Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
1,427
questions
3
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Fredholm property of linearization of Floer map
I am reading Audin and Damian's book "Morse theory and Floer homology". In Proposition 8.1.4 which reveals the transversality property of moduli space of solutions of Floer equation, the ...
6
votes
1
answer
131
views
Symplectic diffeomorphism of the cylinder moving a point to 0
I am currently reading though part of Zehnder's Lectures on Dynamical Systems. In Chapter VII, I have found myself in the following situation:
$Z(1)$ is a subset of standard symplectic space $(\...
7
votes
1
answer
376
views
+200
Goldman symplectic form vs Weil–Petersson symplectic form
I'm confused about the exact multiplicative factor that relates Goldman symplectic form on the $\operatorname{SL}(2,\mathbb R)$-character variety and the Weil–Petersson symplectic form on Teichmüller ...
2
votes
0
answers
135
views
Spectrum of an almost Hamiltonian matrix
I have a complex-valued block matrix $N=\begin{bmatrix}
A & B \\
C & -A^*
\end{bmatrix}$, where $A$ is diagonal, $B=B^*$, and $C$ is rank-1 but not Hermitian.
If $C$ were Hermitian, $N$ would ...
1
vote
1
answer
176
views
Special Darboux chart for tranverse Lagrangians
In the notes of Fukaya-Oh-Ohta-Ono (Lagrangian intersection Floer theory), Chapter 10 §54.1, it is stated:
Let $L_{1}$ and $L_{2}$ be a pair of oriented Lagrangian submanifolds in $(M, \omega)$ that ...
2
votes
1
answer
109
views
Symplectic compatification of a cotangent bundle, or of a neighbourhood of its zero section
Take a closed manifold $\mathcal{L}$ and endow its cotangent bundle $T^*\mathcal{L}$ by the standard symplectic form $\omega = -d\lambda$, $\lambda$ being the Liouville form.
I was wondering if it was ...
4
votes
0
answers
74
views
Pre-Symplectic Mapping
I have been studying symplectic integrators and their pre-symplectic extensions for dissipative systems. According to França, Jordan, and Vidal - On dissipative symplectic integration with ...
1
vote
0
answers
36
views
Displaceability questions in the symplectic 2-sphere for level sets of a Morse function
Consider the symplectic $2$-sphere $S^2$ with the canonical symplectic form $\omega$. A subset $A$ is called displaceable if there exists $H:S^2\rightarrow\mathbb{R}$ smooth such that $\Phi_H^{1}(A)\...
0
votes
1
answer
216
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How to express the Euler-Lagrange equation in arbitrary coordinates where $\omega \neq \sum dp_i \wedge dq_i$
I posted my questions in a previous post MO, but it seems that a more refined version for question on the Euler-Lagrange equation is needed. So, I post my question again.
In standard symplectic ...
0
votes
1
answer
100
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How does the symplectic form $\omega$ manifests itself in the Euler-Lagrange equation? + Extreme confusion with time
Let $\omega$ be a symplectic manifold on $\mathbb{R}^n$ and the smooth function $H : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a Hamiltonian. For $p,q \in \mathbb{R}^n$ let us assume that
\...
1
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0
answers
64
views
Hyperkähler quotient of left $\operatorname{SU}(2)$-action on $(\mathbb{C}^2)^m \cong \mathbb{H}^m$
The natural $\operatorname{SU}(2)$-action on quaternions $\mathbb{H}\cong\mathbb{C}^2$ is hyperkähler. Extending it naturally to $\mathbb{H}^m$, one can make a hyperkähler quotient (where $\mu=(\mu_I,\...
0
votes
1
answer
163
views
Density of the set of convex polygons in the Banach-Mazur distance
Is the set of convex polygons dense in the set of convex domains in $\mathbb{R}^2$, for the Banach-Mazur distance?
Any insight for a negative or positive answer is very much welcome!
4
votes
0
answers
120
views
Coulomb branches which are not of cotangent type
To each $3d \, N=4$ supersymmetric quantum field theory $\mathcal{T}$, there is a related space called the Coulomb branch of this theory, $\mathcal{M}_C(\mathcal{T})$ (it is a piece of the moduli ...
1
vote
1
answer
84
views
Derivative of the symplectomorphism evaluated at a point of the zero section of the cotangent bundle
It might be an easy question, possibly not worth posting here. In the proof of the Lagrangian neighborhood theorem, the authors have written the expression for the derivative of a symplectomorphism at ...
2
votes
0
answers
73
views
Can Coulomb branches have symplectic resolutions?
My question is about Coulomb branches of a $3D$ $\mathcal{N}=4$ supersymmetric gauge theory, in the sense of Bravermann, Finkelberg and Nakajima Towards a mathematical definition of Coulomb
branches ...
0
votes
0
answers
54
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existence of moment maps for non-nef toric varieties
The noncompact toric variety $X_1 = \operatorname{Tot} \mathcal{O}(-1) + \mathcal{O}(-1) \to \mathbb{CP}^1$, the total space of the sum of two line bundles over the complex line, is defined as the ...
6
votes
1
answer
226
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Question about a remark on quantization of Coulomb branches
I will follow the definition of Coulomb branches of $3d$ $\mathcal{N}=4$ gauge theories from the paper by Braverman, Finkelberg and Nakajima, Towards a mathematical definition of Coulomb branches of 3-...
6
votes
0
answers
188
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Does this pseudo-holomorphic triangle contribute to the product $\mu_2$ in Lagrangian Floer cohomology?
I'm computing the product map $$\mu_2 : CF(L_0,V)\otimes CF(V,L_1)\to CF(L_0,L_1)$$ in Seidel's exact triangle for this specific case:
This is a genus 2 surface, and I color-coded the three (...
0
votes
0
answers
20
views
The effect that applying a nodal slide has on the fibers above the eigenline
So recently I've come across the following question posed to me by myself. Suppose I have an almost toric fibration that was obtained from a Delzant polyope by applying a nodal trade. Now in this ...
3
votes
1
answer
300
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In which ways did geometric flows and variational methods from Riemannian geometry enter the symplectic world?
I am interested to learn about the role of geometric analytic methods for solving problems in symplectic geometry, In particular, I would like to know what results heavily rely on this machinery (incl....
1
vote
0
answers
37
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Potential function in the smoothing of toric degenerations when not collapsing all $-2$-Spheres
In the paper "An-type singularity and nondisplaceable Lagrangian tori", https://arxiv.org/pdf/1710.11221.pdf, by Sun the author proves that when taking the toric degeneration of a semi-Fano ...
1
vote
0
answers
114
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Smooth action on cotangent space of the plane
Assuming that $S^1$ acts on $\mathbb{R}^2$ by smooth maps (which are diffeomorphisms), the induced action on the cotangent bundle given by
$$g\cdot(x,\xi)=(g\cdot x,\varphi^∗_{g^{−1}}\xi)$$
acts via ...
1
vote
1
answer
213
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Question about the paper "Infinitely many monotone Lagrangian tori in del Pezzo surfaces"
In the paper "Infinitely many monotone Lagrangian tori in del Pezzo surfaces" by Renato Vianna, the author constructs an infinite amount of non-symplectomorphic monotone Lagrangian tori in ...
1
vote
0
answers
56
views
Lagrangian Floer theory for negative monotone symplectic manifolds and Lagrangians
In the paper "Floer cohomology of Lagrangian intersections and pseudo-holomorphic Disks I", Oh shows that for a compact monotone Lagrangian $L$ in a closed monotone symplectic manifold $M$ ...
2
votes
0
answers
127
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What is the topology on the space of differential forms $\Omega^2(M)$?
I have posted this question on MSE one year ago, but till now I did not received an answer. Therefore I have decided to post it here.
I have difficulty in understanding the meaning of "A ...
1
vote
0
answers
171
views
Is the Poincare Birkhoff theorem valid if we change the volume form of the annulus region?
Is the Poincare-Birkhoff theorem valid if we change the volume form of the annulus region?
Note: A possible approach could be the following: Is it true to say that the answer is affirmative ...
2
votes
1
answer
156
views
Is a simple J-holomorphic curve injective everywhere except for finitely many points?
Let $(M^{2n},J)$ be an almost complex manifold and $(\Sigma,j)$ a closed Riemann surface. Suppose $u: \Sigma \to M$ is a simple, nonconstant, $J$-holomorphic curve. Can we prove that the set
$$
Z:=\{z\...
3
votes
1
answer
272
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Image of the moment map
Let $(M^{2n}, \omega ,\mathbb T)$ be non-compact symplectic manifold with an effective and Hamiltonian torus $\mathbb T$-action.
Suppose its moment map $\mu$ is proper and the fixed point set of $\...
0
votes
0
answers
152
views
Structure maps of $\mathcal{A}_\infty$-bimodules
For Fukaya categories there are functors naturally induced by symplectomorphisms. Twisted versions of symplectic homology (fixed point Floer homology), open-closed maps and bimodules can be defined. ...
3
votes
0
answers
53
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Complex structures compatible with a symplectic toric manifold
Let $(M^{2m},\omega)$ be a compact symplectic manifold with equipped with an effective Hamiltonian torus $\mathbb T^m$ action.
Suppose $J_0$ and $J_1$ are two $\mathbb T^m$-invariant compatible ...
3
votes
1
answer
410
views
Integrability of Schroedinger's equation
Consider the periodic nonlinear Schrödinger equation
$$-i \partial_t u + \Delta u = f(|u|)u, \qquad u=u(t,x) \in \mathbb{C}, \; t\in \mathbb{R}, \; x\in \mathbb{T}^n,$$
where $\mathbb{T}:= \mathbb{R}/\...
2
votes
0
answers
63
views
Why should we restrict the multiplicitiy of hyperbolic orbit to be one in Embedded contact homology?
Embedded contact homology(abbreviated by ECH) is a Floer type theory specially designed for three dimensional contanct manifolds(or generally, manifold with stable Hamiltonian structure) invented by ...
4
votes
0
answers
85
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A couple of questions about the moduli space of annuli with some marked points on the boundary components
I'm trying to work out an answer for my previous question and I'm stuck with the following issue:
In the paper Deformations of Bordered Riemann surfaces and associahedral polytopes by Devadoss, Heath ...
3
votes
0
answers
62
views
Smooth Hamiltonian diffeomorphisms form a Baire space
Let $S$ be a closed surface equipped with an area form $\omega$. In Corollary 1.2 of this paper, Asaoka and Irie demonstrated that Hamiltonian diffeomorphisms which have a dense set of periodic points ...
0
votes
0
answers
77
views
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 ...
1
vote
0
answers
142
views
Invariants associated to a principal bundle whose total space is a symplectic manifold acted symplectically by group structure
The following question - proposal came to my mind about 4 years ago but I did not find any solution to this question and did not find any answer via e-personal comunication with some ...
2
votes
0
answers
248
views
When is $f^*:T^*M\to T^*M$ an ergodic map for a diffeomorphism $f:M\to M$?
Let M be a differentiable manifold and $f:M \to M$ be a diffeomorphism. Then $f$ induces a natural map $f^* :T^*M \to T^*M$.
The pull back map $f^*$ is a symplectomorphism wrt the ...
3
votes
0
answers
71
views
Can a semisimple orbit always be identified with a cotangent bundle?
Let $H$ be a semisimple element of the Lie algebra $\mathfrak{g}$ of a semisimple Lie group $G$, and let $M:=\mathrm{Ad}_G(H)\subset\mathfrak{g}$ be the corresponding adjoint orbit. If we choose a ...
2
votes
1
answer
56
views
Displaceability of the sublevels below the Mane critical value
Recently I have been reading the paper "Symplectic topology of Mané's critical values" by Cielibak, Frauenfelder and Paternain. I am mostly interested in the part of the paper regarding the ...
2
votes
0
answers
59
views
Confusion about proof of $C^0$ bounds for Floer curves on cotangent bundles
I have trouble understanding the proof of theorem 5.4 from Cielibak's article "Pseudo-holomorphic curves and periodic orbits on cotangent bundles". At the bottom of page 267 he defines a ...
2
votes
0
answers
31
views
Kähler quotients for generic $\xi\in \mathfrak{g}^*$
In this question I intentionally omit words like "(non)compact" because I am not sure about the precise setting where this question makes sense.
Let $M$ be a symplectic manifold, $G$ a Lie ...
5
votes
0
answers
155
views
Is the wrapped Fukaya category a symplectomorphism invariant?
Say, let $\phi\colon W_1\to W_2$ be a symplectomorphism of Weinstein manifolds(or with stronger assumption that $W_1$ is Liouville homotopic equivalent to $W_2$, but with non-compact support), do they ...
6
votes
0
answers
186
views
Is Heegaard-Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$?
Is Heegaard Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$? I am interested in the relationship between the theories
...
19
votes
3
answers
939
views
How big can a wedge of $n$ 2-forms in $\mathbb{R}^{2n}$ be?
$\def\RR{\mathbb{R}}$Let $\omega$ be a $2$-form on $\RR^{2n}$, where $\RR^{2n}$ has the usual Euclidean norm. The comass of $\omega$ is defined to be $\max_{|u|, |v| \leq 1} \omega(u,v)$. Here is the ...
2
votes
0
answers
111
views
Reference for Morse-Bott vector fields
I'm looking for a reference for the following result:
Let $X$ be a vector field on a manifold $M$ and let $C$ be a submanifold of $M$ formed by critical points of $X$. Assume that the Hessian of $X$ ...
1
vote
0
answers
76
views
Product structures in Rabinowitz Floer homology
Let $(M,d\lambda)$ be a compact exact symplectic manifold and $\overline{M}$ its symplectic completion. For simplicity we can think of $\overline{M}$ has a cotangent bundle and $\partial M$ the sphere ...
5
votes
0
answers
164
views
Topological obstruction for making a submanifold Lagrangian
Let $M$ be a manifold of symplectic type (we do not fix a particular symplectic form) and $N \subset M$ a submanifold of half the dimension. Is there a topological obstruction on the pair $(M, N)$ for ...
4
votes
0
answers
97
views
Examples of non-equivariant momentum maps
What are examples of non-equivariant momentum maps?
Off the top of my hat, I know about the following examples:
the action of translations of a symplectic vector space (yielding the Heisenberg group ...
1
vote
1
answer
93
views
isotropy of the cotangent lift of a group action
I asked this question in stack exchange but have not received an answer, so I am posting it here.
Given a group action on a manifold (e.g. configuration space of coordinates), cotangent-lift it to the ...
4
votes
0
answers
131
views
Deformation quantization of an integrable system
What is known about lifting n Poisson commuting functions on a 2n-dimensional symplectic manifolds (say R^2n) to Moyal-Weyl commuting functions?