Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

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3
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0answers
71 views

(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-...
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1answer
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
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2answers
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
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1answer
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
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1answer
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 ...
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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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1answer
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
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1answer
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 ...
5
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2answers
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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0answers
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
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1answer
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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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
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1answer
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, ...
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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 ...
3
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0answers
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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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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3answers
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) ...
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2answers
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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0answers
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 ...
2
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1answer
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
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1answer
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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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 ...
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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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1answer
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?
5
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1answer
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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1answer
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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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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1answer
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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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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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 $G$. ...
3
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0answers
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,\...
12
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1answer
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:...
6
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1answer
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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0answers
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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0answers
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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1answer
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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0answers
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 ...
3
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1answer
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: ...
3
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1answer
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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0answers
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
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4answers
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, ...
4
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2answers
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 ...
2
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1answer
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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0answers
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 ...
3
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0answers
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 ...

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