Assume $\mathcal{M}$ is a compact symplectic $2n$-dimensional manifold with a Hamiltonian action of the torus $\mathbb{T}^n$. Given a family of functions $F=(f_1,\ldots,f_n)$ defining an integrable system I can say that the map $F:\mathcal{M}\rightarrow \mathbb{R}^n$ maps the manifold onto the Lie algebra of the torus. If the set of functions define a Delzant polytope on $\mathbb{R}^n$, what would be missing to claim that actually $F$ defines a moment map for this torus action?
$\begingroup$
$\endgroup$
2
-
$\begingroup$ I changed $2n-$dimensional to $2n$-dimensional, fixing this conspicuous typographical error. $\endgroup$– Michael HardyCommented Aug 9 at 18:43
-
1$\begingroup$ Consider the example of the $2$-sphere with its standard area form as symplectic struture. Now let $F = (f_1):S^2\to\mathbb{R}$ be any non-constant smooth function. I assume that meets your criterion as an integrable system, and the image of $F$ is an interval in $\mathbb{R}$. However, in general the flow of the Hamiltonian vector field on $S^2$ that corresponds to $f_1$ will not be periodic. You need that flow to be periodic or it won't come from a circle action. It's easy to construct higher dimensional examples by taking products. $\endgroup$– Robert BryantCommented Aug 10 at 11:07
Add a comment
|