Polygons in Minkowski three space and parabolic Higgs bundles of rank 2 on $ \mathbb{C}{{\mathbb{P}}^1} $

Abstract

Consider the moduli space of parabolic Higgs bundles (E, Φ) 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 \( \left( {E,\varPhi } \right)\mapsto \left( {E,-\varPhi } \right) \). We study the fixed point locus of this involution. In [GM], 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.