Hyperpolygon spaces and their cores

Abstract

Given an n-tuple of positive real numbers (α 1 ,.., an), Konno (2000) defines the hyperpolygon space X(a), a hyperkahler analogue of the Kahler variety M(a) parametrizing polygons in R 3 with edge lengths (α 1 ,..., an). The polygon space M(α) can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, X(a) is the hyperkahler quiver variety defined by Nakajima. A quiver variety admits a natural C*-action, and the union of the precompact orbits is called the core. We study the components of the core of X(α), interpreting each one as a moduli space of pairs of polygons in R 3 with certain properties. Konno gives a presentation of the cohomology ring of X(a); we extend this result by computing the C*-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.