Higgs Bundles and Flat Connections Over Compact Sasakian Manifolds

Given a compact Kähler manifold X, there is an equivalence of categories between the completely reducible flat vector bundles on X and polystable Higgs $$(E, \theta )$$ with $$c_1(E)= 0= c_2(E)$$ (Simpson in J Am Math Soc 1(4):867–918, 1988; Corlette Differ Geom 28:361–382, Uhlenbeck Yau Commun Pure Appl 39:257–293, 1986; Donaldson Duke 54(1):231–247, 1987). We extend this to context Sasakian manifolds. prove that manifold, category semi-simple it basic trivial first second Chern classes. also any stable bundle over admits Hermitian metric satisfies Yang–Mills–Higgs equation.