The entire geometric formulations of the BRST and the anti-BRST structures are worked out in presence of the Nakanishi–Lautrup field. It is shown that in the general form of gauge fixing mechanisms within the Faddeev–Popov quantization approach, the anti-BRST invariance reflects thoroughly the classical symmetry of the Yang–Mills theories with respect to gauge fixing methods. The Nakanishi–Lautrup field is also defined and worked out as a geometric object. This formulation helps us to introduce two absolutely new topological invariants of quantized Yang–Mills theories, so-called the Nakanishi–Lautrup invariants. The cohomological structure of the anti-BRST symmetry is also studied and the anti-BRST topological index is derived accordingly.

中文翻译：

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量化杨-米尔斯规范理论中反 BRST 对称性的几何和拓扑

BRST 和反 BRST 结构的整个几何公式是在 Nakanishi-Lautrup 场的存在下制定的。结果表明，在 Faddeev-Popov 量化方法中规范固定机制的一般形式中，反 BRST 不变性彻底反映了 Yang-Mills 理论关于规范固定方法的经典对称性。Nakanishi-Lautrup 场也被定义为一个几何对象。这个公式有助于我们引入两个全新的量子化杨-米尔斯理论的拓扑不变量，即所谓的Nakanishi-Lautrup 不变量. 还研究了反 BRST 对称性的上同调结构抗BRST拓扑指数是据此推导出来的。