Anti-holomorphic involutions of the moduli spaces of Higgs bundles

TL;DR

This paper investigates anti-holomorphic involutions on moduli spaces of Higgs bundles over Riemann surfaces, exploring fixed points, their relation to orbifold fundamental group representations, and connections with branes, extending previous work in the field.

Contribution

It introduces a comprehensive analysis of anti-holomorphic involutions on Higgs bundle moduli spaces, linking fixed points to orbifold fundamental groups and branes, generalizing prior research.

Findings

Characterization of fixed point loci under anti-holomorphic involutions

Relationship between fixed points and orbifold fundamental group representations

Insights into the connection with branes in the moduli space

Abstract

We study anti-holomorphic involutions of the moduli space of principal $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both $X$ and $G$. We analyze the fixed point locus in the moduli space and their relation with representations of the orbifold fundamental group of $X$ equipped with the anti-holomorphic involution. We also study the relation with branes. This generalizes work by Biswas--Garc\'{\i}a-Prada--Hurtubise and Baraglia--Schaposnik.