Critical values of moment maps on quantizable manifolds

Details Critical values of moment maps on quantizable manifolds Critical values of moment maps on quantizable manifolds

2007-11-02

arxiv.org/abs/0711.0358v1

Mathematics

Symplectic Geometry

10 pages, comments are wellcome

Scientific paper

Let $M$ be a quantizable symplectic manifold acted on by $T=(S^1)^r$ in a Hamiltonian fashion and $J$ a moment map for this action. Suppose that the set $M^{T}$ of fixed points is discrete and denote by ${\alpha}_{pj}\in{\mathbb Z}^r$ the weights of the isotropy representation at $p$. By means of the $\alpha_{pj}$'s we define a partition ${\mathcal Q}_+$, ${\mathcal Q}_-$ of $M^T$. (When $r=1$, ${\mathcal Q}_{\pm}$ will be the set of fixed points such that the half of the Morse index of $J$ at them is even (odd)). We prove the existence of a map $\pi_{\pm}:{\mathcal Q}_{\pm}\to{\mathcal Q}_{\mp}$ such that $J(q)-J(\pi_{\pm}(q))\in I_{\mp}$, for all $q\in {\mathcal Q}_{\pm}$, where $I_{\pm}$ is the lattice generated by the $\alpha_{pj}$'s with $p\in{\mathcal Q}_{\pm}.$ We define partition functions $N_p$ similar to the ones of Kostant \cite{Gui} and we prove that $\sum_{p\in{\mathcal Q}_+}N_p(l)=\sum_{p\in{\mathcal Q}_-}N_p(l)$, for any $l\in{\mathbb Z}^r$ with $|l|$ sufficiently large.

Affiliated with

Also associated with

No associations

Rating

Critical values of moment maps on quantizable manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Critical values of moment maps on quantizable manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Critical values of moment maps on quantizable manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-437999