Involutions of rank 2 Higgs bundle moduli spaces

TL;DR

This paper investigates involutions on the moduli space of rank 2 Higgs bundles over a genus 2 curve, analyzing fixed points via Prym varieties and their relation to fundamental group representations.

Contribution

It characterizes fixed points of specific involutions on Higgs bundle moduli spaces using Prym varieties and links these to representation spaces.

Findings

Fixed points described by Prym varieties.

Involutions relate to tensoring with 2-torsion points.

Connections established with fundamental group representations.

Abstract

We consider the moduli space of rank 2 Higgs bundles with fixed determinant over a smooth projective curve X of genus 2 over the complex numbers, and study involutions defined by tensoring the vector bundle with an element $\alpha$ of order 2 in the Jacobian of the curve, combined with multiplication of the Higgs field by $\pm 1$. We describe the fixed points of these involutions in terms of the Prym variety of the covering of $X$ defined by $\alpha$, and give an interpretation in terms of the moduli space of representations of the fundamental group.