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Hyperpolygons and moduli spaces of parabolic Higgs bundles

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If you have a question about this talk, please contact Mustapha Amrani.

Moduli Spaces

Given an $n$tuple of positive real numbers $lpha$ we consider the hyperpolygon space $X(lpha)$, the hyperkhler quotient analogue to the Khler moduli space of polygons in $mathbb{R}3$. There exists an 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. This allows us to prove that hyperpolygon spaces $X(lpha)$ undergo an elementary transformation in the sense of Mukai as $lpha$ crosses a wall in the space of its admissible values. We describe the resulting changes in the core of $X(lpha)$ as well as the changes in the nilpotent cone of the corresponding moduli spaces of PHBs. If time permits, we will explain how to obtain explicit formulas for the computation of the intersection numbers of the core components of $X(lpha)$ and of the nilpotent cone components of the corresponding moduli spaces of PHBs. As a final application, we describe the cohomology ring structure of these moduli spaces of PHBs and of the components of their nilpotent cone. This is joint work with Leonor Godinho.

This talk is part of the Isaac Newton Institute Seminar Series series.

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