This is an exciting development in LQG. They have a proposal for how to generalize the ideas of spin network and spin foam so that the network vertices are made of chunks of noncommutative space instead of ordinary space. I'd be glad if anybody who's looked at the paper and wants to volunteer to explain any bits and pieces, or ask questions, would do so. Basically it's just a matter of DIFFERENT LABELING of the vertices and edges of the network. A chunk of spectral (i.e. Alain Connes style) geometry is given by a rudimentary spectral triple which can be denoted by a pair (A,H) of an star-algebra A represented on a hilbertspace H. They can be finite dimensional and the fancier aspects of a spectral triple are assumed to vanish--so there is just this rudimentary label (A, H). That Alain Connes pair (A,H) is what labels a vertex in a Marcoli van Suijlekom "gauge network". In usual LQG you have a network that is labeled by other stuff. There is an interpretation which Eugenio Bianchi (among others) has worked out where the vertex labels can be thought of as describing QUANTUM (i.e. fuzzy) POLYHEDRA. These polyhedra can't decide how their actual faces are shaped so they are blurry chunks of ordinary space. So the difference now with the Marcolli-van Suijlekom version is the vertices of the network are labeled with blurry chunks of Alain Connes-type space. But very rudimentary because within each chunk the "Dirac operator" which serves as a substitute metric in spectral geometry is taken to be trivial. =====semantic note====== Don't be put off by the mathematically correct term for "network" that they use. They call the network a QUIVER. Among mathematicians one often distinguishes between a directed graph (at most one edge between any pair of vertices) and a directed multigraph which can have several "arrows" or directed edges going between any pair of vertices. And some mathematicians call that a quiver. But the LQG people were already using quivers as the basis for their spin networks---they just called the quivers by a different name. The LQG people have always been using LABELED QUIVERS to define the quantum states of geometry and to form an orthonormal basis for their Hilbert space of quantum states. Personally I find the word "quiver" distasteful and I wish that the responsible mathematical authorities would provide a different name for directed graphs which can have multiple edges. I'm inclined to think we ought to be able to simply call them GRAPHS, as long as no confusion can arise. But I see the point---if you define a graph restrictively it will correspond to a matrix of zeros and ones---or if directed, to a matrix where the entries are -1, 0, or +1. And matrixes are the apple pie and motherhood of mathematics, so the restrictive definition of graph is forced by a mathematical sense of righteousness. The less restrictive idea of a graph, or network, or "quiver" is two sets E and V with two maps called source and target, namely s:E→V and t:E→V The basic message here is don't be put off by the fact that these authors, in the matter of a few terminologies, do not sound like ordinary physics folks. What they are talking about is real physics---it's just a few words like "quiver" and "functor" that sound a bit on the fancymath side. ==end of semantic note== For me, square one of the paper comes near the top of page 10. The second paragraph there is where they define X the space of representations of a directed graph Γ in a label category C. This label category is all the possible rudimentary chunks of noncommutative space. Crazy Alain Connes polyhedra. A "representation" is in effect a labeling. And there is a group G defined there on page 10 too, in the second paragraph. I think of this group as a kind of gauge equivalence group that is going to be factored out. Now jump to the bottom of page 11 where they begin section 2.3 "Gauge Networks" with the words "The starting point for constructing a quantum theory is to construct a Hilbert space inspired by [a paper by Baez about spin networks]..." You can see them going for the L2 space of square integrable functions that EVERYBODY uses except that it is the L2 defined on this excellent space X and on X/G. This is cool and it was what was destined to happen I have some other things to do but will try to get back to this later today. If you look at the Marcolli van Suijlekom paper (which I think is very important) please comment. I think there is a typo on page 28, in the conclusions section---will indicate later. The link is January 3480------that is, http://arxiv.org/abs/1301.3480 .