Polygons in Minkowski three space and parabolic Higgs bundles of rank two on CP^1

TL;DR

This paper explores the fixed points of an involution on the moduli space of rank two parabolic Higgs bundles on CP^1, linking them to Minkowski 3-space polygons and revealing their connected components.

Contribution

It identifies the fixed point locus of the involution with moduli spaces of polygons in Minkowski 3-space, providing new geometric insights.

Findings

Connected components of fixed point locus characterized

Fixed point locus identified with Minkowski 3-space polygons

Enhanced understanding of moduli space structure

Abstract

Consider the moduli space of parabolic Higgs bundles (E,\Phi) of rank two on CP^1 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)\mapsto (E,-\Phi). 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.