论文标题
量规变换的异构体
Isometries from gauge transformations
论文作者
论文摘要
在四个维度中，可以将自旋连接的手性部分用作编码几何形状的主要对象。然后，从该连接的曲率中恢复该度量。我们解决了如何在这种“纯连接”形式主义中识别等异形的问题。我们表明，从量规变换参数中恢复了等法，这些参数满足了沿着矢量场产生等轴测的连接的谎言衍生物的要求是量规变换。该要求只能作为一阶微分方程重写，仅涉及量规变换参数。一旦找到满足该方程的量规变换，等轴测载体场就会恢复代数。我们列出了用于确定异构体的新形式主义的示例，也证明了一般陈述：紧凑型歧管上的负面联系没有对称性。这是众所周知的Riemannian几何陈述的精确“纯连接”类似物，即在紧凑型歧管上没有杀死的矢量场。
In four dimensions one can use the chiral part of the spin connection as the main object that encodes geometry. The metric is then recovered algebraically from the curvature of this connection. We address the question of how isometries can be identified in this "pure connection" formalism. We show that isometries are recovered from gauge transformation parameters satisfying the requirement that the Lie derivative of the connection along a vector field generating an isometry is a gauge transformation. This requirement can be rewritten as a first order differential equation involving the gauge transformation parameter only. Once a gauge transformation satisfying this equation is found, the isometry generating vector field is recovered algebraically. We work out examples of the new formalism being used to determine isometries, and also prove a general statement: a negative definite connection on a compact manifold does not have symmetries. This is the precise "pure connection" analog of the well-known Riemannian geometry statement that there are no Killing vector fields on compact manifolds with negative Ricci curvature.
