Poincare gauge gravity from nonmetric gravity

TL;DR

The paper analyzes gravity theories with independent solder form and connection, allowing torsion and nonmetricity, and shows that via Cartan structure equations and field redefinitions the general linear gauge theory effectively reduces to Poincaré gauge theory with a Lorentzian, metric-compatible sector. By introducing a minimal nonmetricity structure equation and enforcing integrability, the authors construct a maximal extension whose Maurer–Cartan equations are those of the conformal group $\mathfrak{so}(p+1,q+1)$, revealing a deep link between nonmetricity, torsion, and conformal geometry. The key result is that the mixed-symmetry part of nonmetricity can be absorbed into an altered torsion, and that the sum of original torsion and traceless nonmetricity corresponds to the special conformal curvature within a conformal gauge gravity framework. This work suggests that gravity with a general linear connection can be recast as a conformal gauge theory, unifying nonmetricity and torsion under conformal symmetry and offering new avenues for exploring conformal gravity formulations and higher-spin couplings.

Abstract

We consider general linear gauge theory, with independent solder form and connection. These spaces have both torsion and nonmetricity. We show that the Cartan structure equations together with the defining equation for nonmetricity allow the mixed symmetry components of nonmetricity to be absorbed into an altered torsion tensor. Field redefinitions reduce the structure equations to those of Poincare gauge theory, with local Lorentz symmetry and metric compatibility. In order to allow recovery the original torsion and nonmetric fields, we replace the definition of nonmetricity by an additional structure equation and demand integrability of the extended system. We show that the maximal Lie algebra compatible with the enlarged set is isomorphic to the conformal Lie algebra. From this Lorentzian conformal geometry, we establish that the difference between the field strength of special conformal transformations and the torsion and is given by the mixed symmetry nonmetricity of an equivalent asymmetric system.