Questions tagged [moment-map]

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existence of moment maps for non-nef toric varieties

The noncompact toric variety $X_1 = \operatorname{Tot} \mathcal{O}(-1) + \mathcal{O}(-1) \to \mathbb{CP}^1$, the total space of the sum of two line bundles over the complex line, is defined as the ...
jj_p's user avatar
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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 ...
Dorado Toro's user avatar
6 votes
1 answer
370 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 ...
Mira's user avatar
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1 answer
141 views

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 $\...
user avatar
14 votes
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436 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 ...
Hugh Thomas's user avatar
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3 votes
1 answer
161 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 ...
Malkoun's user avatar
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5 votes
1 answer
557 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 $...
Ali Taghavi's user avatar
7 votes
1 answer
273 views

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, ...
Maria's user avatar
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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 $G$. ...
B K's user avatar
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6 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 =...
Gaussler's user avatar
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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 $\mathbb{R}^{n*}$ ...
BrianT's user avatar
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473 views

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" ...
Student's user avatar
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2 votes
1 answer
169 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 ...
UserUser's user avatar
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1 answer
290 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 \...
Jeremy's user avatar
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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 ...
Mtheorist's user avatar
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1 answer
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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 $0$, ...
DLIN's user avatar
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2 votes
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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 ...
Pedro's user avatar
  • 269
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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^...
Brightsun's user avatar
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$TSU(n)$ 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 ...
Olorin's user avatar
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20 votes
2 answers
3k views

Origin of the name ''momentum map''

Why is the momentum map in the differential geometry of symmetries called the ''momentum'' (or ''moment'') map?
Arnold Neumaier's user avatar
1 vote
1 answer
80 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 ...
Olorin's user avatar
  • 501
3 votes
1 answer
362 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 ...
Olorin's user avatar
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2 votes
1 answer
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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 ...
Olorin's user avatar
  • 501
2 votes
1 answer
126 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 ...
Olorin's user avatar
  • 501
2 votes
1 answer
119 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(...
Olorin's user avatar
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1 vote
0 answers
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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 $l\colon G \times G/K \to G/K$ ...
Olorin's user avatar
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2 votes
0 answers
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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 ...
Olorin's user avatar
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6 votes
2 answers
382 views

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, $M_s$ ...
Nate Ackerman's user avatar
20 votes
2 answers
3k views

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 ...
Zhaoting Wei's user avatar
  • 8,657
8 votes
3 answers
3k 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 = \...
Simon Rose's user avatar
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2 votes
0 answers
357 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 ...
Ben Webster's user avatar
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