The Symplectic Geometry of Polygons in the 3-Sphere

Treloar, Thomas

Canadian journal of mathematics, Tome 54 (2002) no. 1, pp. 30-54

Résumé

We study the symplectic geometry of the moduli spaces ${{M}_{r}}={{M}_{r}}\left( {{\mathbb{S}}^{3}} \right)$ of closed $n$ -gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of $n$ conjugacy classes in $\text{SU}\left( 2 \right)$ by the diagonal conjugation action of $\text{SU}\left( 2 \right)$ . Here the fusion product of $n$ conjugacy classes is a Hamiltonian quasi-Poisson $\text{SU}\left( 2 \right)$ -manifold in the sense of $\left[ \text{AKSM} \right]$ . An integrable Hamiltonian system is constructed on ${{M}_{r}}$ in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on ${{M}_{r}}$ relates to the symplectic structure obtained from gauge-theoretic description of ${{M}_{r}}$ . The results of this paper are analogues for the 3-sphere of results obtained for ${{M}_{r}}\left( {{\mathbb{H}}^{3}} \right)$ , the moduli space of $n$ -gons with fixed side-lengths in hyperbolic 3-space $\left[ \text{KMT} \right]$ , and for ${{M}_{r}}\left( {{\mathbb{E}}^{3}} \right)$ , the moduli space of $n$ -gons with fixed side-lengths in ${{\mathbb{E}}^{3}}\left[ \text{KM}1 \right]$ .

Détail

DOI : 10.4153/CJM-2002-002-1

Mots-clés : 53D

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