Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles

TL;DR

This paper establishes an isomorphism between hyperpolygon spaces and moduli spaces of parabolic Higgs bundles, enabling detailed analysis of their geometric structures, transformations, and intersection rings.

Contribution

It introduces a novel isomorphism linking hyperpolygon spaces with moduli spaces of parabolic Higgs bundles, revealing their geometric transformations and cohomological properties.

Findings

Hyperpolygon spaces are isomorphic to moduli spaces of parabolic Higgs bundles.

Elementary transformations of hyperpolygon spaces correspond to wall crossings in parameter space.

Explicit formulas for intersection numbers and cohomology rings are derived.

Abstract

Given an $n$-tuple of positive real numbers $\alpha$ we consider the hyperpolygon space $X(\alpha)$, the hyperk\"{a}hler quotient analogue to the K\"ahler moduli space of polygons in $\mathbb{R}^3$. We prove the existence of 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 isomorphism allows us to prove that hyperpolygon spaces $X(\alpha)$ undergo an elementary transformation in the sense of Mukai as $\alpha$ crosses a wall in the 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 the nilpotent 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 a recursion relation in $n$ for their intersection numbers and use it to obtain explicit formulas for the computation of these numbers. Using our isomorphism, we obtain similar formulas for each connected component of the nilpotent cone of the corresponding moduli spaces of parabolic Higgs bundles thus determining their intersection rings. As a final application of our isomorphism we describe the cohomology ring structure of these moduli spaces of parabolic Higgs bundles and of the components of their nilpotent cone.