# Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

1,159
questions

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votes

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### (symplectic) $h$-cobordism from $S^1\times S^2$ to itself

I ran into an oriented smooth $h$-cobordism from $S^1\times S^2$ to itself in my project. I wish to argue that it is diffeomorphic/homeomorphic to the product.
From this question 4-dimensional h-...

**1**

vote

**1**answer

51 views

### Existence of Liouville vector fields on symplectic manifolds

Let $(M, \omega)$ be a symplectic manifold. A vector field $V: M \to TM$ is Liouville if $L_{X}
\omega=\omega$. The existence of a Liouville vector field implies that $(M, \omega)$ is exact: the one-...

**5**

votes

**2**answers

336 views

### Translation of Marsden-Weinstein-Meyer into classical mechanics language

The Marsden-Weinstein-Meyer theorem is expressed in a too general way to be understood by a mean square physicist, as me. However, if we limit the scope to a Hamiltonian mechanics, it should be ...

**3**

votes

**1**answer

72 views

### Obstruction to the existence of an invariant symplectic connection

Let $M$ be a symplectic manifold with a symplectic action of a Lie algebra $\mathfrak{g}$. I am interested whether there exists a $\mathfrak{g}$-invariant symplectic connection on $M$. Where does the ...

**6**

votes

**1**answer

199 views

### An extension of symplectomorphism group

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$.
We consider the ...

**2**

votes

**0**answers

61 views

### Conley Zehnder index for Floer homology of a symplectomorphism

I'm trying to get some intuition for the Conley-Zehnder index in the setting of Floer homology of a symplectomorphism $\phi : (M,\omega) \to (M,\omega)$. Let's assume that $\phi$ only has non-...

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vote

**1**answer

157 views

### How are Poisson brackets and the variational principle related?

In the lecture Space and spaces, Segal argues that the origin of non-commutativity in classical mechanics “which is encoded in the Poisson Bracket” is the fact that the evolution of classical states ...

**13**

votes

**1**answer

280 views

### How not to use J-holomorphic curves [closed]

The field of symplectic topology is filled with subtle traps for the unwary, particularly when it comes to the analysis of $J$-holomorphic curves. So that the next generation of symplectic topologists ...

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votes

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349 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 $...

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votes

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39 views

### Explicit formula for the moment map of toric manifold

Let $P$ be a Delzant polytope in $M\otimes{\mathbb R}\cong \mathbb R^n$, and it is well-known that we can associate to it a toric manifold $X=X_P$ with the moment map $\pi: X\to P$.
I would like to ...

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78 views

### Perturbed Cauchy-Riemann equations in fixed point Floer Homology and their mapping cylinder version

I'm writing you this question because I'm slightly confused on how to go back and forth the perturbed Cauchy-Riemann equations (CR) and their mapping cylinder version in the case of fixed Floer ...

**2**

votes

**1**answer

142 views

### A metric naturally arise from the Euclidean symplectic structure?

For $n>1$ let $\omega=\sum_{i=1}^n dx_i\wedge dy_i$ be the standard symplectic structure on $\mathbb{R}^{2n}=\mathbb{R}^n \times \mathbb{R}^n$.
We define the following distribution $D$ on $\mathbb{...

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votes

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135 views

### Differential geometry of Donaldson-Thomas invariants

The Donaldson-Thomas invariants are defined by Thomas in the paper A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, following the proposal in Gauge theory in higher ...

**4**

votes

**1**answer

211 views

### Mirzakhani's hyperbolic method generalized to moduli space of stable maps

I've been learning about Mirzakhani's use of hyperbolic geometry to compute Weil-Petersson volumes of moduli space of curves, and the application to proving Virasoro constraints for a point. Why have ...

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51 views

### Proove the equivalence of two Lagrangians

Consider Lagrangian
$$
L_1= q\dot{\alpha}+\alpha^2
$$
The Euler-Lagrange equations for $q$ and $\alpha$ read
$$
\dot{q}=2\alpha
$$
$$
\dot{\alpha}=0
$$
These two equations can be combined, ...

**6**

votes

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136 views

### Two questions on history of symplectic geometry in the 80's

I have a question about the history of two important results from the eighties in symplectic geometry. In both cases it seems that important results were developed (almost) simultaneously by ...

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53 views

### Isomorphism of certain irreducible representations over finite fields

We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V) $ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...

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55 views

### Local contractibility of group of symplectomorphisms for open manifolds

It is well know that for a closed symplectic manifold $(M, \omega)$ the group of symplectomorphisms in locally contractible. The gist of this proof goes as follows.
Given a $\psi \in \operatorname{...

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votes

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59 views

### Mirror symmetry for $C^*$

The Liouville manifold $T^*S^1$ is said to be "mirror" to the complex variety $C^*$. (see for instance lecture 7 here: http://math.columbia.edu/~topology/Eilenberg_lectures_fall_2016)
This is ...

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vote

**3**answers

91 views

### Strictly isotropic and strictly coisotropic submanifolds

Let $M$ be a $2n$-dimensional symplectic manifold. A question: are there special terms for isotropic submanifolds of $M$ of dimensions $<n$ (i.e., isotropic submanifolds that are not Lagrangian) ...

**7**

votes

**2**answers

398 views

### Classification of symplectic resolutions

A. Okounkov said, "symplectic resolutions are Lie algebras of the 21st century." Is there a conjecture on the classification of symplectic resolutions? Do Braverman-Finkelberg-Nakajima Coulomb ...

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votes

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42 views

### A clarification on why the injectivity radius is involved in Lemma 10.7 of Compactness results in Symplectic Field Theory by B.-E.-H.-W.-Z

I'm trying to understand why in the following lemma (Lemma 10.7 of [BEHWZ]), the upper bound on the $L^{\infty}$-norm of the differential is given in terms of the injective radius w.r.t to a specific ...

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votes

**1**answer

87 views

### Lifting one parameter subgroup $e^{t K}$ to the universal cover of $\mathrm{Sp}(2N,\mathbb{R})$

I would like to lift an arbitrary one-parameter subgroup $e^{t K}$ with $K\in\mathfrak{sp}(2N,\mathbb{R})$ to the universal cover $\widetilde{\mathrm{Sp}}(2N,\mathbb{R})$ (or at least its two-fold ...

**5**

votes

**1**answer

191 views

### Descent of vector bundle along branched cover of curve

Suppose $\pi:C'\to C$ is a branched cover of compact Riemann surfaces such that the associated extension of function fields is Galois with group $G$ -- so that $\pi$ presents $C$ as the quotient $C'$ ...

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votes

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94 views

### Relationship between canonical commutation relations and projective representations?

$\DeclareMathOperator\CCR{CCR}\DeclareMathOperator\Im{Im}\DeclareMathOperator\PU{PU}$Let $V$ be a real vector space equipped with an antisymmetric bilinear form $\omega$. Recall that there is a $C^\...

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89 views

### Organizing mirror pairs

At a maximally vague and naive level, mirror symmetry asks the following question: given a complex manifold $(X, I)$, is there a symplectic manifold $(M, \omega)$ and an equivalence between the ...

**9**

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191 views

### Infinity local systems

I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems".
From what I've been told, given a good cover $\{U_i\}$ of $X$, ...

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votes

**1**answer

71 views

### Derivation of an uncertainty principle from the symplectic non-squeezing theorem

Is there a derivation of an uncertainty principle or uncertainty-type principle from the symplectic non-squeezing theorem?

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votes

**1**answer

247 views

### What is symplectic rigidity?

What is an explanation for what the theory of symplectic rigidity is and what kind of questions it can answer? I was led to this after reading about the symplectic non-squeezing theorem of Gromov.

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votes

**1**answer

2k views

### Is a symplectic camel actually prohibited from passing through the eye of a needle?

Gromov's symplectic nonsqueezing theorem asserts that in the symplectic space ${\bf R}^{2n}$ with canonical coordinates $p_1,\dots,p_n,q_1,\dots,q_n$, and two radii $0 < r < R$, it is not ...

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vote

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46 views

### Representable functors and symplectic co-tangent bundles

I've been banging my head against something that I feel should follow from abstract non-sense, and I hope someone here can set me straight.
Let $\mathcal{M}$ be the category of smooth manifolds, with ...

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vote

**1**answer

84 views

### Chart in $1$-parameter family of Lagrangians in a Kähler manifold

Let $(X,\omega,J)$ be a complex $n$-dimensional Kähler manifold ($\omega$ Kähler form, $J$ complex structure) and $L \subset X$ be a closed real-analytic Lagrangian submanifold. Furthermore, let $L_{t}...

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votes

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85 views

### Symplectic geometry connects random density matrices?

This question arises from studying the following papers: Christandl et al. '14 and Mejia et al. '16.
These two papers use a connection between symplectic geometry and reduced density matrix. In ...

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votes

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53 views

### Infinitesimal orbit type decomposition of Hamiltonian $G$-manifolds

Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a symplectic manifold $M$ with momentum map $\mu:M\to \mathfrak{g}^\ast$, where $\mathfrak{g}$ is the Lie algebra of $G$. ...

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68 views

### Formality and symplectic forms on a smooth manifold

I saw one paper which asks this question.
"Let $(M,\omega,J)$ be a Kähler manifold. Then does $M$ admit a symplectic structure $\sigma$ of non-hard Lefschetz type?".
I was wondering whether I could ...

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161 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,\...

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votes

**1**answer

449 views

### Is there an area-preserving diffeomorphism of the disk which is nowhere conformal?

This question is a cross-post; it is related to this former question of mine. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk.
Does there exist a smooth volume-preserving diffeomorphism $f:...

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votes

**1**answer

242 views

### Topology of the projective symplectic group

Consider the projective symplectic group $\mathrm{PSp}(n+1)$ over $\mathbb{C}$. This parametrizes $(n+1)\times (n+1)$ symplectic matrices modulo scalar multiplication.
Is $\mathrm{PSp}(n+1)$ ...

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votes

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191 views

### What is rigidity of Hirzebruch, and Witten genera?

I would like to find some good references (or any insight) that would help me to understand a few articles mentioning rigidity of Hirzeburch genus. One of the consequences of this phenomenon is that ...

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votes

**0**answers

73 views

### Augmentations of wrapped Floer cochains

Let $M$ be a closed, simply-connected spin manifold and let $F_b \subset T^*M$ be the cotangent fiber over a point $b \in M$. Let $CW^*(L,L)$ be the $A_{\infty}$-algebra of wrapped Floer cochains over ...

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votes

**1**answer

238 views

### “Sameness” of dg and A-infinity categories

Let $k$ be a field.
A folklore theorem states that dg-categories (over $k$), $A_{\infty}$-categories (over $k$) and stable ($k$-linear) $(\infty, 1)$-categories are "the same" (see for example
...

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62 views

### Derived manifold and real virtual dimension

In https://arxiv.org/pdf/1504.00690.pdf, it seems like the "derived manifold structure" given on a certain complex analytic space seems to have the real virtual dimension the same as the complex ...

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votes

**1**answer

105 views

### Pseudo-holomorphic disk which is constant along boundary

Let $(M,J,\omega)$ be a symplectic manifold with a compatible almost complex structure, $D$ be the closed unit disk in $\mathbb{C}$, and $u:(D,i)\to (M,J)$ be a $(J,i)$-holomorphic map.
Question: ...

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votes

**1**answer

205 views

### Viterbo restriction map surjective on Weinstein neighbourhood

In a Liouville manifold $M$ having a Liouville subdomain $i: N \hookrightarrow M$, there is the so-called Viterbo restriction map in symplectic cohomology $$SH^*(i): SH^*(M)\rightarrow SH^*(N).$$
In ...

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votes

**0**answers

136 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 ...

**10**

votes

**4**answers

949 views

### Reading list for Equivariant Cohomology

I was applying equivaraint cohomology, in particular in the symplectic setting, for some time, but I feel like I am missing some nice books/course notes/articles. Could you advise me some literature, ...

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votes

**2**answers

168 views

### Contactomorphisms have in general no fixed points

Let $(V, \xi = \ker \alpha)$ be a cooriented contact manifold. A contactomorphism of $(V, \xi)$ is a diffeomorphism $\phi$ which preserves the contact structure $\xi$ and its coorientation. In other ...

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votes

**1**answer

89 views

### Comparing the minimal Chern number and the cup-length of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. One can define its minimal Chern number $N_M$ as:
$$
N_M := \text{inf} \lbrace k > 0 \ |\ \exists A \in H_2(M; \mathbb{Z}), \langle c_1, A \rangle = k \...

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votes

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129 views

### Representations of $\mathbb Z^2$ in ${\rm Symp}(S^2)$

Suppose $f_1$ and $f_2$ are two commuting symplectomorphisms of the sphere $\mathbb S^2$, of orders different from $2$. Is it possible to deform the pair $(f_1,f_2)$ to the pair of identity maps via a ...

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votes

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141 views

### boundary connect sum of Ganatra-Pardon-Shende

In Section 3.4 of https://arxiv.org/pdf/1809.03427.pdf, Ganatra-Pardon-Shende define the boundary connnect sum of two exact conical Lagrangians in a Liouville domain. In particular, in Figure 10, they ...