论文标题
四维时空中的一般相对论和杨米尔斯理论的拓扑指数
Topological indices of general relativity and Yang-Mills theory in four-dimensional space-time
论文作者
论文摘要
该报告使用共同的数学框架（Chern-Weil理论的主捆绑包）研究了四维时空的总体相对论和阳米尔斯理论。通过将几个主束用量规对称性扭曲了几个主束，因此描述了整个理论，这是由于带有GL（4）对称性的纤维束的描述。 除了主要连接外，我们还将Hodge Dual连接引入Lagrangian，以使量规场具有独立于Bianchi身份的动力学。我们表明，当Z2级式操作员一般存在于捆绑包的总空间中时，双工上层结构将出现在捆绑包中。 Dirac操作员使用一维Clifford代数出现在次级超空间中，并提供了Atiyah-Singer索引定理的拓扑索引。 尽管通常在椭圆型歧管中讨论拓扑指数，但该报告使用新型方法Theta-metric空间在双曲线型时空歧管中对其进行处理。 Theta-Metric同时处理欧几里得和Minkowski的空间，并定义了Minkowski时空中的拓扑指数。
This report investigates general relativity and the Yang-Mills theory in four-dimensional space-time using a common mathematical framework, the Chern-Weil theory for principal bundles. The whole theory is described owing to the fibre bundle with the GL(4) symmetry by twisting several principal bundles with the gauge symmetry. In addition to the principal connection, we introduce the Hodge-dual connection into the Lagrangian to make gauge fields have dynamics independent from the Bianchi identity. We show that the duplex superstructure appears in the bundle when a Z2-grading operator exists in the total space of the bundle in general. The Dirac operator appears in the secondary superspace using the one-dimensional Clifford algebra, and it provides topological indices from the Atiyah-Singer index theorem. Though the topological index is usually discussed in the elliptic-type manifold, this report treats it in the hyperbolic-type space-time manifold using a novel method, the theta-metric space. The theta-metric treats the Euclidean and Minkowski spaces simultaneously and defines the topological index in the Minkowski space-time.
