Classification of the automorphism and isometry groups of Higgs bundle moduli spaces

TL;DR

This paper comprehensively classifies various automorphism and isometry groups of Higgs bundle moduli spaces on Riemann surfaces, revealing deep geometric symmetries and their dependence on the underlying surface's properties.

Contribution

It provides a complete determination of automorphism groups of Higgs bundle moduli spaces across multiple geometric structures, including complex, symplectic, Kähler, quaternionic, and hyper-Kähler.

Findings

Automorphism groups as complex analytic varieties identified

Holomorphic symplectomorphism groups characterized

Isometry group determined based on surface properties

Abstract

Let $\mathcal{M}_{n,d}$ be the moduli space of semi-stable rank $n$, trace-free Higgs bundles with fixed determinant of degree $d$ on a Riemann surface of genus at least $3$. We determine the following automorphism groups of $\mathcal{M}_{n,d}$: (i) the group of automorphisms as a complex analytic variety, (ii) the group of holomorphic symplectomorphisms, (iii) the group of K\"ahler isomorphisms, (iv) the group of automorphisms of the quaternionic structure, (v) the group of hyper-K\"ahler isomorphisms. When $n$ and $d$ are coprime we show that $\mathcal{M}_{n,d}$ admits an anti-holomorphic isomorphism if and only if the corresponding Riemann surface admits such a map. We then use this to determine the isometry group of $\mathcal{M}_{n,d}$.