There is a paper by Martin Reuter (one of the main AsymSafe QG authors) which explores possible connections with LQG. I have only a slight familiarity with it, not enough to recommend, or give any useful advice. But I'll give the link, a bit of the abstract and some excerpts
http://arxiv.org/pdf/1301.5135.pdf
Einstein–Cartan gravity, Asymptotic Safety, and the running Immirzi parameter
J.-E. Daum and M. Reuter
Institute of Physics, University of Mainz Staudingerweg 7, D-55099 Mainz, Germany
Abstract
In this paper we analyze the functional renormalization group flow of quantum gravity on the Einstein-Cartan theory space. ...
...
... Nevertheless, we do find evidence for the existence of at least one non-Gaussian renormalization group fixed point which seems suitable for the Asymptotic Safety construction in a setting where the spin connection and the vielbein are the fundamental field variables.
==quote page 6 and following==
In the literature many generalizations of classical Einstein-Cartan theory with ac- tions more complicated than SHP[e,ω] have been considered [43,46]. In particular in the context of Loop Quantum Gravity (LQG) the so-called Holst action SHo[e,ω] plays an important role [47,48]. It contains an additional term that exists only in 4 dimensions; its prefactor is the dimensionless Immirzi parameter γ. This term is typical of Einstein- Cartan theory; it vanishes for vanishing torsion and, as a result, does not exist in metric gravity. Remarkably, the vacuum field equations implied by SHo[e, ω] do not depend on γ, even though the part of the action it multiplies is not a surface term. Indeed, in presence of fermions coupled to gravity in a non-minimal way, the Immirzi term induces a CP violating four-fermion interaction that might be interesting for phenomenological reasons, in the cosmology of the early universe, for instance [49–51].
The Holst action is of central importance for several modern approaches to the quantization of gravity [52]. This includes canonical quantum gravity on the basis of Ashtekar’s variables [53], Loop Quantum Gravity [54], spin foam models [55], and group field theory [56]. In LQG, for instance, γ makes its appearance in the spectrum of area and volume operators. It was also believed to determine the entropy of black holes since the standard semiclassical result (S = A/4G) obtained for a single value of γ only. This picture was questioned recently, however [57]. At least the kinematical level of LQG suggests that γ constitutes a fixed parameter which labels physically distinct quantum theories. In this respect γ might be comparable to the Θ-parameter of QCD which, too, is absent from the classical equations of motion, but nevertheless leads to observable quan- tum effects. Contrary to the Immirzi parameter, Θ does however multiply a topological invariant which spoils the analogy to some extent.
There is an obvious tension between this picture of a universal, constant value of γ, fixing for instance the absolute size of quantized areas of volumes, on the one hand, and the framework of RG flow equations and Asymptotic Safety on the other. Setting up a FRGE for the theory space TEC, one of the infinitely many couplings parametrizing a generic action is the Immirzi parameter. A priori it must be treated as a “running”, i. e. scale-dependent quantity γ ≡ γ
k; there is no obvious general principle (nonrenormalization theorem) that would forbid such a scale dependence. For this reason the renormalization behavior of the Immirzi parameter will be one of the main themes in the following.
The purpose of the present paper is twofold: First, we are going to construct a general framework which allows the nonperturbative calculation of coarse graining flows in Einstein-Cartan gravity; ...
==endquote==
- [53] A. Ashtekar, Lectures on non-perturbative canonical gravity, World Scientific, Singapore (1991);
A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 21 (2004) R53.
- [54] Th. Thiemann, Modern Canonical Quantum General Relativity, Cambridge University Press, Cambridge (2007).
- [55] A. Perez, Class. Quant. Grav. 20 (2003) R43.
- [56] D. Oriti, in: Approaches to Quantum Gravity, D. Oriti (Ed.), CUP, 2009; L. Freidel, Int. J. Theor. Phys. 44 (2005) 1769.
- [57] E. Bianchi, arXiv:1204.5122.
