Hodge bundles with Galois symmetry

In his 1994 paper on Moduli of Representations, Carlos Simpson used the scaling action on Higgs fields to show that moduli space of semistable Higgs bundles of fixed rank and degree are connected. There are several contexts in which one might want to study a similar question for Higgs bundles with some kind of additional symmetry. In joint work with Tommaso Scognamiglio (Heidelberg University), we look at the case of Higgs bundles over a real algebraic curve and explain how to extend Simpson’s method in this context, at least when the rank and degree are coprime. We will see in particular, the role played by the modular interpretation of Galois fixed points in order to be able to count the number of connected components of this space as a subset of the moduli space of Higgs bundles.