Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

Filter by
Sorted by
Tagged with
6 votes
1 answer
842 views

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 ...
Theo Johnson-Freyd's user avatar
6 votes
2 answers
1k views

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 ...
Jan Weidner's user avatar
  • 12.8k
6 votes
1 answer
398 views

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 ...
Theo Johnson-Freyd's user avatar
6 votes
1 answer
379 views

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 $n$-...
skr's user avatar
  • 512
6 votes
1 answer
136 views

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 ...
user142700's user avatar
6 votes
0 answers
191 views

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 (...
Riccardo's user avatar
  • 1,998
6 votes
0 answers
191 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 ...
contingent's user avatar
6 votes
0 answers
211 views

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 ...
John Baldwin's user avatar
6 votes
1 answer
234 views

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 ...
Bedovlat's user avatar
  • 1,939
6 votes
0 answers
160 views

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 ...
inkievoyd's user avatar
  • 508
6 votes
0 answers
261 views

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 ...
Someone's user avatar
  • 761
6 votes
0 answers
154 views

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 ...
Riccardo's user avatar
  • 1,998
6 votes
0 answers
170 views

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{...
cr1t1cal's user avatar
  • 755
6 votes
0 answers
490 views

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 ...
TheWildCat's user avatar
6 votes
0 answers
151 views

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 ...
S.D.'s user avatar
  • 492
6 votes
0 answers
266 views

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,\...
Riccardo's user avatar
  • 1,998
6 votes
0 answers
328 views

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 ...
Shaoyun Bai's user avatar
6 votes
0 answers
148 views

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 ...
Riccardo's user avatar
  • 1,998
6 votes
0 answers
168 views

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 ...
Francis Harry's user avatar
6 votes
0 answers
269 views

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 ...
aglearner's user avatar
  • 14k
6 votes
0 answers
326 views

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. ...
Mimi's user avatar
  • 61
6 votes
0 answers
114 views

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 ...
Yaniv Ganor's user avatar
  • 1,873
6 votes
0 answers
156 views

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(\...
Allen Knutson's user avatar
6 votes
0 answers
314 views

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 ...
wonderich's user avatar
  • 10.3k
6 votes
0 answers
459 views

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 ...
CuriousUser's user avatar
  • 1,420
6 votes
0 answers
316 views

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:...
Theo Johnson-Freyd's user avatar
6 votes
0 answers
423 views

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 ...
jhgfd's user avatar
  • 175
6 votes
0 answers
454 views

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 ...
Ali Taghavi's user avatar
6 votes
0 answers
502 views

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 ...
user avatar
6 votes
0 answers
253 views

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 ...
JHM's user avatar
  • 2,254
6 votes
0 answers
259 views

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(...
Chris Woodward's user avatar
6 votes
0 answers
431 views

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 ...
Coffee's user avatar
  • 601
6 votes
0 answers
310 views

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$.) ...
Yunhyung Cho's user avatar
  • 1,037
6 votes
0 answers
558 views

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 ...
Vijay's user avatar
  • 61
6 votes
0 answers
653 views

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 ...
Ben Webster's user avatar
  • 43.9k
6 votes
0 answers
497 views

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 ...
agt's user avatar
  • 4,246
5 votes
3 answers
1k views

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, ...
Cheuk Hwang's user avatar
5 votes
1 answer
546 views

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 \...
user avatar
5 votes
2 answers
647 views

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 ...
Antonijo's user avatar
5 votes
2 answers
630 views

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 $\...
James Mracek's user avatar
5 votes
2 answers
1k views

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 $\...
user12998's user avatar
  • 112
5 votes
2 answers
1k views

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.
nicolas's user avatar
  • 683
5 votes
1 answer
677 views

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 ...
user21485's user avatar
5 votes
2 answers
482 views

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 $...
Puzzled's user avatar
  • 8,842
5 votes
2 answers
378 views

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 $\omega_g$...
Ali Taghavi's user avatar
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 ...
issoroloap's user avatar
5 votes
1 answer
763 views

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 ...
L. Kern's user avatar
  • 51
5 votes
1 answer
1k views

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 ...
confused's user avatar
5 votes
2 answers
1k views

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 ...
Daniel Pomerleano's user avatar
5 votes
1 answer
293 views

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 ...
geometer's user avatar
  • 713

1
8 9
10
11 12
29