标题： 共形引力作为变形的拓扑场理论 显示英文标题

标题： Conformal Gravity as a Deformed Topological Field Theory

摘要： 在麦考德尔-曼苏里广义相对论形式中，自旋连接和共标架变量被结合到一个称为麦考德尔-曼苏里连接的单个李代数值连接中，$\omega$。 从连接$\omega$的曲率形式$F$和一个辅助场$B$，可以通过构造一个作用泛函来将广义相对论表述为一种变形的拓扑场理论，该作用泛函的变分产生一组与壳上爱因斯坦方程等价的场方程。 在本文中，我们表明当麦考德尔-曼苏里连接的基本长度尺度被视为一个动力学变量——一个宇宙学标量场——时，从所得作用泛函的变分得到的场方程在壳上等价于共形爱因斯坦方程。 通过卡坦几何的视角，我们随后讨论了广义相对论与其共形变换对应物之间的显著几何差异。 具体来说，对于后者，我们表明时空中的点在无限小范围内由齐性空间（限制在一点）近似，其半径由宇宙学标量场的值参数化。 显示英文摘要

摘要： In the MacDowell-Mansouri formulation of general relativity, the spin connection and coframe variables are incorporated into a single Lie algebra-valued connection called the MacDowell-Mansouri connection, $\omega$. From the curvature form $F$ of $\omega$ and an auxiliary field, $B$, one may formulate general relativity as a deformed topological field theory by constructing an action functional whose variation yields a set of field equations that are equivalent to the Einstein equations on shell. In this article, we show that when the fundamental length scale of the MacDowell-Mansouri connection is regarded as a dynamical variable -- a cosmological scalar field -- the field equations obtained from the variation of the resulting action are equivalent to the conformal Einstein equations on shell. Through the lens of Cartan geometry, we then discuss a notable geometrical difference between general relativity and its conformally transformed counterpart. Specifically, for the latter, we show that points in spacetime are infinitesimally approximated by homogeneous spaces (restricted to a point) whose radii are parameterised by the value of the cosmological scalar field.