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Given an $n$-tuple of positive real numbers $\alpha$ we consider thehyperpolygon space $X(\alpha)$, the hyperk\"{a}hler quotient analogue to theK\"ahler moduli space of polygons in $\mathbb{R}^3$. We prove the existence ofan isomorphism between hyperpolygon spaces and moduli spaces of stable,rank-$2$, holomorphically trivial parabolic Higgs bundles over $\mathbb{C}\mathbb{P}^1$ with fixed determinant and trace-free Higgs field. Thisisomorphism allows us to prove that hyperpolygon spaces $X(\alpha)$ undergo anelementary transformation in the sense of Mukai as $\alpha$ crosses a wall inthe space of its admissible values. We describe the changes in the core of$X(\alpha)$ as a result of this transformation as well as the changes in thenilpotent cone of the corresponding moduli spaces of parabolic Higgs bundles.Moreover, we study the intersection rings of the core components of$X(\alpha)$. In particular, we find generators of these rings, prove arecursion relation in $n$ for their intersection numbers and use it to obtainexplicit formulas for the computation of these numbers. Using our isomorphism,we obtain similar formulas for each connected component of the nilpotent coneof the corresponding moduli spaces of parabolic Higgs bundles thus determiningtheir intersection rings. As a final application of our isomorphism we describethe cohomology ring structure of these moduli spaces of parabolic Higgs bundlesand of the components of their nilpotent cone.

Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles