The symplectic geometry of polygons in Euclidean space (Q679309)

Let \(M_r\) be the moduli space of \(n\)-gons with fixed side lengths \((r_1, \dots, r_n)\) in Euclidean space. The space \(M_r\) is naturally homeomorphic to \(\widetilde M_r/SO (3) \), where \(\widetilde M_r =\{u\in (S^2)^n: \sum r_ju_j =0\}\). The authors give \(M_r\) a symplectic and a complex analytic structure by the following philosophy: whenever a compact Lie group \(G\) and its complexification \(G^\mathbb{C}\) operate on a complex manifold \(M\) with a given \(G\)-invariant Kähler form, one obtains a \(G\)-equivariant moment map \(\mu\) and \( \mu^{-1} (0)/G \approx G^\mathbb{C} \mu^{-1} (0)/G^\mathbb{C}\); the left hand side is a ``symplectic space'' and the right hand side a ``complex analytic space''. By beautiful and natural geometric constructions, the authors produce functions which Poisson-commute and make a dense open subset \(M_r'\) of \(M_r\) into a toric variety. They also identify functions which give action-angle coordinates on a dense open subset \(M_0\) of \(M_r'\). There is some overlap with the work of \textit{A. Klyachko} [Aspects Math. E 25, 67-84 (1992; Zbl 0820.51016)].