Are There Higher-Dimensional Counterparts to the Braid Group?

[atyy]

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!

Thanks! I started a new thread for new questions.

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?

 

[torsten]

Thanks! I started a new thread for new questions.

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?

Yes, braiding is an effect of codimension-2 embeddings. In 3 dimensions one needs 1D objects which are attached to surfaces (points in the surfaces).
So, the generalization to n-dimensional branes: the branes have to be embedded in a n+2 dimensional space.
But I'm not shure about the relations in this higher-dimensional braid group. The relations of the 3D braid group should go over the higher-dim case but there must be additional relations.
In case of surfaces (or membranes( you can check Lee Rudolph's
Braided surfaces and Seifert ribbons for closed braids Comment. Math. Helv. 59 (1983), 1-37.
I know also of a book (Saito?) about higher-dim braids.