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.

给定一个紧凑凯勒歧管X，对完全还原的平坦矢量束之间类别的等价X和polystable希格斯束\（（E，\ THETA）\）上X与\（C_1（E）= 0 = C_2 （E）\）（辛普森（J. Math Math Soc）1（4）：867-918，1988;柯莱特（Corlette）于J Differ Geom 28：361-382，1988;乌伦贝克和丘（Uhle​​nbeck and Yau）于Commun Pure Appl Math 39：257-293，1986;唐纳森（Dukeson）在杜克大学Math J 54（1）：231–247，1987年）。我们将类别的等价关系扩展到紧致的Sasakian流形的上下文。我们证明在紧致的Sasakian流形上，半简单平面向量束的类别与上面具有平凡的第一和第二基本Chern类的多稳态基本Higgs束的类别之间是等效的。我们还证明，在紧凑的Sasakian流形上的任何稳定的基本Higgs束都可以接受满足Yang–Mills–Higgs方程的基本Hermitian度量。