Questions tagged [sg.symplectic-geometry]
Hamiltonian systems, symplectic flows, classical integrable systems
1,427
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How can I see the "missing" Poisson center when the rank of the Poisson structure drops?
Recall that a Poisson algebra is a commutative algebra $A$ along with a bracket $\lbrace,\rbrace: A^{\otimes 2} \to A$ which is a Lie bracket and which is also a derivation in each variable. The ...
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2
answers
1k
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Two versions of hamiltonian reduction
Given a symplectic manifold $X$ with nice $G$ action, equivariant momentmap $\mu$ and
invariant $\chi \in \mathfrak{g}^*$ which is a regular value of $\mu$.
There are two ways to form the ...
6
votes
1
answer
398
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Is there a theory of differential equations for smooth correspondences?
This question is very closely related to another one I just asked. The general question is to what extent there is a theory of differential equations for smooth correspondences (between a smooth ...
6
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1
answer
379
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Homology and cohomology of free loop spaces
String topology, as well as Hochschild (co)homology, give a rich perspective on the homology and cohomology of a free loop space $LM$ of a manifold $M$.
Let $k$ be a field and let $M$ be $n$-...
6
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answer
136
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Existence of isotopy preserving the action
Let $\gamma_1$ and $\gamma_2$ be simple closed curves in $R^4.$ Let $\lambda= x_1 dy_1+ x_2dy_2.$ Suppose that $\int_{\gamma_1} \lambda= \int_{\gamma_2} \lambda.$
I am looking for a reference for ...
6
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191
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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 (...
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191
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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
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211
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Iterating exact triangles (particularly in Floer homology)
There are several different Floer-homological invariants of 3-manifolds (and knots). The most prominent of these are Heegaard Floer homology, monopole Floer homology, and instanton Floer homology. It ...
6
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1
answer
234
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From time-dependent Hamiltonians to time-dependent symplectic/Poisson structures
Let $(M,\{.,.\})$ be a smooth Poisson manifold, and let $H\in C^\infty(M\times\mathbb{R},\mathbb{R})$.
Question: Does there exist $H_0\in C^\infty(M,\mathbb{R})$ and smooth parameter-dependent Poisson ...
6
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160
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Symplectic cohomology of $T^* \mathbb{CP}^2$
I'm looking for an explanation for why the symplectic cohomology $SH^*(T^* \mathbb{CP}^2,\mathbb{Z})$ is 2-torsion (I heard this in passing; perhaps it's not even true!). By a clever argument that I ...
6
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261
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Computation of the Fredhom index in Floer theory
I have been reading Salamon's lecture notes on Floer homology, and to compute the Fredholm index of operators they use the fact that the spectral flow of $A(s)$ is the Fredholm index. Now in the proof ...
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 ...
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Computing Gromov-Witten invariant of $4$ lines in $\mathbb{C}P^3$
I'm trying to understand what the number of genus 0 curves through four lines in $\mathbb{C}P^3$ is i.e $Gr_{0,4}^{\mathbb{C}P^3, L}(PD(L),PD(L),PD(L),PD(L))$ where $L$ is the class of a line $\mathbb{...
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490
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Hamiltonian dynamics on cotangent bundle
I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
6
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151
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Logarithmic Darboux theorem
Let $X$ be a smooth complex analytic manifold and $D$ be a normal crossing divisor. Suppose that there is a complex analytic logarithmic symplectic structure on $X$.
Is there a Darboux like theorem ...
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266
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Is there an symplectic field theory compactness theorem applicable in the context of Floer cohomology of a symplectomorphism?
Is there any reference in the literature about results regarding symplectic field theory (SFT) compactness for a neck-stretch in the context of Floer homology of a symplectomorphism $\phi \colon (M,\...
6
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328
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Hochschild cohomology of (generalizations) of Khovanov's arc algebra
Backgroud: In his seminal paper A functor-valued invariant of tangles, Khovanov (among many other things) introduced the arc algebra $H^{n}$ and several functors between $H^{n}$ and $H^{m}$ related to ...
6
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148
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Maslov index of pair of paths in $\mathcal{L}(2n)$ and its relation with the Maslov index of a loop in $\mathcal{L}(2n)$
I'm reading [RS] and I was wondering what kind of connection there is between the Maslov index for a pair of paths $\lambda_0,\lambda_1 \colon [a,b] \to \mathcal{L}(2n)$ as defined in [RS] and the ...
6
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168
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Introduction to the Adler-van Moerbeke theory
Is there a good introduction to the Adler-van Moerbeke theory of solving completely integrable systems by linearizing the flow on the Jacobian of an algebraic curve, for someone with a background in ...
6
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269
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Varying a $J$-holomorphic sphere in a symplectic $4$-manifolds
I am certain that the following result holds, but was not able to find a reference. Do you know one? Or maybe you can give a short proof?
Statement. Let $(M^4,\omega)$ be a compact symlectic manifold ...
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326
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Lagrangian up to Hamiltonian in cotangent bundle
I want to understand the folklore conjecture that, in a CY manifold, Lagrangians up to Hamiltonian isotopies are represented by special Lagrangians by examining cotangent bundle and Hodge theory.
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6
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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 ...
6
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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(\...
6
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314
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Freed-Hopkins-Lurie-Teleman topological boundary conditions v.s. Lagrangian subspace
This question concerns the comparison of topological boundary conditions of TQFTs on a manifold with some boundary.
For example, we can consider defining the TQFT on a $D^3$ ball with a topological ...
6
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459
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Symplectic orbifolds
I will start by saying that I am not a symplectic topology. However, in my research I now have on my hands a symplectic 4-orbifold, which I would like to understand better. Certain results for ...
6
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316
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Is there any work on "super Fukaya categories"?
There is a well-established notion of "supermanifold", and in the world of supergeometry it makes sense to talk about symplectic structures. Actually, there are various kinds of symplectic structures:...
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423
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Handle attachment in symplectic category
It is known that for an exact symplectic manifold $(M,\omega_M)$ with a convex boundary $(\partial M,\theta_M)$, where $d\theta_M=\omega_M$ (usually called a Liouville domain), one can attachment to ...
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454
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An algebraic Hamiltonian vector field with a finite number of periodic orbits (2)
Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...
6
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0
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502
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When a Spherical variety is $K$-stable
Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical.
My question is When a Spherical variety is $K$-stable? Is ...
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253
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Concrete almost-complex structures on $3 \#CP^2$
The connect sum $X:=CP^2\# CP^2 \# CP^2$ supposedly supports almost-complex structures, i.e. endomorphisms $J$ of the tangent bundle such that $J^2=-id$. The existence of these almost-complex ...
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259
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Unobstructed Lagrangian tori
Let $X$ be a compact symplectic manifold, $U$ a Darboux chart, and $L$ a standard Lagrangian torus in $U$. Is $H^1(L)$ weakly unobstructed in the sense of Fukaya-Oh-Ohta-Ono (that is, does $b \in H^1(...
6
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431
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Laplacians associated to symplectic cohomologies
I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...
6
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310
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Examples of non-Kahler symplectic manifolds.
Hi.
I want to find an example of symplectic (smooth compact) manifold $(M,\omega)$ whose Betti numbers are not unimodal, i.e.
$b_i(M) \leq b_{i+2}(M)$ fails for some $i < n$. ($ \dim{M} = 2n$.)
...
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558
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Jones Polynomial and Quantum Field Theory
I am trying Witten's paper but unable to re-produce the computations presented in the paper.
I tried few things on internet but all these tutorials explicitly don't show the calculations and refer to ...
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653
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Are conical symplectic resolutions Mori dream spaces?
This is one of these questions where it's tempting to just leave it at the title, but let me try to define the objects in question.
A conical symplectic resolution is a projective resolution of ...
6
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0
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497
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How to prove that a certain action is hamiltonian?
I posted this question here on math.stackexchange.com. I have not had an answer, and I thought it could be more appropriate here.
(Please, If you judge this my opinion is wrong, then I will delete ...
5
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3
answers
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Is there a version of Seiberg-Witten-Floer or Heegard-Floer homology for 3-manifolds with boundary?
Recently, the Seiberg-Witten-Floer homology created by Kronheimer and Mrowka has important applications in Taubes' proofs of Weinstein conjecture and Arnold Chord Conjecture. Also, Cagatay Kutluhan, ...
5
votes
1
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546
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Recognizing the stablizer of a degenerate three forms in six dimension
Define $Stab^{+}(\Omega )$={ $\phi \in GL^{+}(V)$ : $\phi^{*}\Omega=\Omega$ }.
we say three-form $\Omega\in\wedge^{3}V^{*}$ is non-degenerate , if $i_X\Omega\neq 0$ for all $X\in V$-{0}
Let $V\cong \...
5
votes
2
answers
647
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What information is required for SYZ mirror symmetry?
Let $X$ be a Calabi-Yau threefold. The Strominger-Yau-Zaslow conjecture suggests that $X$ should have a special Lagrangian $T^3$-fibration (when $X$ lies near a large complex structure limit) and a ...
5
votes
2
answers
630
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How to find faces of polytope defined by a Weyl orbit
A few days ago I asked the following question at MSE and received no answer. I thought I would try here.
Let $\xi$ be an integral dominant weight of an irreducible root system $\Delta$, and let $\...
5
votes
2
answers
1k
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Contact manifolds that are not cooriented
One always sees the definition of a contact manifold $(X,\xi)$ as an odd dimensional manifold with a hyperplane distribution $\xi$ which can locally be expressed as $\xi = \ker \alpha$ for a 1-form $\...
5
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2
answers
1k
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Applications of Floer homology
Can somebody tell me of other applications of Floer homology besides the proof of the Arnold conjecture.
Every answer would be appreciated.
5
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1
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677
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Mirror to the dualizing sheaf
I wonder - is there a general characterization of the object in the Fukaya category that is mirror to the dualizing sheaf on the other side?
This question has two cases:
1. CY
2. Non-CY
In 1. what ...
5
votes
2
answers
482
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Picard group of symplectic group modulo orthogonal group
Let $Sp(2n)$ be the group of complex symplectic $2n\times 2n$ matrices, and $O(2n)$ the group of complex orthogonal $2n\times 2n$ matrices.
Consider $Sp(2n)\cap O(2n)\subset Sp(2n)$ and the quotient $...
5
votes
2
answers
378
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Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces
I apologize in advance if this question has an obvious answer.
Let $(M,g)$ be a Riemannian manifold.
Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$...
5
votes
1
answer
747
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 ...
5
votes
1
answer
763
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Canonical bundle of the Lagrangian Grassmannian
I work through the paper On branched coverings of some homogeneous space of Kim and Manivel and I came across the definition of the canonical bundle of the Lagrangian Grassmannian $\mathbb{LG}_n$, the ...
5
votes
1
answer
1k
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Relative Gromov-Witten Invariants
A central issue in defining relative GW-invariants on a symplectic manifold is the possibility that a sequence of relative pseudoholomorphic curves can degenerate in such a manner, that components lie ...
5
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2
answers
1k
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Generator of a Fukaya category with certain properties
There is an algebraic theory that I'm thinking of trying to develop and I wanted to know if it had any real world prevalence --- I'd like to know an example of a generator L of a Fukaya category on a ...
5
votes
1
answer
293
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Half-dimensional torus fibration vs Lagrangian torus fibration
Assume we have a closed symplectic manifold $M$ which is the total space of a smooth fibration by half-dimensional tori. Can we infer that $M$ is the total space of a smooth fibration by Lagrangian ...