STEGR in internal-space formulation: Formalisms, primary constraints, and possible internal symmetries

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 is 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.