TITLE: Non-Abelian Hodge theory
QUESTION [12 upvotes]: Let $X$ be a compact Riemann surface. I would like to find a somehow complete reference for the proof of the so called non-Abelian Hodge correspondence relating Dolbeaut, Betti and Higgs bundle moduli spaces.
I've tried to read the original articles by Hitchin (1987) or Simpson (1990) but it seems to me that I've not found somehow a complete reference showing a precise proof of all the statements involved (for the case of a curve).
Does anyone know a book or notes related to this?

REPLY [5 votes]: I recently attented a nice online talk by Pengfei Huang and he indicated two sources:

the first chapter of his own phd Non-abelian Hodge theory and some specializations - TEL - Thèses en ligne

Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles and complex variations of Hodge structure by Alberto Garcia-Raboso, Steven Rayan (see Introduction to Nonabelian Hodge Theory | SpringerLink for the published version)


I am not sure to which extent they can be called complete references, but as general references on the subject, I think they certainly deserve to be mentionned.