Symplectic form on hyperpolygon spaces

TL;DR

This paper provides a canonical construction of a family of hyperpolygon spaces, enabling explicit computation of the Higgs symplectic form and proving the isomorphism with the moduli space is a symplectomorphism.

Contribution

It offers a new canonical construction of hyperpolygon spaces and demonstrates that the isomorphism with the moduli space preserves the symplectic structure.

Findings

Computed the Higgs symplectic form explicitly

Proved the isomorphism is a symplectomorphism

Provided an alternative construction of the hyperpolygon family

Abstract

In [GM], a family of parabolic Higgs bundles on $CP^1$ has been constructed and identified with a moduli space of hyperpolygons. Our aim here is to give a canonical alternative construction of this family. This enables us to compute the Higgs symplectic form for this family and show that the isomorphism of [GM] is a symplectomorphism.