What he does in this talk makes a bridge between Asymptotic Safety and Loop/Foam QG. He opens with a discussion of GFT. Here is slide #155
==quote 155==
Group Field Theory
Group field theory (Boulatov, 1992) lies at the crossroads between loop quantum gravity and simplicial quantum gravity. It generalizes matrix field theory and is also an attempt to sum both over discretized metric and topology of space-time.
*The information about the metric of a manifold is encoded in the hologomies along closed curves. Therefore the fields live on a group G
D (typically G = SO(D-1, 1)). They are G-gauge invariant.
*The simplest group field theories are topological. The interaction glues D+1
(D-1)-dimensional simplexes into a D dimensional simplex. Hence the theory has phi
D+1-type interaction. Feynman amplitudes correspond to BF theory, and they coincide with the spin foams of loop quantum gravity.
==endquote==
==quote 164==
Group field theory is non-local and its renormalization is again expected to involve
*A new scale decomposition (with a half-direction, hence uv/ir mixing?),
*A new locality principle ("triangularity") that should hold only for special classes of graphs,
*A new power counting.
Some non-topological theories of this type could be renormalizable in dimension 4 and form a consistent field theory of quantum gravity.
==endquote==
He cites Freidel-Gurau-Oriti 0905.3772 and discusses their results in 3D GFT, which he refers to as a "warm-up" exercise.
Then he refers to some work on GFT scaling bounds. Magnen-Noui-Rivasseau-Smerlak 0906.5477. This leads to a scale parameter appropriate to simplicial or graph-based formalism such as GFT and Loop/Foam. A scale parameter along which the coupling constants can, in effect, run. This is important, to do renormalization flow one must have an appropriate concept of the scale. If it is not some naive distance measure from some pre-established metric, then maybe it can be a measure of the complexity of the detail that one can see with the microscope. A kind of "fine versus coarse" measure. (

! )
Then he moves on, and suddenly he is at slide #182
==quote 182==
EPRLS-FK Models
The BF theory in four dimensions is not general relativity. The difference is expressed through Plebanski constraints. The problem is in a sense to relax the exact gluing of D-1 simplexes in order to allow for the propagating degrees of freedom of 4D gravity to be reflected in an appropriate way in the group field theory propagator.
Works by Engle-Pereira-Rovelli, Freidel and Krasnov, Livine and Speziale, led in 2007 to a model which implements better the Plebanski constraints.
In group field theory language the vertex of this model is made of two 15j symbols like in BF theory. But the Plebanski constraints add two special intertwiners f in the middle of each propagator. There are also natural normalizing factors d
j+ and d
j- for each face.
At large spins the vertex plus its half propagators obeys the desired semi-classical asymptotic limit (Conrady-Freidel, Barrett et al., 2009).
==endquote==
==quote 191==
Asymptotic safety in GFT?
The elements responsible for asymptotic safety of the Grosse-Wulkenhaar model are also present in the EPRLS-FK model:
*The model seems to have the right power counting (Pereira-Rovelli-Speziale 2009)
*It is non-local in group space.
*There is an auxiliary parameter, the Immirzi parameter gamma with interesting enhanced symmetry at gamma = 1.
Therefore we might hope for asymptotic safety in some model of this type.
It is unclear how such a possible asymptotic safety would be related to the fixed points seen in the Functional Renormalization Group (FRG) studies.
==endquote==