The Symplectic Geometry of Polygons in the 3-sphere

TL;DR

This paper explores the symplectic geometry of moduli spaces of closed polygons in the 3-sphere, establishing their structure via reduction of conjugacy classes, constructing integrable systems, and relating to gauge theory, extending known results from hyperbolic and Euclidean spaces.

Contribution

It introduces a symplectic structure on polygon moduli spaces in the 3-sphere through conjugacy class reduction, constructs an integrable bending system, and links the structure to gauge-theoretic descriptions, extending prior hyperbolic and Euclidean results.

Findings

Moduli spaces have symplectic structures via conjugacy class reduction.

An integrable Hamiltonian system with bending flows is constructed.

The symplectic structure relates to gauge-theoretic descriptions.

Abstract

We study the symplectic geometry of the moduli spaces $M_r=M_r(\s^3)$ 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 $C_r^n$, by the diagonal conjugation action of SU(2). Here $C_r^n$ is a quasi-Hamiltonian SU(2)-space. 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 $M_r(\h^3)$, the moduli space of n-gons with fixed side-lengths in hyperbolic 3-space \cite{KMT}, and for $M_r(\E^3)$, the moduli space of n-gons with fixed side-lengths in $\E^3$