Hamiltonian systems, symplectic flows, classical integrable systems

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6
votes
2answers
498 views

Cotangent bundle lift theorem

Let $M$ be a smooth manifold and $T^\ast M$ be its cotangent bundle. Consider the tautological 1-form $\theta$ on $T^\ast M$ ($\theta=\sum y_i dx^i$ in local canonical coordinate systems). A ...
30
votes
8answers
4k views

What is a Lagrangian submanifold intuitively?

What are good ways to think about Lagrangian submanifolds? Why should one care about them? More generally: same questions about (co)isotropic ones. Answers from a classical mechanics point of view ...
10
votes
4answers
2k views

How to see the Phase Space of a Physical System as the Cotangent Bundle

Two things today motivated this question. First, the professor said that in a lecture Thurston mentioned Any manifold can be seen as the configuration space of some physical system. Clearly we ...
12
votes
3answers
875 views

How transitive are the actions of symplectomorphism groups ?

This question is motivated by the classical fact from differential geometry : Let $M$ be a smooth manifold of dimension at least $2$. Then for any $n$ the diffeomorphism group $\textrm{Diff}(M)$ ...
3
votes
1answer
227 views

Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...
44
votes
4answers
3k views

Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...
29
votes
1answer
790 views

What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ ...
24
votes
4answers
2k views

When is a symplectic manifold equivalent to a cotangent bundle?

Let $X$ be a differentiable manifold. Its cotangent bundle $T^*X$ carries a canonical 1-form $ \alpha$ whose exterior differential $\omega = d\alpha$ endows $T^*X$ with the structure of a symplectic ...
13
votes
3answers
903 views

When is a coadjoint orbit an integrable system (in a weak sense explained below)?

Let $X$ be an affine holomorphic symplectic variety of dimension $2n$, with the associated Poisson bracket { , }. Let's say it's an integrable system when there are $n$ algebraically independent ...
11
votes
2answers
2k views

What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
8
votes
3answers
964 views

Intuition for Levi-Civita connection via Hamiltonian flows

A Euclidean metric on a manifold $X$ defines a function on the symplectic space $T^*X$ whose Hamiltonian flow gives geodesics. Is there a similar interpretation of the Levi-Civita connection?
4
votes
1answer
375 views

Quantization of symplectic vector space and choice of lagrangian subspaces

My question is related to Geometric Quantization. I don't undrestand the philosophy of following assertion If $(V,\omega)$ be a symplectic vector space then the quantizations of $V$ ...
4
votes
3answers
717 views

Why can we define the moment map in this way (i.e. why is this form exact)?

Given a symplectic manifold $(X, \omega)$ and a group $G$ acting on $X$ preserving the symplectic form, we define the moment map $\mu : X \to \mathfrak{g}^*$ so that $$ \langle d\mu(v), \xi\rangle = ...
12
votes
2answers
928 views

Why is every symplectomorphism of the unit disk Hamiltonian isotopic to the identity?

That is, for any symplectomorphism $\psi: D^2 \to D^2$, there should be a time-dependent Hamiltonian Ht on D2 such that the corresponding flow at time 1 is equal to $\psi$. I found this in claim a ...
11
votes
2answers
594 views

What are the implications of torsion in H^2 for geometric quantization?

Given a real manifold $M$ with symplectic $2$-form $\omega$, one can ask whether the cohomology class $[\omega] \in H^2(M;{\mathbb R})$ lies in the image of $H^2(M;{\mathbb Z})$. If so, one can ask ...
4
votes
2answers
789 views

complexified kahler form

In mirror symmetry one usually considers a complexified kahler form $B+iw$ instead of kahler form $w$ itself.(Or their moduli) Here is the question: What does $B$ correspond to? what kind of ...
2
votes
1answer
351 views

looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e, ($J^1M=T^*M×\mathbb{R}$) Is there something like this identity for higher jet bundle $J^kM$? I editted ...
9
votes
0answers
371 views

Poincaré recurrence and symplectic packings

Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius ...
6
votes
3answers
1k views

Question about Hodge number

Hi. I am studying Hodge theory on Kahler manifolds. I have several questions. Is Hodge number a topological invariant? (I mean, is it independent of the choice of Kahler structure?) If the ...
4
votes
2answers
376 views

symplectic structure of tangent bundle of $\mathbb{S}^{n-1}$

It is well known that $T\mathbb{S}^{n-1}$ is diffeomorphic to $M= f^{-1}(1)$ where $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is $f(z):=\sum_{i=1}^{n} z_{i}^{2}$. Two questions: 1) Is $M$ a ...
3
votes
1answer
469 views

a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?
3
votes
2answers
529 views

Legendrian Tubular Neighborhood Theorem

Given a Lagrangian submanifold $L\subset(M,\omega)$ of a symplectic manifold, we have Alan Weinstein's celebrated Lagrangian tubular neighborhood theorem. I now look for the analog on Legendrian ...
1
vote
1answer
278 views

The space of holomorphic sections are finite dimensional?

I start my question with a definition and some motivation. Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex ...
3
votes
1answer
570 views

Does the preimage of the Slodowy slice in $T^*G/P$ have a name?

Let $G$ be your favorite simple complex Lie group, and $P\subset G$ your favorite parabolic subgroup. We can identify $T^*G/P$ with the space of pairs $$\{(gP,x)\in G/P\times \mathfrak g | x\perp ...
1
vote
3answers
422 views

Reference on generators of subgroups of symplectic groups

We should start with the definition of the symplectic group for an arbitrary ring $R$. The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with ...
6
votes
1answer
180 views

A lagrangian version of the Withney theorem

Let $M$ be a smooth n dimensional manifold. Is there an smooth embedding $f:M \to \mathbb{R}^{2n}$ which image is a Lagrangian submanifold of $\mathbb{R}^{2n}$?
3
votes
1answer
518 views

Symplectic block-diagonalization of a real symmetric Hamiltonian matrix

Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal? Being ...
1
vote
0answers
65 views

Does the twisted product $K^{\mathbb{C}}\times_{Z(k)^{\mathbb{C}}} X^k$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, and suppose that $(X,J,\omega)$ is a compact Kähler manifold on which the group $K$ acts holomorphically such that the group $K$ preserves the Kähler ...