Let \(H(\beta)\) denote the moduli space of parabolic Higgs bundles \((E, \phi)\) where \(E\) is a holomorphically trivial vector bundle of rank \(2\) on the complex projective line \({\mathbb P}^1_{\mathbb C}\) with a complete flag over each of the \(n\) marked points \(x_1, \dots, x_n\) and weights \(0\leq \beta_1(x_i) < \beta_2(x_i) <1\) (at \(x_i\)) and \(\phi\) is a meromorphic \(\mathrm{End}(E)\)-valued \(1\)-form holomorphic outside the marked points such that \(\phi\) has at most a simple pole with a nilpotent residue (with respect to the flag) at each of the marked points. Let \(\sigma\) denote the involution on \(H(\beta)\) defined by \((E, \phi) \mapsto (E, - \phi)\). The hyper polygon space \(X(\alpha)\), with \(\alpha_i = \beta_2(x_i)- \beta_1(x_i)\), is the hyper Kähler quotient of \(T^* ({\mathbb C}^{2n})\) by \(K = (\mathrm{SU}(2) \times \mathrm{U}(1)^n) / ({\mathbb Z}/2{\mathbb Z})\) where \({\mathbb Z}/2{\mathbb Z}\) acts on the each factor by \(-1\).
L. Godinho and A. Mandini [Adv. Math. 244, 465–532 (2013; Zbl 1318.32018)] had established an isomorphism between \(H(\beta)\) and \(X(\alpha)\) with \(\sigma\) corresponding to the involution \(\sigma'\) on \(X(\alpha)\) defined by \([p,q] \mapsto [-p, q] \in T^*({\mathbb C}^{2n})\). The authors study the fixed point set of \(X(\alpha)\) for \(\sigma'\) which enables them to determine the fixed point set of \(H(\beta)\) for \(\sigma\). They show that this fixed point set has a compact component \(M(\beta)\) isomorphic to the moduli space of parabolic vector bundles of rank \(2\) and degree \(0\) on \({\mathbb P}^1_{\mathbb C}\) and other components \(Z_S\) parametrised by certain subsets \(S\) of \(\{1, \dots, n\}\). Explicit description of the components \(Z_S\) is given. Finally each \(Z_S\) is shown to be diffeomorphic to a moduli space of closed polygons in Minkowski \(3\)-space i.e. in \({\mathbb R}^3\) equipped with the Minkowski inner product \(v_1 \circ v_2 = - x_1 x_2 - y_1 y_2 +t_1 t_2\) for \(v_i = (x_i, y_i, t_i), i= 1,2\).