Questions tagged [moment-map]
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30 questions
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Origin of the name ''momentum map''
Why is the momentum map in the differential geometry of symmetries called the ''momentum'' (or ''moment'') map?
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2
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What is the significance that the Springer resolution is a moment map?
Let $\mathcal{B}$ be the flag variety and $\mathcal{N} \subset \mathfrak{g}$ is the nilpotent cone. We know that the Springer resolution
$$
\mu: T^*\mathcal{B}\rightarrow \mathcal{N}
$$
is the moment ...
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14
votes
0
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views
How should we think about the algebraic moment map?
My question is about the "algebraic moment map", as discussed by Frank Sottile in the final section of this paper, or by Bill Fulton in his Introduction to Toric Varieties, where he referes ...
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votes
3
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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
$$
\langle d\mu(v), \xi\rangle = \...
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votes
1
answer
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Question about an example in symplectic geometry
Let M be a coadjoint orbit of dimension 6 of $SU(3)$, and let T be the maximal torus in $SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map associated to the action of T on M, ...
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votes
1
answer
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Is the symplectic quotient $\mu^{-1}(0)/G$ unique up to something?
Given a Hamiltonian action of a compact Lie group $G$ on a symplectic manifold $(M,\omega)$, we may choose a moment map $\mu \colon M\to \mathfrak{g}^* $ and obtain the symplectic reduction $M/\!\!/G =...
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views
The norm-squared of a moment map behaves like a Morse-Bott function
Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. Let $<.,>$ denote a $G$-invariant inner product on $\mathfrak{g}$.
Let $(M,\omega)$ be a symplectic compact manifold endowed with ...
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2
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Accuracy of the truncated Hausdorff moment problem
For a sequence of real numbers $s = (s_i)_{i \in n}$ let $M_s$ be the collection of functions $f:[0,1] \to [0,1]$ such that
$$(\forall i \leq n) \int_0^1 x^i f(x) dx = s_i$$
In other words, $M_s$ ...
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6
votes
1
answer
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views
Explicit local normal form symplectomorphism at torus fixed point of a coadjoint orbit
Let $K$ be a compact, connected (probably also simple) Lie group and with a maximal torus $T$. Regular coadjoint orbits $\mathcal{O}_{\lambda} \cong K/T$, parameterized by a regular element $\lambda \...
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views
Geometric invariants of a Riemannian manifold encoded in certain moment map
Let $(M,g)$ be a Riemannian manifold with isometric group $G=Iso(M,G)$. The metric gives an isomorphism between tangent and cotangent bundle of $M$. So $g$ induce a natural symplectic structure on $...
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Definition of a moment map with physical context
This was originally posted on Math Stack Exchange, but without an answer. I thus move it here, and hope it's not because I express it unclearly.
Suppose $(M,\omega)$ is a symplectic manifold "well" ...
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3
votes
1
answer
165
views
Stuck on a computation with quaternions and moment maps
I am trying to understand an article by Gibbons, Rychenkova and Goto, called "Hyperkähler quotient construction of BPS Monopole Moduli Spaces". I will paraphrase the relevant notions and ...
- 5,215
3
votes
1
answer
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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 ...
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0
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views
How to calculate the exterior derivative on manifolds of smooth mappings?
Let $S$ be a compact finite-dimensional manifold $S$ and $(M, \omega)$ a symplectic manifold. The space of smooth maps from $S$ to $M$, denoted by $\mathcal{M}$, has a canonical infinite-dimensional ...
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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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3
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On two different descriptions of Delzant polytopes
I have seen two different ways of describing a Delzant polytope:
From Canna Da Silva https://people.math.ethz.ch/~acannas/Papers/toric.pdf, a Delzant polytope is a polytope in $\mathbb{R}^{n*}$ ...
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2
votes
1
answer
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views
coisotropic action on $TS^{2n+1}$
Let $S^{2n+1}$ be the $m$-dimensional sphere in $\mathbb{C}^{n+1}$. Endow $S^{2n+1}$ with the standard metric. Let $S^1$ act by multiplication on $S^{2n+1}$. Then $S^1$ and the canonical action of $SU(...
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votes
1
answer
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views
Lines on a toric cubic surface with a line of nodes
Consider a cubic surface cut out by equations $x^2y - z^2w$ inside $\mathbb{P}^3$. This gives a cubic surface with a line of nodes, it is toric and has normalisation $\mathbb{F}_1$, a Hirzebruch ...
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1
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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 ...
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2
votes
1
answer
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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 ...
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An example of Guillemin Sternberg Conjecture
Guillemin Sternberg Conjecture(proved) says that for symplectic manifold $(M,\omega)$ with $[Q,R]=0$ condiction, with compact group action $G$, such that $\mu:M\to \mathfrak g^*$ is regular at $0$, ...
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Why ask for the co-moment map to preserve brackets?
Let $G$ be a Lie group and $(M, \omega)$ a symplectic manifold. An action of $G$ on $M$ is Hamiltonian if it is equipped with a co-moment map $\widetilde{\mu} : \mathfrak{g} \to C^\infty(M)$ which is ...
- 279
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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 ...
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votes
0
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Pulled back foliation is completely integrable
There is a question that arises, while I'm trying to understand Guillemin & Sternbergs paper "On collective complete integrability according to the method of Thimm".
Assume $M$ is a symplectic ...
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2
votes
0
answers
365
views
Are schematic fixed points of a torus action on an affinized twistor deformation flat?
This is a follow-up to some earlier questions about flatness of schematic fixed points of certain deformations. Since I could never come up with good enough hypotheses in those examples, let me try a ...
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1
vote
1
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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 ...
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vote
1
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Moment map in Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks $2$
In the paper Floer cohomology of Lagrangian intersections
and pseudo-holomorphic disks 2, in the part of the preliminaries the author considers a Hamiltonian action of the isometry group $G$ of $\...
user174565
1
vote
0
answers
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Moment map of isometries on Kähler mainfolds
Let us assume we are given a Kähler manifold $M$, equipped with its metric $g_{\imath\bar\jmath}$ and with the associated symplectic form
$$
\Omega = i\, g_{\imath \bar \jmath}dz^\imath \wedge d\bar z^...
- 121
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Momentum Map on cotangentbundle as submersion
Let $N$ be a homogeneous space. Therefore we find a Liegroup $G$ and a isotropy-subgroup $K$ of $G$, such that we can identify $N = G/K$. Then we have a canonical action $l\colon G \times G/K \to G/K$ ...
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1
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What is general expression for the moment map of a Kaehler Hamiltonian G-manifold
A Kaehler Hamiltonian G-manifold is a Kaehler manifold with a Hamiltonian G-action, i.e., a G-action generated by a moment map. In particular, the Killing vector fields which generate the G-action are ...
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