The usual Chern–Simons extension of Einstein gravity theory consists in adding a squared Riemann contribution to the Hilbert Lagrangian, which means that a square-curvature term is added to the linear-curvature leading term governing the dynamics of the gravitational field. However, in such a way the Lagrangian consists of two terms with a different number of curvatures, and therefore not homogeneous. To develop a homogeneous Chern–Simons correction to Einstein gravity we may, on the one hand, use the above-mentioned square-curvature contribution as the correction for the most general square-curvature Lagrangian, or on the other hand, find some linear-curvature correction to the Hilbert Lagrangian. In the first case, we will present the most general square-curvature leading term, which is in fact the already-known re-normalizable Stelle Lagrangian. In the second case, the topological current has to be an axial-vector built only in terms of gravitational degrees of freedom and with a unitary mass dimension, and we will display such an object. The comparison of the two theories will eventually be commented.

中文翻译：

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最全的质量四维拓扑引力

爱因斯坦引力理论通常的陈-西蒙斯扩展包括向希尔伯特拉格朗日量添加平方黎曼贡献，这意味着将平方曲率项添加到控制引力场动力学的线性曲率领先项。然而，以这种方式，拉格朗日量由具有不同曲率数量的两个项组成，因此不是齐次的。为了开发对爱因斯坦引力的齐次陈-西蒙斯校正，我们可以一方面使用上述平方曲率贡献作为最一般平方曲率拉格朗日的校正，或者另一方面，找到一些线性-对希尔伯特拉格朗日函数的曲率校正。在第一种情况下，我们将介绍最一般的平方曲率领先项，它实际上是已知的可重新归一化的 Stelle Lagrangian。在第二种情况下，拓扑流必须是仅根据重力自由度和单位质量维度构建的轴矢量，我们将展示这样一个对象。两种理论的比较最终将被评论。