Hyperpolygons spaces are a family of (finite dimensional, non-compact) hyperkähler spaces, that can be obtained from coadjoint orbits by hyperkähler reduction. Jointly with L. Godinho, we show that these space are diffeomorphic (in fact, symplectomorphic) to certain families of parabolic Higgs bundles. In this talk I will describe this relation and use it to analyse the fixed points locus of a natural involution on the moduli space of parabolic Higgs bundles. I will show that each connected components of the fixed point locus of this involution is identified with a moduli spaces of polygons in Minkowski 3-space.
This is based on
- L. Godinho, A. Mandini, "Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles" Adv. Math. 244 (2013), 465–532
- I. Biswas, C. Florentino, L. Godinho, A. Mandini, "Polygons in the Minkowski three space and parabolic Higgs bundles of rank two on $\mathbb{CP}^1$, Transfom. Groups 18 (2013)
- I. Biswas, C. Florentino, L. Godinho, A. Mandini, "symplectic form on hyperpolygon space", Geom. Dedicata 179 (2015)
- L. Godinho, A. Mandini, "Quasi-parabolic Higgs bundles and null hyperpolygon spaces", arXiv:1907.01937