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Mar11-12, 01:50 PM
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PF Gold
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Matteo Smerlak's PhD thesis is a useful source of background for the 6-page Bonzom-Smerlak letter.
I should give the latter's abstract--didn't do that yet.
Gauge symmetries in spinfoam gravity: the case for "cellular quantization"
Valentin Bonzom, Matteo Smerlak
(Submitted on 24 Jan 2012)
The spinfoam approach to quantum gravity rests on a "quantization" of BF theory using 2-complexes and group representations. We explain why, in dimension three and higher, this "spinfoam quantization" must be amended to be made consistent with the gauge symmetries of discrete BF theory. We discuss a suitable generalization, called "cellular quantization", which
(1) is finite,
(2) produces a topological invariant,
(3) matches with the properties of the continuum BF theory,
(4) corresponds to its loop quantization. These results significantly clarify the foundations - and limitations - of the spinfoam formalism, and open the path to understanding, in a discrete setting, the symmetry-breaking which reduces BF theory to gravity.
6 pages

A concise excerpt from page 1:
==quote 1201.4996==
The purpose of this letter is to argue that there is a good reason for this: when dealing with 2-complexes only, as in the spinfoam formalism, there is no shift symmetry. To identify this symmetry, one must instead resort to an extension of the spinfoam formalism including higher-dimensional cells. This realization paves the way to what we call cellular quantization. This cellular quantization solves problems 1 to 4, and sheds interesting new light on problem 5.
The letter is organized as follows. We start by reviewing the basic properties of the continuum BF theory, emphasizing its gauge symmetries and relationship to analytic torsion. We then describe the “spinfoam quantization” of BF theory, as described e.g. in Baez’s reference paper [5]. We show how to identify the gauge symmetries in a discrete setting and perform a quantization which does preserve the topological features of the continuum theory. Finally we establish that this cellular quantization corresponds to the loop canonical quantization.

Problems 1 through 5, mentioned in the above excerpt, are as follows:
1. Bubble divergences. The original PRO [Ponzano-Regge-Oguri] partition functions are in general divergent. How should one regularize them?

2. Topological invariance. The PRO partition functions are formally invariant under changes of triangulations, up to divergent factors. How can one turn them into finite topological invariants?

3. Relationship to the canonical theory. The connection between the Ponzano-Regge model and loop quantum gravity in 3 dimensions was established in [13]. Can this connection be extended to 4 dimensions and higher?

4. Relationship to the continuum theory. BF theory was quantized in the continuum in [21, 22], and was showed to be related to the Ray-Singer torsion. Are the PRO models similarly related to torsion? (See [14] for a positive answer in certain three-dimensional cases.)

5. Diffeomorphism symmetry. Both the continuum BF action and the Einstein-Hilbert action are diffeomorphism-invariant. What is the fate of this symmetry in the PRO models?

For completeness, here is the abstract of Smerlak's thesis. It doesn't overlap in results, but shares some concepts---therefore is helpful in part simply because it is longer (over 100 pages instead of only 6) and more deliberate. Goes thru some definitions in a less condensed way.
Divergences in spinfoam quantum gravity
Matteo Smerlak
(Submitted on 23 Jan 2012)
In this thesis we study the flat model, the main buidling block for the spinfoam approach to quantum gravity, with an emphasis on its divergences. Besides a personal introduction to the problem of quantum gravity, the manuscript consists in two part. In the first one, we establish an exact powercounting formula for the bubble divergences of the flat model, using tools from discrete gauge theory and twisted cohomology. In the second one, we address the issue of spinfoam continuum limit, both from the lattice field theory and the group field theory perspectives. In particular, we put forward a new proof of the Borel summability of the Boulatov-Freidel-Louapre model, with an improved control over the large-spin scaling behaviour. We conclude with an outlook of the renormalization program in spinfoam quantum gravity.
113 pages. PhD thesis, introduction and conclusion in French, main text in English.

Paper by Ileana Naish-Guzman and John Barrett cited on page 4 ref [14] in connection with the discrete exterior derivative on a cell complex.
Similarly cited was [26] an earlier paper by Bonzom and Smerlak
Additional webstuff about de Rham complex