Hamiltonian systems, symplectic flows, classical integrable systems

**3**

votes

**1**answer

137 views

### Confusion regarding statement of mirror symmetry for elliptic curves

I am a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...

**0**

votes

**1**answer

50 views

### Hamiltonian potential invariant under lie group action?

Let $(M,g)$ be a riemannian manifold and $G$ a Lie group acting on $M$ by isometries.
Taking a $G$-invariant function $f\colon M \to \mathbb{R}$, we have the riemannian gradient $\operatorname{grad} ...

**2**

votes

**0**answers

32 views

### $TSU(n)$ completely integrable with 3 $SU(2)$ invariant functions?

Consider the Lie group $SU(n)$ endowed with the standard bi-invariant metric. Then $SU(n)$ can be viewed as a symmetric space of $K_{n,n} := SU(n) \times SU(n)$.
Define $M := K_{n,n} /SU(n)$. Using ...

**1**

vote

**1**answer

60 views

### Can a Hamiltonian perturbation map a submanifold to another?

Suppose $(M,\omega)$ is a symplectic manifold, $X$ and $X'$ are symplectic submanifolds, and $X$ can be smoothly deformed via symplectic submanifolds to $X'$. Is there a diffeomorphism generated by a ...

**1**

vote

**0**answers

35 views

### Toric structures on projective space

Consider the symplectic manifold $\mathbb P^n$ equipped with the Fubini-Study symplectic form $\omega$. Given $n+1$ "generic" points $z_0,\dots,z_n$ on $\mathbb P^n$, is there an effective Hamiltonian ...

**1**

vote

**0**answers

54 views

### Symplectic gradients whose span doesn't intersect Lie group orbits

I asked a similar question some hours ago. But thinking about my problem, I found a loophole in my arguments, so that my question wasn't the right one. This one is what I wanted to ask:
Let $G$ be a ...

**4**

votes

**1**answer

134 views

### isomorphism of noncommutative tori

I have a kind of vague question.
Two non-degenerate symplectic vector space of same dimension (say $\mathbb{R}^{2n}$) are isomorphic. Then why all noncommutative tori of same dimension aren't ...

**5**

votes

**1**answer

95 views

### multiplicity free chain of Lie subgroups gives a completely integrable system on symplectic manifold

Assume we have a symplectic manifold $N$ and a Hamiltonian action of a Lie group $G$, such that the algebra of all $G$-invariant functions on $N$ is commutative under Poisson bracket.
Furthermore, ...

**2**

votes

**1**answer

62 views

### Set of singular points for momentum map (with coisotropic action)

Let $G$ be a Lie-group acting on a connected symplectic manifold $M'$ in a hamiltonian way, with an $\operatorname{Ad}^*_G$-equivariant momentum map. Assuming $G$ acts properly on $M'$, we can ...

**4**

votes

**1**answer

299 views

### AKSZ sigma models for higher spin

The AKSZ framework constructs 2D sigma models in the BV formalism. Is there a generalization of the AKSZ approach to higher spin?

**3**

votes

**0**answers

129 views

### Invariant functions on the dual Lie algebra

Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$.
...

**1**

vote

**1**answer

141 views

### Integral points - monotone symplectic toric manifolds

Suppose I am given a symplectic toric manifold $(M,\omega,\psi)$ which is also monotone, hence the symplectic form can be rescaled so that $c_1=[\omega]$. Then the moment map can be taken so that its ...

**2**

votes

**1**answer

49 views

### coisotropic submanifolds on poisson manifolds

Let $(M, \{.,.\})$ be a Poissonmanifold and $B$ the corresponding Poissontensor. Now in this context, a embedded submanifold $C \subset M$ is called coisotropic, if $B^\#(TC^\circ) \subset TC$.
For ...

**0**

votes

**1**answer

53 views

### set of coisotropic orbits open and dense, iff group acts locally transitively almost everywhere

I worked now some time with coisotropic actions of Liegroups on manifolds.
But there is one key fact, that I don't understand, although it is very central in my considerations.
Let $(M,\omega)$ be a ...

**3**

votes

**1**answer

330 views

### Show that the symplectic action 1-form on loop space is closed

I am struggling to see how the symplectic action 1-form $\alpha_H$ on the loop space of a symplectic manifold $(M,\omega)$ is closed. It is defined by
$\alpha_H(Y) = \int_0^1 \omega(\dot{x}(t)-X_H,Y)...

**4**

votes

**0**answers

106 views

### Singular symplectic reduction in infinite dimension

In 1991, Sjamaar and Lerman [1] introduced the notion of stratified symplectic spaces. Namely, if $M$ is a symplectic manifold and $G$ a Lie group acting properly (but not necessarily freely) on $M$ ...

**8**

votes

**2**answers

170 views

### Interior periodic points of area preserving homeomorphisms of a pair of pants

A celebrated result of Franks shows that any area preserving homeomorphism of the closed annulus $A$ with at least one periodic point (possibly along the boundary) has infinitely many interior ...

**2**

votes

**0**answers

151 views

### Moment map of equivariant line bundles

I'm reading Szabo's `Equivariant Cohomology and Localization of Path Integrals'. I've stumbled upon an equation I can't make sense of, in the discussion about $G$-equivariant line bundles on ...

**0**

votes

**0**answers

67 views

### Arnold chord conjecture

Recently I've started learning about the Arnold conjecture for the number of Reeb chords on a Legendrian submanifold of a contact manifold. It says that under some conditions, the number of Reeb ...

**5**

votes

**0**answers

126 views

### Deformation quantization of Poisson bracket without star-product

Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$,
$$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are ...

**9**

votes

**1**answer

660 views

### Is there an algebraic description of Yang-Mills measure?

Consider $X=\operatorname{Hom}(\pi,\mathrm{SL}(2,\mathbb C))/\! \!/\mathrm{SL}(2,\mathbb C)$ and $Y=\operatorname{Hom}(\pi,\mathrm{SU}(2))/\mathrm{SU}(2)$, where $\pi$ is a surface group. Note that ...

**15**

votes

**1**answer

1k views

### What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

**2**

votes

**2**answers

112 views

### Symplectic formulation of compressible Euler equation

It has been widely known that the compressible Euler equation can be cast into the Hamiltonian form. For example, in the book "Dubrovin B A, Fomenko A T, Novikov S P. Modern geometry—methods and ...

**2**

votes

**0**answers

119 views

### Abstract VFC vs. what people actually use for Quintic 3-fold

Moduli space of genus $0$ degree $d$ maps in a quintic Calabi-Yau threefold $X$, written as $\overline{\mathcal{M}}_{0,0}(X,[d])$, can be embedded in the corresponding moduli space of $\mathbb{P}^4$, ...

**2**

votes

**1**answer

177 views

### Is there a matrix that converts the gradient of every possible function to gradient of other function?

I have already asked this question on math.stackexchange.com
http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe
Now I ...

**0**

votes

**0**answers

48 views

### Small perturb a continuous map

I am reading the book: Convex Integration Theory by D. Spring and encounter a
question which I subtract as follows.
Let $p: X \rightarrow V$ be a smooth fibre bundle and $\mathcal{R} \subset X^{(1)}$...

**5**

votes

**0**answers

70 views

### In $(\mathbb{R}^4,\omega_{std})$ is positive symplectic area enough to guarantee a pseudoholomorphic disc representative?

I will present my question in the context that I encountered it, although I believe it probably applies in general context.
Consider $\mathbb{R}^4 \cong \mathbb{C}^2$ with the standard symplectic form ...

**5**

votes

**0**answers

61 views

### Symplectic leaves in positive characteristic

I am currently considering a family of filtered algebras over an algebraically closed field of positive characteristic with the property that the associated graded algebra is a finitely generated ...

**6**

votes

**2**answers

236 views

### Flat connections on 3-manifold with boundary

Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}_{flat}(\Sigma)$ and $\overline {\mathcal A}_{...

**4**

votes

**1**answer

188 views

### Deformation long exact sequence of GW theory in the analytical setting

Let $f\!=\!(u\colon (\Sigma,p_1,\ldots,p_k) \to X)$ be an element of the moduli space of genus $g$ $k$-marked degree $A$ $J$-holomorphic maps $\mathcal{M}_{g,k}(X,A,J)$. For simplicity assume $C=(\...

**2**

votes

**1**answer

60 views

### Hofer-Zehnder capacity of toric varieties

Consider the complex algebraic torus $(\mathbb{C}^*)^n$ where $\mathbb{C}^*$ stands for $\mathbb{C} \setminus \{0\}$. Fix a finite set of lattice points $A = \{\alpha_0, \ldots, \alpha_r\} \subset \...

**1**

vote

**0**answers

120 views

### A question about canonical bundle of moduli space of Kahler Einstein metrics

Let $\mathcal M$ be a moduli space of Kahler-Einstein metrics with
negative Ricci curvatures on pairs $(X,D)$. Is the canonical bundle of
$\mathcal M$ nef?
Motivation: If we know the nefness of $...

**1**

vote

**1**answer

46 views

### multiplicity free actions - Guillemin&Sternbergy collective integrability

In this post I already ask a similar question.
Assume $M$ is a symplectic manifold of dimension $2n$. Assume $G$ is a Liegroup, $\mathfrak{g}$ be the Liealgebra and $\mathfrak{g^*}$ the corresponding ...

**5**

votes

**1**answer

341 views

### A criterion for orbits of complex reductive group to be closed

I am having some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following:
Let $G=K_{\Bbb C}$ be a ...

**2**

votes

**0**answers

61 views

### Grading in Lagrangian Floer homology

What are the conditions on a symplectic manifold (M,w) and on a Lagrangian submanifold L so that Lagrangian Floer complex CF(L, f(L)) is Z-graded? Here f is a compactly supported Hamiltonian isotopy. ...

**6**

votes

**1**answer

139 views

### First Chern class vanishes on a Lagrangian submanifold

Suppose $(M,\omega)$ is a symplectic manifold and $L \subset M$ is a compact Lagrangian submanifold. Is it the case that the first Chern class $c_1(TM) \in H^2(M)$ vanishes when restricted to the ...

**9**

votes

**1**answer

131 views

### Geometric quantization: why are the prequantum operators self-adjoint?

I'm reading a bit about geometric quantization and, among the axioms of this construction, is one requiring that the operator $\hat f = -\textrm i \hbar \nabla _{X_f} + f$ associated to the classical ...

**3**

votes

**0**answers

251 views

### Do real polarization and Kahler polarization of character varieties of closed surfaces give equivalent representations of the Mapping Class Group?

This is a question about the Witten--Reshetikhin--Turaev representations of the mapping class group of a closed surface $\Sigma_g$. For simplicity, we'll stick to the case $G=SU(2)$.
These ...

**4**

votes

**0**answers

72 views

### Maslov class of a diagonal

Let $(M,\omega)$ be a symplectic manifold. Which condition on $M$ guarantees that the diagonal of $(M \times M, (\omega,-\omega))$ has a vanishing Maslov class? $H^1(M,\mathbb{Z})=0$ is enough, but I ...

**1**

vote

**1**answer

103 views

### Polynomials pulled back by momentum maps

Let $G$ be a Lie group acting Hamiltonian on some real analytic symplectic manifold $(M, \omega)$, with an $G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^*$.
Assuming I can find ...

**1**

vote

**0**answers

119 views

### Atiyah-Guillemin-Sternberg Theorem for current

The Atiyah-Guillemin-Sternberg theorem says that the image of a moment map of a toric action is independent of the choice of symplectic form in the cohomology class. So all is well for a smooth ...

**9**

votes

**0**answers

318 views

### Why do we study symplectic geometry? [closed]

What is the motivation behind studying smooth manifolds with a non-degenerate closed two-form?
The subject certainly originated from physics, but is there a deeper reason for why it is still an ...

**3**

votes

**0**answers

198 views

### Matsushita theorem on framed variety (X,D)

I have a question about fibrations on Irreducible log holomorphic symplectic manifolds. Lets give some introduction
Motivation; A holomorphic symplectic manifold (HSM) is a $2n$-dimensional compact K\...

**5**

votes

**0**answers

136 views

### Symplectic invariance of Hodge numbers?

Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$.
My ...

**4**

votes

**1**answer

163 views

### How to understand Taubes' moduli space of holomorphic curves?

Let $(X, \omega)$ be a closed symplectic 4-manifold. Let $\mathcal{C}=(C_i, mi_i)$ be a holomorphic current in $X$, where $C_i$ is a somewhere injective $J$-holomorphic curve in $X$ and $m_i$ is ...

**2**

votes

**0**answers

49 views

### Order of metaplectic operator

I have a weak background on this subject.
Suppose $S$ be a $2m \times 2m$ symplectic matrix of order $n$. Suppose $W_S$ be the corresponding metaplectic operator on $\mathcal{S}(\mathbb{R}^m)$, the ...

**2**

votes

**1**answer

88 views

### multiplicity-free action on $SO(n+1)/SO(n-1)$

I'm trying to show that the Lie group $G=SO(n+1) \times SO(2)$ acts multiplicity-free on the cotangentbundle $T^* (SO(n+1)/SO(n-1))$.
That means:
1)
There exists an $\operatorname{Ad}^*_G$-...

**3**

votes

**0**answers

59 views

### Principal orbits for hamiltonian actions

Let $G$ be a compact Lie group which acts by symplectomorphisms on a symplectic manifold $(M,\omega)$. Futhermore let $\mu \colon M \to \mathfrak g$ be a moment map for this action. Denote by $\Omega =...

**6**

votes

**0**answers

407 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 ...

**8**

votes

**3**answers

2k views

### Almost Complex Structures: 'Tame' versus 'Compatible'

Consider a symplectic manifold $(M,\omega)$ and the space of almost complex structures. These are $J:TM\to TM$ with $J^2=-\text{id}$. A given $J$ is $\omega$-tame when $\omega(v,Jv)>0$, and $J$ is $...