Collective superintegrable systems from the Guillemin--Sternberg torus action

Abstract

We present a novel approach to the superintegrability of collective Hamiltonians invariant under a Hamiltonian action of a connected semisimple compact Lie group, $G$, on a symplectic manifold, $M$. By exploiting a Hamiltonian torus action that goes back to Guillemin and Sternberg [GS,1983], we demonstrate that the functional dimensions of $\mathfrak{H}:={\mathcal{J}}^{\ast }(C^{\infty }({\mathfrak{g}}^{\ast }{)}^G)$, where $\mathcal{J}:M\to {\mathfrak{g}}^{\ast }$ is the momentum map of the $G$ action, and its centralizer $\mathfrak{F}$ in $C^{\infty }(M)$ satisfy the equality $\mathrm{ddim}(\mathfrak{H})+\mathrm{ddim}(\mathfrak{F})=\dim (M)$. Together with a non-triviality condition, this ensures that the Abelian Poisson algebra $\mathfrak{H}\subset C^{\infty }(M)$ represents a superintegrable system, and it also follows that the momentum map of the GS torus action yields action variables for the system. Our work provides a new insight into collective superintegrability complementing earlier results of Bolsinov and Jovanović.

12 pages, v2: corrected the wording of Lemma 2.11 and added a few sentences