Polygons in Minkowski three space and parabolic Higgs bundles of rank 2 on P^1

Consider the moduli space of parabolic Higgs bundles (E; Phi) of rank two on CP1 such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by (E; Phi) -> (E; -Phi). We study the fixed point locus of this involution. In [CM], this moduli space with involution was identified with the moduli space of hyperpolygons equipped with a certain natural involution. Here we identify the fixed point locus with the moduli spaces of polygons in Minkowski 3-space. This identification yields information on the connected components of the fixed point locus.