The SU(2)-character variety of the closed surface of genus 2

TL;DR

This paper explores the symplectic geometry of the SU(2)-representation variety for a genus 2 surface, using Goldman flows and involutions to analyze its structure and fixed points.

Contribution

It introduces a novel analysis of the moduli space of SU(2) representations for genus 2 surfaces, including the use of Goldman flows and antisymplectic involutions.

Findings

Identification of subsets of the moduli space with subsets of projective space

Definition and analysis of two antisymplectic involutions

Characterization of fixed point sets of the involutions

Abstract

We study the symplectic geometry of the SU(2)-representation variety of the compact oriented surface of genus 2. We use the Goldman flows to identify subsets of the moduli space with corresponding subsets of $\mathbb P^3(\mathbb C)$. We also define and study two antisymplectic involutions on the moduli space and their fixed point sets.