It is interesting to show how the 8 slices of E7s concentric hull projection are represented in the 18 (9×2 H4+H4Φ) slices E8’s concentric hull projection. You can even do the same with E6 which shows an icosahedron and dodecahedron from the 6-demicube and 2 irregular 20-vertex polytopes from 40 of the 60 D6 vertices, for a total of 72 vertices.

]]>I’ll take a look at that paper on the arXiv. I notice that it mentions M-theory and 11d spacetime. This makes me wonder, quite idly, whether you could find an interesting model where instead of the 600-cell you used the root polytope in 7 dimensions, the root polytope in 8 dimensions, or something like that. These have even more symmetries than the 600-cell! For example, the root polytope has 696729600 symmetries. On top of that, the 240 vertices of this polytope correspond in a very sneaky way to two copies of the 600-cell’s 120 vertices.

But, I don’t have any concrete ideas of how you’d get a higher-dimensional field theory and discretize space to be one of these delightful higher-dimensional polytopes.

]]>That link just goes to a picture; did you mean this instead?

]]>Hi, the BMN model can be found here: https://arxiv.org/abs/hep-th/0202021

It was all the rage when I was about to graduate from Berkeley. All the stringers were talking about it. It provided a further evidence and elaboration of the holographic correspondence.

It would be interesting to compute quantities in the gauge theory from a nonperturbative, numerical approach, which is why a few of us in the lattice community are interested in it. Of course, we’re also looking for excuses to work with curved spaces.

]]>Neat! It would be nice if starting with the vertex quandle you could *build* the edge and face quandles by some systematic construction, in a way that works for all 3-dot Coxeter diagrams. (And similarly for Coxeter diagrams with more dots.)

Cool! I like the idea of the 600-cell as an approximation to I don’t know about the BMN model.

(By the way, you were using $\latex instead of $latex to begin your math comments, so I fixed that.)

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