What exactly is a Cartan radius vector (and its role in Poincaré gauge theories) I am studying approaches to gravity where the Poincaré group is "gauged". The original motivation of this is to understand what is meant on the statement that "Teleparallel gravity is a gauge theory of the translation group". The standard references are highly confusing and imprecise.
The situation with Poincaré-based theories is also messy with lots of papers using very "physicist-y" math where the geometric meaning or even the validity of construction is questionable. I have also found some papers where more rigorous mathematics is employed, however in this case, I have difficulty translating between the two languages.
I suspect I can clarify a great deal of my (mis)understanding if I understand properly what a "Cartan radius vector" is.



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*I am doing a bit of a "translation work" here, so it is also possible I completely misunderstand my references, but it seems to me a "naive" approach to gauging the Poincaré group is to work (at least initially) in flat Minkowski spacetime, (with general curvilinear coordinates $x^\mu$ is necessary) and we are given four functions $y^a$ on the space, which are interpreted as flat/inertial/Cartesian coordinates. In this case a holonomic, orthonormal vielbein is given by $$ \theta^a=\mathrm dy^a. $$ Under a Poincaré transformation with constant coefficients $y^{\prime a}=\Lambda^a_{\ b}y^b+\tau^a$, the vielbein transforms as $$ \theta^{\prime a}=\Lambda^a_{\ b}\theta^b, $$ however under a Poincaré transformation with point-dependent coefficients, this is not the case.
$$\\$$We can save the day however by considering the inertial coordinates $y^a$ as some kind of section of an affine bundle, and introduce the affine connection $$ \mathscr Dy^a=\mathrm dy^a+\Gamma^a_{\ b}y^b+B^a, $$ and define $\theta^a=\mathscr Dy^a$. $$ \\ $$ From this point on, however it gets fuzzy, because teleparallel gravitists (see for ex. Aldrovandi, Pereira) tend to use this expression to define the vielbein. $$ \\ $$ But in for example Metric-affine gauge theory of gravity by Hehl et al. it is stated that $y^a$ is the "Cartan radius vector" if $B^a=0$, and also that in order to have the $(\theta^a,\Gamma^a_{\ b})$ double as a Cartan-connection, we must have (here apparantly only the linear part of the conenction is used) $$ Dy^a=\mathrm dy^a+\Gamma^a_{\ b}y^b=0. $$

*A bit later in the same Hehl paper, it is stated that the Cartan radius vector is defined by the equation (they used the notation $\xi$ for what I called $y$ before as well) $$ D\xi^a=\theta^a. $$ Here apprantly it is a linear object, not an affine one, and the covariant derivative $D$ is linear, and is claimed that the above equation is not totally integrable in general, but if integrated along an infinitesimal loop, it gives essentially affine holonomy of the form $$ \Delta\xi^a=\frac{1}{2}\left(R^a_{\ b\mu\nu}\xi^b+T^a_{\ \mu\nu}\right)\mathrm dx^\mu\wedge\mathrm dx^\nu. $$

*Based on what I have read about Cartan connections, one can describe a Cartan connection modelled on $G/H$ by having a $G$-fiber bundle $(E,\pi,M,G/H,G)$ with typical fiber $G/H$, an Ehresmann $G$-connection on $E$ specified by a vertical projector $\mathrm v:TE\rightarrow VE$, and a section $s:M\rightarrow E$ such that the pullback $$ s^\ast\mathrm v|_x:T_xM\rightarrow V_{s(x)}E\simeq\mathfrak g/\mathfrak h $$ is an isomorphism. $$\\$$ Here the section $s$ has interpretation of specifying the point of contact between the model geometry $E_x$ and the manifold $M$, and the last condition states that at the point of contact the tangent space of the model geometry must be isomorphic to the tangent space of the base geometry. $$ \\ $$ In fibred coordinates $(x^\mu,y^a)$ for $E$, we can write the connection as $$ \mathrm v=\partial_a\otimes\left( \mathrm dy^a+\Gamma^a(x,y) \right). $$ In case we have $G=\text{ISO}(3,1)$ and $H=\text{SO}(3,1)$, the model space is $G/H\simeq\mathbb R^4$ the affine Minkowski space, and the connection is $$ \mathrm v=\partial_a\otimes(\mathrm dy^a+\Gamma^a_{\ b}(x)y^b+B^a(x)), $$ since $G$ is an affine group, and the pullback condition is that $$ s^\ast\mathrm v=\partial_a\otimes(\mathrm ds^a(x)+\Gamma^a_{\ b}(x)s^b(x)+B^a(x)) $$ is nondegenerate. But this is basically the affine covariant derivative of $s$.



So my question is, how are the objects $y^a$, $\xi^a$, $s$ defined in my bullet points related? What is it we actually mean under a Cartan radius vector? What is its interpretation?
It is clear to me that my $y^a$ in the first bullet point is basically $s$ (in the last bullet point), however confusingly, Hehl says that our affine connection is a Cartan connection if $dy^a+\Gamma^a_{\ b}y^b=0$, which seems to me that i) is impossible to be integrated in general, ii) is in conflict with the more abstract definition in the third bullet point, where for the connection to be Cartan it is enough that $dy^a+\Gamma^a_{\ b}y^b+B^a$ is nondegenerate (which is consistent with the interpretation of $\mathscr D y^a$ as a vielbein).
But I also know that a Cartan connection is, from another point of view, basically a coframe and a linear connection together, and $B^a$ is not in general a coframe in terms of transformation properties, as it has been elucidated by Hehl.
I basically would like to clarify this mess into something coherent. References for papers treating Poincaré gauge gravity with mathematic rigour, consistency and geometric clarity is something I also would like.
 A: While I didn't follow every claim you made, such as the contrast of affine vs linear, briefly establishing the gauge-theoretic structure of metric-affine gravity will help. The spin connection 1-form, the vierbein 1-form, and the metric 0-form in the frame field formulation combined with differential forms for metric-affine formulations of gravity lead to curvature, torsion, and nonmetricity gravitational field strengths. The affine linear group AL(4,R) has 20 generators associated with 6 Lorentz boosts, 4 translations, and 10 shear/dilation transformations of GL(4,R)/Spin(3,1). Torsion relates to translational holonomy and is not in the general linear group, but found in the affine linear group. Contrasting general linear transformations vs affine linear transformations may help with clarity of communication.
Kroner demonstrated that nonmetricity relates to zero-dimensional defects. Kleinert works through a metric-affine formulation of path integrals for spaces with curvature and torsion. Even though Kleinert doesn't discuss nonmetricity directly, he differentiates the vierbein from the frame field based on if the pullback is applied at a single point or in a local neighborhood. It is as if he has a frame field formulation on top of a frame field formulation, roughly. Maybe Kleinert is secretly encoding nonmetricity in this way, as he could have studied an affine formulation to discuss curvature and torsion alone. Kleinert also explores multivalued fields, which is another complication. This vaguely reminded me of how you had xi and x with different derivatives leading to the vierbein as a local displacement field. The difference of those two vector seems to relate to different assumptions about the underlying gauge theoretic structure.
While I haven't read how Hehl refers to a Cartan radius, I've seen this more clearly used in the work of Nikodem Poplawski around 2009. He implemented a toroidal model of the electron with a Cartan radius, which is set by the mass of the particle and contains the Planck mass. This stems from work of Trautman. Here, I would encourage you to consider that classical gravity admits renormalization. Scherk and Schwarz proved that string theory and supergravity require torsion. The conventional approach in supergravity is to constrain torsion to vanish in vacuum, implying it is a type of "gravitational magnetization". However, Ortin's book on Gravity and Strings outlines how supergravity truly contains Poincare gauge gravity and Einstein-Cartan theory. In Einstein-Cartan theory, which is ideal for coupling of Dirac fermions to gravity, the torsion is non-propagating, yet the spin density sources torsion inside matter. The action is simply the Einstein-Hilbert. We can now ignore Poincare gauge gravity or metric-affine gravity for a moment to obtain intuition for a Cartan radius in Einstein-Cartan theory.
General relativity is non-renormalizable, which was recently formulated by Cheung and Remmen to lead to a nonlinear sigma model type of theory. Recall that the nonlinear sigma model of pions and nonrenormalizable with four-point interactions, which eventually lead to the electroweak theory with W bosons and trivalent interactions that were renormalizable. A similar thing occurs with torsion, such that spin-spin interactions of fermions in curved spacetime are mediated by torsion. If the torsion cannot propagate through the vacuum, then the only time it affects things is at high energy. The Cartan radius is therefore the characteristic radius that leads to an effective size of the particle due to the spin of the point particle, which creates a type of local displacement field. The vierbein is a local displacement field, which is the gauge gravitational potential of the torsion tensor.
The gauge group of Einstein-Cartan theory is commonly studied as SO(3,1), but higher gauge groups and quotient spaces can be explored from there. The important thing to remember is that the rotations relate to the Lorentz group, which is the propagating gravitational field. Metric teleparallelism (the one that is metric-compatible) has vanishing nonmetricity and treats ISO(3,1)/SO(3,1) as the gauge group, which leads to the vierbein as the dynamical gauge gravitational potential. In 1976, an equivalent formulation of general relativity was claimed to be found with a Lagrangian quadratic in the torsion instead of utilizing curvature. The geometrical trinity of gravity work of recent summarizes this and compares it with general relativity and symmetric teleparallelism, although the reasons for why the theories are all related is a bit obscure. Working through the metric-affine formulation provides additional clarity on the derivation of the Lagrangians. The metric teleparallel theory is close to some of the work of Pereira.
Poincare gauge gravity has various formulations, but it typically unifies the gauge theoretic structure of Einstein-Cartan theory and metric teleparallelism to find the Poincare group as the gauge group, which lead to dynamically propagating waves of curvature and torsion that are independent. In principle, one could prove that the gauge freedom is large enough to rotate dynamics between curvature and torsion in light of the trinity of gravity, but reviewing Kleinert's path integral formulation of metric-affine spaces seems important and overlooked by most of the community. I suspect this relates to why the gauge theoretic structure of metric-affine gravity has been difficult to get right.
To help clarify Hehl's point in relation to integrability conditions, spacetime torsion can lead to noncommutative momentum space, which leads to a discrete momentum spectrum. Poplawski more recently found a UV-complete formulation of QED by introducing torsional regularization based on insights stemming from the Cartan radius idea. While many mathematicians and physicists are latching onto continuously differentiable spacetime, it seems that introducing torsion discretizes momentum, but only at the position of the particles, as the momentum discretization depends on the mass of the particle. Therefore, if vacuum solutions of gravity are considered, the matter can be neglected and I suspect that things are integrable.
Kleinert has also discussed how noncompact gauge groups unify continuous and discrete spectra, such as SO(4,2) for Hydrogen. Note that noncompact gauge groups are ubiquitous in studies of gravity and that the conformal group contains the Poincare group, although the Hydrogen's SO(4,2) (I believe) is a physically different mathematical realization of SO(4,2) than the conformal group itself.
In conclusion, I suggest that the interpretation is that torsion as the field strength of the frame field may or may not propagate in the continuous vacuum. Regardless of this detail, a spinning point particle with mass effectively obtains a finite size due to the vierbein or local displacements, which also discretizes the allowable momentum eigenvalues for matter. For researchers more interested in propagating torsion in the vacuum, the subtlety of integrals vs sums can be avoided. The Poincare gauge gravity and metric-affine gravity formulations with affine linear gauge group allow for the study of propagating torsion as a type of gravitational wave inside matter and the vacuum. The metric-affine formulation with affine linear gauge group is linear because it is a theory of elasticity, which assumes small displacements. Higher gauge gravity isn't completely formulated yet, but that should lead to Einstein's principle of general covariance to find a relativistic formulation of hydrodynamical gauge gravity.
