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The Symplectic Geometry of Polygons in the 3-Sphere

Published online by Cambridge University Press:  20 November 2018

Thomas Treloar
Thomas Treloar*
Affiliation:
Department of Mathematics University of Arizona Tucson, Arizona 85721 U.S.A., email: treloar@math.arizona.edu
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Abstract

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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 ${{\mathbb{E}}^{3}}\left[ \text{KM}1 \right]$ .

Keywords

53D

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 1 , 01 February 2002 , pp. 30 - 54
Copyright
Copyright © Canadian Mathematical Society 2002

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