We have recently proposed in [1] quantization of pure 4D Einstein gravity through hypersurface foliation, and observed that the 4D Einstein gravity becomes renormalizable once all (or most) of the unphysical degrees of freedom are removed. In this work, we confirm this observation from a more mathematical angle. In particular, we show that the physical state condition arising from the shift vector constraint connects with the requirement that the manifold admit “totally geodesic (TG) foliation” in foliation theory. The TG foliation, in turn, makes it possible to view the 4D manifold as abelian fibration over a 3D base. Associating the abelian fibration with the 4D diffeomorphism leads to reduction of the 4D manifold to 3D, thereby, realizing and generalizing the holography of ’t Hooft. ar X iv :1 40 6. 07 53 v1 [ gr -q c] 3 1 M ay 2 01 4

中文翻译：

---

基于叶理的量化的数学基础

我们最近在 [1] 中提出了通过超表面叶理对纯 4D 爱因斯坦引力进行量化，并观察到一旦所有（或大部分）非物理自由度被移除，4D 爱因斯坦引力就会变得可重整化。在这项工作中，我们从更数学的角度确认了这一观察结果。特别是，我们证明了由位移矢量约束引起的物理状态条件与流形理论中流形允许“完全测地线（TG）叶理”的要求有关。反过来，TG 叶理可以将 4D 流形视为 3D 基底上的阿贝尔纤维化。将阿贝尔纤维化与 4D 微分同胚相关联导致 4D 流形减少到 3D，从而实现和推广 't Hooft 的全息。ar X iv :1 40 6. 07 53 v1 [ gr -qc] 3 1 May 2 01 4