In this talk I will describe the geometry of the moduli space of polygons and of its hyperkaehler analogue, the so called hyperpolygon space. In particular I will discuss how these spaces are isomorphic to certain moduli spaces of stable, rank-2, parabolic and parabolic Higgs (respectively) bundles. If time permits, I will briefly describe two applications of this result. On one way, using the study of variation of moduli spaces of parabolic Higgs bundles over a curve, we can describe the dependence of hyperpolygon spaces X(a) and their cores on the choice of the parameter a showing that, when a wall is crossed, the hyperpolygon space undergoes an elementary transformation in the sense of Mukai. On the other way, we can take advantage of the geometric description of the core components of a hyperpolygon space to obtain explicit expressions for the computation of the intersection numbers of the nilpotent cone components of the above moduli space of parabolic Higgs bundles. This is joint work with Leonor Godinho, arXiv:1101.3241.
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