Holonomy spinfoams: convergence of the partition function

[marcus]

This looks like a significant step forward. The paper is clearly written and gives a brief historical account of progress in spin foams over the past half-dozen years or so: an understandable review that places its results in context.
The authors, Hellmann and Kaminski, discover a problem with the approach they are using and propose a work-around.
I think a briefer version giving results without detailed proofs is either available or will soon be available. Similar results were also achieved by Muxin Han (Marseille CPT) and are cited here.

http://arxiv.org/abs/1307.1679
Holonomy spin foam models: Asymptotic geometry of the partition function
Frank Hellmann, Wojciech Kaminski
(Submitted on 5 Jul 2013)
We study the asymptotic geometry of the spin foam partition function for a large class of models, including the models of Barrett and Crane, Engle, Pereira, Rovelli and Livine, and, Freidel and Krasnov.
The asymptotics is taken with respect to the boundary spins only, no assumption of large spins is made in the interior. We give a sufficient criterion for the existence of the partition function. We find that geometric boundary data is suppressed unless its interior continuation satisfies certain accidental curvature constraints. This means in particular that most Regge manifolds are suppressed in the asymptotic regime. We discuss this explicitly for the case of the configurations arising in the 3-3 Pachner move. We identify the origin of these accidental curvature constraints as an incorrect twisting of the face amplitude upon introduction of the Immirzi parameter and propose a way to resolve this problem, albeit at the price of losing the connection to the SU(2) boundary Hilbert space.
The key methodological innovation that enables these results is the introduction of the notion of wave front sets, and the adaptation of tools for their study from micro local analysis to the case of spin foam partition functions.
63 pages, 5 figures

 

[Evan Fox]

This is a very interesting paper that has provided a great deal of insight into the asymptotic geometry of spin foam models. It is great to see that Hellmann and Kaminski have discovered an issue with the approach they are using and proposed a work-around. It is also good to know that similar results were achieved by Muxin Han (Marseille CPT) and are cited here. I am looking forward to the briefer version of this paper that should be available soon, so that I can get an understanding of the results without going into the detailed proofs.