2021/09/14 by Laurent Baulieu, Baulieu, Laurent
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematics and Applications #Noncommutative and Quantum Gravity Theories #Relativity and Gravitational Theory
paper · pdf · doi:10.48550/arxiv.2109.06681
openalex publication_date 2021/09/14 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
This work shows the existence of a d>2 dimensional covariant "Beltrami vielbein" that generalizes the d=2 situation. Its definition relies on a sub-foliation ΣADMd-1=Σd-3×Σ2 of the Arnowit--Deser--Misner leaves of d-dimensional Lorentzian manifolds \cal Md. Σ2 stands for the sub-foliating randomly varying Riemann surfaces in \cal Md. The "Beltrami d -bein" associated to any given generic vielbein of \cal Md is systematically determined by a covariant gauge fixing of the Lorentz~symmetry of the latter. It is parametrized by \fracd(d+1)2 independent fields belonging to different categories. Each one has a specific interpretation. The Weyl invariant field sector of the Beltrami d-bein selects the (d(d-3))/(2) physical local degrees of freedom of d>2 dimensional gravity. The components of the Beltrami d-bein are in a one to one correspondence with those of the associated Beltrami d-dimensional metric. The Beltrami parametrization of the Spin connection and of the Einstein action delivers interesting expressions. Its use might easier the search of new Ricci flat solutions classified by the genus of the sub-manifold Σ2. A gravitational "physical gauge" choice is introduced that takes advantage of the geometrical specificities of the Beltrami parametrization. Further restrictions simplify the expression of the Beltrami vielbein when \cal Md has a given spatial holonomy. This point is exemplified in the case of d=8 Lorentzian spaces with G2⊂ SO(1,7) holonomy. The Lorentzian results presented in this paper can be extended to the Euclidean case.