Goldman systems and bending systems

TL;DR

This paper demonstrates an isomorphism between the moduli space of parabolic bundles on the projective line and the polygon space, revealing their equivalence as complex and symplectic manifolds with integrable systems under certain conditions.

Contribution

It establishes a novel isomorphism between two geometric structures, connecting moduli spaces and polygon spaces as integrable systems.

Findings

Moduli space of parabolic bundles is isomorphic to polygon space.

Both spaces are complex and symplectic manifolds with integrable systems.

Isomorphism holds when stability parameters are small.

Abstract

We show that the moduli space of parabolic bundles on the projective line and the polygon space are isomorphic, both as complex manifolds and symplectic manifolds equipped with structures of completely integrable systems, if the stability parameters are small.