The n-Category Café

Thanks, Rod. Fixed.

I believe the claim at the moment is that the diagram represents maximal invariant central extensions, but that it doesn’t claim those shown are the only possible ones.

We’ve been having some discuss at the nForum and elsewhere. It does seem to be a form of Whitehead tower.

Urs writes

It was clear all along that the Cayley-Dickson construction knows something about supersymmetry and the stringy spacetimes, but this left open two problems: why consider star-algebras and their CD-doubles in the first place, and why stop the CD-process at some point?

Now with the bouquet, these two questions are answered. We see (that’s how I view it anyway) that those algebras are not the truly fundamental agent here. While they happen to neatly encode the crucial relations, the true fundamental concept is the progression of universal invariant (higher) central extensions of super Lie algebras. That this happens to be accompanied by division algbras for parts of the journey is a useful fact, but division algebras are not conceptually what drives this process.

I posed to them the challenge to work out what happens with other superpoints as starting points, such as $\mathbb{R}^{0|3}$ and $\mathbb{R}^{0|4}$, and given that the Whitehead construction itself is functorial, whether there is something to be said for growing the towers from all superpoints together with mappings.

As for the former point, it’s clear that one will not meet with ordinary spacetimes, but for $\mathbb{R}^{0|3}$ there’s the intriguing possibility of meeting up with the Albert algebra. Urs wondered whether that 27-dimensional algebra might connect to bosonic M-theory.

For more information about the higher portions of the tree, there is also T-Duality from super Lie n-algebra cocycles for super p-branes. It does seem that beyond 11d, we won’t find ordinary spacetimes:

the bosonically 12 dimensional $\mathbb{R}^{9+(1+1),1|32}$ is a super Lie algebra, but not a super-Minkowski Lie algebra, hence not a super-symmetry algebra in the sense of spacetime supersymmetry…This is consistent with the observations and assertions in the literature that F-theory does not have a straightforward spacetime interpretation and, furthermore, that the alternative seems to emanate from superalgebras, but not of the usual type.

If I’m allowed a moment of self-congratulation, it’s fun to see that this work is realising dreams that in part started the Café, about Klein 2-geometry and then Cartan 2-geometry. Of course, now we know we need higher Cartan geometry.

Here are several papers that suggest that there exists a fundamental 27D theory.

http://arxiv.org/abs/hep-th/0012037

http://arxiv.org/abs/0807.4899

http://arxiv.org/abs/hep-th/0104081

http://arxiv.org/abs/hep-th/0104050

http://arxiv.org/abs/hep-th/0110106