(adsbygoogle = window.adsbygoogle || []).push({}); Thanks! I started a new thread for new questions. space_cadet said: ↑In GFT the basic building blocks are (n-1)-simplices (a 0-simplex is a point, a 1-simplex is a line-segment, 2-simplex is a triangle, 3-simplex is a tetrahedron .... and so on), which are glued together to form a simplicial complex (a discretized manifold), whose dynamics is given in terms of group elements assigned to each of the n faces of the (n-1)-simplex (see e.g. arXiv:0710.3276v1) The "field" is then taken to be a complex valued functions acting on these (n+1) group elements:

$$ \phi(g_1, g_2, \ldots, g_{n}) : G^n \rightarrow \mathbb{C} $$

Now, given (n+1) copies of a (n-1)-simplex, one can glue these together along their respective faces to form a n-simplex, e.g. for n=3, given four triangles (a triangle is a 2-simplex), one can glue them together along their edges to form a tetrahedron (which is a 3-simplex). One can write down an action for such a theory (see reference above) and explicitly compute various observable quantities. The resulting theory describes the dynamics of an n-dimensional manifold in terms of its constituent (n-1)-simplices.

The connection with the braid proposal arises from the observation that,a priori, there is no restriction on the form of the group G which is used to label faces of the simplices. G could be SU(2), SL(2,C) or even SL(2,Z) (the modular group) or B_3 (the three-stranded braid group). For instance, if one can write down a GFT action for 2-simplices, with edges labeled by representations of B_3, such an action would describe the dynamics of a manifold constructed by gluing the edges of triangles using 3-strand braids. This is the essence of the relationship I see between GFTs and the braid model.It may or may not turn out to technically feasible.

If you have further questions a new thread might be best, since this reply already takes this thread off-topic!

Let me start with one I don't even know makes sense: are there counterparts to the braid group for higher dimensional objects like membranes?

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# GFT & Braid Group

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