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Is anyone following the efforts of Connes-style noncommutative geometers to understand spin foams, e.g., this preprint?
I would if I were not overwhelmed by the involved mathematics.
Thanks, David.
I haven’t been following this, but I looked at the article you point to just now.
After a first reading I see a collection of possibly useful observations but have a bit of trouble tying them into a coherent picture. I may have to read it again. or maybe somebody here can give me a hint.
Among the things puzzling me is this: I certainly understand that one can build a 2-category whose objects are branched covers of the 3-sphere equipped with a Wilson graph aka spin network structure on a graph representing it, etc, and of course one can then form the 2-category algebra and look for 1-paranmeter families of automorphisms. But how is this really related to any theory of gravity?
It is a neat observation that all 3-manifolds are given by branched covers of the 3-spehere encoded by a graph on the 3-sphere and some group representation, and it reminds one of spin notworks, yes. But the point of a spin network in theories of gravity is that it encodes a functional on the space of connections on the manifold that it is embedded in. How does the spin graph on the 3-sphere encode such a functional on a conection on the 3-manifold encoded by the corresponding branched cover of the 3-sphere, though? Or a single one might be lifted, but it seems we’d just get a very restrictive class of networks on the coverin 3-manifold.(?)
And why should we then next be interested in functions on the space of such “topspin networks”? As spin networks are themselves to be thought of as functionals on the configuration space of a gauge theory, we have now functions on these functionals. I suppose I can think of this somehow as passing to some space of dual states, maybe, but i am lacking a clear picture of what might be going on. And it seems the article doesn’t dwell much on completing this physical picture but is more into playing around with the mathematical structures.
A similar problem I have towards the end of the article: I am very fond of the Chamseddine-Connes-Marcolli et al model mentioned there, but I am left with less than a vague idea on what exactly that now has to do with the constructions in the article. Maybe I need to re-read it. It looks like it would help me if I read the work by Paschke et al that is being mentioned. Maybe I should do that and then report back.
That’s just my first impression on a first reading. Maybe somebody else can say more.
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