STEGR in Internal-Space Formulation: Formalisms, Primary Constraints, and Possible Internal Symmetries

TL;DR

The paper develops an internal-space gauge formulation of Symmetric Teleparallel Equivalent to GR (STEGR) by recasting STEGR within metric-affine gauge theory and introducing three formalisms distinguished by vanishing-torsion conditions. It derives the canonical structure for Formalism 1 and Formalism 2, including primary constraints and their PB-algebras, and identifies two potential internal symmetries: a translation symmetry for Formalism 1 and a local GL$(n+1;\mathbb{R})$-type symmetry for Formalism 2, with translation potentially broken or absent depending on the formalism. The analysis reveals that a priori translation symmetry can be present in Formalism 1 but is not compatible with the auxiliary torsion terms, while Formalism 2 allows GL$(n+1;\mathbb{R})$ symmetry but lacks a translation generator; Formalism 3, involving Stückelberg fields, is left for future work due to its bi-metric characteristics. The work sets the stage for a full Dirac-Bergmann treatment of secondary constraints and their algebras, aiming to clarify the internal symmetry structure and its relation to spacetime diffeomorphisms in STEGR.

Abstract

We establish the theories of Symmetric Teleparallel Equivalent to General Relativity (STEGR) in the internal-space and investigate possible internal-space symmetries among primary constraint densities in the theories. First of all, we revisit STEGR in terms of the gauge approach to gravity and formulate it in the internal-space set-up. We find three possible formalisms according to the vanishing-torsion property. Then, we investigate possible internal-space symmetries in each formalism. We find that in our formulation there are two possible symmetries. One satisfies the translation symmetry but broken in the local symmetry provided by the general linear group which contains the local Lorentz symmetry. The other satisfies the latter symmetry but is absent in the former symmetry. Finally, we conclude this work and show future perspectives.