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Given a compact K\"ahler manifold $X$, there is an equivalence of categoriesbetween the completely reducible flat vector bundles on $X$ and the polystableHiggs bundles $(E,\, \theta)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC},\cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories tothe context of compact Sasakian manifolds. We prove that on a compact Sasakianmanifold, there is an equivalence between the category of semi-simple flatbundles on it and the category of polystable basic Higgs bundles on it withtrivial first and second basic Chern classes. We also prove that any stablebasic Higgs bundle over a compact Sasakian manifold admits a basic Hermitianmetric that satisfies the Yang--Mills--Higgs equation.

Higgs bundles and flat connections over compact Sasakian manifolds