Real spectrum compactification of Hitchin components, Weyl chamber valued lengths, and dual spaces

Abstract: The main result of this article is that Hitchin representations over real closed field extensions $\mathbb{F}$ of $\mathbb{R}$ correspond precisely to those representations of the fundamental group of a closed surface into $\textrm{PSL}(n,\mathbb{F})$ that are conjugate to $\mathbb{F}$-positive representations, i.e. representations that admit an equivariant limit map from the set of fixed points in the boundary of the universal cover of the surface into the set of full flags in $\mathbb{F}^n$ satisfying specific positivity properties. As the theorem treats general real closed fields, and not only the reals, the tools of analysis are not available. Instead, our proof is based on the Tarski-Seidenberg transfer principle and a multiplicative version of the Bonahon-Dreyer coordinates. We use this result to prove that $\mathbb{F}$-positive representations form semi-algebraically connected components of the space of all representations, that consist entirely of injective and discrete representations, which are positively hyperbolic and weakly dynamics-preserving over $\mathbb{F}$. Furthermore, we show how to associate intersection geodesic currents to $\mathbb{F}$-positive representations, and conclude with applications to the Weyl chamber length compactification and to dual spaces of geodesic currents.