Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [moment-map]

Questions related to smooth actions of Lie groups (typically on symplectic or Poisson manifolds) which can be described by a Hamiltonian, called the moment map.

Questions related to smooth actions of Lie groups (typically on symplectic or Poisson manifolds) which can be described by a Hamiltonian, called the moment map.

36 questions
7
votes
1 answer

Cohomology ring of a configuration space

Consider the following configuration space of triples of points. $$\begin{align}C &= \left\lbrace (z_1,z_2,z_3) \in (\mathbb C^*)^3, z_1 \ne z_2, z_1 \ne z_3, z_2 \ne z_3\right\rbrace \\&\phantom{abcde}\setminus \left\lbrace (z_1,z_2,z_3) \in…
user79456
  • 369
5
votes
1 answer

Cotangent lift of an action and its effect on the moment covector

If I have an action of a Lie group on a configuration space。 $G\to \text{Diff}(M)$, $g \mapsto \rho_g$, $\rho_g : q \mapsto \rho_g(q)$ (for example a rotation). Then when we consider the phase spaces $T^*M$, we provide it with the action : $G\to…
roi_saumon
  • 4,406
5
votes
1 answer

A query about Atiyah's proof of the convexity of moment map

This is about proving connectedness of level sets of moment map of a $\mathbb T^n$ $\implies$ convexity of image of moment map for $\mathbb T^{n+1}$ action. I am following Ana Cannas's wonderful lecture notes but I got stuck at a point in the proof…
Soham
  • 2,053
5
votes
1 answer

Explicit computation moment map complex projective space

Consider the hamiltonian action of $T^2$ on $\mathbb{CP}^2$ : $$ \varphi: ((e^{i\theta_1},e^{i\theta_2}),[z_0:z_1:z_2]) \longmapsto [z_0:e^{i\theta_1}z_1,e^{i\theta_2}z_2].$$ I've read that its moment map is $$ \mu (z_0,z_1,z_2) = -\tfrac{1}{2}…
Reb
  • 387
4
votes
1 answer

Understand the moment map

I'm studying moment map via the book Geometry of Four-Manifolds, in this book, the authors give several ways to understand (co-)moment map: For given symplectic group action $G$ (and correspondent Lie algebra $\mathfrak{g}$) and symplectic manifold…
taiat
  • 1,217
4
votes
1 answer

Symplectic Reduction of 3-D Chern Simons Theory

So, I'm new to gauge theories and symplectic reduction and was trying to analyze the Chern Simons theory in three dimensions. I have a few questions regarding the steps towards reduction. First off, is it necessary for the bundle to be…
MrHolmes
  • 515
4
votes
1 answer

Showing that the moment map $\langle \mathbf{J}(z),\xi\rangle=\mathbf{i}_{\xi_P}(\Theta)(z)$ is equivariant

I am studying through Introduction to mechanics and symmetry by Marsden and Ratiu, specifically the chapter on Momentum maps, and wanted some confirmation as to whether my argument for the following problem is correct. I have added quite a bit of…
J.V.Gaiter
  • 2,659
4
votes
0 answers

Fundamental vector field and moment map for action on mathematical pendulum.

I am trying to compute the moment map, for an action. As I understand it if I have a configuration space which is given by a manifold $M$ and an action $\rho_g \colon M \to M$, it induces an action on the phase space $(\rho_{g^{-1}})^* : T^*M \to…
roi_saumon
  • 4,406
4
votes
3 answers

Moment maps unitary group acting on matrices

I am reading the fifth chapter of "An introduction to extremal Kaehler metrics" by Gabor Szekelyhidi. At the very beginning of that chapter, the author describes moment maps and Hamiltonian action. Here is a short description: Suppose that a…
Salvatore
  • 1,302
3
votes
1 answer

Canonical coordinates and tautological one-form: about a paragraph in the Wikipedia article on the Tautological One-form

As @peek-a-boo wrote in one of his answer, "the word "momentum" gets thrown around more often than candy during Halloween". I found two definitions of momentum generalized coordinates I want to reconcile one way or another. We go with the usual…
brunoh
  • 2,649
3
votes
0 answers

Why do we construct Lagrangian submanifolds after symplectic reductions

I am learning about Hamilton-Jacobi actions, symplectic reductions and Lagrangian submanifolds and I am trying to understand the relation between these concepts. I have read that Lagrangian submanifolds are physically interesting as they can be…
Spida
  • 373
3
votes
1 answer

Questions about the moment map

Let $(M,\omega)$ be a symplectic manifold endowed with a hamiltonian action of a torus $T$. Let $\mu : M \longrightarrow {Lie(T)}^*,$ be a moment map associated to this action. Let $S_M =\bigcap\limits_{m \in M} Stab(m)$, and $s_m$ be its lie…
Maria
  • 179
2
votes
2 answers

Noether‘s Theorem and Moment Maps

Noether‘s Theorem says that every continuous symmetry of a physical system (i.e., a Lie group action on phase space ${\bf R}^{2n}$ preserving a Hamiltonian $H$) leads to a conservation law (i.e., a function $I$ satisfying $\left\{I,H\right\}=0$). In…
user39082
  • 2,174
2
votes
2 answers

Image of a coadjoint orbit of SU(3) under a moment map

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, then the image of the moment map is a hexagon…
Maria
  • 179
2
votes
1 answer

Definition of Hamiltonian Action from Ana Cannas da Silva's book

I've been learning about Lie Groups, Lie Algebras and moment maps and my aim is to get to Sympletic Reduction. Currently I'm struggling with a few details of the definition of hamiltonian action from Ana da Cannas Silva's book on Sympletic Geometry.…
Paulo Mourão
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