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Hamiltonian systems, symplectic flows, classical integrable systems

4
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0answers
48 views

GSO projection and $H^d(M, \mathbb{Z}_2)$

This follows up the comment which suggests that asking the later 2nd part of subquestion in "GSO (Gliozzi-Scherk-Olive) projection and its Mathematics" as a new different question GSO (Gliozzi-Scherk-...
6
votes
0answers
130 views

GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?

GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...
2
votes
0answers
62 views

Global symplectic reduction

Let $M$ be a symplectic manifold equipped with a hamiltonian action of a compact Lie group $G$ with moment map $\mu\colon M\to \mathfrak g^*$. Assume $c\in \mathfrak g^*$. Then the symplectic ...
4
votes
0answers
84 views

Bound on critical points of Lefschetz fibration over the disk with prescribed monodromy

Let $\phi$ be a right-veering diffeomorphism of a surface $\Sigma$ of genus $g$ and $r$ boundary components. Suppose that the diffeomorphism is freely periodic so if $M$ is the associated open book ...
8
votes
0answers
284 views

What is the mirror of an algebraic group?

Background: Kontsevich's homological mirror symmetry conjecture posits the existence of pairs $(X,\check X)$ with an equivalence of dg/$A_\infty$-categories $$\mathcal F(X)=\mathcal D^b(\check X)$$ ...
8
votes
1answer
180 views

Weinstein neighborhood theorem for Lagrangians with Legendrian boundary

$\require{AMScd}$ Weinstein's neighborhood theorem says that every Lagrangian has a standard neighborhood. The more precise statement goes like this. Theorem 1: (Lagrangian Neighborhood Theorem) ...
5
votes
1answer
119 views

Symplectic Lefschetz fibrations in terms of factorization in symplectic mapping class group

There is a well-known theorem stating that there is a bijection between diffeomorphism classes of Lefschetz fibrations over $S^2$ whose general fiber is a closed orientable surface $\Sigma_g$ of genus ...
5
votes
0answers
63 views

Quantum homology of $(S^2 \times S^2,\omega_{FS}\oplus \omega_{FS})$ and Poincare duality

I am having some issues computing Poincare duality for the quantum homology $QH(M)$ when $(M,\omega)=(S^2 \times S^2,\omega_{FS}\oplus \omega_{FS})$. I am using the simple novikov ring $\Lambda$ ...
0
votes
0answers
54 views

Effective classes in toric Kähler manifolds

In an article about toric manifolds, I have seen the following notions, which I don't understand. We view a symplectic toric manifold $(M,\omega)$ as a Kähler manifold with Kähler form $\omega$, and ...
8
votes
1answer
920 views

What is the current status of the Arnold conjecture?

Let $(M, \omega)$ be a symplectic manifold. V.Arnold conjectured that the number of fixed points of a Hamiltonian symplectomorphism is bounded below by the number of critical points of a smooth ...
2
votes
1answer
201 views

An orientable compact even dimensional manifolds whose all even cohomologies do not vanish but it does not admit any symplectic structure

What is an example of an orientable compact $2n$ dimensional manifold $M$ whose all even dimensional De Rham cohomology groups $H_{\mathrm{DeR}}^{2i}(M)$ are nonzero, but $M$ does not admit any ...
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0answers
113 views

Existence of an “almost” skew-symmetric matrix

Let $A\in\mathbb{R}^{3\times 3}$ be a matrix of the form $$ A=\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ -a_{12} & a_{22} & a_{23} \\ -a_{13} & -a_{23} & a_{33} \end{bmatrix} $...
6
votes
1answer
189 views

Physical intuition behind prequantization spaces

Given a symplectic manifold $(M,\omega)$ with integral symplectic form, that is $$\omega \in \text{Im}(H_2(M,\mathbb{Z}) \to H_2(M,\mathbb{R})),$$ one can form a so-called prequantization space, that ...
3
votes
0answers
126 views

Where can I find good surveys on Symplectic and Contact geometry

Are there any good survey articles in symplectic and contact geometry, which focus on the "big picture", i.e how this discipline fits into the mathematical world ? In the symplectic case : I am ...
5
votes
1answer
169 views

Non-Hamiltonian actions in physics

I was reading the following article when I came across the interesting sentence "non-Hamiltonian [symplectic group] actions also occur in physics" I took a cursory look at the article cited but ...
3
votes
2answers
298 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 $\omega_g$...
3
votes
0answers
66 views

Existence of harmonic symplectic structure on symplectic Riemannian manifold

This post is an expanded version of this MSE post. Assume that $(M, \omega)$ is a symplectic manifold which is equiped with a Riemannian metric. Is there a symplectic structure $\...
15
votes
0answers
406 views

What are hyperkähler metrics used for?

It seems that a lot of effort has been devoted to endow holomorphic-symplectic manifolds with hyperkähler metrics. It started with Calabi [4] with $T^*\mathbb{CP}^n$. Other examples include coadjoint ...
2
votes
0answers
76 views

The mirror of the Landau--Ginzburg model given by elliptically fibered K3

Let $f:X\rightarrow \mathbb{P}^1$ be an elliptically fibered K3 surface. Choose a coordinate on $\mathbb{P}^1$ and consider $X\backslash f^{-1}(\infty)\rightarrow \mathbb{C}$ as a Landau--Ginzburg ...
0
votes
1answer
85 views

Symplectic submanifolds of the tangent bundle $TM$ which have the form of a vector or fiber bundle

Let $(M,g)$ be a Riemannian manifold which admit a non vanishing vector field.(That is $\chi(M)=0$ when $M$ is a compact manifold). We pull back The symplectic structure of the ...
5
votes
2answers
241 views

Complex Analytic Structure on Moduli Space of Stable Maps

Suppose $(X,\omega,J)$ is a compact Kähler manifold, and $\beta\in H_2(X,\mathbb Z)$ is given. Then, we can form the space $\overline{\mathcal M}:=\overline{\mathcal M}_{0,0}(X,\beta)$ of stable maps $...
8
votes
1answer
460 views

Does every manifold admit a Lagrangian Riemannian metric?

Let $(M,g)$ be a Riemannian manifold. The $LC$ connection associated to the metric gives an $n$ dimensional distribution $D$ for $TM$. Let $\omega$ be the symplectic structure of $TM$ which is ...
21
votes
2answers
648 views

Proof of Giroux's correspondence

It is extensively used and cited the following statement due to Giroux: Given a closed $3$-manifold $M$, there is a $1:1$ correspondence between oriented contact structures on $M$ up to isotopy and ...
20
votes
1answer
475 views

What can we say about the Cartesian product of a manifold with its exotic copy?

Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$. Is it true that $M\times M$ is diffeomorphic to $M\times M^E$? I am ...
3
votes
0answers
169 views

Existence of compact leaf for certain foliation of a symplectic manifold

Is there a symplectic manifold $(M,\omega)$ equipped with a Riemannian metric such that $d^* \omega \wedge dd^*\omega=0$ and there is a compact leaf for the foliation of $M\setminus S$ defined by $d^*...
5
votes
1answer
177 views

Isotrivial Monodromy

Let $X\to \Delta$ be a projective family, smooth over $\Delta^*$, such that all fibers over $t\in \Delta^*$ are isomorphic. Does the monodromy representation factor through the algebraic automorphism ...
4
votes
0answers
67 views

The ¨irreducible¨ representation variety of surface group

Let S be a closed surface of genus larger than 1, G be a compact, simply connected simple Lie group with finite center. Consider the representation variety M(S,G)=Rep($\pi_1$(S), G). Witten´s Formula ...
7
votes
1answer
118 views

Special Cases of Duistermaat-Heckman Formula

The Duistermaat Heckman localization formula states how integrals over symplectic spaces with Hamiltonian $U(1)$ group actions. $$ \int_M \frac{\omega^n}{n!} e^{-\mu} = \sum_{x_i \text{ fixed}} \frac{...
5
votes
1answer
138 views

Can I bring the Kirillov 2-form on coadjoint orbits to adjoint orbits?

I tried asking this question on stackexchange and received no response. Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirillov 2-form. Can I ...
2
votes
0answers
109 views

space of $\omega$-tame almost complex structures and $\mathrm{Diff}(M)$

Let $(M,\omega)$ be a symplectic manifold, and $J$ is an almost complex structure on $M$. $J$ is said to be $\omega$-tame if $$ \omega(v, Jv) >0 $$ for all non-zero $v\in TM$. It is commonly said ...
8
votes
1answer
159 views

Moyal $\star$-product inverse?

On a 2n-dimensional phase-space with coordinates $x$ and $p$, the Moyal product can be written explicitly as $$g(x,p) \star h(x,p) = g(x,p) e^{\frac{i}{2}\left( \overleftarrow{\partial_x} \cdot \...
10
votes
1answer
246 views

Is there a classification of polynomial Poisson brackets?

As an example, consider the following Poisson bracket on ${\mathbb R}^n$: $$\{x_i, x_{i+1}\} = x_ix_{i+1}(x_i+x_{i+1}),\\ \{x_i, x_{i+2}\} = x_ix_{i+1}x_{i+2}.$$ The indices are taken modulo $n$, and ...
4
votes
0answers
235 views

A cohomology associated to a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. Let $$\Omega_{\omega}^k(M)=\{\alpha \in \Omega^{k}(M)\mid \alpha \wedge \omega \;\;\text{is an exact form}\}$$ Then we have a chain comlex$$\...
2
votes
0answers
91 views

Reference request: explicit equivariant localization formula on toric complete intersections

This post is about an equivariant integration formula in a famous paper https://arxiv.org/pdf/alg-geom/9701016.pdf by Alexander Givental, where the author presented the formula without proof or ...
8
votes
0answers
356 views

Determinant as a Hamiltonian

Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$...
6
votes
2answers
553 views

Is $\Bbb S^2 \times \Bbb S^4$ symplectic?

I'm playing around with products $M = \Bbb S^{n_1} \times \Bbb S^{n_2}$, and a quick computation using the Künneth formula tells us that if $(n_1,n_2)$ is not $(1,1)$ or $(2,4)$, $M$ is not symplectic ...
9
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0answers
382 views

Floer cohomology from mapping spaces of $\infty$ categories

There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at ...
10
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0answers
307 views

Symplectic structures in rigid geometry

Let $K$ be a non-archimedean valued field (with any further adjectives attached as necessary). I'm looking for references or information about symplectic structures on rigid $K$-spaces. For example, ...
2
votes
0answers
59 views

Effective actions by non-commutative groups have non-commuting fundamental vector fields?

I have a bit of a contradiction in my brain and I was hoping once again that excellent Mathoverflow community could help me out :) Let $\rho_g$ be the action associated to a non-abelian Lie Group $G$ ...
0
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0answers
72 views

On the measure of regular and chaotic regions in a phase space

Consider a Hamiltonian, non-linear, dynamical system associated to $H(\vec{q},\vec{p})$. Assume that the number of effective degrees of freedom is relatively small, say $D=3,4,5$. Now choose a certain ...
3
votes
2answers
180 views

Non-trivial examples of overtwisted contact structures

Are there any non-trivial examples of overtwisted contact structures on closed contact $3$-manifolds? By non-trivial I mean any examples besides the trivial one $\xi = \ker (\cos(r)dz - r \sin(r)d\...
3
votes
1answer
120 views

What is symplectic cut of a 4-ball?

Lerman's symplectic cut construction applied on 4-ball by collapsing its boundary 3-sphere along the $\mathbb{S}^1$ orbits of Hopf fibration gives a closed 4-dimensional symplectic manifold. ...
4
votes
1answer
193 views

Symplectic forms and sign of eigenvalues

This question has come out while reading J. Moser "New Aspects in the Theory of Stability of Hamiltonian Systems". I'm particularly interested to the Appendix, where one investigates the stability of ...
5
votes
0answers
138 views

Dimension of linear complex-symplectic reduction

Let $(V,\omega)$ be a finite-dimensional complex-symplectic vector space and $G$ be a complex reductive group acting linearly on $V$ by preserving $\omega$. Then, there is a moment map $$\mu:V\to\...
1
vote
1answer
224 views

A Lie algebra associated to a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold of dimension $2n$ with the volume form $\omega^n.$ In this question we associate a Lie algebra $L(M,\omega)$ to $(M,\omega)$. Then we are interested ...
2
votes
0answers
73 views

how to understand the manifold with boundary jet bundle and cotangent bundle with boundary

Suppose that $M\subset (W^{2n},\omega)$ is an $n$-dimensional manifold with smooth boundary $\partial M$, where $(W,\omega)$ is a $2n$-dimensional Kähler manifold and boundary with contact type ...
12
votes
1answer
141 views

Recovering topological invariants of symplectic manifold from the group of Hamiltonian diffeomorphisms?

There is a famous result of Banyaga stating that if two closed symplectic manifolds $(M_1, \omega_1)$ and $(M_2, \omega_2)$ have isomorphic groups of Hamiltonian diffeomorphisms $\mathrm{Ham}(M_1, \...
10
votes
0answers
309 views

What is a derived Kähler manifold?

From what I understand, there exists a notion of derived $\mathbb{C}$-analytic space. Let $T_{an}$ be the pregeometry in the sense of Lurie whose underlying $\infty$-category is the category of open ...
7
votes
2answers
303 views

Question about Obstruction Bundle Gluing paper of Hutchings-Taubes

I'm trying to learn about Embedded Contact Homology. To understand the proof of $d^2=0$, I started by watching Hutchings' lectures on Obstruction Bundle Gluing on YouTube (1, 2, 3) and have now ...
5
votes
0answers
192 views

Arithmetic symplectic geometry via mirror symmetry?

Homological mirror symmetry in the classical setting relates the bounded derived category of coherent sheaves on a Calabi-Yau manifold to the split-closure of the derived Fukaya category of the mirror ...