5
votes
Accepted
Question about an example in symplectic geometry
I'll preserve your notation: $M$ is the coadjoint orbit of a regular semisimple element $X \in \mathfrak t^*$ (which you seem to also call $x$). I also assume we're working in characteristic $0$, or ...
- 12.9k
5
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Is the symplectic quotient $\mu^{-1}(0)/G$ unique up to something?
Given a Hamiltonian group action, moment maps may only differ by constant addition. So you seem to be comparing the reduced spaces at different levels. Let me state the two extreme cases.
When $G$ is ...
- 1,398
4
votes
Accepted
The norm-squared of a moment map behaves like a Morse-Bott function
As requested I am submitting my comment as an answer.
The desired statement is proved in Gradient flow of the norm squared of a moment map by Eugene Lerman who attributes the proof to Duistermaat.
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2
votes
Accepted
Lines on a toric cubic surface with a line of nodes
The cubic scroll in $\mathbb{P}^4$ is isomorphic to $\mathbb{F}_1$ and its torus-invariant divisor has three line components and one conic component. The linear projection $\mathbb{P}^4 \...
- 39.3k
2
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Explicit local normal form symplectomorphism at torus fixed point of a coadjoint orbit
Here is a partial answer.
The map $\mathfrak{t}^{\perp} \to T_{\lambda}\mathcal{O}_{\lambda}$, $X \mapsto ad_X(\lambda)$, endows $\mathfrak{t}^{\perp}$ with a linear symplectic form
$$ \omega_0(X,Y) ...
- 401
1
vote
Accepted
Moment map in Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks $2$
The choice of $\xi$ amounts to choosing a Hamiltonian circle action on $\mathbb{CP}^n$ by isometries. I.e. choosing a suitable 1-parameter subgroup of the $PU(n+1,\mathbb{C})$.
Consider the ...
- 6,995
1
vote
Accepted
Stuck on a computation with quaternions and moment maps
I was able to finally prove that
$$d(\omega.d\mathbf{r}) = - \frac{1}{2r^3}(d\mathbf{r}\,\mathbf{r} \wedge d\mathbf{r}).$$
In the process, I have learned a lot. The main issue for me was that I was ...
- 5,215
1
vote
Geometric invariants of a Riemannian manifold encoded in certain moment map
Yes, all has been solved by Jean-Marie Souriau: see Barbaresco, F.; Gay-Balmaz, F. Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on ...
1
vote
Definition of a moment map with physical context
$\def \cG {\cal G}$
$\def \RR {\mathbf R}$
The moment map is one of the most important constructions in symplectic mechanics. There are at least two main reasons for that:
The moment map $\mu : M \to ...
- 2,289
1
vote
Accepted
An example of Guillemin Sternberg Conjecture
Since you are looking for an official source, I recommend the book Symplectic Fibrations and Multiplicity Diagrams by Guillemin, Lerman and Sternberg. It has a lot to say about symplectic reduction on ...
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