotes the dimension of a maximal positive-definite subspace of $H^2(X;\mathbb R)$ under the ...
5
votes
2
answers
365
views
Manifold of mappings between $M$ and $N$, with non-compact source $M$
EDIT: Let $M$ and $N$ are two smooth manifold and suppose $N$ is compact but $M$ is not necessarily compact. For my purpose, I just need to consider the case $M=\mathbb R \times S^1$ or $\mathbb R \...
4
votes
1
answer
440
views
Smooth manifold with affine structure: aspherical?
I wonder if a smooth manifolds $M$ which admits an affine structure must necessarily satisfy $\pi_2(M)=0$.
By affine structure I mean an atlas all of whose change of coordinates maps are affine maps. ...
11
votes
2
answers
428
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Fredholm property about $L^p$-extension $(p\neq 2)$ of differential operators
The following is a well-known result for elliptic operators.
Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact ...
7
votes
0
answers
137
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Could we extend the star product on a Poisson manifold from its ring of smooth functions to its de Rham complex?
Let $M$ be a smooth manifold with a Poisson bracket $\{-,-\}$. Kontsevich proved that there exists a deformation quantization of $M$, i.e. let $C^{\infty}(M)[[\hbar]]=C^{\infty}(M)\otimes_{\mathbb{R}}\...
9
votes
2
answers
763
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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 ...
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$,...
4
votes
0
answers
190
views
A quantity associated with a Riemannian surface
Assume that $E$ is a Riemannian vector bundle, then its structure group is reduced to $O(n)$. Then the structure group of $E \oplus E$ is reduced to $D(O(n) \oplus O(n)) \subset Sp(2n)$ ...
8
votes
1
answer
778
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How to construct the mirror partner of a blowup?
Question: Let's assume we have a pair $(X,\check{X})$ that are mirror dual to each other in the sense of Homological mirror symmetry (EDIT: this does not have to be CY n-folds, but can also be a Fano ...
5
votes
0
answers
528
views
Darboux's Theorem
I read proofs of Darboux's theorem that a symplectic form $\omega$ on M locally is symplectomorphic to a standard symplectic form $\omega_{0}= \sum dp \wedge dq$ in Rolf Berndt's An Introduction to ...
18
votes
1
answer
1k
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Categorification of Floer homology
Floer homology associates a vector space $HF^\ast(L_1,L_2)$ to any pair of Lagrangian submanifolds $L_1,L_2$ inside a symplectic manifold $X$. By a categorification of Floer homology, I mean a ...
4
votes
0
answers
319
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Higher genus Cohen-Jones-Segal's conjecture?
Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of the stablization-evaluation morphism
$$(ev,\Phi):\...
3
votes
0
answers
149
views
Integral Homology of GIT Quotients
Is there any example of GIT quotients of linear actions on a Euclidean space ${\mathbb C}^n$ that satisfy the following conditions?
The quotient is compact and smooth.
The homology of the quotient ...
3
votes
1
answer
188
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Constructing a Hamiltonian $S^1$-action on a neighborhood of a symplectic divisor
Let $M^{2n}$ be a symplectic manifold and let $M^{2n-2}$ be a symplectic submanifold. How to construct a non-trivial Hamiltonian $S^1$-action on $M^{2n-2}$ on a small neighborhood of $M^{2n-2}$, that ...
11
votes
1
answer
865
views
Monopole Floer Homology vs. Heegaard-Floer theory
I have a (possibly very naive) question: what is the relation between Monopole Floer Homology and Heegaard-Floer theory? (both known and conjectured)
Is there some version of Atiyah-Floer conjecture ...
2
votes
0
answers
134
views
What is the difference between the Fukaya category and the Fukaya-Floer category?
What is the difference between the Fukaya category and the Fukaya-Floer category? It seems that the former is defined in terms of flat $U(1)$ line bundles and the latter in terms of flat unitary ...
3
votes
1
answer
194
views
Why are the toric fibers of a toric manifold Lagrangian submanifolds?
How does one know in general that the toric fibers of a Kahler toric manifold are Lagrangian submanifolds? I can roughly fathom this for e.g. a circle in $\mathbb{C}P^1$, but how about more general ...
3
votes
0
answers
264
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Classical analogue of the theorem of equivalence of the S-matrix
In quantum field theory there is a statement called the equivalence theorem of the S-matrix. S-matrix is invariant under reparametrization of the field. Is there in classical mechanics, the analogous ...
5
votes
1
answer
333
views
Is there a relationship between the moduli space of spatial polygons and the moduli space of labeled points?
It is well known that the set of all polygons with consecutive side lengths $l_1, \dots, l_n$ in $\mathbb{R}^3$, considered up to rigid motions, is a compact complex manifold. Of course, I am assuming,...
3
votes
1
answer
188
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Index of linearized operator for symplectic vortex equations
In reference [1], the index of the linearized operator for the symplectic vortex equations is computed on page 27-28.
The first step of the proof says that the operator
\begin{equation}\tag{1}
\...
3
votes
1
answer
344
views
Is it difficult or easy to find non-symplectomorphic symplectic forms on a manifold?
If people think this question is a little basic I will move it to stackexchange, but I decided to try here first. So it is clear that if two symplectic forms are not cohomologous then they cannot be ...
5
votes
1
answer
926
views
Monotone symplectic manifolds with Hamiltonian actions are Kähler?
I am wondering if the following is true:
Let $(M,\omega)$ be a compact symplectic manifold which is also monotone, i.e. $c_1(TM)=\lambda [\omega]$.
Moreover assume that it admits a Hamiltonian ...
3
votes
0
answers
94
views
Can one choose a sufficiently generic path of a.c.s such that only "codimension 1" bubbling occurs?
Consider a symplectic manifold $(M,\omega)$ of dimension $2n$ (closed or open with bounded geometry). Let $L\subset M$ be a compact Lagrangian submanifold (not necessarily connected).
Consider two ...
0
votes
2
answers
496
views
A relation between gradient vector field and Hamiltonian vector field
Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$.
Assume that $\omega$ is an arbitrary symplectic structure on $U$.
Is there a Riemannian ...
2
votes
0
answers
169
views
Question on Hori, Iqbal and Vafa's 'D-branes and Mirror Symmetry'
In the paper mentioned above, on page 19, the physics of A-type supersymmetry is related to a Lagrangian submanifold $\gamma$ of a Kaehler manifold $X$. In particular, the phrase "...holomorphic ...
2
votes
0
answers
58
views
Configurations of minimal vectors for a 4-dimensional symplectic lattice
The possible configurations of minimal vectors for a 4-dimensional lattice are known for ages, but what about symplectic lattices ? If a 4-dimensional symplectic lattice $\Lambda$ has two minimal ...
4
votes
0
answers
233
views
Lagrangian submanifold of Poisson manifolds
Let $V$ be a finite dimensional vector space.
Let $\psi\in \Lambda^2V$ be a (possibly degenerate) $2$-vector. Then $\psi$ defines a map $V^*\rightarrow V$. Let $U\subset V$ denote the image of this ...