Higgs Bundles and Flat Connections Over Compact Sasakian Manifolds

Abstract

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