The Symplectic Geometry of Polygons in Euclidean Space (original) (raw)

We study the symplectic geometry of moduli spaces M r of polygons with fixed side lengths in Euclidean space. We show that M r has a natural structure of a complex analytic space and is complexanalytically isomorphic to the weighted quotient of (S 2 ) n constructed by Deligne and Mostow. We study the Hamiltonian flows on M r obtained by bending the polygon along diagonals and show the group generated by such flows acts transitively on M r . We also relate these flows to the twist flows of Goldman and Jeffrey-Weitsman. Contents 1 Introduction 2 2 Moduli of polygons and weighted quotients of configuration spaces of points on the sphere 5 3 Bending flows and polygons. 15 4 Action--angle coordinates 21 This research was partially supported by NSF grant DMS-9306140 at University of Utah (Kapovich) and NSF grant DMS-9205154, the University of Maryland (Millson). 1 5 The connection with gauge theory and the results of Goldman and Jeffrey-Weitsman 24 6 Transitivity of bending ...