We consider moduli spaces of polygons in Minkowski 3-space with a fixed number of sides lying in the future time-like cone and the others in the past (where each side has a fixed Minkowski length) and study their relation with the fixed-point set of a natural involution on the space of rank-2 parabolic Higgs bundles over ${\mathrm{CP}}^1$.

This is joint work with I. Biswas, C. Florentino and A. Mandini.