The Symplectic Geometry of Polygons in the 3-sphere

Abstract. We study the symplectic geometry of the moduli spaces Mr = Mr(S ) 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 SU(2), denoted C r , by the diagonal conjugation action of SU(2). Here C n r is a quasi-Hamiltonian SU(2)-space. An integrable Hamiltonian system is constructed on Mr in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on Mr relates to the symplectic structure obtained from gauge-theoretic description of Mr. The results of this paper are analogues for the 3-sphere of results obtained for Mr(H ), the moduli space of n-gons with fixed side-lengths in hyperbolic 3-space [KMT], and for Mr(E ), the moduli space of n-gons with fixed side-lengths in E [KM1].