Automorphisms of order three of the moduli space of Spin-Higgs bundles

Abstract

In this work we consider a family of ${\rm Spin}$ complex groups constructed in \cite{anton-article} which have outer automorphisms of order three. We define an action of ${\rm Out}({\rm Spin}(n,\mathbb{C}))\times\mathbb{C}^*$ on the moduli space of ${\rm Spin}$-Higgs bundles and we study the subvariety of fixed points of the induced automorphisms of order three. These fixed points can be expressed in terms of some kind of Higgs pairs associated to certain subgroups of ${\rm Spin}(n,\mathbb{C})$ equipped with a representation of the subgroup. We further the study for the simple case, $G={\rm Spin}(8,\mathbb{C})$.