Mathematical foundation of foliation-based quantization

Abstract

We have recently proposed in \cite{Park:2014tia} the 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". 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.