Involutions and higher order automorphisms of Higgs bundle moduli spaces

TL;DR

This paper investigates automorphisms of Higgs bundle moduli spaces, especially involutions, and describes their fixed point subvarieties, linking them to representation spaces in complex and real forms of Lie groups.

Contribution

It characterizes fixed point subvarieties of automorphisms of Higgs bundle moduli spaces, including involutions, and relates them to subgroups and real forms of complex Lie groups.

Findings

Fixed point subvarieties are hyperk"ahler or Lagrangian depending on involution type.

Involutions correspond to moduli spaces of representations in certain subgroups or real forms.

Explicit descriptions for SL(n,C) and Spin(8,C) cases.

Abstract

We consider the moduli space $\mathcal{M}(G)$ of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. This is a hyperk\"ahler manifold homeomorphic to the moduli space $\mathcal{R}(G)$ of representations of the fundamental group of $X$ in $G$. In this paper we study finite order automorphisms of $\mathcal{M}(G)$ obtained by combining the action of an element of order $n$ in $H^1(X,Z)\rtimes \mbox{Out}(G)$, where $Z$ is the centre of $G$ and $\mbox{Out}(G)$ is the group of outer automorphisms of $G$, with the multiplication of the Higgs field by an $n$th-root of unity, and describe the subvarieties of fixed points. We give special attention to the case of involutions, defined by the action of an element of order $2$ in $H^1(X,Z)\rtimes\mbox{Out}(G)$ combined with the multiplication of the Higgs field by $\pm 1$. In this situation, the subvarieties of fixed points are hyperk\"ahler submanifolds of $\mathcal{M}(G)$ in the (+1)-case, corresponding to the moduli space of representations of the fundamental group in certain reductive complex subgroups of $G$ defined by holomorphic involutions of $G$; while in the (-1)-case they are Lagrangian subvarieties corresponding to the moduli space of representations of the fundamental group of $X$ in real forms of $G$ and certain extensions of these. We illustrate the general theory with the description of involutions for $G=\mbox{SL}(n,\mathbb{C})$ and involutions and order three automorphism defined by triality for $G=\mbox{Spin}(8,\mathbb{C})$.