Hyperpolygon spaces and their cores

Abstract:

Given an $n$-tuple of positive real numbers $(\alpha _1,\ldots ,\alpha _n)$, Konno (2000) defines the hyperpolygon space $X(\alpha )$, a hyperkähler analogue of the Kähler variety $M(\alpha )$ parametrizing polygons in $\mathbb {R}^3$ with edge lengths $(\alpha _1,\ldots ,\alpha _n)$. The polygon space $M(\alpha )$ can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, $X(\alpha )$ is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural $\mathbb {C}^*$-action, and the union of the precompact orbits is called the core. We study the components of the core of $X(\alpha )$, interpreting each one as a moduli space of pairs of polygons in $\mathbb {R}^3$ with certain properties. Konno gives a presentation of the cohomology ring of $X(\alpha )$; we extend this result by computing the $\mathbb {C}^*$-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.