I am interested in references that treat teleparallel gravity in a mathematically rigorous manner, especially in regards to it being a “gauge theory of the translation group”.

The standard reference is the book by Aldrovandi and Pereira, however this book uses extremely sloppy mathematics at times, and is extremely confusing.

The paper “Einstein Lagrangian as the translational Yang-Mills Lagrangian ” by Cho edges closer to what I am looking for, but it still seems to suffer from issues (it considers a principal translation bundle, which seems to be problematic). Some of these issues are outlined here (without reference to the Cho paper).

Probably the closest I’ve seen to what I am looking for is the paper “Geometric meaning of the Poincaré group gauge theory” by Pilch, however this is rather short, not very detailed and somewhat unclear to me.

Basically, I am looking for a mathematically precise, principal bundle-based treatment of teleparallel gravity as a gauge theory. Based on the “naive gauging” done by phyicists I expect the connection would be an affine connection whose translational part is dynamical, and its linear $ O(3,1)$ part is flat (Weitzenböck connection) and is somehow determined by the translational part (we have $ W^\kappa_{\mu\nu}=B^\kappa_a\partial_\mu B^a_\nu$ in the local formulation, where $ W$ is the Weitzenböck connection in a holonomic frame, and $ B_\mu^a$ is the parallel covielbein).

The specifics of this however elude me, especially that while in ordinary GR a solder form $ \theta$ on the principal Lorentz bundle is essentially the same “globally” as an orthonormal covielbein, a solder form cannot simply correspond to the parallel Weitzenböck tetrad, as the latter is not invariant under arbitrary point-dependent Lorentz transformations.

The coupling to matter fields is also unclear to me, especially if we wish to keep the concept of “Lagrangian not invariant under gauge transformations is made invariant by the introduction of a connection”, which actually makes sense even for nontrivial principal bundles, as evidenced by the formulation present (for “ordinary”) gauge theories by David Bleecker in “Gauge theory and variational principles”.

I kinda wish the references (if they exist) should be fairly detailed and didactic, as I feel that teleparallel gravity is relatively well defined albeit somewhat artificial and ad-hoc when the gauge theory interpretation is not shoehorned into it, however in the usual physicists’ formulations it becomes rather confused if the gauge theory interpretation is forced. It is clarity I seek.

Although the question is primarily about teleparallel gravity, I am interested in any formulation of gravity equivalent to GR which can be described as a gauge theory on a principal fibre bundle, so references to mathematically precise formulations of “Poincaré gauge theory gravity” or “Lorentz group gauge theory gravity” are also very much welcome on my part, although I am much more ignorant about these approaches than I am about teleparallelism.