Higgs bundles on Sasakian manifolds

We extend the Donaldson-Corlette-Hitchin-Simpson correspondence between Higgs bundles and flat connections on compact K"ahler manifolds to compact quasi-regular Sasakian manifolds. A particular consequence is the translation of restrictions on K"ahler groups proved using the Donaldson-Corlette-Hitchin-Simpson correspondence to fundamental groups of compact Sasakian manifolds.