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This paper by http://arxiv.org/abs/1110.2115" [Broken]. But that other post is trying to do too many things at once.

What I want to do here is still rather ambitious, but perhaps it can be done. I want to compare and contrast the (2+1) dimensional field theories constructed by Cecotti et al, with the constructions of (2+1) dimensional gravity coming out of LQG. In this forum we have sought to compare "loops and strings" via (2+1) dimensional gravity https://www.physicsforums.com/showthread.php?t=493321".

What caught my attention is the reference (page 31 of Bonzom and Laddha) to "Pachner moves", local changes to the triangulation of a manifold which produce a new triangulation of the same manifold. Of course they commonly show up in quantum gravity. But they are also playing a role in http://arxiv.org/abs/1110.2115" [Broken]. Turn to pages 66 and 67, and there they are.

suprised said in my earlier thread that, just because Cecotti et al use braids, and Bilson-Thompson also uses braids, doesn't necessarily mean that the two uses have anything in common, and I agree absolutely. However, if you think from a braid-centric perspective - from the perspective of a "quantum braid theorist" - if there are two ways to make a quantum field theory out of braids, there probably is something to be said about the relationship between those two ways of proceeding. They may not lie on the ends of a continuum - they may be completely orthogonal - but there would still be something to say.

We don't have any working LQG models based on braids; the existing LQG constructions which employ Bilson-Thompson's braid set (that I'm aware of) fall short of being "models", in my opinion. However, Pachner movesareemployed in a number of LQG models. So I think they provide another tractable point of technical comparison between loops and strings. There probably is a unified meta-theory of "how to build QFTs from Pachner moves"; you can see part of it http://sbseminar.wordpress.com/2008/12/05/tqfts-via-planar-algebras/".

Now let me state the obvious difference between how the Pachner moves apply in LQG and in Cecotti et al. In LQG, a Pachner move is an elementary process in Feynman's sense, something that has an amplitude. A history of the quantum geometry might consist of many, many concatenated Pachner moves, and then this history would form one contribution to a path integral.

In Cecotti et al, the Pachner move is playing quite a different role, at least initially. We start with M5-branes (which are 5+1 dimensional) compactified on a triangulated 2-manifold. That leaves 3+1 large dimensions. Then we consider one of the 3 large spacelike dimensions, and suppose that the 2-manifold (the small compact space attached at each point in 3+1 space) changes along that direction. You could visualize this by thinking of our familiar 3-dimensional space, picking some cosmic direction in it (like north celestial pole to south celestial pole), and supposing that in the infinite cosmic north, the extra dimensions form a sphere, and in the infinite cosmic south, the extra dimensions form a torus, and they change somewhere in between. In Cecotti et al, the extra dimensions change in an analogous fashion, within a particular slice of 3-dimensional space of finite thickness. This actually creates a domain wall - 3-dimensional space is divided in two - and there are excitations trapped on that domain wall. The (2+1)-dimensional field theory describes the interactions of the excitations trapped on the wall, and its properties can be derived by considering how the triangulation of the compact dimensions changes from one side of the wall to the other side.

THIS IS WHERE THE PACHNER MOVES SHOW UP! They describe spacelike changes to the geometry of the compact extra dimensions.

Hopefully it is apparent how different this is to the use of Pachner moves in LQG. There, the moves are timelike, they build up histories of the quantum geometry, and you have superpositions of those histories. Here, the Pachner moves describe spacelike variations in the geometry of the extra dimensions, and they seem somewhat "classical" - they are a way to describe a background space, from which properties of the field theory on that space (e.g. the particle spectrum) can be derived in a combinatorial way.

It might seem that these are indeed two completely orthogonal ways to obtain a QFT from a set of Pachner moves. In one, a set of Pachner moves defines a fixed background geometry, and the multiplets of the field theory arise from different ways you can wrap branes around that geometry. In the other, a set of Pachner moves defines one contribution to a path integral. In one, a Pachner "move" has a set of brane configurations associated with it (branes hanging off or wrapped around that part of the geometry). In the other, a Pachner move has an amplitude associated with it.

However, what if we think about a theory which has all these features at once? That is, the Pachner moves describe spacelike or timelike variations in a geometry, but there are strings or branes associated with this geometry (they may simply offer a dual description of it), and these geometries enter into superpositions and therefore have amplitudes. I find it especially conceivable that there may be some way, building on Cecotti et al, to go back from 3d to 4d, but rather than considering a 4d space split in half by a fixed domain wall, instead we consider a 4d space in which the compact 2-manifold is varying quantum-mechanically. It's a bit strange, from the standard LQG perspective, to be using Pachner moves to describe six dimensions (4+2) rather than just four dimensions; but I am reminded of the twistor string, a topological string that lives in six (bosonic) dimensions, and which nonetheless gives you amplitudes for gauge theory in four dimensions. Maybe http://arxiv.org/abs/1108.0369" [Broken] have a role to play here.

I have again drifted off into complicated speculations at the end here - whose true assessment surely requires difficult and original technical work - rather than just sticking to the details of the paper initially under discussion. But I think there's a real opportunity for a conceptual breakthrough here, in understanding "loops vs strings". I would again suggest that the key is to start with the Pachner moves, and think in terms of introducing extra structure. One way of proceeding leads to a particular (2+1)+(2+1)-dimensional geometry, and a (2+1)-dimensional field theory arising from it; the other way of proceeding is the LQG way, and gives rise to a model of gravity in (2+1) or (3+1) dimensions; two paths through the forest of mathematical possibility. If we can get a high enough view, we might find that the divergence of those two paths is not as severe as it seems.

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# More on Braids, Walls, and Mirrors

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